valuation of counterparty risk for commodity derivatives

Hochschule Darmstadt - University of Applied Sciences

Faculty of Mathematics and Natural Sciences

Valuation of Counterparty Risk forCommodity Derivatives

submitted by: Michael NoePeterstraße 464683 Einhausen

student’s admission number: 708259

submitted at: July 8, 2011

supervisor: Prof. Dr. Marcus R.W. MartinProf. Dr. Martina Bohmer

Statement of Authorship

Except where reference is made in the text of this thesis, this thesis contains no materialpublished elsewhere or extracted in whole or in part from a thesis presented by mefor another degree or diploma. No other person’s work has been used without dueacknowledgement in the main text of the thesis. This thesis has not been submittedfor the award of any other degree or diploma in any other tertiary institution.

All sentences or passages quoted in this thesis from other people’s work have beenspecifically acknowledged by clear cross-referencing to author, work and page(s). Anyillustrations which are not the work of the author of this thesis have been used with theexplicit permission of the originator and are specifically acknowledged. I understandthat failure to do this amounts to plagiarism and will be considered grounds for failurein this thesis and the degree examination as a whole.

............................... ...............................(Place, Date) (Signature)

I

Abstract

For several years the trading with commodities at the international capital marketsplay an increasingly important role. In exchange-traded derivatives the default risk ofthe counterparty does not matter, because the Clearing House is the central counter-party between the trading partners. However, as most derivatives are traded over thecounter, the counterparty risk plays a very important role. There are many historicalexamples in which a trading partner has defaulted until maturity of the contract. Inthe financial sector, this took place especially in the financial crisis of 2007/08, hereare some very large banks defaulted, such as Bear Stearns, Leman Brothers, AIG,Washington Mutual, and more. The energy company Enron held a variety of energyderivatives, and went in 2001 in a major scandal in the bankruptcy.

In this work, commodity models are calibrated to market data for WTI crude oiland wheat. These models are both One-Factor and Two-Factor models, each with adeterministic shift. As default process, a intensity model with a stochastic intensityfunction was adopted. This default process is calibrated to quoted CDS spreads.

With these models the Credit Valuation Adjustment is calculated for wheat and crudeoil swaps. Furthermore, it was examined how sensitive the Credit Valuation adjust-ment reacts on changes in CDS spreads. Another topic is the model risk, here are theeffects investigated, which follow from a not optimal model selection.

III

Zusammenfassung

Seit einigen Jahren spielt der Handel mit Rohstoffen an den Internationalen Kapi-talmarkten eine immer wichtigere Rolle. Bei borslich gehandelten Derivaten spielt dasAusfallrisiko des Kontrahenten keine Rolle, da hier das Clearinghouse der Borse alsZentraler Kontrahent zwischen den Handelspartnern auftritt. Da allerdings die meis-ten Derivate ausserborslich gehandelt werden, spielt das Kontrahentenrisiko eine sehrwichtige Rolle. Es gibt viele historische Beispiele, bei denen ein Handelspartner biszum Laufzeitende des Kontraktes ausgefallen ist. Im Finanzbereich traf dies beson-ders in der Finanzkrise von 2007/08 zu, hier sind einige sehr große Banken ausgefallen,wie z.B. Bear Stearns, Leman Brothers, AIG, Washington Mutual, und weitere. DasEnergieunternehmen Enron hielt eine vielzahl von Energiederivaten und ging 2001 ineinem großen Skandal in die Insolvenz.

In dieser Arbeit werden Rohstoffmodelle an Marktdaten fur West Texas Intermediate(WTI) Rohol und Weizen kalibriert. Diese Modelle sind sowohl Ein-Faktor als auchZwei-Faktor Modelle jeweils mit einem deterministischen Shift. Als Ausfallprozess wirdein Intensitatsmodell mit stochastischer Intensitatsfunktion angenommen. Dieser Aus-fallprozess wird mittels quotierten CDS-Spreads kalibriert.

Mit diesen Modellen wird das Credit Valuation Adjustment fur Weizen- und Ol-Swapsberechnet. Des Weiteren wurde untersucht wie sensitiv der Credit Valuation Adjust-ment auf anderung der CDS-Spreads reagiert. Ein weiteres Themengebiet ist dasModellrisiko: Hier werden die Auswirkungen untersucht, die die Folgen einer nichtoptimalen Modellwahl sind.

V

Contents

1. Introduction 11.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2. Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3. Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4. Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2. Counterparty Risk 3

3. Financial Derivatives 73.1. Credit Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1.1. Credit Default Swap . . . . . . . . . . . . . . . . . . . . . . . . . 73.1.2. Credit Default Swaptions . . . . . . . . . . . . . . . . . . . . . . 10

3.2. Commodity Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.1. Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.2. Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.3. Call- and Put-Options . . . . . . . . . . . . . . . . . . . . . . . . 15

4. The Default Process 174.1. Approaches for Modeling the Default Time . . . . . . . . . . . . . . . . 174.2. The Intensity Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.2.1. Deterministic Intensity function . . . . . . . . . . . . . . . . . . . 194.2.2. Stochastic Intensity function . . . . . . . . . . . . . . . . . . . . 20

4.3. Approach to Simulate the Default Time . . . . . . . . . . . . . . . . . . 214.4. Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.4.1. Calibrate the Deterministic Intensity Function . . . . . . . . . . 304.4.2. Calibrate the Stochastic Intensity Function . . . . . . . . . . . . 34

5. Commodities and Commodity Models 395.1. Properties of Commodity Prices . . . . . . . . . . . . . . . . . . . . . . . 395.2. The Theory of Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.3. Models for Commodity Prices . . . . . . . . . . . . . . . . . . . . . . . . 43

5.3.1. The Schwartz One Factor Model . . . . . . . . . . . . . . . . . . 435.3.2. The Gibson-Schwartz model . . . . . . . . . . . . . . . . . . . . . 475.3.3. The Smith-Schwartz model . . . . . . . . . . . . . . . . . . . . . 48

VII

Contents

5.4. Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.4.1. The Schwartz One-Factor model . . . . . . . . . . . . . . . . . . 625.4.2. The Gibson-Schwartz model . . . . . . . . . . . . . . . . . . . . . 645.4.3. The Smith-Schwartz model . . . . . . . . . . . . . . . . . . . . . 655.4.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6. Implementation 716.1. Implementation of the Cox-Ingersoll-Ross Process . . . . . . . . . . . . . 71

6.1.1. Euler scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.1.2. Milstein scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.1.3. Euler-Implicit scheme . . . . . . . . . . . . . . . . . . . . . . . . 826.1.4. Milstein-Implicit scheme . . . . . . . . . . . . . . . . . . . . . . . 846.1.5. Alfonsis E family . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.1.6. Exact Simulation Algorithm . . . . . . . . . . . . . . . . . . . . . 896.1.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.2. Implementation of the Default Process . . . . . . . . . . . . . . . . . . . 936.2.1. Implementation with deterministic intensity . . . . . . . . . . . . 936.2.2. Implementation with the stochastic intensity . . . . . . . . . . . 95

6.3. Implementation of the Commodity Models . . . . . . . . . . . . . . . . . 1006.3.1. Implementation of the Schwartz One-Factor model . . . . . . . . 1006.3.2. Implementation of the Gibson-Schwartz model . . . . . . . . . . 1026.3.3. Implementation of the Smith-Schwartz model . . . . . . . . . . . 103

7. Results 1077.1. Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097.2. Analysis of the Model Risk . . . . . . . . . . . . . . . . . . . . . . . . . 1137.3. Analysis of the Effects of Correlations . . . . . . . . . . . . . . . . . . . 115

8. Conclusion 1178.1. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178.2. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Appendix 119

A. Stochastic Calculus 121A.1. Poisson Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121A.2. Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122A.3. Ornstein-Uhlenbeck Process . . . . . . . . . . . . . . . . . . . . . . . . . 123

B. Implementation in Matlab 125

VIII

1. Introduction

During the last ten years, trading derivatives has significantly increased both in vol-ume and total amount. The most important aspect in trading derivatives was the socalled market risk which means the change of the price of the derivative in a fixed timeperiod as a result of changing market risk factors as interest rates, FX rates etc. Inthe last years also the risk that a counterparty cannot fulfill his payment obligationwas considered because of insolvency as particularly observed as a consequence of theGreat Financial Crisis 2007/08. The countperarty risk means the risk that the coun-terparty gets insolvent up until maturity.

In this thesis the counterparty risk for swaps on crude oil and wheat is valued. Forthis purpose models for commodities are presented and calibrated to market data forthe chosen commodity. With the model that fits best further analysis is performed.Furthermore, a default process is chosen and is calibrated to market data as well.

1.1. Motivation

Counterparty risk is a dynamic field of research and over the last few years, researchershave been spending a lot of effort for pricing and hedging of the exposure caused bythe counterparty risk. With the just recent events as Bear Stearns in spring 2008 andLehman Brothers in September 2008, this topic gains significant interest in the broadpuplic. Trading commodities and commodity derivatives, for reason of both to hedgeoneself against price fluctuations or to speculate with specific price movements is animportant section of the financial markets. Enron traded until its insolvency in 2001 alarge number of future contracts on natural gas. This causes the author to on the onehand consider the valuation of the counterparty risk and on the ohter hand to takethe commodity derivatives into consideration.

1.2. Objectives

The author aims at valueing the counterparty risk for various commodity derivatives.The achieve this goal, the following steps have to be taken:

• Find the appropriate models for the defaults.

1

1. Introduction

• Find the appropriate models to represent the commodity prices well.

• Calibrate the models to market data.

• Implement the models.

• Value the counterparty risk for commodity derivatives.

• Examine the sensitivity with which the result reacts to changes of the main riskdrivers of the default model.

• Analyze the impact of the choice to apply an inappropriate model for the com-modities that it has on the whole result.

This thesis offers the reader a survey of different ways to model defaults and commodityprices. Additionally, the models are implemented and calibrated.

1.3. Related Work

Brigo, Chourdakis, and Bakkar present in [BCB08] a model to value the counterpartyrisk for oil swaps. The author of this thesis extends this work in considering the sub-jects of calibration and impementation.

Cesari, Aquilina, Charpillon, Filipovic, Lee, and Manda provide in [CAC+09] anoverview concerning the modelling and implementation of counterparty risk. Com-pared to this thesis, the evaluation measure for the counterparty risk differs fromthat in their book. Moreover, the modelling of commodity prices is also not furthermentionned in their book.

1.4. Outline of the thesis

The thesis is organized as follows: Chapter 2 is devoted to introduce the notion andbasic concepts on counterparty risk. In chapter 3 we review the economics and pricingof fnancial derivatives. The next chapter 4 holds various default model possibilities.In that effect, in chapter 5 , basic characteristics of the commodities and their modelpossibilities are shown. Chapter 6 contains a presentation of almost all in this thesismentionned models and the ways to implement them. In the following chapter 7 , thecounterpartyrisk is determined. Further the consequences of choosing a wrong modelis discussend and finally the sensitivity of the results is studied. In the final chapter 8, the focus lies upon the summary methods presented in this thesis and their results.Finally, other topics for further research work are recommended.

2

2. Counterparty Risk

This chapter defines the counterparty risk and explains a way to value it.The counterparty risk is the risk, that the counterparty becomes insolvent during thelifetime of the derivative.

In [Noe11] the author states:

If a derivative is traded at the stock exchange, there is no counterpartyrisk, as the clearing house acts for both parties as a so-called central coun-terparty. The counterparty risk is repealed with so-called margin accounts.These margin accounts are managed by the clearing house for both parties.Any change in market value is immediately settled via so-called margin-calls.

Most derivatives are nowadays still traded over the counter (OTC), hence the coun-terparty risk plays a significant role for them. The counterparty risk became knownin particular throughout the financial crisis. Many derivatives could not be completedbecause one of the parties defaulted (e.g. Bear Stearns, Lehman Brothers and so on).Basel III lays out strong regulatory incentives for transfering OTC Derivatives to cen-tral counterparties. Nontheless a certain portion of tailor-made derivatives will stillremain on an OTC-traded basis.

The sum of all discounted cash flow from time t to time T is named with Π(t, T ) = Π(t)and therefore on the defaultable cash flows are named with ΠD(t, T ) = ΠD(t). Thenet present value NPV (t) of a derivative corresponds to the expected discounted cash-flows of the derivative, it is NPV (t) = E (Π(t, T )|Ft) = Et (Π(t, T )).

The failure-prone cash flow can be decomposed as on the one hand in that part wherethe counterparty survives and on the other hand in that part where it defaults. Math-ematicaly, it can be stated as follows:

ΠD(t) = 1τ>TΠ(t, T )

+ 1t<τ≤T

[Π(t, τ) + P (t, τ)

(RR (NPV (τ))

+ − (−NPV (τ))+)]

Where τ is the default time, and the recovery fraction RR is called recovery rate.Therefore LGD = 1−RR is the loss given default and P (t, T ) is the discount factor

3


between t and T . In this work the interest rates and the LGD are assumed to bedeterministic.

Before proceeding, one must define the probability space:

For a given intervall I := [0, T∗] with T∗ ∈ (0,∞), the probability space is (Ω, A, F , Q),with σ-Algebra A and filtration F := (Ft)t∈I . Where F describres the information ofthe general market (e.g. risk free interest rates). And Q is the risk neutral probabilitymeasure.

The value of a fail-prone derivative can be expressed as the value of the derivativewithout default risk diminished valuation adjustment for default risk. This adjust-ment is called Credit Valuation Adjustment (CVA). Mathematically it is expressed asshown in the following theorem (see [BCB08, Page 4]):

Theorem 1 (The Credit Valuation Adjustment). At valuation time t, and on τ > t,the price of the payoff under counterparty risk is

Et[ΠD(t)

]= Et [Π(t, T )]− LGD · Et

[1t<τ≤T · P (t, τ) · (NPV (τ))+

]where LGD is the loss given default and it is assumed to be deterministic. The CVAis given by the second term of the equation.

A similar version of the following proof can be found in [BM04, Page 7].

Proof.

By definition we have

Et[ΠD(t)

]= Et

[1τ>TΠ(t, T )

+1t<τ≤T(Π(t, τ) + P (t, τ) ·

(RR(NPV (τ))+ − (−NPV (τ))+

))]= Et

[1τ>TΠ(t, T )

]+ Et

[1t<τ≤T ·Π(t, τ)

]+ Et

[1t<τ≤T · P (t, τ) ·RR · (NPV (τ))+

]− Et

[1t<τ≤T · P (t, τ) · (−NPV (τ))+

]In the next step the second and the fourth term are transformed:

= Et[1t<τ≤T ·Π(t, τ)

]− Et

[1t<τ≤T · P (t, τ) · (−NPV (τ))+

]= Et

[1t<τ≤T ·Π(t, τ)− 1t<τ≤T · P (t, τ) · (−NPV (τ))+

]

4

By definition it is NPV (τ) = Eτ [Π (τ, T )]

= Et[1t<τ≤T ·Π(t, τ)− 1t<τ≤T · P (t, τ) · (−Eτ [Π (τ, T )])+

]Since (−f)+ = f− we have

= Et[1t<τ≤T ·Π(t, τ)− 1t<τ≤T · P (t, τ) · (Eτ [Π (τ, T )])−

]and because of (f)− = − (f − (f)+) we obtain

= Et[1t<τ≤T ·Π(t, τ) + 1t<τ≤T · P (t, τ) ·

(Eτ [Π (τ, T )]− (Eτ [Π (τ, T )])+

)]At the time τ , Π (τ, T ) is Fτ -measurable, so it follows: Eτ [Π (τ, T )] = Π (τ, T )

= Et[1t<τ≤T ·Π(t, τ) + 1t<τ≤T · P (t, τ) ·

(Π (τ, T )− (Eτ [Π (τ, T )])+

)]= Et

[1t<τ≤T ·

(Π(t, τ) + P (t, τ) ·

(Π (τ, T )− (Eτ [Π (τ, T )])+

))]= Et

[1t<τ≤T ·

(Π(t, τ) + P (t, τ) ·Π (τ, T )− P (t, τ) · (Eτ [Π (τ, T )])+

)]Hence, Π(t, T ) = Π(t, τ) + P (t, τ) ·Π (τ, T ) yields

= Et[1t<τ≤T ·

(Π(t, T )− P (t, τ) · (Eτ [Π (τ, T )])+

)]= Et

[1t<τ≤T ·

(Π(t, T )− P (t, τ) · (NPV (τ))+

)]The results of the transformation are used in the above equation:

Et[ΠD(t)

]= Et

[1τ>TΠ(t, T )

]+ Et

[1t<τ≤T · P (t, τ) ·RR · (NPV (τ))+

]+ Et

[1t<τ≤T ·

(Π(t, T )− P (t, τ) · (NPV (τ))+

)]= Et

[1τ>TΠ(t, T )

]+ Et

[1t<τ≤T · P (t, τ) ·RR · (NPV (τ))+

]+ Et

[1t<τ≤T ·Π(t, T )

]− Et

[1t<τ≤T · P (t, τ) · (NPV (τ))+

]= Et

[1τ>TΠ(t, T )

]+ Et

[1t<τ≤T ·Π(t, T )

]+ Et

[1t<τ≤T · P (t, τ) · (RR− 1) · (NPV (τ))+

]with RR− 1 = −LGD we obtain

= Et[1τ>TΠ(t, T )

]+ Et

[1t<τ≤T ·Π(t, T )

]− Et

[1t<τ≤T · P (t, τ) · LGD · (NPV (τ))+

]

5


LGD is assumed as constant, therefore

= Et[1τ>TΠ(t, T )

]+ Et

[1t<τ≤T ·Π(t, T )

]− LGD · Et

[1t<τ≤T · P (t, τ) · (NPV (τ))+

]= Et

[1τ>TΠ(t, T ) + 1t<τ≤T ·Π(t, T )

]− LGD · Et

[1t<τ≤T · P (t, τ) · (NPV (τ))+

]= Et [Π(t, T )]− LGD · Et

[1t<τ≤T · P (t, τ) · (NPV (τ))+

]which proves the theorem.

Brigo, Chourdakis, and Bakkar show in [BCB08, Page 4] an approximation formulafor the CVA for t = 0 by discretizing using a time grid T0, T1, . . . , Tb = T ,

Et[ΠD(0, Tb)

]= Et [Π(0, Tb)]− LGD

b∑j=1

E[1Tj−1<τ≤TjP (0, τ) (Eτ [Π(τ, Tb)])

+]

≈ Et [Π(0, Tb)]− LGDb∑j=1

E[1Tj−1<τ≤TjP (0, Tj)

(ETj [Π(Tj , Tb)]

)+](2.1)

For further reading [BM05], [Gre09], and [CAC+09] are suggested.

6

3. Financial Derivatives

Derivatives are financial products, whose value or payoff profile depend on another fi-nancial asset or product which are called underlying of the derivative. Underlyings cane.g. be equities, interest rates, credits, indices, foreign exchange rates, commodities orsimilar. Most derivatives are still traded over the counter . The buyer of a derivativeholds the long position while the seller holds the short position.

In this chapter, the derivatives used in this thesis are presented such as commod-ity derivatives and credit derivatives. For further information about derivatives ingeneral [Hul03], [Pfe04], [HDK02], and [Rei10] are recommended.

3.1. Credit Derivatives

Credit derivatives are derivatives whose value depend on the payment behavior of aloan or pool of loans. Hull defines credit derivatives (see [Hul03, Page 637] as follows:

Credit derivatives are contracts where the payoff profile depends on thecreditworthiness of one or more commercial or sovereign entities.

In this chapter the most liquidly traded credit derivative, the Credit Default Swap(CDS), and call options on a CDS are introduced. For further information aboutcredit derivatives in general, the author refers to [MRW06], [Hul03] and [BM05].

3.1.1. Credit Default Swap

As mentionned previously the Credit Default Swap is one of the most popular creditderivatives. It can be seen as a credit default insurance. The buyer of a CDS caninsure a bond against default, but also can speculate on the insolvency of the borrower.The seller of a CDS is also named protection seller and the buyer is called protectionbuyer (see [Noe11, Page 4]). The protection seller insures the protection buyer againsta default of the bond, for doing this he gets paid a premium until maturity (T ) oruntil the bond default time in regular intervals for this. This premium is a percentageof the insured notional (N) and is named CDS-spread (s).

The fair price of the CDS at inception is-as-usual-zero, hence protection and pre-mium leg (see the following equations (3.1) and (3.2) both equations can be found in

7


[Mar10, Page 114]) have to have the same present value. They represent the presentvalues of payments by the protection sellers and protection buyers. The most impor-tant parameters for the pricing are the survival probabilities G(t) and the recovery rateRR.

PV Prem(0, T, s0) = s0 ·Nv∑j=1

∆jP (t, tj) ·G(tj) +

tj∫tj−1

(u− tj−1) · P (t, u)d(1−G(u))

(3.1)

PV Prot(0, T ) = 1τ>t ·N ·T∫t

P (t, u) · (1−RRu)d(1−G(u)) (3.2)

The RRu is assumed as constant

= 1τ>t ·N · (1−RR)

T∫t

P (t, u)d(1−G(u))

The Credit Basis Point Value (CBPV) corresponds to the discounted survival proba-bilties until maturity.

CBPV (t, T ) =

T∫t

P (t, u)G(u)du (3.3)

≈m∑j=1

∆jP (0, tj) ·G(tj) +

tj∫tj−1

(u− tj−1) · P (0, u) d (1−G(u))

Where ∆j stands for the year fraction between tj and tj−1, and the termtj∫

tj−1

(u− tj−1)·

P (0, u) d (1−G(u)) stands for the accrued premium from the final payment until thedefault time. If the accrued premium is omitted for simplificity, then:

CBPV (t, T ) =

m∑j=1

∆jP (0, tj) ·G(tj) (3.4)

The present value for the buyer of the CDS is the difference between protection legand premium leg:

PV CDSbuy (0, T, s) = PV Prem(0, T, s)− PV Prot(0, T )

= PV Prem(0, T, s)− s · CBPV (0, T ) (3.5)

8


And from the perspective of the protection sellers, the present value is presented asfollows:

PV CDSsell (0, T, s) = −PV CDSbuy (0, T, s)

= s · CBPV (0, T )− PV Prem(0, T, s) (3.6)

To specify the fair CDS spread, equation (3.5) or equivalently equation (3.6) is setequal to zero. This leads to the following equation:

s0(tm) =

LGD ·T∫t

P (0, u) d (1−G(u))

m∑j=1

∆jP (0, tj) ·G(tj) +tj∫

tj−1

(u− tj−1) · P (0, u) d (1−G(u))

(3.7)

Martin, Reitz and Wehn provide in [MRW06, Equation (4.17), page 175] an approxi-mation for the last equation in discrete time:

s0(tm) ≈(1−RR) ·

m∑j=1

P(

0,tj−1+tj

2

)· [G(tj−1)−G(tj)]

m∑j=1

[∆jP (0, tj) ·G(tj) +

∆j

2 · P(

0,tj−1+tj

2

)(G(tj−1)−G(tj))

] (3.8)

If the fair CDS spread (s0) is not paid, but rather an alternative spread (sA) is paidthen a so-called upfront payment has to be paid. These upfront payments correspondto the present value of a CDS with the alternative spread which can be mathematicalexpressed as follows:

UCDSbuy (0, T ) = PV Prot(0, T )− sA · CBPV (0, T ) (3.9)

As a consequence of the financial crisis of 2007/08, the CDS spreads have been stan-dardized, which led to the

• Standard North American Corporate1 (SNAC), with standard CDS spreads of100 and 500 basis points (bp) and the

• Standard European Corporate2 (STEC), with standard CDS spreads of 25bp,100bp, 500bp, and 1000bp.

1http://www.cdsmodel.com/assets/cds-model/docs/Standard%20Corporate%20CDS%20Contract%

20Specification%20-%20May%2012,%202009.pdf2http://www.cdsmodel.com/assets/cds-model/docs/Standard%20Corporate%20CDS%20Contract%

20Specification.pdf

9

http://www.cdsmodel.com/assets/cds-model/docs/Standard%20Corporate%20CDS%20Contract%20Specification%20-%20May%2012,%202009.pdf

http://www.cdsmodel.com/assets/cds-model/docs/Standard%20Corporate%20CDS%20Contract%20Specification%20-%20May%2012,%202009.pdf

http://www.cdsmodel.com/assets/cds-model/docs/Standard%20Corporate%20CDS%20Contract%20Specification.pdf

http://www.cdsmodel.com/assets/cds-model/docs/Standard%20Corporate%20CDS%20Contract%20Specification.pdf


Furthermore, a constant RR of 40% for senior corporate debt and Western EuropeanSovereigns and 20% for subordinated corporate debt is assumed (see [Mar10, Slide xl]).

With these standards the fair CDS spreads are no longer quoted, the pricing is basedon quoted upfront payments.

For further readings to the Credit Default Swap [MRW06] and [Hul03] are recom-mended.

3.1.2. Credit Default Swaptions

A European call option on a CDS is the CDS option or Credit Default Swaption(CDSw). With a CDSw, the right is acquired to buy at a future point in time Ta > 0a CDS with CDS spread K as strike and maturity at the time Tb. The Notional ofthis option is 1 (see [BM05, Page 844]).

The value of such an option is:

PV CDSw(t, Ta, K) = E

(1τ>TaP (t, Ta)

[PV CDS (Ta, Tb, sa, b)

−PV CDS (Ta, Tb, K)

]+

|Ft

)(3.10)

As stated above, the filtration F describres the information of the general market. Sothe filtration H models the information about the default time. Together, the twofiltations result in the filtration G := (Gt)t∈I with Gt := σ (Ft, Ht).

The payoff profile at the time Ta can be formed as follows:

h(Ta) =[PV CDS (Ta, Tb, sa, b)− PV CDS (Ta, Tb, K)

]+

10


Because sa, b is the fair CDSspread at time Ta, it follows: PV CDS (Ta, Tb, sa, b) = 0

=[−PV CDS (Ta, Tb, K)

]+=[−(PV Prem(Ta, Tb, K)− PV Prot(Ta, Tb)

)]+=[PV Prot(Ta, Tb)− PV Prem(Ta, Tb, K)

]+= max

0, PV Prot(Ta, Tb)− PV Prem(Ta, Tb, K)

= max

0, 1τ>t · (1−RR)

Tb∫Ta

P (Ta, u)d(1−G(u))−K · CBPV (Ta, Tb)

(3.11)

In the next step the forward CDS spread sa, b is required for the period from Ta untilTb:

0 = 1τ>t · (1−RR)

Tb∫Ta

P (Ta, u)d(1−G(u))− sa, b · CBPV (Ta, Tb)

sa, b · CBPV (Ta, Tb) = 1τ>t · (1−RR)

Tb∫Ta

P (Ta, u)d(1−G(u))

sa, b = 1τ>t · (1−RR)

Tb∫Ta

P (Ta, u)d(1−G(u))

CBPV (Ta, Tb)(3.12)

With the forward CDS spread shown in equation (3.12), the payoff profile from equa-tion (3.11) can further be transformed (see [MRW06, Page 189]):

h(Ta) = max

0, 1τ>t · (1−RR)

Tb∫Ta

P (Ta, u)d(1−G(u))−K · CBPV (Ta, Tb)

= max 0, sa, bCBPV (Ta, Tb)−K · CBPV (Ta, Tb)= max 0, (sa, b −K) · CBPV (Ta, Tb)= (sa, b −K) · CBPV (Ta, Tb)+

Because CBPV (Ta, Tb) > 0, it follows:

= CBPV (Ta, Tb) sa, b −K+

11


Returning to the initial valuation equation (3.10):

PV CDSw(t, Ta, K) = E(1τ>TaP (t, Ta) · h(Ta)

)= E

(1τ>TaP (t, Ta) · CBPV (Ta, Tb) sa, b −K+ |Gt

)The accrued premium after the last payment until the default time is neglected (seeequation (3.4)):

= E(

1τ>TaP (t, Ta) · CBPV (Ta, Tb) sa, b −K+ |Gt)

The risk neutral filtration G will be replaced by filtration F without information ofthe default (see [BM05, Page 728]).

=1τ>t

Q (τ > t|Ft)E(

1τ>TaP (t, Ta) · CBPV (Ta, Tb) sa, b −K+ |Ft)

Since Ft ⊂ FTa it follows: E(1τ>Ta|Ft

)= E

(E(1τ>Ta|FTa

)|Ft)

=1τ>t

Q (τ > t|Ft)E(E(1τ>Ta|FTa

)P (t, Ta) · CBPV (Ta, Tb) sa, b −K+ |Ft

)It is: E

(1τ>Ta|FTa

)= Q (τ > Ta|FTa)

=1τ>t

Q (τ > t|Ft)E(P (t, Ta) · CBPV (Ta, Tb)Q (τ > Ta|FTa) sa, b −K+ |Ft

)The probability measure will be changed by the numeraire B(see [BM05, Page 30] or[BM05, Page 849])

=1τ>t

Q (τ > t|Ft)EB(B(t)

B(Ta)· CBPV (Ta, Tb)Q (τ > Ta|FTa) sa, b −K+ |Ft

)

The probability measure will be changed by the numeraire CBPV (see [BM05, Page30] or [BM05, Page 849])

=1τ>t

Q (τ > t|Ft)Ea, b

(CBPV (t, Tb)Q (τ > t|Ft)

CBPV (Ta, Tb)Q (τ > Ta|FTa)

·CBPV (Ta, Tb)Q (τ > Ta|FTa) sa, b −K+ |Ft

)=

1τ>t

Q (τ > t|Ft)Ea, b

(CBPV (t, Tb)Q (τ > t|Ft) sa, b −K+ |Ft

)

12

3.2. Commodity Derivatives

At the time t, Q (τ > t|Ft) is Ft-measurable, so it follows: Ea, b [Q (τ > t|Ft) |Ft] =Q (τ > t|Ft)

=1τ>t

Q (τ > t|Ft)Q (τ > t|Ft)Ea, b

(CBPV (t, Tb) sa, b −K+ |Ft

)= 1τ>tEa, b

(CBPV (t, Tb) sa, b −K+ |Ft

)At the time t, CBPV (t, Tb) is Ft-measurable, so it follows: Ea, b

[CBPV (t, Tb)|Ft

]=

CBPV (t, Tb)

= 1τ>tCBPV (t, Tb)Ea, b(sa, b −K+ |Ft

)

It is assumed, that the CDS spread follows the equation dsa, b(t) = σa, bsa, bdWa, b(t).

Furthermore, W a, b(t) is a Brownian Motion under Qa, b. Hence following the Black76equation:

PV CDSw(t, Ta, K) = 1τ>tCBPV (t, Tb) · [s0Φ (d1)−KΦ (d2)]

mit

d1, 2 : =log(x0

x

)± (1

2 σ2a, b (Ta − t)

σa, b√Ta − t

A similar derivation of the valuation formula for CDSw can be found [BM05, Page848] and [MRW06, Page 189].

For further information the author refers to [Jam03], [BM05], and [MRW06].


Commdodity derivatives are the first derivatives traded in history. As stated by Aris-totle in [AriBC, Book 1, chapter 11]

There is the anecdote of Thales the Milesian and his financial device,which involves a principle of universal application, but is attributed to himon account of his reputation for wisdom. He was reproached for his poverty,which was supposed to show that philosophy was of no use. According tothe story, he knew by his skill in the stars while it was yet winter that

13


there would be a great harvest of olives in the coming year; so, having alittle money, he gave deposits for the use of all the olive-presses in Chiosand Miletus, which he hired at a low price because no one bid against him.When the harvest-time came, and many were wanted all at once and of asudden, he let them out at any rate which he pleased, and made a quantityof money.

For further readings about commodity derivatives [BPR09], [Ruj08], [Gem05a], and[Gem08] are recommended

3.2.1. Forward

A forward is a contract for the delivery of the underlying S at an agreed price at aspecified time T in the future. Forwards are traded OTC. The price is called the for-ward price. The fair forward price corresponds to the expected value of the underlyingat time T and is calculated as follows

F (t, T ) = E(S(T )|S(t)), t < T (3.13)

The present value of such a contract with a forward price K at a time t is equal to thediscounted difference between the fair forward price at time t and K:

NPVt = (F (t, T )−K) · P (t, T ) (3.14)

The fair forward price is concluded when the present value of the equation (3.14) is zero.

Forwards are used to hedge against price increases or decreases.

In addition to the forward, there is the future, which in contrast to the forward isstandardized and traded on exchanges.

3.2.2. Swap

A swap is a contract for the regular exchange of the underlying against a fixed amount.This amount is called the swap rate as it refers to a certain notional amount. Theexchange dates are ti with i = 1, . . . , n. Swaps are the most activly traded derivativesand are usually traded OTC. The most swaps are traded in the field of interest rates.

Swaps are combinations of various forwards with the same forward rate. According tothis, the NPV is the sum of the NPVs of the corresponding forwards with notionalsNj (j = 1, . . . , n) and forward price K:

NPVt =

n∑i=1

Nj (F (t, ti)−K) · P (t, ti) (3.15)

14


To determine the fair swap rate, equation (3.15) must be set to zero and solved forSt,T :

St,T =

n∑j=1

Nj · F (t, Tj) · P (t, Tj)

n∑j=1

Nj · P (t, Tj)(3.16)

Because of the regular exchange of underlying against the swap rate, these kinds ofderivatives are used to hedge against price fluctuations.

3.2.3. Call- and Put-Options

With call and put options the buyer gets the right but not the duty to buy or sell ata fixed point of time t and a future with maturity T at a fixed price K. This price iscalled strike price.

Black developed closed-form expressions for pricing of such options under the assump-tion that future prices have the same lognormal property as the spot prices.

c0 = e−rt · (F0(T ) · Φ(ε)−KΦ(ε− σF (t, T ))) (3.17)

p0 = e−rt · (KΦ(−ε+ σF (t, T ))− F0(T ) · Φ(−ε)) (3.18)

ε =log (F0(T )/K) + 1

2σ2F (t, T )

σF (t, T )

Where c0 is the price of a call, p0 is the price for a put, and σF is the volatility of theforward.

For further information the author refers to [Bla76].

15

4. The Default Process

To introduce the default process in this chapter, the basics necessary to understandthis process are first explained followed by a short presentation of the common waysto model the default itself in section 4.1. Building on top of the previous explanations,the intensity models in section 4.2 are shown. In section 4.3 the reader gets to knowthe process of the simulation of default times. The final topic in this chapter dealswith the way models are calibrated.

4.1. Approaches for Modeling the Default Time

In this section two models which generate the default times are presented. The twomost applied models for this are on the one hand the firm value model and on theother hand the intensity model.

The most famous model of the firmvalue models is the Merton model . In this model,the firm value St at the time of t of a firm is interpreted as a geometric Brownianmotion. Furthermore it is assumed that the liability N is constant and that it owns amaturity T . The equity capital ECT is the positive difference between the asset andthe liability. Article 19 of the German Insolvenzverordnung means, that a company isinsolvent as soon as the liability exceeds the asset. This is mathematically expressedas follows:

ECT = (ST −N)+

(4.1)

The equity capital is therefore either St − N or in case of insolvency, it equals zero.Since St is assumed to follow a geometric Brownian motion and N is constant in thiscase, the equation can also be interpreted as the payoff profile of an European calloption. The corporate is insolvent by the time t the buyer of this option performs it.

The intensity models use the Poisson processes to exogenously model the default event.The first jump of the Poisson process can be interpreted as the default time. The inten-sity functions λ(t) are not only used in the Poisson processes but also in the intensitymodels. The intensity functions allow the models to be easily calibrated to marketdata.

Applying the classical Merton model, the value of the equity follows a continuous

17


process. With this fact the greatest weakness of this model is found: In [Mar10, Seite79] the author comments the following:

The most significant disadvantage of the classical firmvalue model lieswithin the fact that the received credit spreads (and with it the defaultprobabilities), especially for shorter duration compared to the spreadsquoted at the market, are way too small! The reason for this is that in thismodel the value of the company as a geometric Brownian motion changesonly continuously. Summarized you can say:

It is possible to predict the insolvency.

A sudden unpredictable jump of default is impossible in this model, eventhough if it happened it would result in generating special bigger CDSspreads in shorter duration!

Since the intensity models do not have this disadvantage, it serves as a default processin this thesis. More information on the firmvalue models one can find in [MRW06]and [BOW08] and for additional information about the intensity models the reader isadvised to have a look at section 4.2, [Mar10], and [DS03].

4.2. The Intensity Model

In the previous section the subject of intensity models was briefly introduced and willbe discussed in more detail in this section. Compared to the firmvalue models, the de-fault time is deterministic and unexpected. In this thesis a difference is made betweenthe deterministic models and those which are both deterministic and stochastically.Both of them will be presented in this section.

As mentioned previously, the intensity models model the default time as the firstjump of a Poisson process. This default time is exponentially distributed with theparameter function Λ(t) under the risk neutral probability measure Q. The functionΛ(t) is also named intensity function. The intensity function can either be modeleddeterministically or stochastically. Both cases are relevant in this thesis. As statedearlier the default time τ is distributed exponentially and therefore it follows:

Q(τ ≤ t) = 1− exp −Λ(t)

According to this equation shown above the survival probabilities result immediately.

Q(τ > t) = exp −Λ(t)

18

4.2. The Intensity Model

In [BM05] the author further presents this possibility:

Λ(τ) =: ξ ∼ Exp(1)

According to the inverse function it results in:

τ = Λ−1(ξ)

It follows the generalized inverse of the cumulative intensity function

= inf

t ∈ R+ :

t∫0

λ(u)du ≥ ξ

The next subsections deal with the various ways to model the intensity function. Insection 4.4 the reader is introduced with the fact how the default processes can becalibrated on market data.

4.2.1. Deterministic Intensity function

The presentation by means of a deterministic intensity function is a very simple ap-proach to intensity modeling. The intensity function is stated as follows:

λ(t) =

n∑i=1

1[ti−1, ti)(t) · λi (4.2)

In the following step the integrated intensity function is used so it results in:

Λ(t) =

t∫0

λ(u)du

λ(t) is a deterministic jump process with jumpstime ti, jumphight λi and therefore itfollows:

Λ(t) =

n∑i=1

λi · (ti − ti−1)

Furthermore, t0 = 0 and tn = t.

Further details may be found in [Mar10] and [MRW06].

19


4.2.2. Stochastic Intensity function

In case of a stochastic intensity function, an advanced Cox-Ingersoll-Ross Process(CIR++) as it is presented in [BM05] is used. In this case the default time canbe interpreted as the first jump of a Cox process. The modeling of the default timeaccording to this method conforms to the current market practice, so Martin in [Mar10,Page 101]. It consist of a classical Cox-Ingersoll-Ross Process (CIR) as it is defined inDefinition 1 and a deterministic function. The CIR process originates from short-ratemodeling and was introduced from John C. Cox, Jonathan E. Ingersoll and StephenA. Ross in [CIR85]. The reader can find the following definition in [BM05, Page 64,equation (3.21)]:

Definition 1 (The Cox-Ingersoll-Ross-Process). It is assumed that r(t) is a stochasticprocess which is defined as follows:

dr(t) = κ (θ − r(t)) dt+ σ√r(t)dW (t)

r(0) = r0

The above mentioned equation behaves as follows:

W (t) is a Brownian motion and κ is the mean-reversion rate, θ is the mean-reversionlevel and σ is the volatility. These parameters are positive constants. Furthermore,2κθ > σ2 guarantees that r(t) is greater than zero for all t > 0.

The zero bond price can be given in closed-form in the CIR model which is neededlater on this thesis (see [BM05, Page 66]):

Properties 1 (Properties for defintion 1). The price of a zero bond at the time of twith the time-to-maturity T − t, of which the short rate is modeled with a CIR processis expressed as:

P (t, T ) = A(t, T ) · exp −B(t, T ) · r(t)

A(t, T ) =

[2 · h exp(κ+h)·(T−t)/2

2h+ (κ+ h)(exp(T − t)h − 1)

]2κθ/σ2

B(t, T ) =2(exp(T − t)h − 1)

2h+ (κ+ h)(exp(T − t)h − 1)

h =√κ2 + 2σ2

In this thesis, the CIR process is not used to model a short-rate but intensities, thefollowing parameters are named differently:

r(t) ⇒ y(t)

θ ⇒ µ

σ ⇒ ν

20

4.3. Approach to Simulate the Default Time

With the CIR process y(t) there will be stochastic modeling with mean reversionproperty in this thesis and with deterministic function ψ(t) the model can be fittedexactly to the survival probabilities. The stochastic intensity function defines as shownbelow:

λ(t) = ψ(t) + y(t) (4.3)

The cumulative intensity is, therefore, given as

Λ(t) =

t∫0

λ(u)du

=

t∫0

(ψ(u) + y(u)) du

=

t∫0

ψ(u)du+

t∫0

y(u)du

= Ψ(t) + Y (t) (4.4)

For more information on the CIR process the reader may have a look at [CIR85],[RSM04], and [BM05]. The CIR++ process is described in more detail in [BM05].

4.3. Approach to Simulate the Default Time

In this section, the actual process of simulating default processes is explained. It startswith a simple example and is extended to an applied case.

Example 1. Let an intensity model be given with constant intensity λ.

Q(τ > t) = E (exp −λ · t)Q(τ ≤ t) = 1− E (exp −λ · t)

Generate U ∼ U(0, 1)

U = 1− exp −λ · τ

Solve the equation for τ

τ = − log(1− U)

λ

21


Since the intensity in the previous example was chosen to be constant, it is possibleto use a closed-form expression in order to simulate the default processes. If this is notthe case and there are at least non-constant deterministic intensities, using an analyticformula is not possible. In the latter, the default times have to be solved iteratively:

Example 2. Let there be an intensity model with variable intensity as a default process:Let Λ(t) =

∫ t0λ(u)du be the cumulative intensity function.

Q(τ > t) = E (exp −Λ(t))Q(τ ≤ t) = 1− E (exp −Λ(t))

Simulate U ∼ U(0, 1)

U = 1− exp −Λ(t)− log(1− U) = Λ(t)

Find the smallest 0 ≤ t0 ≤ T which fulfills the following equation: Λ(t0) ≥ − log(1−U)

⇒ if a t0 was found, set τ = t0, otherwise set τ = T + ∆t. Where ∆t > 0

In the previous mentioned example it is shown, how a default time is determined incase of a non constant intensity function. The process in example 2 is used in orderto generate default times both for deterministic intensity function and for stochasticintensity function.

4.4. Calibration

In section 4.2 a variety of possibilities were presented to model intensity function. Be-low, different methods to calibrate the intensity function are demonstrated. In section4.4.1 the calibration for the deterministic intensity function, thus in section 4.4.2 thethe stochastic intensity function is calibrated. To be able to start the necessary sur-viving probabilities by means of CDS-Market, data has to be determined.There are two possibilities to specify the survival probabilities:

1. Based on either quoted CDS-Spreads,

2. or quoted Upfront-Payments.

The first possibility was market practice until 2009. With the fair CDS-Spreads fromthe market the survival probabilities can be calculated (see Theorem 2). In 2009, theCDS-Contracts were standardized, so that they are traded with standardized CDSSpreads ssmall. If the standardized spread differs to the fair spread, an Upfront-payment will be paid (see theorem 3). The survival probabilities can be determinedwith these Upfront Payments. This approach describes the current market practice.

22

4.4. Calibration

Theorem 2 (Bootstrapping the Survival Probabilities from market CDS-spreads).The implementation calculates the survival probabilities iteratively and starts with thesmallest time bucket. There are piecewise constant intensities. They are calculatedusing the following equation:

G(tm) =(1−RR) · (Am − Em)− s0(tm) · (Cm +Dm)

Bm

Furthermore the variables A, B, C, D, and E are determined iteratively, with thefollowing approach:

for m = 1:

A1 = P

(0,t0 + t1

2

)·G(t0),

B1 = s0(t1)

[∆1 · P (0, t1)− ∆1

2· P(

0,t0 + t1

2

)]+ (1−RR) · P

(0,t0 + t1

2

),

C1 = 0,

D1 =∆1

2· P(

0,t0 + t1

2

),

E1 = 0,

G1 =(1−RR) · (A1 − E1)− s0(t1) · (C1 +D1)

B1;

for m > 1:

Am = (RR− 1) · P(

0,tm−1 + tm

2

)− ssmall ·

(∆mP (0, tm)− ssmall · ∆m

2· P(

0,tm−1 + tm

2

)),

Bm =

m∑j=1

[(1−RR) · P

(0,tj−1 + tj

2

)+ ssmall · ∆j

2· P(

0,tj−1 + tj

2

)]·G(tj−1),

Cm =

m−1∑j=1

[(RR− 1) · P

(0,tj−1 + tj

2

)− ssmall ·

(∆jP (0, tj)− ssmall ·

∆j

2· P(

0,tj−1 + tj

2

))]·G(tj)

Proof.

Let s0(tm) given by the formula (3.8):

s0(tm) ≈(1−RR) ·

m∑j=1

P(

0,tj−1+tj

2

)· [G(tj−1)−G(tj)]

m∑j=1

[∆jP (0, tj) ·G(tj) +

∆j2· P(

0,tj−1+tj

2

)(G(tj−1)−G(tj))

]

23


It is: Nm = (1−RR) ·m∑j=1

P(

0,tj−1+tj

2

)· [G(tj−1)−G(tj)] and

Dm =m∑j=1

[∆jP (0, tj) ·G(tj) +

∆j2· P(

0,tj−1+tj

2

)(G(tj−1)−G(tj))

]

=NmDm

First of all, the denominator is transformed:

Dm =

m∑j=1

[∆jP (0, tj) ·G(tj) +

∆j

2· P(

0,tj−1 + tj

2

)(G(tj−1)−G(tj))

]=

=m∑j=1

[∆jP (0, tj) ·G(tj) +

∆j

2· P(

0,tj−1 + tj

2

)·G(tj−1)− ∆j

2· P(

0,tj−1 + tj

2

)·G(tj)

]

=

m∑j=1

[G(tj) ·

(∆jP (0, tj)−

∆j

2· P(

0,tj−1 + tj

2

))+

∆j

2· P(

0,tj−1 + tj

2

)·G(tj−1)

]

=

m∑j=1

[G(tj) ·

(∆jP (0, tj)−

∆j

2· P(

0,tj−1 + tj

2

))]+

m∑j=1

[∆j

2· P(

0,tj−1 + tj

2

)·G(tj−1)

]

= G(tm) ·(

∆mP (0, tm)− ∆m

2· P(

0,tm−1 + tm

2

))+

m∑j=1

[∆j

2· P(

0,tj−1 + tj

2

)·G(tj−1)

]

+

m−1∑j=1

[G(tj) ·

(∆jP (0, tj)−

∆j

2· P(

0,tj−1 + tj

2

))]

Then, the numerator is transformed:

Nm = (1−RR) ·m∑j=1

P

(0,tj−1 + tj

2

)· [G(tj−1 −G(tj))] =

= (1−RR) ·

[m∑j=1

P

(0,tj−1 + tj

2

)·G(tj−1)−

m∑j=1

P

(0,tj−1 + tj

2

)·G(tj)

]=

= (1−RR) ·

[m∑j=1

P

(0,tj−1 + tj

2

)·G(tj−1)−

m−1∑j=1

P

(0,tj−1 + tj

2

)·G(tj)

]

− (1−RR) ·[P

(0,tm−1 + tm

2

)·G(tm)

]

24

4.4. Calibration

The initial equation is multiplied with the denominator and generates the following:

s0(tm)

[G(tm) ·

(∆m · P (0, tm)− ∆m

2· P(

0,tm−1 + tm

2

))]+ s0(tm)

m−1∑j=1

[G(tj) ·

(∆jP (0, tj)−

∆j

2· P(

0,tj−1 + tj

2

))]

+ s0(tm)

m∑j=1

[∆j

2· P(

0,tj−1 + tj

2

)·G(tj−1)

]

= (1−RR) ·

[m∑j=1

P

(0,tj−1 + tj

2

)·G(tj−1)−

m−1∑j=1

P

(0,tj−1 + tj

2

)·G(tj)

]

− (1−RR) ·[P

(0,tm−1 + tm

2

)·G(tm)

]

The terms on the left side are summarized:

s0(tm)

[G(tm) ·

(∆m · P (0, tm)− ∆m

2· P(

0,tm−1 + tm

2

))]+ (1−RR) ·

[P

(0,tm−1 + tm

2

)·G(tm)

]= (1−RR) ·

[m∑j=1

P

(0,tj−1 + tj

2

)·G(tj−1)−

m−1∑j=1

P

(0,tj−1 + tj

2

)·G(tj)

]

− s0(tm)

m−1∑j=1

[G(tj) ·

(∆jP (0, tj)−

∆j

2· P(

0,tj−1 + tj

2

))]

− s0(tm)

m∑j=1

[∆j

2· P(

0,tj−1 + tj

2

)·G(tj−1)

]

25


The terms on the left side of the equation are further summarized

G(tm)

[s0(tm) ·

(∆m · P (0, tm)− ∆m

2· P(

0,tm−1 + tm

2

))]+G(tm) ·

[(1−RR) · P

(0,tm−1 + tm

2

)]= (1−RR) ·

[m∑j=1

P

(0,tj−1 + tj

2

)·G(tj−1)−

m−1∑j=1

P

(0,tj−1 + tj

2

)·G(tj)

]

− s0(tm)

m−1∑j=1

[G(tj) ·

(∆jP (0, tj)−

∆j

2· P(

0,tj−1 + tj

2

))]

− s0(tm)

m∑j=1

[∆j

2· P(

0,tj−1 + tj

2

)·G(tj−1)

]

Some substitutions are performed:

Am =

m∑j=1

P

(0,tj−1 + tj

2

)·G(tj−1)

Bm = s0(tm)

[∆m · P (0, tm)− ∆m

2· P(

0,tm−1 + tm

2

)]+ (1−RR) · P

(0,tm−1 + tm

2

)Cm =

m−1∑j=1

[G(tj) ·

(∆jP (0, tj)−

∆j

2· P(

0,tj−1 + tj

2

))]

Dm =

m∑j=1

[∆j

2· P(

0,tj−1 + tj

2

)·G(tj−1)

]

Em =

m−1∑j=1

P

(0,tj−1 + tj

2

)·G(tj)

G(tm) ·Bm = (1−RR) · (Am − Em)− s0(tm) · (Cm +Dm)

Solved to G(tm):

G(tm) =(1−RR) · (Am − Em)− s0(tm) · (Cm +Dm)

Bm

Theorem 3 (Bootstrapping the Survival Probabilities from market Upfront Pay-ments). The implementation calculates the survival probabilities iteratively and starts

26

4.4. Calibration

with the nearest time. They are calculated using the following equation:

G(tm) =Bm + U(tm)− Cm

Am

Furthermore the variables A, B, and C are determined iteratively, with the followingapproach:

for m = 1:

A1 = (RR− 1) · P(

0,t0 + t1

2

)− ssmall ·

(∆1P (0, t1)− ∆1

2· P(

0,t0 + t1

2

)),

B1 =

[(1−RR) · P

(0,t0 + t1

2

)+ ssmall · ∆1

2· P(

0,t0 + t1

2

)]·G(t0),

C1 = 0,

G(t1) =B1 + U(t1)− C1

A1;

for m > 1:

Am = (RR− 1) · P(

0,tm−1 + tm

2

)− ssmall ·

(∆mP (0, tm)− ∆m

2· P(

0,tm−1 + tm

2

)),

Bm = Bm−1 +

[(1−RR) · P

(0,tm−1 + tm

2

)+ ssmall · ∆m

2· P(

0,tm−1 + tM

2

)]·G(tm−1),

Cm = Cm−1 +

[(RR− 1) · P

(0,tm−2 + tm−1

2

)

−ssmall ·(

∆m−1P (0, tm−1)− ∆m−1

2· P(

0,tm−2 + tm−1

2

))]·G(tm−1),

Gm =Bm + U(tm)− Cm

Am

Proof.

PV Prot(t, T ) = PV Prem(t, T )

PV Prot(t, T ) = CBPV (t, T ) ·(ssmall + ε

)PV Prot(t, T ) = CBPV (t, T ) · ssmall + CBPV (t, T ) · ε

In the following t0 := t and tm := T :

27


First of all, the left side of the equation is transformed

PV Prot(t, T ) =

= (1−RR) ·m∑j=1

P

(0,tj−1 + tj

2

)· [G(tj−1)−G(tj)]

= (1−RR) ·m∑j=1

P

(0,tj−1 + tj

2

)·G(tj−1)− (1−RR) ·

m∑j=1

P

(0,tj−1 + tj

2

)·G(tj)

The right side of the equation is transformed

PV Prem(t, T ) =

= CBPV (t, T ) ·(ssmall + ε

)= CBPV (t, T ) · ssmall + CBPV (t, T ) · ε

CBPV (t, T ) · ε is equal to the Upfront-Payment U(tm)

= ssmall ·m∑j=1

∆jP (0, tj) ·G(tj) +∆j

2· P(

0,tj−1 + tj

2

)(G(tj−1)−G(tj)) + U(tm)

= ssmall ·m∑j=1

∆jP (0, tj) ·G(tj) + ssmall ·m∑j=1

∆j

2· P(

0,tj−1 + tj

2

)·G(tj−1)

− ssmall ·m∑j=1

∆j

2· P(

0,tj−1 + tj

2

)·G(tj) + U(tm)

= ssmall ·m∑j=1

(∆jP (0, tj)−

∆j

2· P(

0,tj−1 + tj

2

))·G(tj)

+ ssmall ·m∑j=1

∆j

2· P(

0,tj−1 + tj

2

)·G(tj−1) + U(tm)

So far, the left and right side of the starting equation was transformed separately. In thenext step, these two equations are equated and solved for the survival probability.

(1−RR) ·m∑j=1

P

(0,tj−1 + tj

2

)·G(tj−1)− (1−RR) ·

m∑j=1

P

(0,tj−1 + tj

2

)·G(tj)

= ssmall ·m∑j=1


∆j

2· P(

0,tj−1 + tj

2

))·G(tj)

+ ssmall ·m∑j=1

∆j

2· P(

0,tj−1 + tj

2

)·G(tj−1) + U(tm)

28

4.4. Calibration

The equation is transformed into:

− (1−RR) ·m∑j=1

P

(0,tj−1 + tj

2

)·G(tj)

− ssmall ·m∑j=1


∆j

2· P(

0,tj−1 + tj

2

))·G(tj) =

= (1−RR) ·m∑j=1

P

(0,tj−1 + tj

2

)·G(tj−1) + ssmall ·

m∑j=1

∆j

2· P(

0,tj−1 + tj

2

)·G(tj−1) + U(tm)

G(tj) will be factorized on the left side

m∑j=1

[(RR− 1) · P

(0,tj−1 + tj

2

)− ssmall ·

(∆jP (0, tj)−

∆j

2· P(

0,tj−1 + tj

2

))]·G(tj) =

=

m∑j=1

[(1−RR) · P

(0,tj−1 + tj

2

)+ ssmall · ∆j

2· P(

0,tj−1 + tj

2

)]·G(tj−1) + U(tm)

The summation will be split on the left side:

G(tm) ·[(RR− 1) · P

(0,tm−1 + tm

2

)− ssmall ·

(∆mP (0, tm)− ∆m

2· P(

0,tm−1 + tm

2

))]+

m−1∑j=1

[(RR− 1) · P

(0,tj−1 + tj

2

)− ssmall ·

(∆jP (0, tj)−

∆j

2· P(

0,tj−1 + tj

2

))]·G(tj) =

=

m∑j=1

[(1−RR) · P

(0,tj−1 + tj

2

)+ ssmall · ∆j

2· P(

0,tj−1 + tj

2

)]·G(tj−1) + U(tm)

G(tm) will be factorized on the left side

G(tm) ·[(RR− 1) · P

(0,tm−1 + tm

2

)− ssmall ·

(∆mP (0, tm)− ∆m

2· P(

0,tm−1 + tm

2

))]=

=

m∑j=1

[(1−RR) · P

(0,tj−1 + tj

2

)+ ssmall · ∆j

2· P(

0,tj−1 + tj

2

)]·G(tj−1) + U(tm)

−m−1∑j=1

[(RR− 1) · P

(0,tj−1 + tj

2

)− ssmall ·

(∆jP (0, tj)−

∆j

2· P(

0,tj−1 + tj

2

))]·G(tj)

29


Some substitutions are performed:

Am = (RR− 1) · P(

0,tm−1 + tm

2

)− ssmall ·

(∆mP (0, tm)− ∆m

2· P(

0,tm−1 + tm

2

))Bm =

m∑j=1

[(1−RR) · P

(0,tj−1 + tj

2

)+ ssmall · ∆j

2· P(

0,tj−1 + tj

2

)]·G(tj−1)

Cm =

m−1∑j=1

[(RR− 1) · P

(0,tj−1 + tj

2

)− ssmall ·

(∆jP (0, tj)−

∆j

2· P(

0,tj−1 + tj

2

))]·G(tj)

G(tm) ·Am = Bm + U(tm)− Cm

The equation is solved for G(tm)

G(tm) =Bm + U(tm)− Cm

Am

4.4.1. Calibrate the Deterministic Intensity Function

In this section two options to calibrate the deterministic intensity function are pre-sented. The first possibility is shown in theorem 4. To apply this method, the defaulttimes are to be known.

Theorem 4 (Bootstrapping of the Intensity function). For given survival probabilitiesG1, . . . , Gm for the points in time t1, . . . , tm, the intensity function with piecewiseconstant intensities can be determined in this way:

λi =− log (Gi) +Hi

∆i

Hi = Hi−1 − λi−1 ·∆i−1

=

i−1∑j=0

λj ·∆j

Where we let t0 = 0, H1 = 0, and ∆i = ti − ti−1 for i = 1, . . . , m.

Proof.

Gi = Q(τ ≥ ti)

By definition we have:

= exp −Λ(ti)

30

4.4. Calibration

Because of Λ(ti) =ti∫0

λudu we have:

= exp

−ti∫

0

λudu

Since

ti∫0

λudu =i∑

j=0

λj ·∆j with ∆i = ti − ti−1 we obtain

= exp

−i∑

j=0

λj ·∆j

= exp

−λi ·∆i +

i−1∑j=0

λj ·∆j

Solving the equation for λi yields:

λi =

− log (Gi)−i−1∑j=0

λj ·∆j

∆i

By definition and setting: Hi = −i−1∑j=0

λj ·∆j we arrive at

=− log (Gi) +Hi

∆i

which proves the theorem.

A crude approximation for the intensity function returns the following theorem (see[MRW06, Page 179]):

Theorem 5 (The Credit-Triangle). Under the assumption of a constant LGD, a flatyield curve, and a flat CDS spread curve, it is possible to approximate the intensity bythe following formula:

λ ≈ s

LGD.

31


The proof for theorem 5 can be found in [MRW06, Page 179]. Below is an intensityfunction determined, using the mid spreads from table 4.1, the yield curve from figure4.1, and assuming a constant LGD of 60%.

Maturity (in years) 0.5 1 2 3 4 5 7 10Bid Spread (in bps) 22.6 23.0 33.1 50.8 72.4 87.0 98.6 97.0Ask Spread(inbps) 25.6 25.0 36.0 53.8 76.4 87.0 102.6 97.0Mid Spread (in bps) 24.1 24.0 34.6 52.3 74.4 87.0 100.6 97.0

Table 4.1.: CDS-spreads for various maturities of Deutsche Bank (Data fromBloomberg, as of May 3, 2011)

1 2 3 4 5 6 7 8 9 10

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

Maturity in years

Yie

ld (

in p

erce

nt)

Figure 4.1.: Yield Curve for German Federal Bonds (Data from Deutsche Borse, as ofMay 12, 2011)

Using theorem 2 it is possible to derive the required survival probabilities. Theresults are shown in the following table 4.2.

Maturity (in years) 0.5 1 2 3 4 5 7 10Survival probility (in percent) 99.80 99.60 98.86 97.41 95.11 92.90 88.75 85.05

Table 4.2.: Survival probabilities for various maturities of Deutsche Bank (Calculatedusing theorem 2)

With the survival probabilities, and theorem 4 it is possible to define the intensity

32

4.4. Calibration

function, the results are presented below in table 4.3.

Maturity (in years) 0.5 1 2 3 4 5 7 10Intensity 0.0040 0.0040 0.0075 0.0148 0.0241 0.0236 0.0229 0.0134

Table 4.3.: Default intensities calibrated from market survival probabilities

Now, the intensity function is approximated applying the Credit-Triangle.

Maturity (in years) 0.5 1 2 3 4 5 7 10Intensity 0.0040 0.0040 0.0075 0.0147 0.0239 0.0234 0.0229 0.0141

Table 4.4.: Default intensities determined with the Credit-Triangle

This approximation is compared with the reference curve from table 4.3:

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)

Figure 4.2.: Relative Deviation between both intensity functions

It is clearly recognizable that the approximation by the credit triangle is good, thedeviations are between 0.4% in the short term and 1.75% in the long run. Therefore,the Credit-Triangle is only a rule of thumb.

33


4.4.2. Calibrate the Stochastic Intensity Function

This section presents a way to calibrate the stochastic intensity function to marketdata.

In this case, both the stochastic and the determinstic part of the intensity functionmust be calibrated. In the first step, the stochastic part is calibrated using marketCDS spreads. For stochastic part of the intensity function the following vector beta tcalibrated to market data:

βt = κ, y(t), µ, ν

As shown above, using the CDS spreads, one can define the values of the deterministicintensity function λt for various time points t. By definition, the survival probabilities

are E(

exp

−

t∫0

λudu

), where λu follows a CIR process. Therefore, the expected

model survival probabilities are equivalent to the price of a zero bond with the sameparameters. As seen in the following derivation, the intensities of the deterministic casecorrespond to the negative logarithmic derivative of the zero bond price (see [BM05,Page 794]).

PCIR(0, t) = E

exp

−t∫

0

λudu

− log(PCIR(0, t)

)=

t∫0

λudu

The equation is derived:

− ∂

∂tlog(PCIR(0, t)

)=

∂

∂t

t∫0

λudu

It is the fundamental theorem of calculus:

− ∂

∂tlog(PCIR(0, t)

)= λt

Now, the above shown equation is discretized:

−(log(PCIR(0, t+ ∆t)

)− log

(PCIR(0, t)

))∆t

= λt (β0)

34

4.4. Calibration

To calibrate the stochastic part, the following optimization problem must be solved:

β0 = minβ0

E((

λt − λt (β0))2)

(4.5)

= minβ0

1

N

N∑i=1

(λt − λt (β0)

)2

(4.6)

Where N corresponds to the number of given CDS spreads for different time points.The optimal parameter vector β0, of course must meet the requirements of the CIRprocess.

For the mid CDS spreads from table 4.1 and the above procedure, the optimal param-eter vector β0 is determined. The minimized differences for this parameter vector areshown in Figure 4.3 and the parameter vector is given in table 4.5. The result for thisparameter combination is a Mean Square Error (MSE) in the amount of 0, 0000238242.

0 1 2 3 4 5 6 7 8 9 100

0.005

0.01

0.015

0.02

0.025

Market implied deterministic intensitiesBest approximation

Figure 4.3.: Market implied deterministic intensity function compared to the best ap-proximation using a CIR process

y0 κ µ ν0.0004302 0.5428162 0.0182458 0.0000266

Table 4.5.: Calibrated Parameters for the CIR process

After the stochastic part of the intensity function is calibrated, below the determinis-tic part is produced. With this deterministic part the default process will be calibrated

35


exactly to the market implied survival probabilities (see Theorem 6, [BCB08, Page 6]).

Theorem 6 (Calibrating the deterministic Shift of the CIR++). For given marketimplied survival probabilities Q(τ > t)market and zero bond prices PCIR(0, t), one canset the deterministic function of the CIR++ model using the following equation:

Ψ(t) = log

(PCIR(0, t)

Q(τ > t)market

)Proof.

Per definition is:

Q(τ > t)market = E (exp −Λ(t))

According to equation (4.4):

= E (exp − (Ψ(t) +X(t)))= E (exp −Ψ(t) · exp −X(t))= E (exp −Ψ(t)) · E (exp −X(t))

Since X(t) is a CIR process it is: E (exp −X(t)) = PCIR(0, t)

= E (exp −Ψ(t)) · PCIR(0, t)

Since Ψ(t) is not stochastic, it follows:

= exp −Ψ(t) · PCIR(0, t)

The equation is solved for Ψ(t):

Ψ(t) = − log

(PCIR(0, t)

Q(τ > t)market

)

Using the data from table 4.2, 4.5, and theorem 6 the nodes are calculated andshown in the following table:

36

4.4. Calibration

t 0.5 1 2 3 4 5 7 10Ψ(t) -0.000511 -0.0033 -0.0021 0.0064 0.0133 0.0241 0.0101

Table 4.6.: Nodes for the deterministic function Ψ(t)

With these nodes it is possible to interpolate the function. One can interpolate thefunction constant or linear between the nodes.

0 1 2 3 4 5 6 7 8 9 10−0.005

0

0.005

0.01

0.015

0.02

0.025

t

Ψ(t

)

(a) Function Ψ(t) with konstant interpola-tion

0 1 2 3 4 5 6 7 8 9 10−0.005

0

0.005

0.01

0.015

0.02

0.025

t

Ψ(t

)

(b) Function Ψ(t) with linear interpolation

Figure 4.4.: Calibrated function Ψ(t) in two versions

In the later section, these two interpolation methods are compared to each otherand evaluated.For further reading [BM06] is recommended.

37

5. Commodities and CommodityModels

To gain understanding of the various Commodity Models one first needs to grasp thefundamental properties of Commodities. These properties are presented in section 5.1.In section 5.2 the Theory of Storage is introduced. And last but not least in section5.3 several models to simulate the commodity prices are discussed.

5.1. Properties of Commodity Prices

In this section, characteristics of commodities are presented. Not all properties dis-cussed apply to each commodity.The initially shown property is the seasonality , which means that the price, the volatil-ity or other risk factors depend on the season. The seasonality is one of the majordifferences between stocks, bonds, other conventional asset classes and commodities,which is confirmed by Borokova and Geman in [BG06, Page 168]:

Traditional pricing, hedging and volatility modelling techniques from eq-uity and fixed income markets are not directly applicable to commodities,whose specific properties need to be taken into account. One such char-acteristic feature is seasonality. Seasonality in the spot prices of manyagricultural and energy commodities-naturally arising from seasonal pat-terns in supply (e.g., harvest) and demand (e.g., cold weather)...

and by Richter and Sørensen in [RS02, Page 3]:

Seasonality is known to be one of the empirical characteristics that makecommodities strikingly different from stocks, bonds, and other conventionalfinancial assets (...), and is especially important for agricultural commodi-ties with a seasonal harvesting pattern.

Heating oil is the most typical case for seasonality in the spring prices fall because ofweak demand, and in the Autumn, the prices rise due to rising demand. Borokovaand Geman present in [BG06, Page 168] a similar excample for the gas and electricitymarket:

39

5. Commodities and Commodity Models

In the case of energy commodities such as electricity and natural gas,prices are driven by seasonal demand: for instance, in the UK there isalways a higher demand for gas and electricity during winter months, re-sulting in a price premium for futures expiring then.

Furthermore, there are seasonalities in the agricultural commodities market as alsoshown by Geman and Borokova in [BG06, Page 168]:

For agricultural commodities (e.g., soybean, wheat, coffee, cocoa), pricesare driven by seasonal supply, hence they are generally higher before theharvest. Seasonal forward curves are not observed in the crude oil market,which is a world market.

For further information to this certian property [RS02], [BPR09], and [BG06] are rec-ommended.The next characteristic of commodities is the mean reverting property. This charac-teristic in this context represents the fact that prices or other risk factors move backto their long-term means. Deng writes in [Den00, Page 4]:

The most noticeable price behavior of energy commodities is mean-reverting. When the price of a commodity is high, its supply tends toincrease thus putting a downward pressure on the price; when the spotprice is low, the supply of the commodity tends to decrease thus providingan upward lift to the price.

Moreover, Routledge, Seppi and Spatt confirm that many Commodities have the meanreverting characteristic (see [RSS00, Page 1]):

Spot and future prices are mean reverting for many commodities.

For more information about the above mentionned characteristic the author refers to[BCSS95], [Pin99] and [Gem05b].

A phenomenon in the commodity market are the spikes, which occur in particularin the energy sector. At a spike, the price jumps up and returns a little later tothe old level. Spikes occur for example in short-term shortage of supply. Borgmanndiscusbject in his dissertation ([Bor04, Page 122]):

Spikes result from an extreme imbalance between supply and demandon a short-term basis. The most common reason for a reduced supplyof electricity are power plant outages. An increased demand is causedfor example by extreme climatic conditions. In the electricity market thedemand is almost inelastic because it is a derived demand.

Spikes are the greatest risk drivers in the electricity market as shown by De Jong andHuismann in [Hd03, Page 6]:

40

5.2. The Theory of Storage

We focus so much on spikes, because they are the main source of risk inelectricity markets and the main value-driver of options.

Spikes can also be found in food prices as shown by Piese and Thirtle in [PT09].For further information [Den00], [Gem05a], [CFG08], [HM01] and [BS01] are recom-mended.The last discussed property is the storability . There is a distinction between thestorable commodities (e.g. coal, crude oil) and not storable commodities (e.g. elec-tricity, weather, perishable food). For storable commodities the Theory of Storageplays an important role, which is presented in the following section 5.2. For furtherinformation of this property [Tur82], [Gem05a], and [BM10] are recommended.

5.2. The Theory of Storage

According to the classical theory the Future price follows the following equation (see[Hul03, Equation (3.16), page 58]):

Ft = St exp(rf + u)(T − t) (5.1)

Here Ft the current Future price for the commodity denotes for a time T in the future.St is the current spot price of commodities, rf is the risk-free interest rate and ucorresponds to the storage costs.However, the equality is not given for commodities. At this point Kaldor’s Theory ofStorage attaches (see [Kal39]). A good representation of this theory can be found in[Ben04, Page 5]:

The Theory of Storage [...], states that the spread between spot and fu-tures prices is determined by fundamental supply-and-demand conditions.Specifically, the behavior of commodity futures and spot prices are relatedto storage costs, inventory levels and convenience yields.

According to this, the equation (5.1) is extended by the Convenience Yield (see [Hul03,Equation (3.21), page 60]):

Ft = St exp(rf + u− δ)(T − t) (5.2)

Where δ is the convenience yield . It can be determenistic or stochastic. Pindyckdescribes the properties of the convenience yield as follows (see [Pin01, Page 16]):

Note that the convenience yield obtained from holding a commodity isvery much like the dividend obtained from holding a company’s stock.

Other important features of the convenience yield are shown by Hull in [Hul03, Page60]:

41


The convenience yield reflects the market’s expectations concerning thefuture availability of the commodity. The greater the possibility that short-ages will occur, the higher the convenience yield. If users of the commodityhave inventories, there is very little chance of shortages in the near futureand the convenience yield tends to be low. On the other hand, low inven-tories tend to lead to high convenience yields.

Another important term in this context is the cost of carry , it is defined in [Hul03,Page 60] as follows:

The relationship between futures prices and spot prices can be summa-rized in terms of the cost of carry. This measures the storage cost plusthe interest that is paid to finance the asset less the income earned on theasset. For a non-dividend-paying stock, the cost of carry is r, because thereare no storage costs and no income is earned; for a stock index, it is r− q,because income is earned at rate q on the asset. [...] for a commodity withstorage costs that are a proportion u of the price, it is r + u; and so on.

Since a commodity achieves no returns, it follows that c = rF + u where c denotes thecost of carry. The futures price in equation (5.2) can be shown as it is stated below:

Ft = St exp(c− δ)(T − t) (5.3)

For further information to the theory of storage, the convenience yield and the cost-of-carry [Kal39], [Ben04], [AM08], [Hul03], [WW05], and [SRN08] are suggested.

If the spot price S0 is less than the futures price F0 then the market situation iscalled contango. This is the case if the convinience yield is greater than the storageand financing costs (cost of carry). In this market situation the forward curve is up-ward sloping.If the value of a Future with a short maturity is higher than with a long maturity,then the market is in a situtaion which is called backwardation. This is the case if theconvenience yield is smaller than the storage and financing costs (cost of carry). Thismarket condition can for example arise in case of supply shortages. In this case theforward curve is downward sloping. Ribeiro and Hodges write for this purpose (see[RH04, Page 7]):

According to this, the curve is in backwardation if the convenience yieldδ is greater than (rf + u); the curve is in contango if the convenience yieldis less than (rf + u), that is the situation if the forward curve increaseswith time to maturity.

For further information to contango and backwardation the author advises [Cha85],[Roc67], and [RH04].

42

5.3. Models for Commodity Prices


In this section various modelling possibilities for commodity prices are demonstrated.First, in subsection 5.3.1 a one-factor model by Schwartz is exhibited. Afterwards, insubsection 5.3.2 the Gibson-Schwartz model is presented. The last model discussed isthe Smith-Schwartz model and also various modifications are presented.

5.3.1. The Schwartz One Factor Model

First, we present a one-factor model according to Schwartz to generate spot prices(see [Sch97, Page 926]). Now this model is called the Schwartz One-Factor model(Schwartz-F1 model).

Definition 2. In the Schwartz One-Factor model the spot price is modelled using thefollowing equation:

dS(t) = κ (µ− log (S(t)))S(t)dt+ σS(t)dW (t)

Where κ > 0 is the mean reversion rate with which the process returns to the mean-reversion level µ and σ > 0 is the volatility.

With the Ito lemma, the equation can be simplified as follows:

dS(t) = κ (µ− log (S(t)))S(t)dt+ σS(t)dW (t)

It is X(t) = logS(t), with the lemma of Ito follows:

dX(t) =∂ expX(t)

∂S(t)dS(t) +

1

2

∂2 expX(t)∂S(t)2

· (S(t) · σ1)2dt

=1

S(t)dS(t)− 1

2

1

S(t)2· (S(t) · σ)

2dt

By definition it is: dS(t) = κ (µ− log (S(t)))S(t)dt+ σS(t)dW (t)

=1

S(t)(κ (µ− log (S(t)))S(t)dt+ σS(t)dW (t))− 1

2

1

S(t)2· (S(t) · σ)

2dt

= (κ (µ−X(t)) dt+ σdW (t))− 1

2σ2dt

=

(κ (µ−X(t))− 1

2σ2

)dt+ σdW (t)

=

(κ (µ−X(t))− κ 1

2κσ2

)dt+ σdW (t)

= κ

((µ− σ2

2κ

)−X(t)

)dt+ σdW (t)

43


Letting α = µ− σ2

2κ , we obtain

= κ (α−X(t)) dt+ σdW (t)

Because of the simplifications, the stochastic differential equation (SDE) has turned tojust an Ornstein-Uhlenbeck process. The solution is known for this process (see sectionA.3):

X(t) = X(s) · exp −κ(t− s)+ α (1− exp−κ(t− s)) + σ exp −κ · (t− s)t∫s

exp κ · u dWu

Furthermore, the distribution of X(t) is shown below.

Theorem 7 (Distribution of the Schwartz One-Factor model). The Distribution ofX(t) for a given X(s) follows a Normal distribution:

X(t)|X(s) ∼ N(X(s) · exp −κ(t− s)+ α (1− exp−κ(t− s)) , σ

2

2κ(1− exp−2κ(t− s))

)Proof.


dX(t) = κ (α−X(t)) dt+ σdW (t)

follows Ornstein-Uhlenbeck process, therefore we yield

E(X(t)|X(s)) = X(s) · exp −κ(t− s)+ α (1− exp−κ(t− s))V(X(t)|X(s)) = Cov(X(t), X(t)|X(s))

=σ2

2κ(1− exp−2κ(t− s))

Therefore we have

X(t)|X(s) ∼ N(X(s) · exp −κ(t− s)+ α (1− exp−κ(t− s)) , σ

2

2κ(1− exp−2κ(t− s))

)The distribution of X(t) can be determined with its properties:

44


Corollary 1 (Distribution of the spot prices). The spot price S(t) for a given X(s)follows a Lognormal distribution:

S(t)|X(s) ∼ LogN (E(X(t)|X(s)), V(X(t)|X(s)))

With expected value:

E(S(t)|χ(s), ξ(s)) = exp

X(s) · exp −κt+ α (1− exp−κt) +

1

2V(X(t)|X(s))

In the equation (3.13) it is demonstrated that the Forward (or Future) price with

maturity T − t is equal to the expected value of the spot price at the same point intime. The result is the expected value of the commodity by the Schwartz-F1 model:

F (t, T ) = exp

X(t) · exp −κT+ α (1− exp−κT) +

1

2V(X(T )|X(t))

(5.4)

Because, X(T ) is normally distributed, it follows that log (F (t, T )) is also Normaldistributed with a variance of σ2

F :

σ2F (t, T ) =

σ2

2κ(1− exp−2κ(T − t)) e−2kχ(T−t)

Thus the Black76-Equation can be applied for the pricing of a European option on aForward:


while for a European Put-option it is:


with ε =log(F0(T )/K)+ 1

2σ2F (t, T )

σF (t, T ) .

For further information to this model [Sch97] is recommded.

The advanced Schwartz One-Factor Model

A modification of the classical Schwartz one-factor Model is called the advanced SchwartzOne-Factor model . This modification adds an deterministic function to the factor X.With this deterministic function, the model can be calibrated to the market forwardcurve.

45


Definition 3 (Advanced Schwartz One Factor Model). The spot price follows

log(St) = X(t) + ϕ(t)

In the advanced Schwartz one-factor model the factor X(t) has the same distributionas in the classical Schwartz One Factor Model. In the following the distribution of S(t)is shown: this model has the same distribution as the classical Schwartz one-factormodel with the only difference that to the expected value of X(t) the deterministicfunction ϕ(t) is added.

Corollary 2 (Distribution of the spot prices). The spot price S(t) for a given X(s)follows a Lognormal distribution:

S(t)|X(s) ∼ LogN (E(X(t)|X(s)) + ϕ(t), V(X(t)|X(s)))

This distribution holds the following expected value:


X(s) · exp −κt+ α (1− exp−κt) + ϕ(t) +

1

2V(X(t)|X(s))

In equation (3.13) it is shown that the Forward (or Future) price with the maturity

T − t is equal to the expected value of the spot price at the same point of time. Itfollows the expected value of the commodity by the Schwartz-F1 model:

F (t, T ) = exp

X(t) · exp −κT+ α (1− exp−κT) + ϕ(T ) +

1

2V(X(T )|X(t))

(5.7)

Because, X(T ) is Normal distributed, it follows that log (F (t, T )) is also Normal dis-tributed with variance σ2

F :

σ2F (t, T ) =

σ2

2κ(1− exp−2κ(T − t)) e−2kχ(T−t)



And for a European Put-option the formular is:



2σ2F (t, T )

σF (t, T ) .

46


5.3.2. The Gibson-Schwartz model

The Gibson-Schwartz model was initialized by Gibson and Schwartz in [GS90]. In theliterature it is also called Stochastic Convenience Yield Model .The spot price of the commodity is subject to a Two-Factor model, where the first fac-tor S(t) models the spot price and follows a geometric Brownian motion with stochas-tic drift term. A part of this stochastic drift term is the second factor δ(t) which isan Ornstein-Uhlenbeck process. The convenience yield is modelled with this secondfactor.

Definition 4 (The Gibson-Schwartz model). The spot price S(t) and the convenienceyield δ(t) are defined as follows:

dS(t) = (µ− δ(t))S(t)dt+ σ1S(t)dW1(t)

dδ(t) = κ (α− δ(t)) dt+ σ2dW2(t)

dW1(t)dW2(t) = ρdt

Where σ1, σ2, and κ > 0.

The meaning of the parameter is exhibited in the following table 5.1:

Parameter Model descriptionµ Long-term total returnσ1 Volatility of the spot pricedW1 Spot price process incrementsκ Mean-reverting coefficientα Long-term cenvenience yieldσ2 Volatility of the convenience yielddW2 Convenience yield process incrementsρ Correlation between the stochastic processes

Table 5.1.: Denotation of the parameters of the Gibson-Schwartz model

The first equation can by simplified by the transformation S(t) = expX(t) andapplying the lemma of Ito, as it is shown below:

dS(t) = (µ− δ(t))S(t)dt+ σ1S(t)dW1(t)

It is X(t) = logS(t) , with Ito’s Lemma we obtain:

dX(t) =∂ expX(t)

∂S(t)dS(t) +

1

2

∂2 expX(t)∂S(t)2

· (S(t) · σ1)2dt

=1

S(t)dS(t)− 1

2

1

S(t)2· (S(t) · σ1)

2dt

47


By definition it is: dS(t) = (µ− δ(t))S(t)dt+ σ1S(t)dW1(t)

= ((µ− δ(t)) dt+ σ1dW1(t))− 1

2σ2

1dt

=

(µ− δ(t)− 1

2σ2

1

)dt+ σ1dW1(t)

As an illustrative interpretation of this model Schwartz wrote in [Sch97, Page 928]:

In this model the commodity is treated as an asset that pays a stochasticdividend yield δ.

The price of a Future or Forward under the Gibson-Schwartz model is given by (see[Sch97, Page 928])

F (t, T ) = S · exp

−δ 1− exp−κ(T − t)

κ+A(t, T )

(5.10)

where

A(t, T ) =

(µ− α+

1

2

σ22

κ2− σ1σ2ρ

κ

)+

1

4σ2

2

1− exp−2κ(T − t)κ3

+

(ακ+ σ1σ2ρ−

σ22

κ

)1− exp−κ(T − t)

κ2

The derivation of the above equation can be found in [Bje91].For further information the author refers to [GS90], [HHK09], [Sch97], [RH04], and[LC03].

5.3.3. The Smith-Schwartz model

The next model we would like to present for modelling commodity prices was intro-duced by Smith and Schwartz in [SS00]. Here it is called Smith-Schwartz model andin the literature it is also known as Short-Term/Long-Term Model .In contrast to the Gibson-Schwartz model, this model does not use a convenienceyield. The spot price consists of the sum of two stochastic processes. The first processξ(t) models the long-term equilibrium level and follows a geometric Brownian motion.Whereas the second factor χ(t) models the short-term fluctuation at the equilibriumlevel. This second factor follows an Ornstein-Uhlenbeck process and has according tothis the mean reverting property.

48


Definition 5 (The Smith-Schwartz model). The spot price is defined is:

log(St) = χ(t) + ξ(t)

Where both factors follow the stochsatic differential equation system

dχ(t) = −kχ · x(t)dt+ σχdWχ

dξ(t) = µξ · dt+ σξdWξ

dWχdWξ = ρχ, ξdt

With kχ, σχ, σξ > 0, and ρχ, ξ ∈ [−1, 1].

The meaning of the parameters are shown in the following table 5.2:

Parameter Model descriptionκ Short-term mean-reversion rateσχ Short-term volatilitydWχ Short-term process incrementsµξ Equilibrium drift rateσξ Equilibrium volatilitydWξ Equilibrium process incrementsρχ, ξ Correlatin in increments

Table 5.2.: Denotation of the parameters of the Smith-Schwartz model (see [SS00,Table 1])

The common distribution from both factors can be determined by solving the dif-ferential equation system from definition 5:

Theorem 8 (Distribution of the Smith-Schwartz model). The common distributionof the factors χ(t) and ξ(t) for given χ(s) and ξ(s) underlie a bivariate Normal distri-bution:[χ(t)ξ(t)

]∣∣∣∣χ(s), ξ(s)

∼ N

([χ(s) exp (−kχ (t− s))ξ(s) + µξ (t− s)

],

[σ2χ

2kχ(1− exp (−2kχ(t− s))) Covχ, ξ(s, t)

Covχ, ξ(s, t) σ2ξ (t− s)

])Covχ, ξ(s, t) = ρχ, ξ

σχσξkχ

(1− exp(−kχ(t− s)))

Where kχ, σχ, σξ > 0, and ρχ, ξ ∈ [−1, 1]

The results of the theorem 8 is proved below. A similar proof can be found in thethesis of Roriz Soares de Carvalho (see [Ror10]).

49


Proof.


dχ(t) = −kχ · χ(t)dt+ σχdWχ

describes an Ornstein-Uhlenbeck process, since we obtain

E(χ(t)|χ(s)) = χ(s) · exp−kχ(t− s)V(χ(t)|χ(s)) = Cov(χ(t), χ(t)|χ(s))

According to the properties of the Ornstein-Uhlenbeck process the following statementfollows

=σ2χ

2kχ(1− exp−kχ2 (t− s))

Therefore follows:

χ(t)|χ(s) ∼ N

(χ(s) · exp −kχ · (t− s) ,

σ2χ

kχ · 2(1− exp −kχ · 2 · (t− s))

)

By definition we have:


Is a geometric Brownian motion, hence follows:

E(ξ(t)|ξ(s)) = ξ(s) + µξ · (t− s)V(ξ(t)|ξ(s)) = σ2

ξ · (t− s)

therefore follows:

ξ(t)|ξ(s) ∼ N(ξ(s) + µξ · (t− s), σ2

ξ · (t− s))

50


Covχ, ξ(s, t) = Cov (χ(t), ξ(t)|χ(s), ξ(s))

= E ((χ(t)− E(χ(t)|χ(s))) · (ξ(t)− E(ξ(t)|ξ(s))) |χ(s), ξ(s))

= E

(χ(s) · exp −kχ · (t− s)+ σχ exp −kχ · tt∫s

exp kχ · u dWχ(u)− (χ(s) · exp −kχ · (t− s))

· (ξ(s) + µξ · (t− s) + σξ · (Wξ(t)−Wξ(s))− (ξ(s) + µξ · (t− s))) |χ(s), ξ(s)

)

= E

σχ exp −kχ · tt∫s

exp kχ · u dWχ(u)

· (σξ · (Wξ(t)−Wξ(s))) |χ(s), ξ(s)

= σχ exp −kχ · tσξ · E

t∫s

exp kχ · u dWχ(u)(Wξ(t)−Wξ(s))


t∫s


t∫s

dWξ(u)


t∫s

t∫s

exp kχ · u dWχ(u)dWξ(u)

According to the definition 5 it is true that dWχdWξ = ρχ, ξdt


t∫s

exp kχ · u ρχ, ξdu

= σχ exp −kχ · tσξρχ, ξ · E

t∫s

exp kχ · u du

Solving the integralt∫s

exp kχ · u du = 1kχ· (exp kχ · t − exp kχ · s)

= σχ exp −kχ · tσξρχ, ξ · E(

1

kχ· (exp kχ · t − exp kχ · s)

)

51


Because of 1kχ· (exp kχ · t − exp kχ · s) is deterministic it follows:

=σχσξρχ, ξ

kχ· (exp −kχ · t · exp kχ · t − exp −kχ · t · exp kχ · s)

=σχσξρχ, ξ

kχ· (1− exp −kχ · (t− s))

Therefore underlying χ(t) and ξ(t) the following common Normal distribution:[χ(t)ξ(t)

]∣∣∣∣χ(s), ξ(s)

∼ N

([χ(s) exp (−kχ (t− s))ξ(s) + µξ (t− s)

],

[σ2χ


Covχ, ξ(s, t) σ2ξ (t− s)

])Covχ, ξ(s, t) = ρχ, ξ

σχσξkχ

(1− exp(−kχ(t− s)))

The common distribution was shown in the last theorem, as the properties of theNormal distribution follow the distribution of log(S(t)) = χ(t) + ξ(t):

Corollary 3 (Distribution of the Prices). The spot price S(t) for known χ(s) and ξ(s)with s < t can be described with the following distribution:

S(t)|χ(s), ξ(s) ∼ LogN (E(log(S(t))|χ(s), ξ(s)), V(log(S(t))|χ(s), ξ(s)))

E(log(S(t))|χ(s), ξ(s)) = χ(s) · exp−kχ(t− s)+ ξ(s) + µξ · (t− s)

V(log(S(t))|χ(s), ξ(s)) =σ2χ

2kχ(1− exp−kχ2 (t− s)) + σ2

ξ · (t− s) + 2 · Covχ, ξ(s, t)


χ(s) · exp−kχ(t− s)+ ξ(s) + µξ · (t− s) +

1

2V(log(S(t))|χ(s), ξ(s))

In equation (3.13) it is shown that the Forward (or Future) price with maturity of

T − t is equal to the expected value of the spot price at the same point in time. Theexpected value of the commodity by the Smith-Schwartz model results in:

F (t, T ) = exp

χ(t) · exp−kχ(T − t)+ ξ(t) + µξ · (T − t) +

1

2V(log(S(T ))|ξ(t), ξ(t))

(5.11)

Because, χ(t) and ξ(t) are Normal distributed, so log (F (t, T )) is also Normal dis-tributed with variance σ2

F :

σ2F (t, T ) =

σ2χ

2kχe−2kχ(T−t) (1− exp−kχ2 (t− s)) + σ2

ξ · (t− s)

+ 2 · ρχ, ξσχσξkχ

e−kχ(T−t) (1− exp(−kχ(t− s)))

52




And for a European Put-option it is:



2σ2F (t, T )

σF (t, T ) .

The Smith-Schwartz model can be transformed into the Gibson-Schwartz model asshown from Smith and Schwartz in [SS00, Page 899]:

Relationships between the variables:

χ(t) =1

κ(δ(t)− α)

ξ(t) = X(t)− χ(t) = X(t)− 1

κ(δ(t)− α)

The following table 5.3 contains the parameter transformations from the Smith-Schwartzmodel to the Gibson-Schwartz model.

Smith-Schwartz model Gibson-Schwartz modelκ κσχ σ2/κdWχ dW2

µξ(µ− α− 1

2σ21

)σξ

(σ2

1 + σ22/κ2 − 2ρσ1σ2/κ

)1/2dWξ (σ1dW1 − (σ2/κ) dW2)

(σ2

1 − σ22/κ2 − 2ρσ1σ2/κ

)−1/2

ρχ, ξ (ρσ1 − σ2/κ)(σ2

1 + σ22/κ2 − 2ρσ1σ2/κ

)−1/2

λχ λ/κλξ µ− r − λ/κ

Table 5.3.: Relationships between the parameters (see [SS00, Table 1]

For further information to the Smith-Schwartz model [SS00], [CS02], and [Ror10]are recommended.

53


The advanced Smith-Schwartz model

A modification of the classical Smith-Schwartz model is called advanced Smith-Schwartzmodel (see [BCB08]). This modification adds a deterministic function to the sum ofboth factors. With this deterministic function, the model can be calibrated to fit themarket forward curve.

Definition 6 (Advanced Smith-Schwartz model). The spot price follows

log(St) = χ(t) + ξ(t) + ϕ(t)

where both factors follow the stochastic differential equation system.




with the assumption that kχ, σχ, σξ > 0, and ρχ, ξ ∈ [−1, 1] holds.

In the advanced Smith-Schwartz model the factors χ(t) and ξ(t) have the samecommon distribution as in the classical Smith-Schwartz model in theorem 8. In thefollowing the distribution of S(t) is shown: this model has the same distribution as theclassical Smith-Schwartz model with the only difference that to the expected value ofχ(t) + ξ(t) the deterministic function ϕ(t) is added.

Theorem 9 (Distribution of the Prices). The spot price S(t) for known χ(s) and ξ(s)with s < t can be described with the following distribution:


E(log(S(t))|χ(s), ξ(s)) = χ(s) · exp−kχ(t− s)+ ξ(s) + µξ · (t− s) + ϕ(t)


2kχ(1− exp−kχ2 (t− s)) + σ2

ξ · (t− s) + 2 · Covχ, ξ(s, t)


χ(s) · exp−kχ(t− s)+ ξ(s) + µξ · (t− s) + ϕ(t)

+1


In equation (3.13) it is shown that the Forward (or Future) price for maturity T − tis equal to the expected value of the spot price at the same point of time. It follows

54


the expected value of the commodity by the advanced Smith-Schwartz model:

F (t, T ) = exp

χ(t) · exp−kχ(T − t)+ ξ(t) + µξ · (T − t) + ϕ(T )

+1

2V(log(S(T ))|χ(t), ξ(t))

(5.14)

Because, χ(T ) and ξ(T ) are Normal distributed, the consequence is that log (F (t, T ))is also Normal distributed with variance σ2

F :

σ2F (t, T ) =

σ2χ

2kχe−2kχ(T−t) (1− exp−kχ2 (t− s)) + σ2

ξ · (t− s)

+ 2 · ρχ, ξσχσξkχ

e−kχ(T−t) (1− exp(−kχ(t− s)))

Thus, the Black76-Equation can be applied for pricing of a European option on aForward:


And for an European Put-option it is:



2σ2F (t, T )

σF (t, T ) .

For further information to the advanced Smith-Schwartz model [BCB08] is advised.

The advanced Smith-Schwartz model with Seasonality

An extension of the previously presented advanced Smith-Schwartz model providesthe possibility to fit the model to seasonalities. This model was introduced by Back,Prokopczuk, and Rudolf in [BPR09, Page 9]. In this thesis their approach will becalled advanced Smith-Schwartz model with Seasonality (aSSmwS).

The spot price underlies the sum of two stochastic processes and a deterministicfunction. The first factor ξ(t) models the long-term equilibrium level and followsa geometric Brownian motion. Whereas the second factor χ(t) models the short-termfluctuation at the equilibrium level. This second factor follows an Ornstein-Uhlenbeckprocess and has according to this the mean reverting property. The seasonality is mod-elled by the volatility of the first factor, this is the volatility of the equilibrium pricein the classical Smith-Schwartz model σξ is replaced with the following term σξe

ψ(t).

55


Definition 7 (The advanced Smith-Schwartz model with Seasonality). The spot pricefollows

log(S(t)) = χ(t) + ξ(t) + ϕ(t)

where both factors follow the stochsatic differential equation system:

dχ(t) = −kχ · x(t)dt+ σχdWχ

dξ(t) = µξ · dt+ σξeψ(t)dWξ


ψ(t) = θ sin (2π(t+ α))

With kχ, σχ, σξ > 0, ρχ, ξ ∈ [−1, 1], θ > 0, and α ∈ [−0.5, 0.5].

By solving the differential equation system from the definition 7 the common distri-bution of both factors is formed:

Theorem 10 (Distribution of the advanced Smith-Schwartz model with Seasonality).The common distribution of the factors χ(t) and ξ(t) for given χ(s) and ξ(s) orienteersitself on a bivariate Normal distribution:

[χ(t)ξ(t)

]∣∣∣∣χ(s), ξ(s)

∼ N

[χ(s) exp (−kχ (t− s))ξ(s) + µξ (t− s)

],

σ2χ


Covχ, ξ(s, t) σ2ξ

t∫s

e2ψ(u)du

where

Covχ, ξ(s, t) = ρχ, ξσχσξ

t∫s

exp −kχ · (t− u) eψ(u)du

Proof.

By definition we know that


describes an Ornstein-Uhlenbeck process. Hence, we immediately infeer

E(χ(t)|χ(s)) = χ(s) · exp−kχ(t− s)

V(χ(t)|χ(s)) = Cov(χ(t), χ(t)|χ(s)) =σ2χ

2kχ(1− exp−kχ2 (t− s))

56


Therefore, it follows:

χ(t)|χ(s) ∼ N(χ(s) · exp −kχ · (t− s) ,

σ2χ

kχ · 2(1− exp −kχ · 2 · (t− s))

)

By definition,

dξ(t) = µξ · dt+ σξeψ(t)dWξ

describes a geometric Brownian motion, hence it follows:

E(ξ(t)|ξ(s)) = ξ(s) + µξ · (t− s)

V(ξ(t)|ξ(s)) = σ2ξ

t∫s

e2ψ(u)du

Therefore we obtain

ξ(t)|ξ(s) ∼ N

ξ(s) + µξ · (t− s), σ2ξ

t∫s

e2ψ(u)du

Covχ, ξ(s, t) = Cov (χ(t), ξ(t)|χ(s), ξ(s))

= E ((χ(t)− E(χ(t)|χ(s))) · (ξ(t)− E(ξ(t)|ξ(s))))

= E

(χ(s) · exp −kχ · (t− s)+ σχ exp −kχ · tt∫s

exp kχ · u dWχ(u)− (χ(s) · exp −kχ · (t− s))

·

ξ(s) + µξ · (t− s) + σξ

t∫s

eψ(u)dWξ(u)− (ξ(s) + µξ · (t− s))

|χ(s), ξ(s)

)

= E

σχ exp −kχ · tt∫s


·σξ t∫

s

eψ(u)dWξ(u)

|χ(s), ξ(s)


t∫s


t∫s

eψ(u)dWξ(u)|χ(s), ξ(s)

since dWχdWξ = ρχ, ξdt we derive further


t∫s

ρχ, ξ exp kχ · u eψ(u)du|χ(s), ξ(s)

= ρχ, ξσχ exp −kχ · tσξ · E

t∫s

exp kχ · u eψ(u)du|χ(s), ξ(s)

57


Because oft∫s

exp kχ · u eψ(u)du is determinsitic it follows:

= ρχ, ξσχ exp −kχ · tσξt∫s

exp kχ · u eψ(u)du

= ρχ, ξσχσξ

t∫s


Therefore underlying χ(t) and ξ(t) the common Normal distribution:

[χ(t)ξ(t)

]∣∣∣∣χ(s), ξ(s)

∼ N

[χ(s) exp (−kχ (t− s))ξ(s) + µξ (t− s)

],

σ2χ


Covχ, ξ(s, t) σ2ξ

t∫s

e2ψ(u)du

Covχ, ξ(s, t) = ρχ, ξσχσξ

t∫s


The common distribution of both factors was shown in the last theorem, with theproperties of the Normal distribution we can easily derive the distribution of

log(S(t)) = χ(t) + ξ(t) + ϕ(t) :

Corollary 4 (Distribution of the Prices). The spot price S(t) for known χ(s) and ξ(s)with s < t can be described with the following distribution:


E(log(S(t))|χ(s), ξ(s)) = χ(s) · exp−kχ(t− s)+ ξ(s) + µξ · (t− s) + ϕ(t)


2kχ(1− exp−kχ2 (t− s)) + σ2

ξ

t∫s

e2ψ(u)du

+ 2ρχ, ξσχσξ

t∫s


Consequently, we have


χ(s) · exp−kχ(t− s)+ ξ(s) + µξ · (t− s) + ϕ(t)

+1


58

5.4. Calibration

In equation (3.13) it is shown that the Forward (or Future) price with the maturityT − t is equal to the expected value of the spot price at the same point of time.The consequence is the expected value of the commodity spot price by the advancedSmith-Schwartz model with seasonality:

F (t, T ) = exp


+1


(5.17)

Since the χ(t) and ξ(t) are Normal distributed, log (F (t, T )) has to be Normal dis-tributed as well with the variance of σ2

F (see [BPR09, Page 11]):

σ2F (t, T ) = σ2

ξ

t∫0

e2θ sin(2π(u+α))du+σ2χ

2kχe−2kχ(T−t) (1− e−2kχt

)

+ 2ρχ, ξσχσξe−kχ(T−t)

t∫0

eθ sin(2π(u+α))e−kχ(t−u)du

Thus the Black76-Equation can again be applied for the pricing of a European optionon a Forward:


And for a European Put-option it is:



2σ2F (t, T )

σF (t, T ) .

For further information the author refers to [BPR09].

5.4. Calibration

The parameter vector β can be determined using the least squares approach for theMSE between market and model prices for exchange traded financial products oncommodities. Here we will focus on wheat and WTI crude oil to demonstrate the cal-ibration. These financial products are European Call and Put options and Forwards.For wheat there are several European Put and Call options traded liquidity, while forcrude oil there is primarly a very liquid market for futures. The future curves used to

59


calibrate the models are shown in figure 5.1 and in figure 5.2.

0 1 2 3 4 5 6 7105

106

107

108

109

110

111

112

113

114

time (in years)

For

war

d P

rice

/ bar

rel

Figure 5.1.: Forward Prices for Crude Oil (WTI) (Data from Bloomberg, May 3, 2011)

0 0.5 1 1.5 2 2.5740

760

780

800

820

840

860

880

900

920

940

time (in years)

For

war

d P

rice

/ bus

hel

Figure 5.2.: Forward Prices for Wheat (Data from Bloomberg, May 3, 2011)

And the data for options on wheat futures is shown in table 5.4.

60

5.4. Calibration

Strike Excercise Date Maturity Bid price Ask Price Mid Price Type790 1m 2m 25,250 28.375 26.8125 Call795 1m 2m 23,125 26.250 24.6875 Call800 1m 2m 21,125 24.250 22.6875 Call805 1m 2m 19,250 22.375 20.8125 Call810 1m 2m 17,500 20.625 19.0625 Call815 1m 2m 16.000 19.125 17.5625 Call820 1m 2m 14.500 17.625 16.0625 Call790 2m 38m 43.250 46.375 44.8125 Call800 2m 38m 39.000 42.125 40.5625 Call810 2m 38m 35.000 38.125 36.5625 Call820 2m 38m 31.500 34.625 33.0625 Call750 1m 2m 13.750 16.875 15.3125 Put755 1m 2m 15.500 18.625 17.0625 Put760 1m 2m 17.375 20.500 18.9375 Put765 1m 2m 19.500 22.625 21.0625 Put770 1m 2m 21.625 24.750 23.1875 Put775 1m 2m 23.875 27.000 25.4375 Put780 1m 2m 26.375 29.500 27.9375 Put785 1m 2m 28.875 32.000 30.4375 Put790 1m 2m 31.125 35.250 33.1875 Put795 1m 2m 34.000 38.125 36.0625 Put800 1m 2m 37.000 41.125 39.0625 Put805 1m 2m 40.125 44.250 42.1875 Put810 1m 2m 41.375 49.500 45.4375 Put815 1m 2m 45.000 53.125 49.0625 Put820 1m 2m 48.500 56.625 52.5625 Put750 2m 38m 30.125 33.250 31.6875 Put760 2m 38m 34.500 37.625 36.0625 Put770 2m 38m 39.125 42.250 40.6875 Put780 2m 38m 44.250 47.375 45.8125 Put790 2m 38m 49.500 52.625 51.0625 Put800 2m 38m 54.750 58.875 56.8125 Put810 2m 38m 60.875 65.000 62.9375 Put820 2m 38m 67.250 71.375 69.3125 Put

Table 5.4.: Observed market prices for Options on wheat futures

It is:

βt = minβt

∑i=1

Nt

(Pt, i(βt)− Pt, i

)2

(5.20)

61


Where Pt, i are the market prices for options or forwards and Pt, i(βt) are the corre-sponding model-based prices. Furthermore the estimated parameters must fulfill theassumptions for these parameters.

5.4.1. The Schwartz One-Factor model

In this model the following parameters have to be estimated:

β = κ, µ, σ

With the spot price

S0 = exp X0

minimizing equation (5.20) based on the market data for wheat yields the parametersfrom reported in 5.5 with a MSE of 1026.85. The domination portion of the errorcame from the option data. Ignoring these options, the MSE can be dramaticallyreduced to 206.33 and the parameters are similar.

κ µ σ1.2233 6.8432 0.0000

Table 5.5.: Parameter for modelling wheat spot prices in the Schwartz One-Factormodel

Minimizing equation (5.20) and the market data for crude oil follow the parametersfrom table 5.6 with a MSE of 0.0371.

κ µ σ0.7785 4.8025 0.6886

Table 5.6.: Parameter for modelling crude oil spot prices in the Schwartz One-Factormodel

The advanced Schwartz One-Factor model

Above we mentionned that the model can be calibrated to the forward curve by usingthe deterministic function ϕ, this can be done by applying the following theorem:

Theorem 11 (Calibrating ϕ to market data). Given the forward curve T 7→ FM (0, T )from market and using equation (5.14), it is possible for solve the equation to the

62

5.4. Calibration

deterministic function ϕ(T ). The result is:

ϕ(T ) = log(FM (0, T )

)−X0 exp −kχ · T − α (1− exp−κT)− 1

2V(log(S(T ))|X(0))

For the proof of the last theorem, the market forward curve is set equal to theforward price in the advanced Smith-Schwatz model (equation 5.14) for each point intime. This new equation is solved after ϕ for each point in time.

Proof.

We have

FM (t, T ) = exp

X(t) · exp −κT+ α (1− exp−κT) + ϕ(T )

+1

2V(X(T )|X(t))

log(FM (t, T )

)= X(t) · exp −κT+ α (1− exp−κT) + ϕ(T ) +

1

2V(X(T )|X(t))

ϕ(T ) = log(FM (0, T )

)−X0 exp −kχ · T − α (1− exp−κT)

− 1

2V(log(S(T ))|X(0))

Using theorem 11 and the forward curve for wheat, one can calibrate the function ϕ.

0 0.5 1 1.5 2 2.5−0.06

−0.04

−0.02

0

0.02

0.04

0.06

t

phi(t

)

Figure 5.3.: Calibrated function ϕ in the advanced Schwartz One-Factor model towheat market forward prices

63


Using theorem 11 and the forward curve for crude oil, one can calibrate the functionϕ.

0 1 2 3 4 5 6 7−4

−2

0

2

4

6

8x 10

−3

t

phi(t

)

Figure 5.4.: Calibrated function ϕ in the advanced Schwartz One-Factor model tocrude oil market forward prices

5.4.2. The Gibson-Schwartz model

In this model the following parameters need to be estimated:

β = µ, σ1, κ, α, σ2, ρ

In this model there are only model-based equations, therefore solely the future pricesare used for calibration purposes.Minimizing equation (5.20) and the market data for wheat yields the parameters re-ported in table 5.7 with a MSE of 34,565.81.

δ0 µ σ1 κ σ2 α ρ0 31.1067 0.6374 15.1329 0.0000 31.0911 0.9999

Table 5.7.: Parameter for modeling wheat spot prices in the Gibson-Schwartz model

Minimizing equation (5.20) and the market data for crude oil gives the parametersin table 5.8 with a MSE of 1.059.

64

5.4. Calibration

δ0 µ σ1 κ σ2 α ρ0.1424 0.1226 0.0000 0.0156 0.0000 0.0000 0.2797

Table 5.8.: Parameter for modeling crude oil spot prices in the Gibson-Schwartz model

5.4.3. The Smith-Schwartz model


β = χ0, ξ0, kχ, σχ, µξ, σξ, ρχ, ξ

With the spot price

S0 = exp χ0 + ξ0

minimizing equation (5.20) based on the market data for wheat yields the parametersfrom reported in 5.9 with a MSE of 259.76. The domination portion of the error camefrom the option data. Ignoring these options, the MSE can be dramatically reducedto 87.89 and the parameters are similar.

χ0 ξ0 kχ σχ µξ σξ ρχ, ξ3.1979 3.4132 0.3066 1.1232 0.1021 0.0065 -0.0182

Table 5.9.: Parameter for modeling wheat spot prices in the Smith-Schwartz model


χ0 ξ0 kχ σχ µξ σξ ρχ, ξ4.7264 0.0003 0.1425 0.9567 0.2226 0.0002 -0.7416

Table 5.10.: Parameter for modelling crude oil spot prices in the Smith-Schwartz model

The advanced Smith-Schwartz model

Above it was explained, that the model can be calibrated to the forward curve byusing the deterministic function ϕ, this can be done with the following theorem:

Theorem 12 (Calibrating ϕ to market data). Given the forward curve T 7→ FM (0, T )from the market and using the equation (5.14), it is possible to solve the equation tothe deterministic function ϕ(T ). The result is:

ϕ(T ) = log(FM (0, T )

)− χ0 exp −kχ · T − ξ0 − µξ · T −

1


65


For the proof of the last theorem, the market forward curve is set equal to theforward price in the advanced Smith-Schwatz model (equation 5.14) for each point intime. This new equation is solved after ϕ for each point in time.

Proof.

We have

FM (t, T ) = exp


+1


log(FM (t, T )

)= χ(t) · exp−kχ(T − t)+ ξ(t) + µξ · (T − t) + ϕ(T )

+1


ϕ(T ) = log(FM (t, T )

)− χ(t) · exp

−kχ(T − t) − ξ(t)− µξ · (T − t)

−1


For the calibration one needs the market forward curves for crude oil (figure 5.1)and for wheat (figure 5.2). Using theorem 12 and the forward curve for wheat, onecan calibrate the function ϕ.

66

5.4. Calibration

0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

t

phi(t

)

Figure 5.5.: Calibrated function ϕ in the advanced Smith-Schwartz model to wheatmarket forward prices

Using theorem 12 and the forward curve for crude oil, one can calibrate the functionϕ.

0 1 2 3 4 5 6 7−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

t

phi(t

)

Figure 5.6.: Calibrated function ϕ in the advanced Smith-Schwartz model to crude oilmarket forward prices

67


The advanced Smith-Schwartz model with seasonality


β = χ0, ξ0, kχ, σχ, µξ, σξ, ρχ, ξ, θ, α

Furthermore the deterministic shift ϕ must be determined. With the spot price

S0 = exp χ0 + ξ0

minimizing equation (5.20) based on the market data for wheat yields the parametersfrom reported in 5.11 with a MSE of 498.34. The domination portion of the errorcame from the option data. Ignoring these options, the MSE can be dramaticallyreduced to 48.47 and the parameters are similar.

χ0 ξ0 kχ σχ µξ σξ ρχ, ξ θ α5.2148 1.4059 0.2147 1.7855 0.0000 0.2335 -0.4809 1.9386 0.4332

Table 5.11.: Parameter for modeling wheat spot prices in the Smith-Schwartz modelwith seasonality


χ0 ξ0 kχ σχ µξ σξ ρχ, ξ θ α4.7264 0.0003 0.1425 0.9567 0.2226 0.0002 -0.7416 0.0000 -0.0001

Table 5.12.: Parameter for modeling crude oil spot prices in the Smith-Schwartz modelwith seasonality

In the final table 5.12 the reader can see that θ ≈ 0, which implies that there is nosaisonality in the spot prices for crude oil.After calibrating the parameters, now the deterministic function ϕ must be specified.With this function, the model will exactly fit the forward curve observed in the market.This function is determined using theorem 12 and the forward curve.Using the forward curve for wheat one can calibrate the function ϕ:

68

5.4. Calibration

0 0.5 1 1.5 2 2.5−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

t

phi(t

)

Figure 5.7.: Calibrated function ϕ in the advanced Smith-Schwartz model with sea-sonality to wheat market forward prices

5.4.4. Conclusion

In the last subsections the models were calibrated but the results were not compared.This comparison is done in the following table 5.13.

Crude oil WheatSchwartz One-Factor model 0.0371 206.33.Gibson-Schwartz model 1.059 34,565.81Smith-Schwartz model 0.2770 87.89Smith-Schwartz model with seasonality 0.2770 48.47

Table 5.13.: MSE for all models calibrated to both commodities

For crude oil we observe a good fit in all models but the best approximation is gen-erated by the Schwartz One-Factor model. For wheat the fit is not as good. Comparedto crude oil we have to state that the approximations are poor for all models. Thebest of these approximations the Smith-Schwartz model capturing seasonality effects.Particularly amazing in these results is that the most simple model provides the bestfit for crude oil.

Both models will be used with the deterministic shift, to get a better fit to the forwardcurve.

69

6. Implementation

In this chapter some important and interesting issues concerning the actual imple-mentation of numerical and stochastical methods to generate potential paths for theCVA-driving variables are discussed.

6.1. Implementation of the Cox-Ingersoll-Ross Process

In this work the default process including a CIR process is modeled, as explainedin section 4.2.2. In contrast to many other stochastic processes, it is in general notpossible for the CIR process to simulate it by simple discretization the integrals forsmall ∆t. In this section we therefore examine several methods to simulate the CIRprocess. This section follows predominantly the structure of [BM06, Page 797].

The main problem of simulating a CIR process is the square root operation in thestochastic differential equation, which implicitly assumes that the simulated valuesare greater than zero. For each method, a simulation scheme is given. Furthermore, inthe appendix a reference to the implementation in MATLAB is given and finally themean of the simulated values compared to its corresponding expected value is shown.

This comparison is done with the following parameters from [BCB08, Tab. 4] i.e.t = 0, T = 1, x0 = 0.0560, κ = 0.6331, θ = 0.0293, σ = 0.02, and ∆t = 0.01which we will call here Brigo parameters for short, while the parameters from table4.5 will be called Deutsche Bank parameters henceforth. After the presentation ofthese methods a comparison between their features is done. The parametervector iscalled α = x0, κ, θ, σ.

6.1.1. Euler scheme

The so-called Euler scheme (Es) is the most likely approach. The integrals are dis-cretizied for small ∆t:

xα(ti+1) = xα(ti) + κ (θ − xα(ti)) · (ti+1 − ti) + σ√xα(ti) · (W (ti+1)−W (ti)) (6.1)

71

6. Implementation

At the beginning of this section the reader has been indicated, that this concept doesnot work. Since the positivity can not ensured and only rarely a complete continuouspath is generated applying the Euler scheme. The MATLAB code can be found in ap-pendix B. In the next figure 6.1 a typical path generated by the Euler scheme is shown.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.03

0.035

0.04

0.045

0.05

0.055

0.06

Figure 6.1.: A typical path simulated based on the Euler scheme (Brigo parameters)

The following figure 6.2 shows how many paths finally reach the maturity, this onehas been chosen from 0.01 to 5 years. For each maturity 10,000 paths were generated.It is evident that with the chosen parameter combination and longer maturities nopaths ever reaches maturity.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

t

num

ber

of p

aths

Figure 6.2.: Number of continuous paths from 10,000 scenarios of CIR processes, sim-ulated with the Euler scheme (Brigo parameters)

A path is said to “reach maturity” if x(ti) ∈ R for all i ∈ 1, . . . , n as given by equa-

72


tion (6.1). Note that it may happen that x(ti) < 0 which may lead to x(ti+1) ∈ C\Raccording to (6.1). In the latter case, the path does not reach maturity.For maturities up to one year most of the paths reach maturity. For comparison pur-poses the maturity was limited to one year. It compares the expected analytical zerobond prices of the CIR model with the simulated zero bond prices. The results areshown in the following figures 6.3 and 6.4.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.95

0.955

0.96

0.965

0.97

0.975

0.98

0.985

0.99

0.995

1

t

Zer

obon

d P

rice

Expected Zerobond PricesSimulated Zerobond Prices

(a) Expected zero bond price comparedwith simulated zero bond prices

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)

(b) Relative deviation between the ex-pacted and the simulated zero bond price

Figure 6.3.: 10,000 scenarios of CIR processes based on the Euler scheme (Brigoparameters)

73

6. Implementation

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.995

0.996

0.997

0.998

0.999

1

1.001

t

Zer

obon

d P

rices



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−6

−5

−4

−3

−2

−1

0x 10

−3

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)


Figure 6.4.: 10,000 scenarios of CIR processes based on the Euler scheme (DeutscheBank parameters)

It generates useful results for both parameter sets. The deviation between the sim-ulated and the expected prices for the selected period is between -0.05% and +0.25%(Brigo parameters) and between -0.0005% and 0.005% (Deutsche Bank parameters).Therefore the Euler scheme is solely appropriate for maturities up to one year.

The following two concepts can guarantee continuous paths, this happens in combina-tion with the Euler scheme and the absolute value function respictively the indicatorfunction.

In the first modification of the Euler scheme, the classical Euler scheme is used, how-ever the square root of x(ti) is only drawn if x(ti) is positive. This is ensured by theindicator function underneath the root. This approach was presented by Deelstra andDelbaen in [DD98] and is subsequently named as the Euler-Deelstra-Delbaen scheme(EDDs).

xα(ti+1) = xα(ti) + κ (θ − xα(ti)) · (ti+1 − ti)

quad+ σ√xα(ti) · 1xα

(ti)>0

· (W (ti+1)−W (ti)) (6.2)

With this approach, although the positivity is not guaranteed, the paths will not breakoff. The MATLAB code can be found in appendix B. In the next figure 6.5 a typicalpath generated by the Euler-Deelstra-Delbaen scheme is shown.

74


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

Figure 6.5.: One typical path simulated based on the Euler-Deelstra-Delbaen scheme(Brigo parameters)

Since in this method, all paths reach maturity; for comparison prupose the maturitywas set to five years. It compares the expected zero bond prices of the CIR modelwith the simulated zero bond prices. The results are shown in the following figures 6.6and 6.7.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

t

Zer

obon

d P

rices


(a) Expected zero bond prices comparedwith simulated zero bond prices

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−2

0

2

4

6

8

10

12

14

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)

(b) Relative deviation between the ex-pected and the simulated zero bond prices

Figure 6.6.: 10,000 scenarios of CIR processes based on the Euler-Deelstra-Delbaenscheme (Brigo parameters)

75

6. Implementation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.94

0.95

0.96

0.97

0.98

0.99

1

t

Zer

obon

d P

rices



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.016

−0.014

−0.012

−0.01

−0.008

−0.006

−0.004

−0.002

0

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)


Figure 6.7.: 10,000 scenarios of CIR processes based on the Euler-Deelstra-Delbaenscheme (Deutsche Bank parameters)

For the Brigo parameters the results are very good in short-term, but at the ma-turities greater than one year the simulated and expected zero bond prices from theCIR model fall steady apart. And for the Deutsche Bank parameters the results arevery good for all maturities.

The second modification takes the absolute value of the sum gains and the intialvalue as shown below:

xα(ti+1) =∣∣∣xα(ti) + κ (θ − x(ti)) · (ti+1 − ti) + σ

√xα(ti) · (W (ti+1)−W (ti))

∣∣∣ (6.3)

This method was introduced by Diop in [Dio03] and it is referred to in this work asthe Euler-Diop scheme (EDs). In contrast to the first modification to this method,the positivity of the path is guaranteed. The MATLAB code can be found in B. Inthe next figure 6.8 a typical path generated by the Euler-Diop scheme is shown.

76


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.01

0.02

0.03

0.04

0.05

0.06

Figure 6.8.: One typical path simulated based on the Euler-Diop scheme (Brigoparameters)

Since in this method, all paths reach the maturity, to examine the contrast the ma-turity is set to five years. It compares the expected zero bond prices of the CIR modelwith the simulated zero bond prices. The results are shown in the following figures6.9 and 6.10. As with the first modification the results for the Brigo parameters aregood in the short-term, but at maturities greater than one year the simulated andexpected zero bond prices from the CIR model differ steady apart. For the DeutscheBank parameters the results are very good for all maturities.

77

6. Implementation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

t

Zer

obon

d P

rices



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−2

0

2

4

6

8

10

12

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)


Figure 6.9.: 10,000 scenarios of CIR processes based on the Euler-Diop scheme (Brigoparameters)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.94

0.95

0.96

0.97

0.98

0.99

1

t

Zer

obon

d P

rices



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.016

−0.014

−0.012

−0.01

−0.008

−0.006

−0.004

−0.002

0

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)


Figure 6.10.: 10,000 scenarios of CIR processes based on the Euler-Diop scheme(Deutsche Bank parameters)

For the last two modifications of the Euler scheme the probability measurementactually needs to be changed (this is not done here) which is a possibility to explainthe deviations. In the following figure 6.11, there is a comparison between the relativedeviation of the Euler scheme and both extensions. One can see that for maturitiesup to one year the schemes have approximately the same relative deviation.

78


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)

Euler schemeEuler−Deelstra−Delbaen schemeEuler−Diop scheme

Figure 6.11.: Comparison of the relative deviation of the Euler scheme and both ex-tensions for 10,000 scenarios (Brigo parameters)

6.1.2. Milstein scheme

The next discretization approach, is the Milstein scheme (Ms),

xα(ti+1) = xα(ti) + κ (θ − xα(ti)) · (ti+1 − ti) + σ√xα(ti) · (W (ti+1)−W (ti))

+1

4σ2[(W (ti+1)−W (ti))

2 − (ti+1 − ti)]

(6.4)

Neither the Milstein scheme nor the Euler scheme can guarantee the positivity of thegenerated paths. In consequence, negative values can be simulated and the path doesnot reach maturity. For further information for the Milstein scheme the author refersto [Alf08] and [BA04]. The MATLAB code for this method can be found in appendixB. In the next figure 6.12 a typical path generated by the Milstein scheme is shown.

79

6. Implementation

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.04

0.045

0.05

0.055

0.06

Figure 6.12.: One typical path simulated based on the Milstein scheme (Brigoparameters)

The following figure 6.13 shows how many paths reach the maturity, which has beenchosen from 0.01 to 5 years. For each maturity 10,000 paths were generated. It is rec-ognizable that with the given parameter constellation and for longer maturities, thepaths reach only the maturity for a small number of time points. Thus, this schemeis improper, too.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

t

num

ber

of p

aths

Figure 6.13.: Number of continuous paths from 10,000 scenarios of CIR processes,simulated with the Milstein scheme (Brigo parameters)

For maturities up to one year most of the paths reach maturity, for comparison pur-

80


poses the maturity is limited to one year. It compares the expected zero bond pricesof the CIR model with the simulated zero bond prices. The results are shown in thefollowing figures 6.14 and 6.15.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.95

0.955

0.96

0.965

0.97

0.975

0.98

0.985

0.99

0.995

1

t

Zer

obon

d P

rices



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)


Figure 6.14.: 10,000 scenarios of CIR processes based on the Milstein scheme (Brigoparameters)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.995

0.996

0.997

0.998

0.999

1

1.001

t

Zer

obon

d P

rices



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−6

−5

−4

−3

−2

−1

0x 10

−3

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)


Figure 6.15.: 10,000 scenarios of CIR processes based on the Milstein scheme(Deutsche Bank parameters)

As previously described, neither the Euler scheme nor the Milstein scheme doesguarantee positivity of paths.

81

6. Implementation

6.1.3. Euler-Implicit scheme

Brigo and Alfonsi introduce the Euler-Implicit scheme (EIs) in [BA04] by letting

xα(ti+1) =

(σ (W (ti+1)−W (ti)) +

√∆ti

2 (1 + κ · (ti+1 − ti))

)2

(6.5)

∆ti = σ2 (W (ti+1)−W (ti)) + 4

(xαti +

(κθ − σ2

2

)(ti+1 − ti)

)(1 + κ (ti+1 − ti))

It is assumed that 2κθ > σ2 holds. The derivation can be found in [BA04]. Incontrast to the first two methods presented this one ensures the positivity of thepaths. Further, x(ti+1) is an increasing function in dependency of x(ti). This propertyis called Monotonicity by Brigo, and is defined as (see [BM05, Page 799]):

“For a given path (Wti(ω))i, x0 ≤ x0 implies xαti(ω) ≤ xαti(ω) for allti’s.” (...) Indeed if we set δt = xαt − xαt with x0 ≤ x0, we have dδt =δt(−κdt+ σ/(

√xαt +√xαt )dWt

). Thus, δt appears as a Doleans exponential

process and remains positive for all t.

A path generated by the theoretical model does not have this property, so this is aweakness. The MATLAB code can be found in appendix B. In the following figure6.16 a typical path generated by the Euler-Implicit scheme is shown.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

Figure 6.16.: One typical path simulated based on the Euler-Implicit scheme

Since in this method, all paths reach maturity, for comparing purposes the maturitywas set to five years. It compares the expected zero bond prices of the CIR model withthe simulated zero bond prices. The results are shown in the following figures 6.17

82


and 6.18. The deviation between the expected and the simulated zero bond prices arebetween -0.05% and -0.7% (Brigo parameters) and between -0.0005% and -0.0035%(Deutsche Bank parameters).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

t

Zer

obon

d P

rice



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)


Figure 6.17.: 10,000 scenarios of CIR processes based on the Euler-Implicit scheme(Brigo parameters)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.94

0.95

0.96

0.97

0.98

0.99

1

t

Zer

obon

d P

rices



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0x 10

−3

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)


Figure 6.18.: 10,000 scenarios of CIR processes based on the Euler-Implicit scheme(Deutsche Bank parameters)

83

6. Implementation

6.1.4. Milstein-Implicit scheme

Alfonsi introduces in [Alf05] the following approach which we will call Milstein-Implicitscheme (MIs) here:

xα(ti+1) =

(σ2 (W (ti+1)−W (ti)) +

√xαti +

√∆ti

2(1 + κ

2 · (ti+1 − ti)) )2

(6.6)

∆ti =(σ

2(W (ti+1)−W (ti)) +

√xαti

)2

+ 2(

1 +κ

2(ti+1 − ti)

) (κθ − σ2

/4)

(ti+1 − ti)

It is assumed that 4κθ > σ2 and ∆t < 2κ holds. This assumption is not the classical

one of the CIR process. Just as in the previous approach, this one also guaranteesthe positivity of the paths. Furthermore, x(ti+1) is an increasing function dependenedupon x(ti). The MATLAB code can be found in appendix B. In the following figure6.19 a typical path generated by the Milstein-Implicit scheme is shown.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

Figure 6.19.: One typical path simulated based on the Milstein-Implicit scheme

Since, all paths reach the maturity in this method, for comparing reasons the matu-rity was set to five years. It examines the differences of the expected zero bond pricesof the CIR model with the simulated zero bond prices. The results are shown in thefollowing figures 6.20 and 6.21. The deviation between the expected and the simu-lated zerobond prices are between -0.06% and -0.2% (Brigo parameters) and between-0.0005% and 0.01% (Deutsche Bank parameters).

84


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

t

Zer

obon

d P

rice



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.2

−0.18

−0.16

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)


Figure 6.20.: 10,000 simulation of CIR processes with the Milstein-Implicit scheme(Brigo parameters)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

t

Zer

obon

d P

rice



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−2

0

2

4

6

8

10x 10

−3

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)


Figure 6.21.: 10,000 simulation of CIR processes with the Milstein-Implicit scheme(Deutsche Bank parameters)

6.1.5. Alfonsis E family

Another family of schemes for the simulation of CIR processes is also presented in[Alf05], this family is denoted by E(λ). This scheme needs, in contrast to the otherschemes explained before, not solely the classical parameters κ, θ and σ, but also an

85

6. Implementation

additional parameter λ. The concept works as shown below:

xα(ti+1) =

((1− κ

2(ti+1 − ti)

)√xαti +

σ (W (ti+1)−W (ti))

2(1− κ

2 (ti+1 − ti)) )2

+(κθ − σ2

/4)

(ti+1 − ti) + λ[(W (ti+1)−W (ti))

2 − (ti+1 − ti)]

(6.7)

It is assumed that 2κθ > σ2 and 0 ≤ λ ≤ κθ − σ2/4 holds. This assumption is not the

classical one of the CIR process. In the next figure 6.22 a typical path generated bythe E(0) scheme is shown.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

Figure 6.22.: One typical path simulated based on the E(0) scheme

As the previous approaches, this one ensures the positivity of the paths, but hasnot the downside of monoticity. Moreover, this approach is for λ = 0 equivalent tothe MIs, as Alfonsi states in [Alf05]. The MATLAB code can be found in appendixB. Inasmuch as in this method, all paths reach the maturity, for comparing purposesthe maturity was set to five years. It provides a comparison of the expected zero bondprices of the CIR model with the simulated zero bond prices, for various values of λ.The results are shown in the upcoming figures 6.23 and 6.24. For λ = 0 there aredeviations between 0.06% and 0.19% (Brigo parameters) and between 0% and -0.02%(Deutsche Bank parameters), these deviations are similar to the MIs thus, these re-sults correspond to the statement above. For λ = σ2

/8 the deviations are between-0.05% and -10% (Brigo parameters) and between 0% and -0.02% (Deutsche Bankparameters) and last but not least for λ = σ2

/4 the deviations are between -0.05% and-20% (Brigo parameters) and between 0% and -0.02% (Deutsche Bank parameters).

86


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

t

Zer

obon

d P

rices


(a) Expected zero bond prices comparedwith simulated zero bond prices with λ = 0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.2

−0.18

−0.16

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)

(b) Relative deviation between the ex-pacted and the simulated zero bond pricewith λ = 0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.7

0.75

0.8

0.85

0.9

0.95

1

t

Zer

obon

d P

rices


(c) Expected zero bond price comparedwith simulated zero bond prices with λ =σ2/8

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−12

−10

−8

−6

−4

−2

0

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)

(d) Relative deviation between the ex-pacted and the simulated zero bond pricewith λ = σ2/8

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.65

0.7

0.75

0.8

0.85

0.9

0.95

1

t

Zer

obon

d P

rices


(e) Expected zero bond price comparedwith simulated zero bond prices with λ =σ2/4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−25

−20

−15

−10

−5

0

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)

(f) Relative deviation between the ex-pacted and the simulated zero bond pricewith λ = σ2/4

Figure 6.23.: 10,000 simulation of CIR processes with the E(λ) family for variousvalues of λ (Brigo parameters)

87

6. Implementation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.94

0.95

0.96

0.97

0.98

0.99

1

t

Zer

obon

d P

rices


(a) Expected zero bond prices comparedwith simulated zero bond prices with λ = 0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.025

−0.02

−0.015

−0.01

−0.005

0

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)

(b) Relative deviation between the ex-pacted and the simulated zero bond pricewith λ = 0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.94

0.95

0.96

0.97

0.98

0.99

1

t

Zer

obon

d P

rices


(c) Expected zero bond price comparedwith simulated zero bond prices with λ =σ2/8

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.025

−0.02

−0.015

−0.01

−0.005

0

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)

(d) Relative deviation between the ex-pacted and the simulated zero bond pricewith λ = σ2/8

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.94

0.95

0.96

0.97

0.98

0.99

1

t

Zer

obon

d P

rices


(e) Expected zero bond price comparedwith simulated zero bond prices with λ =σ2/4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.025

−0.02

−0.015

−0.01

−0.005

0

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)

(f) Relative deviation between the ex-pacted and the simulated zero bond pricewith λ = σ2/4

Figure 6.24.: 10,000 simulation of CIR processes with the E(λ) family for variousvalues of λ (Deutsche Bank parameters)

88


6.1.6. Exact Simulation Algorithm

The last approach presented was introduced by Andersen, Jackel and Kahl in [AJK10]and is called the Exact Simulation Algorithm (ESs). As shown in [AM08, Page 183],xα(t) follows a non-central chi-squared distribution. This approach generates the non-central chi-squared random numbers:

1. Draw a Poisson random variable N , with mean 12 · x

α(t) · n(t, t+ ∆).

2. Given N , draw a regular chi-square random variable χ2v, with v = d+2N degrees

of freedom.

3. Set xα(t+ ∆) = χ2v ·

exp−κ∆n(t,t+∆) .

With n(t, T ) = 4κe−κ(T−t)

σ2(1−e−κ(T−t))and d = 4κθ/σ2. It is assumed that 2κθ > σ2 holds.

With this approach, the positivity of the paths is guaranteed. Compared to the othermethods presented, it is numerically significantly more expensive which results fromthe need to generate Chi-square distributed random numbers. The MATLAB codecan be found in appendix B. In the following figure 6.25 a typical path generated bythe Exact Simulation Algorithm is shown.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

Figure 6.25.: A typical paths simulated based on the Exact Simulation Algorithm(Brigo parameters)

Given the fact that in this method, all paths reach maturity, for comparing purposesthe maturity was set to five years. It compares the expected zero bond prices of theCIR model with the simulated zero bond prices. The results are shown in the following

89

6. Implementation

figures 6.26 and 6.27. The deviation between the expected and the simulated zerobondprices are between -0.05% and -0.058% (Brigo parameters) and between -0.0005% and-0.009% (Deutsche Bank).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

t

Zer

obon

d P

rice



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.058

−0.057

−0.056

−0.055

−0.054

−0.053

−0.052

−0.051

−0.05

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)


Figure 6.26.: 10,000 scenarios of CIR processes with the Exact Simulation Algorithm(Brigo parameters)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.94

0.95

0.96

0.97

0.98

0.99

1

t

Zer

obon

d P

rices



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−9

−8

−7

−6

−5

−4

−3

−2

−1

0x 10

−3

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)


Figure 6.27.: 10,000 scenarios of CIR processes with the Exact Simulation Algorithm(Deutsche Bank parameters)

90


6.1.7. Conclusion

Above, various methods have been introduced to simulate the CIR process. Thefollowing table 6.1 summarises the properties of the presented approaches:

Positivity Monotonicity Computing timeEs N N 00:00:03Ms N N 00:00:01EDDs N N 00:00:02EDs Y N 00:00:02EIs Y Y 00:00:02MIs Y Y 00:00:03E(λ) Y N 00:00:02ESs Y N 01:40:25

Table 6.1.: Properties of the shown implementation approachse of the CIR process

The computing time corresponds to 10.000 scenarios of the CIR process, a maturityof five years (except for the first and the second scheme here is the maturity limitedat one year) and using the Brigo parameters. Moreover, in the following figure 6.28,the simulated zero bond prices are compared to the expected zero bond prices. Thiscomparison is done only for the following procedures:

1. EIs

2. MIs

3. E(0)

4. ESs

Figure 6.28 below shows, that the Exact Simulation Algorithm generates the bestapproximation of the approaches discussed, while having the clear disadvantage that itrequires the largest computing time by far. The second best approximation is providedby the following two approaches: Milstein Implicit Scheme and E(0).

91

6. Implementation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

t

Zer

obon

d P

rices

Expected Zerobond PricesSimulated Zerobond Prices (Euler Implicit Positivity−Preserving Scheme)Simulated Zerobond Prices (Milstein Implicit Positivity−Preserving Scheme)Simulated Zerobond Prices (E(0))Simulated Zerobond Prices (Exact Simulation Algorithm)

(a) Expected zero bond prices compared with simulated zero bond prices

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)

Euler Implicit Positivity−Preserving SchemeMilstein Implicit Positivity−Preserving SchemeE(0)Exact Simulation Algorithm

(b) Relative deviation between the expacted and the simulated zero bondprices

Figure 6.28.: Overview over the results of the various approaches (Brigo parameters)

92

6.2. Implementation of the Default Process

Therefore, we have to conclude that all schemes presented above provide very goodresults for the Deutsche Bank parameters. For the Brigo parameters, however, onlythe MIs, E(0), and the ESs provide good results.

Because of the results reported in table 6.1 we will apply the E(0) approach unlessotherwise specified.


In section 4.2 the default models are presented, which are used in this work. Further,in section 4.3 an approach for the simulation of the default time is demonstrated andfurthermore it is shown in section 4.4 how the models can be calibrated to marketdata. Now, however, we discuss the following questions:

1. How can the models be implemented?

2. How good are the results of these models?

Both questions are addressed in this section and ultimately answered. For thispurpose, the models are first considered seperately and then the results of the modelsare examined. For comparison purposes the same market data as in section 4.4 is used.

In all these models, the survival probabilities are used. These can be estimated likeshown in section 4.4.

The MATLAB code that refers to the determination of the survival probabilitiescan be found in appendix B.Basically, the various approach differ in the default time generating only with regardto the intensity function used. In the first approach the intensity function is assumedto be deterministic. However, in the second approach it is assumed to be stochasticwith an additive deterministic part.

6.2.1. Implementation with deterministic intensity

The first approach behaves as shown in section 4.4.1. One can determine the intensityfunction by using theorem 4, the intensity function is shown in table 4.3.

With the following algorithm, default times can be generated with a deterministicintensity function:

93

6. Implementation'

&

$

%

Algorithm

1. Simulate U ∼ U(0, 1).

2. Set t = 0.

3. Choose a sufficiently large n.

4. Set ∆t = T/n.

5. Discretize the function Λ(t) =t∫

0

λ(u)du by Λ(t) = tn

n∑i=1

λ(ti),

where tn = t.

6. Check if Λ(t) ≥ − log(1− U) .

a) if so, set τ = t,

b) otherwise increase t by ∆t and go to step 7.

7. Check if t ≤ T : if so, go to step 6, otherwise set τ = T + ∆t.

Using the above algorithm which generated 100,000 default times, they are com-pared in the following figure 6.29 with the market implied default times.

0 2 4 6 8 10 120

500

1000

1500

2000

2500

3000

3500

4000

4500

t

Def

aults

Model DataMarket Data

(a) Comparison between simulated andmarket implied default times

0 2 4 6 8 10 12−4

−3

−2

−1

0

1

2

3

4

5

6

t

Rela

tive D

evia

tion (

in p

erc

ent)

(b) Relative deviations between marketimplied and simulated default times

Figure 6.29.: 100,000 simulated default times compared with marked implied defaulttimes using a deterministic intensity function

The simulated results are good, this shows especially figure 6.29(b). The followingpicture 6.30 compares the simulated survival probabilities with the market survival

94


probabilities from table 4.2.

0 1 2 3 4 5 6 7 8 9 1085

90

95

100

t

Sur

viva

l Pro

babi

lity

(in p

erce

nt)


Figure 6.30.: Survival probabilities, calculated from 100,000 simulated default times

6.2.2. Implementation with the stochastic intensity

The second approach models the intensity function using a stochastic intensity func-tion as described in section 4.4.2. The calibration can be done with theorem 6 usingthe given parameters. The nodes of the deterministic part of the intensity functioncan be found in table 4.6.

Now, the questions is what values should take Ψ(t) between the nodes? In practise,there exist the following approaches:

1. Piecewise constant interpolation vs.

2. piecewise linear interpolation.

With the following algorithm, default times can be generated with a stochastic in-tensity function:

95

6. Implementation'

&

$

%

Algorithm

1. Generate U ∼ U(0, 1).

2. Set t = 0.

3. Choose a sufficently large n.

4. Set ∆t = T/n .

5. Create a path of the CIR Process y until the maturity T .

6. Calculate Y (t) = ∆tn∑i=1

y(ti), with tn = t,

7. Calculate Ψ(t).

8. Check if Ψ(t) + Y (t) ≥ − log(1− U),

a) if so set τ = t,

b) otherwise increase t by ∆t and go to step 8.

9. Check if t ≤ T is; if so, go to step 6, otherwise set τ = T + ∆t.

The constant interpolation of Ψ(t) and the above algorithm which generated 100,000default times, they are compared with the expected default times in the following fig-ure 6.31.

0 2 4 6 8 10 12−500

0

500

1000

1500

2000

2500

3000

3500

4000

4500

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)


(a) Comparison between simulatedand marked implied default times

0 2 4 6 8 10 12−2

0

2

4

6

8

10

t

Rela

tive D

evia

tion (

in p

erc

ent)

(b) Relative deviations between mar-ket implied and simulated defaulttimes

Figure 6.31.: 100,000 simulated default times compared with market implied defaulttimes using a stochastic intensity function with constant interpolation

96


The simulated results seem to fit the market-implied default times very well, espe-cially seen in figure 6.31(b). The following figure 6.32 compares the simulated survivalprobabilities with the survival probabilities shown in table 4.2.

0 1 2 3 4 5 6 7 8 9 1084

86

88

90

92

94

96

98

100

t

Sur

viva

l Pro

babi

lity

(in p

erce

nt)



Here, the result does not fit so good, we can observe are significant discrepanciesbetween the market survival probabilities and the simulated survival probabilites.With linear interpolation of Ψ(t) and the above algorithm generating 100,000 defaulttimes, these are compared with the expected default times in the following figure 6.33.

0 2 4 6 8 10 120

500

1000

1500

2000

2500

3000

3500

4000

4500

t

Def

aults


(a) Comparison between simulatedand market implied default times

0 2 4 6 8 10 12−4

−3

−2

−1

0

1

2

3

4

t

Rela

tive D

evia

tion (

in p

erc

ent)

(b) Relative deviations between mar-ket implied and simulated defaulttimes

Figure 6.33.: 100,000 simulated default times compared with market implied defaulttimes using a stochastic intensity function with linear interpolation

97

6. Implementation

The simulated results fit well, especially presented in figure 6.33(b). The followingfigure 6.34 compares the simulated survival probabilities with the survival probabilitiesfrom table 4.2.

0 1 2 3 4 5 6 7 8 9 1085

90

95

100

t

Sur

viva

l Pro

babi

lity

(in p

erce

nt)



The results are very good, there are no significant differences between the marketsurvival probabilities and simulated survival probabilities. In the following figures6.35(a) and 6.35(b) the relative deviation between the survival probabilities from theDefault Process with deterministic intensity function and the survival probabilitiesfrom the Default Process with stochastic intensity function are shown.

98


0 1 2 3 4 5 6 7 8 9 10-1.5

-1

-0.5

0

0.5

1

t

Rel

ativ

e D

evia

tion

(in p

erce

nt)

(a) Constant interpolation for the deterministic part of the stochasticintensity function

0 1 2 3 4 5 6 7 8 9 10−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

t

Rel

ativ

e D

evia

tion

(in

per

cent

)

(b) Linear interpolation for the deterministic part of the stochasticintensity function

Figure 6.35.: Relative deviation between the survival probabilities from the DefaultProcess with deterministic intensity function and the survival probabili-tiess from the Default Process with stochastic intensity function (100,000Scenarios)

99

6. Implementation

In figure 6.35(a) we can see very well that the relative deviation changes very strongat the nodes of the deterministic part of the stochastic intensity function. This can beexplained by the jumps of the deterministic part of the stochastic intensity function (seefigure 4.4(a)). The contrary can be seen in figure 6.35(b), there are small deviationsfor linear interpolation of the nodes. And contrary to figure 6.35(a), here the deviationdoes not change dependending on the nodes.

6.3. Implementation of the Commodity Models

In section 5.3 the commodity models were presented which are used in this work.Based on these in section 5.4 it is shown how the models can be calibrated to marketdata. Now, however, the following questions are discussed again:

1. How can the models be implemented?

2. How good are the results of these models?

Both questions are addressed in this section and ultimately answered. For thispurpose, the models are first considered seperately and then the results of the modelsare examined. For comparison purposes the same market data as in section 5.4 is used.

6.3.1. Implementation of the Schwartz One-Factor model

The first model presented here was the Schwartz One-Factor model. Paths of thismodel can be generated using the following algorithm:

Algorithm

1. Generate X(t) for each t ∈ (0, T ] by theorem 7,

2. Set S(t) = exp X(t).

To assess the qualiy of the model, one can compare simulated forward prices withmarket prices. The simulated forward prices can be calculated by averaging the sim-ulated spot prices. This comparison is done for crude oil with the correspondingparameters (see table 5.6) and the calibrated function ϕ(t). This comparison is shownin the following figure 6.36.

100


0 1 2 3 4 5 6 7104

106

108

110

112

114

116

t

Pric

e / b

arre

l

Market Forward PricesModel Forward Prices

Figure 6.36.: Simulated forward prices compared to the market forward curve for crudeoil (100,000 Scenarios by the Schwartz One-Factor model)

The quality of the model is acceptable and in-line with the expectations since thesimulated forward prices fluctuate constantly in correspondance to the market prices.

Implementation of the advanced Schwartz One Factor model

The enhancement of the Schwartz One Factor model is the advanced Schwartz OneFactor model, paths of this model can be generated using the following algorithm:'

&

$

%

Algorithm

1. Generate X(t) for each t ∈ (0, T ] by theorem 7,

2. Calculate the values of ϕ(t) for each t ∈ (0, T ] by theorem 11,

3. Set S(t) = exp X(t) + ϕ(t).

To assess the quality of the model, one can compare simulated forward prices withmarket prices. The simulated forward prices can be calculated by averaging the sim-ulated spot prices. This comparison is done for crude oil with the corresponding pa-rameters (see table 5.10) and the calibrated function ϕ(t). This comparison is shownin the following figure 6.40.

101

6. Implementation

0 1 2 3 4 5 6 7104

105

106

107

108

109

110

111

112

113

114

t

Pric

e / b

arre

l


Figure 6.37.: Simulated forward prices compared to the market forward curve for crudeoil (100,000 scenarios by the advanced Schwartz One-Factor model)

The quality of the model is very good and in-line with the expactations since thesimulated forward prices fluctuate constantly in correspondance with the market prices.

6.3.2. Implementation of the Gibson-Schwartz model

The next model presented above was the Gibson-Schwartz model. Paths of this modelcan be generated using the following algorithm:'

&

$

%

Algorithm

1. Generate δ(t) for each t ∈ (0, T ] with definition 4,

2. Generate X(t) for each t ∈ (0, T ],

3. Set S(t) = exp X(t).

To assess the quality of the model, one can compare simulated forward prices withmarket prices. The simulated forward prices can be calculated by averaging the sim-ulated spot prices. This comparison is done for crude oil with the correspondingparameters (see table 5.8) and the calibrated function ϕ(t). This comparison is shownin the following figure 6.38.

102


0 1 2 3 4 5 6 7102

104

106

108

110

112

114

t

Pric

e / b

arre

l


Figure 6.38.: Simulated forward prices compared to the market forward curve for crudeoil (100,000 Scenarios by the Gibson-Schwartz model)

The quality of the model is of a poor quality because the simulated forward pricesare not in-line with the market prices.

6.3.3. Implementation of the Smith-Schwartz model

The next presented model was the Smith-Schwartz model and paths of this model canbe generated using the following algorithm:

Algorithm

1. Generate χ(t) and ξ(t) for each t ∈ (0, T ] with theorem 8,

2. Set S(t) = exp χ(t) + ξ(t).

To assess the qualiy of the model, one can compare simulated forward prices withmarket prices. The simulated forward prices can be calculated by averaging the sim-ulated spot prices. This comparison is done for crude oil with the corresponding pa-rameters (see table 5.10) and the calibrated function ϕ(t). This comparison is shownin the following figure 6.39.

103

6. Implementation

0 1 2 3 4 5 6 7102

104

106

108

110

112

114

t

Pric

e / b

arre

l


Figure 6.39.: Simulated forward prices compared to the market forward curve for crudeoil (100,000 Scenarios by the Smith-Schwartz model)

The quality of the model is acceptable and in-line with the expactations since thesimulated forward prices fluctuate constantly in correspondance with the market prices.Nonetheless, the fluctuations indicate that the number of scenarios are not sufficientto replicate the forward curve in a smooth way.

Implementation of the advanced Smith-Schwartz model

The first enhancement of the Smith-Schwarz model is the advanced Smith-Schwartzmodel and paths of this model can be generated using the following algorithm:'

&

$

%

Algorithm


2. Calculate the values of ϕ(t) for each t ∈ (0, T ] with theorem12,

3. Set S(t) = exp χ(t) + ξ(t) + ϕ(t).

To asses the qualiy of the model, one can compare simulated forward prices with mar-ket prices. The simulated forward prices can be calculated by averaging the simulatedspot prices. This comparison is done for crude oil with the corresponding parameters

104


(see table 5.10) and the calibrated function ϕ(t). This comparison is shown in thefollowing figure 6.40.

0 1 2 3 4 5 6 7100

105

110

115

t

Pric

e / b

arre

l


Figure 6.40.: Simulated forward prices compared to the market forward curve for crudeoil (100,000 scenarios by the advanced Smith-Schwartz model)

The quality of the model is acceptable and in-line with the expactations since thesimulated forward prices fluctuate constantly in correspondance with the market prices.Nonetheless, the fluctuations indicate that the number of scenarios is not sufficient toreplicate the forward curve in a smooth way.

Implementation of the advanced Smith-Schwartz model with seasonality

The second enhancement of the Smith-Schwarz model is the advanced Smith-Schwartzmodel with seasonality, paths of this model can be generated using the following algo-rithm:

105

6. Implementation'

&

$

%

Algorithm


2. Calculate the values of ϕ(t) for each t ∈ (0, T ] with theorem12,

3. Set S(t) = exp χ(t) + ξ(t) + ϕ(t).

To asses the qualiy of the model, one can compare simulated forward prices with mar-ket prices. The simulated forward prices can be calculated by averaging the simulatedspot prices. This comparison is done for wheat with the corresponding parameters(see table 5.11) and the calibrated function ϕ(t). This comparison is shown in thefollowing figure 6.41.

0 0.5 1 1.5 2 2.5740

760

780

800

820

840

860

880

900

920

940

t

Pric

e / b

arre

l


Figure 6.41.: Simulated forward prices compared to the market forward curve forwheat (100,000 scenarios by the advanced Smith-Schwartz model withseasonality)

The quality of the model is very adequate in the short-term but in the long runthe deviation widens. It is in-line with the expactations since the simulated forwardprices fluctuate constantly in correspondance with the market prices. However, thefluctuations indicate that the number of scenarios is not sufficient to replicate theforward curve in a smooth way.

106

7. Results

In this section, the effects of counterparty risk will be evaluated for two swaps. Theresults are scaled with the present value of fixed payments (fixed payer leg).

In the first trade the fair swap rate $903.00 is swapped on a seminual basis against abushel of wheat. The fixed payer leg is worth $ 3,540.96.

The second trade is based on swapping the fair swap rate $107.93 against a barrelof WTI crude oil. The fixed payer leg is $ 506.82.

The following figure 7.1, shows the expected default probabilities for the time intervalfrom each date Ti to the next date Ti+1 with i ∈ 0, . . . , n. These probabilities arecalculated using the default process, which was calibrated to the Deutsche Bank CDSspread data.

0 1 2 3 4 5 6 7 8 9 100

0.005

0.01

0.015

0.02

0.025

t

Def

ault

Pro

babi

litie

s (in

per

cent

)

Figure 7.1.: Expected default probabilities between each point in time

In figure 7.2(a) we can see that the expected exposure increases up to the first ex-change point and then sharply declines before rising again. These local peaks declinein magnitude for as the time to maturity increases. The following illustration 7.2(b)shows the values of the expected exposure weigthed by the default probabilities. These

107

7. Results

weigthed expected exposures are the addends from equation (2.1). We can see thatthese addends, the CVA, have pretty much the same shape as the expected exposure.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

t

Cre

dit V

alua

tion

Adj

ustm

ent (

in p

erce

nt)

Expected Value95%−Quantile99%−Quantile

(a) Expected Exposure in percent of the fixedpayer leg

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

100

200

300

400

500

600

700

800

t

Exp

ecte

d E

xpos

ure

(in p

erce

nt)


(b) Addend of the Credit Valuation Adjust-ment in percent of the fixed payer leg

Figure 7.2.: Expected exposure and Credit Valuation Adjustment for a two-year swapon wheat with semi-annual exchange. Calculated using 100,000 Scenariosgenerated with the advanced Smith-Schwartz model with seasonality

The following table 7.1 shows the average Credit Valuation Adjustment and twohigh quantiles:

Expected Value 95%-Quantile 99%-Quantile20.7974 29.9632 36.9415

Table 7.1.: Expected values and quantiles of the Credit Valuation Adjustments forswaps on wheat (2 years, semi-annually settlement) (in percent of the fixedpayer leg). Calculated using 100,000 Scenarios generated with the advancedSmith-Schwartz model with seasonality

In figure 7.3(a) we can see that the expected exposure increases up to the firstexchange point and then declines sharply before rising again. These local peaks aresmaller against maturity. The following illustration 7.3(b) shows the values of theexpected exposure weigthed with the default probabilities. These weigthed expectedexposures are the addends from equation (2.1), i.e. the CVA. In contrast to the abovecase for wheat, here the addends do not follow the shape of the expected exposure.

108

7.1. Sensitivity Analysis

The explanation can be found in Figure 7.1, here we can see that the default proba-bilities increase dramatically after the first two years.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

10

20

30

40

50

60

70

t

Exp

ecte

d E

xpos

ure

(in p

erce

nt)



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

t

Cre

dit V

alua

tion

Adj

ustm

ent (

in p

erce

nt)



Figure 7.3.: Expected exposure and Credit Valuation Adjustment for a five-year swapon crude oil with annual exchange. Calculated using 100,000 Scenariosgenerated with the advanced Schwartz One-Factor model

The following table 7.2 shows the average Credit Valuation Adjustment and twohigh quantiles:

Expected Value 95%-Quantile 99%-Quantile16.3250 54.8940 86.8378

Table 7.2.: Expected values and quantiles of the Credit Valuation Adjustments forswaps on Crude Oil (5 years, annually settlement) in percent of the fixedpayer leg. Calculated using 100,000 Scenarios generated with the advancedSchwartz One-Factor model


The following sensitivity analysis is performed on calculating the effects of changes inrisk factors (RF ). As a risk measure we use the Value at Risk (VaR). It gives themaximum loss occuring in a defined period with a probability c for a certain holdingperiod. To determine the VaR here, the so-called Delta-normal approach is used (see

109

7. Results

[Pfe04, Theorem 11.1.2]):

V aRRF (c, 1year) =

∣∣∣∣RF · ∂PV∂RF

∣∣∣∣ · σRF · qc (7.1)

Where σRF is the volatility of the risk factor. If there are multiple risk factors, the VaRfor each risk factor is calculated separately and then aggregated using the followingformula (see [Pfe04, Theorem 11.1.2]):

V aR =

√√√√ m∑i=1

m∑j=1

V aRi · ρi, j · V aRj (7.2)

Where m is the number of risk factors, besides ρi, j is the correlation between the i-thand the j-th risk factor.

In this work the main risk factors of CVA are the CDS spreads. For each CDSspread the VaR must be calculated, and then aggregated using the equation (7.2).

V aRCV A (c, 1 year) =

∣∣∣∣s · ∂CV A∂s

∣∣∣∣ · σs · qcThe 30 days volatilities of the CDS spreads are shown in the table 7.3 below:

Maturity Volatility6 months 107.71241 year 106.57302 year 104.30263 year 60.49604 year 44.98135 year 40.97987 year 37.958110 year 37.1443

Table 7.3.: 30-days volatility (in basis points) of the CDS-spreads for various maturitiesof Deutsche Bank (Data from Bloomberg, as of May 3, 2011)

Using the square-root approach, the volatility can be approximated for one year:

σs = V ola1 year ≈ V ola30 days ·√

12

110


Thus, the VaR is calculated for the CVA with 99% of confidence for each risk factorin case of the crude oil swap:

V aRCV A1 (99%, 1 year) =

∣∣∣∣s1 ·∂CV A

∂s1

∣∣∣∣ · σs1 · q99%

= |24bp · (−15, 2874%)| · 107.71bp ·√

12 · 2.33 = 3.20%

V aRCV A2 (99%, 1 year) = |24bp · 0, 6142%| · 106.57bp ·√

12 · 2.33 = 0.13%

V aRCV A3 (99%, 1 year) = |35bp · (−1, 9373%)| · 104.30bp ·√

12 · 2.33 = 0.56%

V aRCV A4 (99%, 1 year) = |52bp · (−10, 6035%)| · 60.50bp ·√

12 · 2.33 = 2.71%

V aRCV A5 (99%, 1 year) = |74bp · 1.3064%| · 44.98bp ·√

12 · 2.33 = 0.35%

V aRCV A6 (99%, 1 year) = |87bp · 0, 0463%| · 40.98bp ·√

12 · 2.33 = 0.01%

Since for the aggregation of the individual VaRs the correlation between risk factorsneeded. Since it is not available, we have to make assumptions about the correlation.The following figure shows the historical CDS spreads. The spreads for different ma-turities usually run in the same direction, therefore we can assume that they are highcorrelated. Because of this presumption in this thesis a correlation of 1 is assumedbetween the risk factors as a conservative rule of thumb. Hence our considerationsyield an upper bound on the VaR.

Figure 7.4.: Historical CDS-spreads for various maturities of Deutsche Bank (Datafrom Bloomberg, as of June 6, 2011)

111

7. Results

With the above assumption about the correlation, the aggregate VaR is determined:

V aR =

√√√√ 6∑i=1

6∑j=1

V aRi · ρi, j · V aRj

=

√√√√ 6∑i=1

6∑j=1

V aRi · V aRj

= 6.97%

The result indicates that the value of the CVAwith 99% probability increases by amaximum of 6.97 percentage points of the fixed payer leg. Especially interesting isthat the sensitivities of the CVA regarding the CDS spreads increase with the matu-rity and decays again towards the end. We can see that this model is very sensitive tothe CDS spreads, the mean of the CVA for this derivative was 16.32% and with thecalculated VaR the maximum CVA with 99% probability is 23.29% of the fixed payerleg.

After the VaR was determined for the CVA for the swap on crude oil, in the followingthe VaR for the CVA is determined for the swap on wheat. Since this swap only hastwo years to maturity, the only relevant risk factors are the CDS spreads with thematurity up to two years. Thus, the VaR is calculated for the CVA with 99% ofconfidence for each risk factor in case of the wheat swap:

V aRCV A1 (99%, 1 year) = |24bp · (−11.9854%)| · 107.71bp ·√

12 · 2.33 = 2.51%

V aRCV A2 (99%, 1 year) = |24bp · 8.3035%| · 106.57bp ·√

12 · 2.33 = 1.71%

V aRCV A3 (99%, 1 year) = |35bp · 21.0976%| · 104.30bp ·√

12 · 2.33 = 6.15%

Corresponding to the case of the swap on crude oil, in the following the VaRis aggre-gated for the swap on wheat:

V aR =

√√√√ 3∑i=1

3∑j=1

V aRi · ρi, j · V aRj

=

√√√√ 3∑i=1

3∑j=1

V aRi · V aRj

= 10.37%

112

7.2. Analysis of the Model Risk

The result indicates that the value of the CVA with 99% probability increases by aconservative maximum of 10.37 percentage points of the fixed payer leg. Especiallyinteresting is that the sensitivities of the CVA regarding to the CDS spreads incresesas the time to maturity increases. We can see that this model is very sensitive tothe CDS spreads, the mean of the CVA for this derivative was 20.08% and with thecalculated VaR the maximum CVA with 99% probability is 30.45% of the fixed payerleg.

7.2. Analysis of the Model Risk

In the following the model risk is determined. Here the effects are shown by performingan analysis with the suboptimal model. For this as optimal model the Schwartz One-Factor model for crude oil is assumed according to our calibration analysis in section5.4.4. The next best model can be found in table 5.13, it is the Smith-Schwartz modelwith seasonality. From the same table, we can see that the MSE for the optimal modelis by a factor of 7.5 smaller in comparison to the next best model. In the previousanalysis, the models were always provided with a deterministic function, so that theexpected values of the model matches exactly the forward curve. To highlight moreclearly the model error, the deterministic function here is omitted. In the figure 7.5below, the expected exposure and the addends of the Credit Valuation Adjustment forthe Schwartz One-Factor model is shown.

113

7. Results

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

10

20

30

40

50

60

70

t

Exp

ecte

d E

xpos

ure

(in p

erce

nt)



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

t

Cre

dit V

alua

tion

Adj

ustm

ent (

in p

erce

nt)



Figure 7.5.: Expected exposure and Credit Valuation Adjustment for a five-year swapon crude oil with annual exchange. Calculated using 100,000 Scenariosgenerated with the Schwartz One-Factor model

In figure 7.6 below, the expected exposure and the addends of the Credit ValuationAdjustment for the Smith-Schwartz model with seasonality is shown.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

50

100

150

200

250

300

350

400

450

t

Exp

ecte

d E

xpos

ure

(in p

erce

nt)



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t

Cre

dit V

alua

tion

Adj

ustm

ent (

in p

erce

nt)



Figure 7.6.: Expected exposure and Credit Valuation Adjustment for a five-year swapon crude oil with annual exchange. Calculated using 100,000 Scenariosgenerated with the Smith-Schwartz model with seasonality

114

7.3. Analysis of the Effects of Correlations

In comparison between the two last figures the expected exposure and the addendsof the Credit Valuation Adjustment is for the suboptimal model about a factor of10 higher than for the optimal model. Further information about the impact on theCredit Valuation Adjustment are shown in the following table 7.4.

Expected Value 95%-Quantile 99%-QuantileSchwartz One-Factor model 16.3446 19.0711 20.3165Smith-Schwartz model with seasonality 73.5095 97.8119 113.5480Relative deviation 349.7479 412.8805 458.8957

Table 7.4.: Expected values and quantiles of the Credit Valuation Adjustments forswaps on Crude Oil(5 years, annually settlement) (in percent of thefixed payer leg). Calculated using 100,000 Scenarios generated with theSchwartz One-Factor model respectively with the Smith-Schwartz modelwith seasonality

In the previous table it can clearly be seen that the estimation error can be verylarge by choosing a non-optimal model. The MSE of the optimal model is 0.0371 whilethe MSE of the next better model is about 750% higher. The deviations of the CVAare not so extreme, but the mean has a deviation of 349.7479%. In the high quantiles,the deviation is higher in the 99% quantile the deviation is 458.8957%.

7.3. Analysis of the Effects of Correlations

Last but not least the effect of correlations between the default process and de Com-modity price process have been investigated. We examined various correlations of -1 to1. However, the investigations the CVA of the wheat swap and of the crude oil swapfound no differences for different correlations. This result is surprising in contrast tothe results of Brigo, Chourdakis, and Bakar in [BCB08], who have shown that theCVA decreases with increasing correlation. The differences between the results of thisthesis and the results in [BCB08], can be explained by the volatility of the defaultprocess. In this thesis the volatility is 0.00266% toward a volatility of 59%.

115

8. Conclusion

In the context of this thesis the counterparty risk was valued for commodity derivatives.

8.1. Summary

Various kinds of possibilities to implement a counterparty risk valuation method wereexamined such as, e.g., for the default modelling, the commodity price modelling andthe CIR process.

For the final valuation of the CVA, the author considered the implemention of theCIR which on the one hand needs the desired characteristics such as no monotonicityand positivity and on the other hand it takes solely an acteptable amount of time tocalculate (see chapter 6). These models were calibrated to market data for forwardsand options. The author achieved this by minimizing the MSE between the market-and the modelprices.

Afterwards, the CVA was calculated under the best model for each individual commod-ity. For the default process there exists a variety of different stochastic or deterministicintensity models. Both of these kinds of models were fitted to market data for CDS.For the valuation that followed, the stochastical intensity model was used because ofits performance across all market segments (in particular index based CDOs like CDXor iTraxx tranches; see [BPT06]). Furthermore, the impacts of seasonalities in thecommodity prices was analyzed.

The model risk, which means the risk to choose the inappropriate model or not thebest model to value the model, was also inspected. Since the CDS spreads cause themost contribution of risk in the CVA valuation, they were also analyzed from theauthor in order to answer the question how sensitive their results react to the CDSspreads.

The outcome of the work in this thesis brought astonishing results when consider-ing the simulation of a CIR process. The Brigo parameters produced very large faultswhen transferring from continuous to the discrete case, in comparison to this, theDeutsche Bank parameters had no difficulties at all in the transformation process.Here the question arises which requirements the parameters would need to fulfill in

117

8. Conclusion

order to only just barely make the transformation from the continuous to the concretecase.

The algorithms shown in this thesis, can be applied or expanded in either programinglanguage. There is also the possibility to simply use the MATLAB code provided bythe author. Furthermore, the reader can also use the attached data to calibrate themodels discussed in this thesis or to apply them to other models for own researchpurposes.

8.2. Outlook

There are further topics related to the subject considered in this thesis that can beconsidered in future research. There exists the possibility to additionally examine themodel calibration. Some of the following different approaches are suggested: there isthe possibility to implement the Newton scheme or other numerical schemes. Anotherapproach can be achieved by extending the commodity models to cope with stochas-tically interest rates. Further commodities can be evaluated to capture seasonalityeffects such as natural gas or soy beans.

In this thesis it was shown, that there exists quite a problem to realize the imple-mention of the CIR. These difficulties result when migrating from the continuous tothe discrete case. These problems where discovered with the Brigo Parameters butnot with the Deutsche Bank Parameters. Now, the question is with which parametersthe migration can barely be done. Furthermore, applying the model calibration forcrude oil the simplest model achieved the best results. It would be interesting to findout if this is also the case for a longer period of time. In this thesis the counterpartyrisk was only examined for swaps. There exists a further the possibility to value otherderivatives. In order to value the counterparty risk, the CVA was used. There existsthe additional option to build on this foundation and determine risk adjusted spreadsor strike prices. In this thesis the counterparty risk was investigated under the as-sumption that the company being evaluatied cannot default. Assuming the opposite,that the company could actually default would be interesting as well. In doing this onewould also consider the default risk of both of the firms. This result would also lead tonoteworthy results stamming from the risk adjusted spreads. This topic is currentlyvery intensively discussed by the International Accounting Supervisor Board (IASB)especially concerning accounting. This topic is also recommended from the author tobe throughly examined as well.

118

Appendix

119

A. Stochastic Calculus

In this chapter, the used stochastic processes are discussed. The chapter starts withthe Poisson process in section A.1 go further with the Brownian Motion in section A.2and ends with the Ornstein-Uhlenbeck process in section A.3.

A.1. Poisson Process

The Poisson process is a stochastic process named after Poisson (1781 – 1840). Itis a counting process N(t) with Poisson distributed increases and intensity Λ(t) =t∫

0

λ(u)du. The time between jumps is exponentially distributed with the same intensity

as the counting process. There are different versions of the Poisson process, they differin the intensity λ(t). With a constant intensity λ(t) := λ it is named as a HomogenousPoisson process.

Definition 8 (Homogenous Poisson Process). A counting process N(t) with constantintensity λ is called a homogenous poisson process, if and only if:

1. N(0) = 0

2. N(t+ u)−N(t) ∼ Pois(λ · u) for t ≥ 0, u > 0

3. The random variables N(ti+1)−N(ti), i ∈ 0, . . . , n− 1, are stochastich inde-pendent for any 0 = t0 < t1 < . . . < tn.

If the intensity is deterministic then it is a inhomogenous Poisson process.

Definition 9 (Inhomogenous Poisson Process). A counting process N(t) with deter-ministic intensity function Λ(t) is called a inhomogenous poisson process, if and onlyif:

1. N(0) = 0

2. N(t+ u)−N(t) ∼ Pois(Λ(t+ u)− Λ(t)) for t ≥ 0, u > 0

3. The random variables N(ti+1)−N(ti), i ∈ 0, . . . , n− 1, are stochastich inde-pendent for any 0 = t0 < t1 < . . . < tn.

121

A. Stochastic Calculus

If the intensity is a stochastic and deterministic then it is a Cox process. The Coxprocess is also known as a double stochastic process. Because in the first step a re-alization of λ will be generated and in the next step the counting varible N will besimulated.

The following figure A.1 shows the relationship between the various versions of thePoisson processes.

Homogeneous Poisson process Constant intensity

Non-homogeneous Poisson process Stochastic intensity

Non-homogeneous Poisson process Deterministic intensity

Cox process Stochastic and deterministic intensity

Figure A.1.: Overview over the different poisson processes

For further reading [CD09], [BD08] and [Pfe09] are suggested.

A.2. Brownian motion

The (geometric) Brownian motion was first discovered by Robert Brown in 1827 inthe field of microbiology. It has been defined as follows by Pfeifer in [Pfe09, Page 4.1]:

Definition 10 (Brownian Motion). A stochastic process X =(

[Ω, F , P] , Xtt∈[ 0,∞ )

)adapted to a filtration Ftt∈I is called a (one-dimensional) Brownian Motion when

1. W (0) = 0 f.s.,

2. W (t+ ∆t)−W (t) ∼ N(0, ∆t), for all s, t ≥ 0,

3. W (t)−W (s) independent from Fs for 0 ≤ s ≤ t,

4. Paths are continuous.

The Brownian Motion is also known as Wiener process. For further information theauthor refers to [KS08], [Bro27], and [Bac00].

122

A.3. Ornstein-Uhlenbeck Process

A.3. Ornstein-Uhlenbeck Process

The Ornstein-Uhlenbeck procress was defined by Ornstein and Uhlenbeck in 1930. Ithas the mean-reverting property and therefore is used in the short-rate modelling.

Definition 11 (The Ornstein-Unlenbeck process). A stochastic process X is calledOrnstein-Uhlenbeck process if it follows the following SDE:

dXt = θ (µ−Xt) dt+ σdWt, X0 = a

Where a, µ ∈ R and θ, σ > 0.

By solving the SDE one arrives the following solution for X(t) (see [MMS09, Page2]):

Xt = a · exp −θt+ µ (1− exp−θt) + σ exp −θ · tt∫

0

exp θ · u dWu

With expected value and variance:

E(Xt) = a · exp−θt+ µ (1− exp−θt)

V(Xt) =σ2

2θ(1− exp−2θt)

For further readings [UO30] and [MMS09] are recommended.

123

B. Implementation in Matlab

Cox-Ingersoll-Ross DVD:\MATLAB\CIRCIRECIREulerCIREulerCountCIREulerDeestraDelbaenCIREulerDiopCIREulerImplicitSchemeCIRExactSimulationAlgorithmCIRMilsteinCIRMilsteinCountCIRMilsteinImplicitSchemeCIRPPZeroBondPriceCIR

Default Time Simulation DVD:\MATLAB\Default TimeBootstrapIntensityFunctionDefaultTimeDefaultTimeDeterministicIntensityNodesForPsipsiintegratedSurvProbDetIntSurvProbsCalibrationTransitionProbabilities

Commodity Models DVD:\MATLAB\Commodity ModelsadvSchwartzF1advSmithSchwartzadvSmithSchwartzSeasonalityadvSmithSchwartzSeasonalityPsiSmithSchwartzSmithSchwartzSeasonalityPhiPhiSchwartzF1SchwartzF1

125

B. Implementation in Matlab

Pricing of Derivatives DVD:\MATLAB\PricinggetNPVForwadvSchwartzF1getNPVForadvSmithSchwartzgetNPVForadvSmithSchwartzSeasonalitygetNPVForSchwartzF1getNPVForSmithSchwartzSeasonalitygetNPVSwapadvancedSchwartzF1getNPVSwapadvancedSmithSchwartzSeasonalitgetNPVSwapSchwartzF1getNPVSwapSmithSchwartzSeasonalitNPVadvSchwartzF1NPVadvSmithSchwartzSeasonalityNPVSchwartzF1NPVSmithSchwartzSeasonalityCreditValuation Adjustment DVD:\MATLAB\CVACVAEEMarket Data DVD:\MATLAB\Market DatagetForwardMarketPricesgetYieldCurveOther Functions DVD:\MATLAB\OthergetDiscountFactorsgetIndexgetLtgetxt

126

List of Tables

4.1. CDS-spreads for various maturities of Deutsche Bank (Data from Bloomberg,as of May 3, 2011) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2. Survival probabilities for various maturities of Deutsche Bank (Calcu-lated using theorem 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3. Default intensities calibrated from market survival probabilities . . . . . 33

4.4. Default intensities determined with the Credit-Triangle . . . . . . . . . 33

4.5. Calibrated Parameters for the CIR process . . . . . . . . . . . . . . . . 35

4.6. Nodes for the deterministic function Ψ(t) . . . . . . . . . . . . . . . . . 37

5.1. Denotation of the parameters of the Gibson-Schwartz model . . . . . . . 47

5.2. Denotation of the parameters of the Smith-Schwartz model (see [SS00,Table 1]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.3. Relationships between the parameters (see [SS00, Table 1] . . . . . . . . 53

5.4. Observed market prices for Options on wheat futures . . . . . . . . . . . 61

5.5. Parameter for modelling wheat spot prices in the Schwartz One-Factormodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.6. Parameter for modelling crude oil spot prices in the Schwartz One-Factor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.7. Parameter for modeling wheat spot prices in the Gibson-Schwartz model 64

5.8. Parameter for modeling crude oil spot prices in the Gibson-Schwartzmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.9. Parameter for modeling wheat spot prices in the Smith-Schwartz model 65

5.10. Parameter for modelling crude oil spot prices in the Smith-Schwartzmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.11. Parameter for modeling wheat spot prices in the Smith-Schwartz modelwith seasonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.12. Parameter for modeling crude oil spot prices in the Smith-Schwartzmodel with seasonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.13. MSE for all models calibrated to both commodities . . . . . . . . . . . 69

6.1. Properties of the shown implementation approachse of the CIR process 91

127

List of Tables

7.1. Expected values and quantiles of the Credit Valuation Adjustments forswaps on wheat (2 years, semi-annually settlement) (in percent of thefixed payer leg). Calculated using 100,000 Scenarios generated with theadvanced Smith-Schwartz model with seasonality . . . . . . . . . . . . . 108

7.2. Expected values and quantiles of the Credit Valuation Adjustments forswaps on Crude Oil (5 years, annually settlement) in percent of thefixed payer leg. Calculated using 100,000 Scenarios generated with theadvanced Schwartz One-Factor model . . . . . . . . . . . . . . . . . . . 109

7.3. 30-days volatility (in basis points) of the CDS-spreads for various ma-turities of Deutsche Bank (Data from Bloomberg, as of May 3, 2011) . . 110

7.4. Expected values and quantiles of the Credit Valuation Adjustments forswaps on Crude Oil(5 years, annually settlement) (in percent of thefixed payer leg). Calculated using 100,000 Scenarios generated withthe Schwartz One-Factor model respectively with the Smith-Schwartzmodel with seasonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

128

List of Figures

4.1. Yield Curve for German Federal Bonds (Data from Deutsche Borse, asof May 12, 2011) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2. Relative Deviation between both intensity functions . . . . . . . . . . . 33

4.3. Market implied deterministic intensity function compared to the bestapproximation using a CIR process . . . . . . . . . . . . . . . . . . . . . 35

4.4. Calibrated function Ψ(t) in two versions . . . . . . . . . . . . . . . . . . 37

5.1. Forward Prices for Crude Oil (WTI) (Data from Bloomberg, May 3, 2011) 60

5.2. Forward Prices for Wheat (Data from Bloomberg, May 3, 2011) . . . . . 60

5.3. Calibrated function ϕ in the advanced Schwartz One-Factor model towheat market forward prices . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.4. Calibrated function ϕ in the advanced Schwartz One-Factor model tocrude oil market forward prices . . . . . . . . . . . . . . . . . . . . . . . 64

5.5. Calibrated function ϕ in the advanced Smith-Schwartz model to wheatmarket forward prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.6. Calibrated function ϕ in the advanced Smith-Schwartz model to crudeoil market forward prices . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.7. Calibrated function ϕ in the advanced Smith-Schwartz model with sea-sonality to wheat market forward prices . . . . . . . . . . . . . . . . . . 69

6.1. A typical path simulated based on the Euler scheme (Brigo parameters) 72

6.2. Number of continuous paths from 10,000 scenarios of CIR processes,simulated with the Euler scheme (Brigo parameters) . . . . . . . . . . . 72

6.3. 10,000 scenarios of CIR processes based on the Euler scheme (Brigoparameters) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.4. 10,000 scenarios of CIR processes based on the Euler scheme (DeutscheBank parameters) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.5. One typical path simulated based on the Euler-Deelstra-Delbaen scheme(Brigo parameters) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.6. 10,000 scenarios of CIR processes based on the Euler-Deelstra-Delbaenscheme (Brigo parameters) . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.7. 10,000 scenarios of CIR processes based on the Euler-Deelstra-Delbaenscheme (Deutsche Bank parameters) . . . . . . . . . . . . . . . . . . . . 76

129

List of Figures

6.8. One typical path simulated based on the Euler-Diop scheme (Brigo pa-rameters) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.9. 10,000 scenarios of CIR processes based on the Euler-Diop scheme(Brigo parameters) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.10. 10,000 scenarios of CIR processes based on the Euler-Diop scheme(Deutsche Bank parameters) . . . . . . . . . . . . . . . . . . . . . . . . 78

6.11. Comparison of the relative deviation of the Euler scheme and both ex-tensions for 10,000 scenarios (Brigo parameters) . . . . . . . . . . . . . . 79

6.12. One typical path simulated based on the Milstein scheme (Brigo param-eters) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.13. Number of continuous paths from 10,000 scenarios of CIR processes,simulated with the Milstein scheme (Brigo parameters) . . . . . . . . . . 80

6.14. 10,000 scenarios of CIR processes based on the Milstein scheme (Brigoparameters) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.15. 10,000 scenarios of CIR processes based on the Milstein scheme (DeutscheBank parameters) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.16. One typical path simulated based on the Euler-Implicit scheme . . . . . 82

6.17. 10,000 scenarios of CIR processes based on the Euler-Implicit scheme(Brigo parameters) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.18. 10,000 scenarios of CIR processes based on the Euler-Implicit scheme(Deutsche Bank parameters) . . . . . . . . . . . . . . . . . . . . . . . . 83

6.19. One typical path simulated based on the Milstein-Implicit scheme . . . . 84

6.20. 10,000 simulation of CIR processes with the Milstein-Implicit scheme(Brigo parameters) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.21. 10,000 simulation of CIR processes with the Milstein-Implicit scheme(Deutsche Bank parameters) . . . . . . . . . . . . . . . . . . . . . . . . 85

6.22. One typical path simulated based on the E(0) scheme . . . . . . . . . . 86

6.23. 10,000 simulation of CIR processes with the E(λ) family for variousvalues of λ (Brigo parameters) . . . . . . . . . . . . . . . . . . . . . . . 87

6.24. 10,000 simulation of CIR processes with the E(λ) family for variousvalues of λ (Deutsche Bank parameters) . . . . . . . . . . . . . . . . . . 88

6.25. A typical paths simulated based on the Exact Simulation Algorithm(Brigo parameters) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.26. 10,000 scenarios of CIR processes with the Exact Simulation Algorithm(Brigo parameters) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.27. 10,000 scenarios of CIR processes with the Exact Simulation Algorithm(Deutsche Bank parameters) . . . . . . . . . . . . . . . . . . . . . . . . 90

6.28. Overview over the results of the various approaches (Brigo parameters) 92

6.29. 100,000 simulated default times compared with marked implied defaulttimes using a deterministic intensity function . . . . . . . . . . . . . . . 94

6.30. Survival probabilities, calculated from 100,000 simulated default times . 95

130

List of Figures

6.31. 100,000 simulated default times compared with market implied defaulttimes using a stochastic intensity function with constant interpolation . 96

6.32. Survival probabilities, calculated from 100,000 simulated default times . 976.33. 100,000 simulated default times compared with market implied default

times using a stochastic intensity function with linear interpolation . . . 976.34. Survival probabilities, calculated from 100,000 simulated default times . 986.35. Relative deviation between the survival probabilities from the Default

Process with deterministic intensity function and the survival prob-abilitiess from the Default Process with stochastic intensity function(100,000 Scenarios) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.36. Simulated forward prices compared to the market forward curve forcrude oil (100,000 Scenarios by the Schwartz One-Factor model) . . . . 101

6.37. Simulated forward prices compared to the market forward curve forcrude oil (100,000 scenarios by the advanced Schwartz One-Factor model)102

6.38. Simulated forward prices compared to the market forward curve forcrude oil (100,000 Scenarios by the Gibson-Schwartz model) . . . . . . . 103

6.39. Simulated forward prices compared to the market forward curve forcrude oil (100,000 Scenarios by the Smith-Schwartz model) . . . . . . . 104

6.40. Simulated forward prices compared to the market forward curve forcrude oil (100,000 scenarios by the advanced Smith-Schwartz model) . . 105

6.41. Simulated forward prices compared to the market forward curve forwheat (100,000 scenarios by the advanced Smith-Schwartz model withseasonality) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.1. Expected default probabilities between each point in time . . . . . . . . 1077.2. Expected exposure and Credit Valuation Adjustment for a two-year

swap on wheat with semi-annual exchange. Calculated using 100,000Scenarios generated with the advanced Smith-Schwartz model with sea-sonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.3. Expected exposure and Credit Valuation Adjustment for a five-yearswap on crude oil with annual exchange. Calculated using 100,000 Sce-narios generated with the advanced Schwartz One-Factor model . . . . . 109

7.4. Historical CDS-spreads for various maturities of Deutsche Bank (Datafrom Bloomberg, as of June 6, 2011) . . . . . . . . . . . . . . . . . . . . 111

7.5. Expected exposure and Credit Valuation Adjustment for a five-yearswap on crude oil with annual exchange. Calculated using 100,000 Sce-narios generated with the Schwartz One-Factor model . . . . . . . . . . 114

7.6. Expected exposure and Credit Valuation Adjustment for a five-yearswap on crude oil with annual exchange. Calculated using 100,000 Sce-narios generated with the Smith-Schwartz model with seasonality . . . . 114

A.1. Overview over the different poisson processes . . . . . . . . . . . . . . . 122

131

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138

Index

AAdvanced Cox-Ingersoll-Ross Process20Alfonsi E family . . . . . . . . . . . . . . . . 85

BBackwardation . . . . . . . . . . . . . . . . . 42Brownian motion . . . . . . . . . . . . . . 122

CCalibration

advanced Schwartz One-Factor model62

advanced Smith-Schwartz model65advanced Smith-Schwartz model with

seasonality . . . . . . . . . . . . . 68Commodtiy Models . . . . . . . . . 59Default Process . . . . . . . . . . . . . 22Deterministic intensity function30Gibson-Schwartz model . . . . . . 64Schwartz One-Factor model . . 62Smith-Schwartz model . . . . . . . 65Stochastic intensity function . . 34Survival probabilities . . . . . . . . 22

Call options . . . . . . . . . . . . . . . . . . . . 15CBPV . . see Credit Basis Point ValueCDS . . . . . . . see Credit Default SwapCDS option . . . . . . see Credit Default

SwaptionCDS-spread . . . . . . . . . . . . . . . . . . . . . 7CDSw . . . see Credit Default SwaptionCIR . see advanced Cox-Ingersoll-Ross

Process

CIR++ . . . . . . . . . . . . . . . see advancedCox-Ingersoll-Ross Process

Clearing house . . . . . . . . . . . . . . . . . . . 3Commdodity derivatives . . . . . . . . . 13Commodities . . . . . . . . . . . . . . . . . . . 39

agricultural . . . . . . . . . . . . . . . . 40Commodity Models . . . . . . . . . . . . . 39Contango . . . . . . . . . . . . . . . . . . . . . . 42Convenience yield . . . . . . . . . . . . . . . 41Cost of carry . . . . . . . . . . . . . . . . . . . 42Counterparty risk . . . . . . . . . . . . . . . . 3Cox process . . . . . . . . . . . . . . . . . . . 122Cox-Ingersoll-Ross Process . . . . . . . 20Credit Basis Point Value . . . . . . . . . . 8Credit Default Swap. . . . . . . . . . . . . . 7Credit Default Swaption . . . . . . . . . 10Credit derivatives . . . . . . . . . . . . . . . . 7Credit Valuation Adjustment . . . . . . 4CVA see Credit Valuation Adjustment

DDefault process . . . . . . . . . . . . . . . . . 17Delta-normal approach . . . . . . . . . 109Derivatives . . . . . . . . . . . . . . . . . . . . . . 7Deterministic intensity function . . . 19

EE(λ) . . . . . . . . . . . see Alfonsi E familyEDDs . . . . see Euler-Deelstra-Delbaen

schemeEDs . . . . . . . . . . see Euler-Diop schemeEIs . . . . . . . . see Euler-Implicit schemeEs . . . . . . . . . . . . . . . see Euler scheme

139

Index

ESs . see Exact Simulation AlgorithmEuler scheme . . . . . . . . . . . . . . . . . . . 71Euler-Deelstra-Delbaen scheme . . . 74Euler-Diop scheme . . . . . . . . . . . . . . 76Euler-Implicit scheme . . . . . . . . . . . 82Exact Simulation Algorithm . . . . . . 89

FFirm value model . . . . . . . . . . . . . . . 17Forward . . . . . . . . . . . . . . . . . . . . . . . 14Forward price . . . . . . . . . . . . . . . . . . 14Future . . . . . . . . . . . . . . . . . . . . . . . . . 14

GGibson-Schwartz model . . . . . . . . . . 47

IImplementation

Commodity Models . . . . . . . . 100Cox-Ingersoll-Ross . . . . . . . . . . 71Default-Process . . . . . . . . . . . . . 93

Intensity functions . . . . . . . . . . . . . . 17Intensity models . . . . . . . . . . . . . . . . 17

JJumps . . . . . . . . . . . . . . . . . . . . . . . . . 40

LLGD . . . . . . . . see Loss Given DefaultLoss given default . . . . . . . . . . . . . . . . 3

MMargin accounts . . . . . . . . . . . . . . . . . 3Mean reverting . . . . . . . . . . . . . . . . . 40Merton model . . . . . . . . . . . . . . . . . . 17Milstein scheme . . . . . . . . . . . . . . . . . 79Milstein-Implicit scheme . . . . . . . . . 84MIs . . . . . see Milstein-Implicit schemeModel risk . . . . . . . . . . . . . . . . . . . . 113Monotonicity . . . . . . . . . . . . . . . . . . . 82Ms . . . . . . . . . . . . . see Milstein scheme

OOrnstein-Uhlenbeck Process . . . . . . 44

Ornstein-Uhlenbeck procress . . . . 123OTC . . . . . . . . . . see Over the counterOver the counter . . . . . . . . . . . . . . . 3, 7

PPoisson process . . . . . . . . . . . . . . . . 121

Homogenous . . . . . . . . . . . . . . 121Inhomogenous . . . . . . . . . . . . . 121

Premium leg . . . . . . . . . . . . . . . . . . . . 7Protection buyer . . . . . . . . . . . . . . . . . 7Protection leg . . . . . . . . . . . . . . . . . . . 7Protection seller . . . . . . . . . . . . . . . . . 7Put options . . . . . . . . . . . . . . . . . . . . 15

RRecovery rate . . . . . . . . . . . . . . . . . 3, 8RR . . . . . . . . . . . . . . . see Recovery rate

SSchwartz One-Factor model . . . . . . 43

advanced . . . . . . . . . . . . . . . . . . 45Schwartz-F1 model . . . . . see Schwartz

One-Factor modelSeasonality . . . . . . . . . . . . . . . . . . . . . 39Sensitivity analysis . . . . . . . . . . . . . 109Short-Term/Long-Term Model . . . . see

Smith-Schwartz modelSmith-Schwartz model . . . . . . . . . . . 48

advanced . . . . . . . . . . . . . . . . . . 54advanced with Seasonality . . . 55

SNAC . see Standard North AmericanCorporate

Spikes . . . . . . . . . . . . . . . . . . . . . . . . . 40Standard European Corporate . . . . . 9Standard North American Corporate 9STEC . . . . . . . see Standard European

CorporateStochastic Convenience Yield Modelsee

Gibson-Schwartz modelStochastic intensity function . . . . . . 20Storability . . . . . . . . . . . . . . . . . . . . . 41Strike price . . . . . . . . . . . . . . . . . . . . 15

140

Index

Survival probabilities . . . . . . . . . . . . . 8Swap . . . . . . . . . . . . . . . . . . . . . . . . . . 14Swap rate . . . . . . . . . . . . . . . . . . . . . . 14

TTheory of Storage . . . . . . . . . . . . 39, 41

UUnderlying . . . . . . . . . . . . . . . . . . . . . . 7Upfront payment . . . . . . . . . . . . . . . . 9

VValue at Risk . . . . . . . . . . . . . . . . . . 109VaR . . . . . . . . . . . . . . see Value at Risk

141

valuation of counterparty risk for commodity derivatives

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