fer financial engineering review engineering... · 2019. 1. 14. · editorial note october 2003...

FER

Risk║Latte

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CONTENTS PAGE Editorial 1 Call for articles 2 A ‘reverse enquiry’ credit market application: the single-tranche Synthetic Collateralised Debt Obligation 7 Moorad Choudry Shout Floors 15 Terry Cheuk and Ton Vorst Generating Correlated Random Variable: Simple Techniques and Implementation 36 Henry Partier Binomial Approach for Lookback Options 48 Mark Ioffe Copyright : Risk Latte Company Limited Hong Kong Managing Editor : Rahul Bhattacharya +852 2251 1905 [email protected] Advertising : [email protected] Published by : Risk Latte Company Limited, Hong Kong

Financial Engineering Review

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Editorial Note October 2003 Welcome readers to the second monthly issue of Financial Engineering Review (FER) for 2003. Financial engineering, as an amalgamated discipline, has recently experienced a surge in interest from both academics and financial market practitioners alike across the Asia Pacific region. Over recent years, this interest has translated into an increasing demand for and creation of financial engineering programmes in a number of universities across the region, as well as a growing trend towards relevant quantitative skills as a prerequisite in recent industry hires. In other developed markets, this phenomenon started some time ago and has now grown into a recognised industry. In Europe and North America, there are now dedicated trade publications, regular conferences, seminars, technical consultancy and software houses, as well as professional bodies and specialist recruitment agencies created to serve the growing demand for qualified financial engineers. If this experience is any guide, there is definite potential to see similar developments across Asia Pacific in the near future. It is in light of these potential future developments that FER was conceived, with the aim of bridging the gap between academia and practice that currently exists in the Asia Pacific region. The main objective of this journal is to provide a much needed forum for academics and practitioners to publish ideas, methodologies and studies. The hope is that the articles published will help provide readers with a better understanding of existing and newly developed financial engineering techniques, as well as their associated commercial implementation and application. An editorial team for FER has recently been assembled, which now consists of academics and practitioners from across the region. There are also other contacts that will serve as referees or independent reviewers of submitted articles as the need arises from time to time. The editorial team will oversee and manage both the content and format of the journal going forward. Interested readers, who would like to contribute to the discussions and the sharing of ideas and information in the FER, are welcome to send their article to the editorial team by following the procedures outlined under a separate section. There will be further developments in terms of the look and feel of this journal over the coming months, as new ideas begin to flow regarding the journal’s content, format and distribution. Therefore, readers should expect a number of improvements as the journal develops into a recognised industry publication in the Asia Pacific region. We trust that you will share in our excitement, in what an opportunity like FER means for all of us involved in the practice and development of financial engineering in this region. We hope you enjoy this month’s issue, and we look forward to your continued support in the future. The Editors Financial Engineering Review

1 Financial Engineering Review

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Call for articles October 2003 Readers who would like to contribute an article in the FER are welcome to send their work to the editors at the following email address, [email protected]. Electronic format could be in MSWord, pdf, or other popular word processing files. Articles could be on any topic in financial engineering, which includes but is not limited to (in no particular order), • Derivative valuations • Term structure modeling • Risk measurement and management • Volatility modeling or forecasting • Fund performance measurement and attribution • Trading or market making strategy studies • Hedging techniques and implementation • Computational/implementation studies • Behavioural finance • Statistical or mathematical techniques applicable to finance and economics The articles may be theoretic or applied in nature. Format of the submitted articles include single spaced paragraphs with formulae that are numerically indexed and referenced in the article. Extensive proofs (if necessary) or computer code should be included as separate appendices to the article. Articles should include an abstract of 100 words or less and a bibliography section. Small tables and figures should be included in the body of the article and captioned in brief. Large tables or figures should be placed under separate appendices but appropriately referenced in the body of the article. We look forward to receiving your article and welcome the opportunity of future publication. The Editors Financial Engineering Review


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Risk Latte Company is a boutique financial engineering firm engaged in the field of risk and derivatives advisory and training in Asia. We help banks, hedge funds and financial institutions with:

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Risk Latte Company Limited Level 25, Bank of China Tower

1 Garden Road, Central Hong Kong

Telephone: (852) 2251 1905 Fax: (852) 2251 1818 e-mail: [email protected] website: www.risklatte.com


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VP, Regional Credit

Our client is a leading international investment bank, with a strong and well-established Asian business. They offer a wide varietyof banking products to a diverse and high quality client base across the region. This company is committed to delivering thehighest quality service to both internal and external clients.

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Interested candidates should forward their resume to Andrew Oliver (quoting reference FER/AO168) by email [email protected] or visit us online at www.michaelpage.com.hk.

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Our Client is a premiere Investment Bank with an enviable presence in the Asia Pacific region. They consistently deliver cuttingedge solutions to the needs of a high quality client base.

They have a senior level opening within their regional risk management function. As a Senior VP , you will lead a team of quantitative valuationanalysts responsible for reviewing and independently verifying the derivatives pricing process and exposure across the Asia Pacific region. Theteam will review and assess risk models and pricing processes across a full range of derivatives products to ensure the firms P&L is accuratelycalculated and reporting this to global management.

Close liaison with senior management on a global basis is required. The function will also review the risk profile of new trading initiatives andproducts, and liaise with the quantitative, product control and front office trading teams in determining appropriate models for new business.

Successful applicants will either have a proven background in control and risk management or, alternatively, will be working in a trading marketrisk management position. Most important requirements will be strong people management, technical and quantitative skills, along withexcellent communication ability.

Interested candidates should forward their resume together with salary details to Breige MacManus (quoting reference FER/BM101) by emailat [email protected] or visit us online at www.michaelpage.co.jp.

Japan Based - Regional CoverageQuantitative Valuations

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A “reverse enquiry” credit market application: the single-tranche synthetic Collateralised Debt Obligation

Moorad Choudhry There is a considerable literature, for example see Tavakoli (2001) amongst others, which illustrates how combining certain aspects of traditional cashflow securitisation technique with credit derivative technology gave rise to so-called synthetic securitisation, also known as unfunded securitisation. In a synthetic transaction, the credit risk of a pool of assets is transferred from an originator to investors, but the assets themselves are not sold.1 In certain jurisdictions, it may not be possible to undertake a cash securitisation due to legal, regulatory, cross-border or other restrictions. Or, it may be that the process will simply take too long under the prevailing market conditions. In such cases, originators use synthetic transactions, which employ some part of the traditional process allied with credit derivatives. However if the main motivation of the originator remains funding concerns, than the cashflow approach must still be used. Synthetic transactions are mainly used for credit risk and regulatory capital reasons, and not funding purposes. Since the inception of the first synthetic deals the market has evolved, with continuing development of newer structures to meet differing originator and investor requirements. For instance a proposed multi-SPV hybrid CDO structure may be considered the “fourth-generation” of such products, following the structures introduced previously (see Choudhry [2003]). This is illustrated in Exhibit 1. Another “fourth-generation” CDO product is the so-called single-tranche CDO, which is the focus of this article.

1st GenerationCashflow

Managed CDO

2nd GenerationStatic funded/partially-funded synthetic CDO

3rd GenerationManaged Synthetic CDO

4th GenerationHybrid multi-SPV

managed CDO

Exhibit 1 Four generations in the development of synthetic CDOs

The single-tranche CDO One of the advantages offered to investors in the synthetic market is the ability to invest at maturities required by the investor, rather than at maturities selected by bond issuers. For instance, exhibit 2 illustrates how while the bond market provides assets at only selected points on the credit curve, synthetic products allow investors to access the full curve.

1 Although the first synthetic transactions were “balance sheet” deals, in which the originating bank transferred the credit risk of a pool of assets it held without actually selling them off its balance sheet, the fact that assets are not actually held means that the originator does actually have to own them in the first place. It may wish to transfer the credit risk for portfolio trading reasons. An analysis of this development in CSOs was given in Choudhry (2002).


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The flexibility of the synthetic CDO or CSO, enabling deal types to be structured to meet the needs of a wide range of investors and issuers, is well illustrated with the tailor-made or “single-tranche CDO” structure.2 This structure has been developed in response to investor demand for exposure to a specific part of a pool of reference credits. With this structure, an arranging bank creates a tailored portfolio that meets specific investor requirements with regard to: portfolio size and asset class;

portfolio concentration, geographical and industry variation;

portfolio diversity and rating;

investment term-to-maturity.

The structure is illustrated at Exhibits 3 and 4, respectively with and without an SPV issuer. Under this arrangement, there is only one note tranche. The reference portfolio, made up of credit default swaps, is dynamically hedged by the originating bank itself. The deal has been arranged to create a risk/reward profile for one investor only, who buys the single tranche note. This also creates an added advantage that the deal can be brought to market very quickly. The key difference with traditional CSOs is that the arranging bank does not transfer the remainder of the credit risk of the reference pool. Instead, this risk is dynamically managed, and hedged in the market using derivatives.

0

1

2

3

4

5

6

7

1y 2y 5y 10y 20y

BondsSynthetic

exhibit 2 hypothetical credit term structure

2 These deals have been arranged by a number of investment banks, including JPMorgan Chase, Bank of America, UBS Warburg and Credit Agricole Indosuez. They are known variously as tailor-made CDOs, tranche-only CDOs, on-demand CDOs, iCDOs and investor-driven CDOs as well as single-tranche CDOs. The author prefers the last one.


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InvestorDemand

Reference poolcashflows andpremiums

Premium and interest

Credit event losspayments

BANK

Mezzanine CLN[AA to BB] rating

Mixed cash and synthetic credit portfolio

Reference asset 1Reference asset 2Reference asset 3

.

.

.Reference asset 50

No credit event: 100% par value on due maturity dateCredit event: [Par - market value] of defaulted reference entities exceeding the subordination

Exhibit 3 Single Tranche CDO I: issue direct from arranging bank

InvestorDemand

Reference poolcashflows andpremiums

Premiums

Contingent payment if credit event in tranche

BANK

Mezzanine CLN[AA to BB] rating

Mixed cash and synthetic credit portfolio

Reference asset 1Reference asset 2Reference asset 3

.

.

.Reference asset 50

No credit event: 100% par value on due maturity dateCredit event: [Par - market value] of defaulted reference entities exceeding the subordination

SPV

Exhibit 4 Single Tranche CDO II: issue via SPV Deal structure The investor in a single-tranche CDO will decide on the criteria of assets in the portfolio, and the subordination of the issued tranche. Typically this will be at the mezzanine level, so for example covering the 4% to 9% loss level in the portfolio. This enables a very favourable risk return profile to be set up: a CDO tranche that is exposed to 4-9% losses has a very low historical risk of default (approximately equivalent to a Moody’s A2 rating) and a high relative return given its tranching, around Libor plus 200 basis points as at May 2003.3 This is the risk/return profile of the mezzanine piece. Exhibit 5 shows the default probability distribution for credit events in a CDO. Exhibit 6 shows the more specific distribution as applicable to the mezzanine tranche.

3 Source: Bloomberg L.P.


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Exhibit 5 Credit loss distribution Exhibit 6 Expected loss distribution for tranched notes Unlike a traditional CDO, a single-tranche CDO has a very simple cashflow “waterfall”. Compared with the waterfall for a cash CDO, a single-tranche waterfall will consist of only agency service and hedge costs, and the coupon of the single tranche itself. Some of the issues the investor will consider when working with the arranging bank to structure the deal include:

Loss on the Underlying Assets

Frequency

Loss on the Underlying Assets

Frequency

Credit Enhancement of the Mezz Tranche 4%

Credit Enhancement of the Senior Tranche 9%

Mezz Tranche Senior TrancheEquity

Losses

Probability

A + B: Probability of losses hitting the junior tranche

B: Probability of losses hitting the senior tranche

A B


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the number of names in the credit portfolio; usually this ranges from 50 to 100 names

the geographical split of the reference names;

the required average credit rating and average interest spread of the portfolio;

the minimum credit rating required in the portfolio.

If the deal is being rated, as with any CDO type the mix of assets will need to meet ratings agency criteria for diversity and average rating. The diversity score of a portfolio is a measure of the diversity of a portfolio based on qualities such as industrial and geographical concentration. It can be defined as the number of equivalent uncorrelated assets in the pool.4 We illustrate an hypothetical portfolio at Exhibit 7, which shows the composition of a generic portfolio for a single-tranche CDO.

Automobile 9.3Banks 15Electronics 14Insurance 11Media 6.8Oli & Gas 6.2Real Estate 5.7Telecoms 12Transport 6Utilities 14

100

Aaa 2Aa2 7Aa3 18 Number of reference assets 50A1 19 Moodys diversity score 46A2 10 Average rating A2A3 24 Average maturity 5 yearsBaa1 12Baa2 8

100

9.3

15

14

11

6.8

6.2

5.7

12

6

14

Automobile

Banks

Electronics

Insurance

Media

Oli & Gas

Real Estate

Telecoms

Transport

Utilities

Aaa Aa2 Aa3 A1 A2 A3 Baa1 Baa2

Exhibit 7 Hypothetical portfolio composition for generic single-tranche CDO

4 Further background on Moody’s diversity score is given at the Appendix in Choudhry (ibid.).


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The position and rating of the issued single tranche is as required by the investor. The subordination of the note follows from the required rating of the investor. For instance, the investor may require an A2 rating for the note. The process followed involves: targeting the required rating on the issued tranche;

setting the required return on the note, and hence determining where the tranche will

lie; defining the percentage of first loss that must occur before the issued tranche is

impacted by further losses; setting the size of the note issue, in line with investor requirements. For instance if the

investor wishes to place $20 million in the note, and the reference pool is $800 million nominal value, this will imply a 2.5% tranche.

As with the previous synthetic CDOs, a single-tranche CDO can be either a static or a managed deal. In a managed deal, the investor can manage the portfolio and effect substitutions if this is part of its requirement. To facilitate this, the deal may be set up with one or more fund managers in place to deal with the investor when substitutions are required by the investor. Alternatively, an investor may leave trading decisions to a fund manager. Advantages of the single-tranche structure For certain investors, the single-tranche CDO presents a number of advantages over the traditional structure. These include: Flexibility: the features of the investment can be tailor-made to suit the investors needs

precisely. The investor can select the composition of the portfolio, the size of the tranche and its subordination level;

Note terms exactly as required: the coupon and maturity of the note are tailor-made for

the investor; Shorter time frame: the deal can be brought to market relatively quickly, and in as

little as four weeks compared to anything from two months to one year for a conventional CSO;

Lower cost of issue: including lower legal costs because of short time to issue and no

protracted marketing effort by the arranger. The flexibility of the single-tranche structure means that the market can expect to see more variations in their arrangement, as more investors evaluate it as an asset class. The market has seen both “static” and “managed” single-tranche CDOs, following experience with traditional CSOs.


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Hypothetical pricing example Exhibit 8 is a simplified illustration of a pricing example for a single-tranche CDO, with market rates as observed on Bloomberg during April 2003. We assume the portfolio is constituted in the following way: Number of credits: 80 Nominal size: €800 million Diversity score: 48 Average rating: BBB+ / Baa1 Minimum rating: BBB- / Baa3 Maturity: 5 years The originating bank structures a single-tranche CDO following investor interest with the following terms: Subordination level: 3.90% (this means that five defaults would be supported, assuming

a 35% recovery rate) Tranche size: €25 million Expected rating: A / A2 Spread: Euribor + 220 basis points


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35% recovery rate

10.30%(8 defaults)

5.60%(5 defaults)

3.90%

Synthetic credit portfolio

A / A2 tranche

Exhibit 8 Single tranche CDO illustrative pricing example References: Choudhry, M., “Combining Securitisation and trading in credit derivatives: an analysis of the managed synthetic CDO”, Euromoney Debt Capital Markets Handbook, Euromoney Publications 2002 Choudhry, M., “The multi-SPV credit hybrid CDO”, submitted Working Paper, Department of Management, Birkbeck, University of London 2003 Tavakoli, J., Credit Derivatives and Synthetic Structures, Wiley 2001 Moorad Choudhry is a Visiting Professor at the Department of Economics, Finance and International Business at London Metropolitan University. His research can be viewed at www.YieldCurve.com


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Shout Floors

Terry H. F. Cheuk∗ Ton C. F. Vorst†

Abstract

It is common to find index funds being marked with a protective floor. It

gives investors the upside potential of the equity market, while protecting

them from possible losses. In this paper, we describe a new type of protective

floors, of which the floor level is not set at inception of the contract. Instead,

during the contract life, the holder can give notice - he shouts - to set at

one point the floor at the prevailing index level. It is shown in the paper

that the optimal shout policy does not depend on the index level, but only on

time-to-maturity. The concept is generalised further to multiple-shout floors.

If the index hits a higher level after the first shout, the holder is allowed to

shout again to reset the floor at the then prevailing (higher) index level. An

efficient numerical method is proposed to price multiple-shout floors.

Keywords: Shout options, numerical methods.

∗Assistant Professor of Finance at The Hong Kong University, School of Business.†Professor, Department of Finance, Erasmus University Rotterdam and Erasmus Center for Finan-

cial Research.


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Shout Floors

� Introduction

It is common to �nd index funds being marketed with a protective �oor�

It gives investors the upside potential of the equity market� while protecting

them from possible losses� Instead of a protective �oor set at inception of the

contract� sometimes the investor� who holds a long position in the underlying

index� is given the possibility to install a protective �oor at the prevailing spot

index level whenever he deems appropriate� without additional payments� At

any time during the life of the shout �oor� the holder can give notice � which

is called to shout � to install a �oor at the prevailing index level� He

determines the timing of installation� but the �oor level is set by the market

at the spot value of the underlying index� The investor can shout only once�

except for the case of multiple shout �oors� which we discuss later on� In this

paper we describe an e�cient methodology to price these so called �single�

shout �oors�

As the holder is allowed to shout immediately� a shout �oor is at least as

valuable as an at�the�money �oor � which is a standard put� We will show

that the optimal shout policy does not depend on the index level� but only on

time�to�maturity� This is a very unusual property� A semi�analytic formula

for the shout �oor price is derived� One could use well�known numerical

methods to solve the semi�analytic formula in order to �nd the optimal time

to shout� We will also derive a lattice model to price this derivative�

There are strong similarities between a shout �oor and a shout option�

�


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as described in Thomas �� A shout call option is a regular call with

the additional feature that the holder can reset the exercise price to the

prevailing higher index level and at the same time receives a payment equal

to the di�erence between the new and old strike price�

Of course� shout �oors and shout options can be generalized to allow

for multiple shout possibilities� If the index hits a higher level after the

�rst shout� the holder is allowed to shout again to reset the �oor at the then

prevailing �higher� index level� Multiple shout options should not be confused

with ladder options� A ladder option is a put option with increasing levels of

exercise prices� Whenever the index rises above the next higher exercise level�

a rung of the ladder� that exercise level becomes the new exercise price� The

di�erence between a ladder option and a multiple shout �oor is the instance

that the exercise levels are set� All the possible exercise levels for a ladder

option are set at the inception of the contract� In contrast� for a multiple

shout option� the exercise levels are determined at the prevailing index level

at each instant the holder shouts� Only the number of possible exercise levels

�i�e� the number of times to shout� is determined at the contract inception�

The number of times to shout should be an important parameter for the value

of this derivative� A shout �oor which allows the holder to shout an in�nite

number of times would have the same payo� as an otherwise similar lookback

put option� hence the same price� In this case he would shout continuously�

Hence he would also shout at the highest level of index� Thus he would be

able to sell the index at maturity against the highest value over the life time�

This is exactly a lookback put option�

Since the optimal shout policy for multiple shout options is determined

�


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by the actual development of the underlying asset price� the value depends

not only on the actual underlying asset price but also on the existing �oor

�from the previous shout� and the remaining number of possibilities to shout�

In this paper we also develop a method to price multiple shout �oors� A trick

that was previously used in Cheuk and Vorst �� to construct a single�

state�variable path�independent binomial model for the prices of lookback

options� is used to reduce the pricing problem to the evaluation of a �xed

number of simple trinomial trees� This �xed number is equal to the number

of shout possibilities the investor has� This is computationally more e�cient

than a three�state�variable model� where one variable is the underlying asset

price� the second is the existing �oor and the �nal variable is the number of

remaining shout possibilities�

� Single�Shout

First� we will consider a shout �oor which allows the holder to shout only

once� A �oor is actually a put option� since it allows the holder to sell the

index for the �oor level at maturity� Using the Black�Scholes formula� a

European put has a price at time t equal to

Put�T � t� St� X� � Xe�r�T�t�N��d�� Ste�q�T�t�N��d��

d� �log St

X� �r � q � �

��T � t�

�pT � t

� ��

d� � d� � �pT � t� � �


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with St the index level at time t� X the strike� T the expiration time� r the

interest rate� q the dividend rate� � the index volatility� and the initial time

is �� We use a continuous dividend yield� Since� we are primarily interested

in shout �oors based on an index� this is not an unrealistic assumption�

For an at�the�money put� it simpli�es to

Put�T � t� St� St� � St

�e�r�T�t�N��d�� e�q�T�t�N��d��

��

d� ��r � q � �

��T � t�

�pT � t

� ��

d� � d� � �pT � t� ��

Using well�known results from option pricing� the value of a derivative is

equal to the risk�neutral expectation of its discounted payo�s� As the holder

will choose an exercise policy to maximize his pro�ts� the value V of a shout

�oor at time zero is given by

V � max��t�T

E

�e�rtPut�T � t� St� St�

�

� max��t�T

E

�e�rtSt


��

� max��t�T

E

�E

�e�rtSt


��t��

� max��t�T

E

�S�e

�qt�e�r�T�t�N��d�� e�q�T�t�N��d��

��

��

d� ��r � q � �

��T � t�

�pT � t

� ��

d� � d� � �pT � t� ��

and E is the expectation operator under the risk�neutral probability measure�

�


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The �rst equality follows from equation �� the second is based on the law

of iterated expectations� since t is stochastic� The third equality follows from

the fact that under the risk�neutral process the expected discounted future

value of the underlying asset is equal to today�s value corrected for dividend

payments� The last line of equation �� can be immediately rewritten as�

V � max��t�T

S�

�e�rT��r�q�tN��d�� e�qTN��d��

��

d� ��r � q � �

��T � t�

�pT � t

� ��

d� � d� � �pT � t� ��

since the expression between brackets in �� is deterministic and hence the

expectation does not play a role�

The above equations show that the optimal shout policy is non�stochastic�

It does not depend on the future index level but only on time� Also� the shout

�oor price is linear in the index level� which is of course not surprising�

The graph of the expression in �� as a function of t� the time to shout�

is plotted in Figure �� for a shout �oor of �ve years� The parameters are

S� � �� T � �� r � �� q � �� The three lines correspond to � � ��

and �� To determine the optimal time to shout one has to �nd the maximum

value of the expression in �� as a function of t� One can �nd the optimal t by

setting the derivative of �� with respect to t equal to zero� This derivative

is known as the theta and involves both the normal density function and the

normal distribution function� There are no analytic expressions for zero�s of

the theta� Hence the zero has to be determined numerically� In the cases

�


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0

1

2

3

4

5

6

7

8

0 1 2 3 4 5Waiting Time

Pri

ce (

$)

Sigma = 0.3Sigma = 0.2Sigma = 0.1

Figure �� Value of a Five�Year Shout Floor versus Waiting Time

to Shout

S� � �� T � �� r � �� q � �� and ��

that we considered� the solution was always unique� For the shout �oor with

the above given parameters and � � �� it is best to wait �� years� i�e� to

shout �� years before maturity for an investor that holds a long position in

the index� �Optimal waiting times for other parameter values are given in

Table �� Although the time to shout is independent of what happens to

the index level� the value of the option at the time of shout clearly depends

on the index level� The shout �oor value is �� In comparison� under the

same parameters� a standard �oor struck at �� is worth only �� while

a lookback option would have a value of �� The large di�erence with a

lookback option is partly due to the long maturity of the option� We will

come back to this di�erence in Section �� The analytic results of the single�

�


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� r q n Analytic Waiting�� Value Time

��

��

��

��

��

��

Table �� Single�Shout Floor Prices

S� � �� and T � ��

shout �oor for di�erent values of �� r and q are given in Table �� columns �

and ��

The last column of Table � gives the optimal points in time to shout�

The optimal time depends on the parameters but not on the underlying

index level� Figure � illustrate the fact that one should shout earlier for

underlying assets with a higher dividend yield� In some cases one has to

shout immediately at issuance of the contract� in which case the shout �oor

price is equal to the price of an at�the�money put� But more generally� for

otherwise identical parameter values� higher dividend yields give rise to lower

expected values for the underlying asset in the risk�neutral world� Hence the

�


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0

1

2

3

4

5

6

0.00 0.01 0.02 0.03 0.04 0.05Dividend Yield

Pri

ce (

$) /

Wai

tin

g T

ime

(Yea

rs) Price

Waiting Time

Figure �� Value of a Five�Year Shout Floor versus Dividend Yield

S� � �� T � �� r � �� and � � ��

underlying asset will decrease in expected value and thus it is more probable

that the underlying asset will go down in value for higher dividend yields�

One has to shout earlier� as the price of a put option always increases with

higher dividend yields�

Of course the shout �oor increases in value with an increase in the volatil�

ity of the underlying asset� �See Figure �� At the same time� one should

shout earlier for a higher volatility� One can pro�t more from higher volatil�

ities � if the remaining time to maturity of an option is longer� Hence� for

higher values of � one does not want to wait too long to pick up the higher

option premium implied by the higher volatility�

�


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0

1

2

3

4

5

6

7

8

0.1 0.2 0.3Sigma

Pri

ce (

$) /

Wai

tin

g T

ime

(Yea

rs) Price

Waiting Time

Figure �� Value of a Five�Year Shout Floor versus Sigma

S� � �� T � �� r � �� and q � ��

� Trinomial Lattices

Alternatively� one can use a lattice model to price shout �oors� We use the

trinomial lattice of Boyle �� Although� also a binomial model could be

used as explained in Thomas �� we use the trinomial model since� in

contrast with the binomial model it has at all times in the tree a node at

exactly the initial level of the spot price� This is essential for the induction

procedure that we will apply later on to price multiple shout options� The

convergence is also faster than for a binomial model� The parameters for our

risk�neutral trinomial model� with n steps in the lattice� are listed below�

where the underlying asset price can go up with a factor u for the up�state�

go down with d � ��u for the down�state� or remain the same for the middle�

�


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state��

�t �T

n� ��

u � e��p�t� ��

M � e�r�q��t� ��

V �M�

�e�

��t � �

��

pu �u�V �M� �M�� M � ��

�u� ��u� � ��

pd �u��V �M� �M�� u��M � ��

�u� ��u� � ��

pm � �� pu � pd� ��

Let F�i�j denote the single�shout �oor price at the trinomial tree node

�i� j�� i�e� time is i�t� and the index is at S�uj� As at�the�money options

with a zero time�to�maturity are worth nothing� the �nal values of a shout

�� can be chosen freely� as long as the resulting probabilities are positive� Omberg

�� argued � � �

�

p�� is optimal�

��


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�oor are

F�n�j � ��

for j � �n� � � � � n�

The backward recursion formula is

F�i�j � max

�Put�T � i�t� S�u

j� S�uj��

e�r�t

�puFi��j�� pmFi��j � pdFi��j��

��

��

for i � �� n� �� and j � �i� � � � � i�

The shout value at each trinomial node is equal to an at�the�money

put� which can be calculated using the Black�Scholes formula� If the holder

decides not to shout� the value of the shout �oor at this node would be the

discounted risk�neutral expected value from the nodes one time step later�

The initial shout �oor price is given by F�� Results of the single�shout

�oor for di�erent values of �� r and q are given in Table �� columns � to ��

The speed of convergence of the trinomial model to the analytic value is very

fast� The model converges already very well to the analytic values with only

�� steps� for the parameter values considered� As stated before� we used the

Black�Scholes formula for the value of an at�the�money put to �nd the shouts

value� Of course� we could also use the put option value calculated through a

binomial or trinomial tree� as suggested in Thomas �� When we did so�

the values did not converge as fast as with the Black�Scholes formula� Since

calculating Black�Scholes values is not very time consuming compared with

��


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a binomial or trinomial tree� we preferred the direct use of Black�Scholes�

� Initial Strike

An interesting extension of the shout �oor is a shout �oor where there is

already an existing �oor� which is called the initial strike� The investor has

the right to shout during the life of the contract to reset the initial strike

to the then prevailing index value� If he declines to shout during the life of

the contract� the shout �oor would have the same payo� as a �oor struck

at the initial strike� As the holder will only consider to shout if the index

has risen above the initial strike� the shout policy in this case depends on

the evolution of the underlying asset price� i�e� the policy is stochastic� Of

course one might wonder whether it is possible to �nd an analytic expression

for the optimal shout time through some dynamic programming procedure�

However� in the previous case the essential trick was the linearity of the value

as a function of the underlying asset price� With an initial strike the optimal

strike strategy not only depends on the underlying asset price but also on the

initial strike� By shouting one gives up the existing put option with the initial

strike price plus the opportunity to shout later on in exchange for a new put

option� Now� this is similar to an American put option where one gives up

the underlying asset plus the opportunity to exercise later on in exchange for

the exercise price� It is well known that there are no analytic solutions to the

pricing of American options� but only very robust approximations� Although

our problem is similar� it is certainly more complex than the American put

option since the value of the put that is given up is a more complex function

��


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Fr q � � ��

��

��

Table �� Single�Shout Floor Prices F with Initial Strike

S� � �� K � �� n � �� and T � �

of time and the underlying asset price than the value of the underlying asset

that is given up with an American put� Also the value of the new put is

a more complex function than the constant exercise value� Van Moerbeke

�� gives an overview of optimal stopping problems that can be solved

analytically for these stochastic processes� The structures of the problems

that can be solved are much simpler than the American put option problem

and certainly than our problem at hand� Hence we have to stick to numerical

procedures to �nd the optimal shout policy� These procedures will also show

that the optimal timing does depend on the underlying price and hence

indeed is more complex than the simple shout �oor�

We can value the single shout �oor with initial strike using the lattice

model developed in the previous section� Only the �nal values in equation

�� need to be modi�ed to re�ect the initial strike� i�e�

�F�n�j � K � S�uj� if S�u

j � K� ��

� �� otherwise� ��

with K equal to the initial strike�

�


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Results of the single�shout �oor with an initial strike struck at �� are

given in Table � for di�erent values of �� r and q�� Comparing Tables � and ��

we see that the single�shout �oor with an initial strike� of course� has a higher

value than a similar shout �oor without an initial strike� Furthermore� the

optimal moment to shout depends on the value of the underlying asset price

with respect to the initial strike� If the asset price remains below the initial

strike one will never shout� Hence� there is no optimal time independent of

the underlying asset price as for the single shout �oor�

� Multiple�Shout

An interesting extension is the case where the holder has the right to shout

twice � that is to set the �oor level twice� The holder shouts when he thinks

protection is warranted� He might shout again at a higher level� if the index

rises� If the holder is given an in�nite number of times to shout� the price

of such a shout �oor should be equal to that of a lookback put option� as

explained in the introduction�

To value a twice�shout �oor� one also needs to know the prices for single�

shout �oors with an initial at�the�money strike for all the nodes of the trino�

mial model� When the holder shouts for the �rst time� he gets a single�shout

�oor with an initial strike struck at�the�money� Hence� one needs the val�

ues of single�shout at�the�money �oors for all possible index level� However�

prices for the single�shout at�the�money �oors are linear in the underlying

�All prices are calculated with a trinomial tree with n � � steps� Convergence asa function of n is slower for shout oors with an initial strike than for products withoutinitial strike�

��


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asset price and hence one only needs the value of a single�shout with at�the�

money initial strike �oor for a �xed level� say the initial asset price S� but for

all moments in time� These can be inferred from the trinomial tree that gives

the values for single�shout �oors with initial strike� To be more speci�c� we

need the prices on the horizontal axis of that trinomial lattice� If we would

have used a binomial tree we would not have a node on the horizontal axis

for every moment in time� but just for every second moment� This is the

main reason for using a trinomial tree

A model for the twice�shout �oor is described below� The price of a single�

shout strike with an initial strike struck at the money is given by �F�i�j� The

price of a twice�shout �oor is given by F�i�j� The �nal values are

�F�n�j � S� � S�uj� if S�u

j � S��

� �� otherwise� ��

F�n�j � �� for all j� ��

In the backward recursion�

�F�i�j � max

�Put�T � i�t� S�u

j� S�uj��

e�r�t

�pu �F�i��j�� pm �F�i��j � pd �F�i��j��

��

��

F�i�j � max

��F�i��

S�uj

S��

e�r�t

�puF�i��j�� pmF�i��j � pdF�i��j��

��

��

��


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The shout value for F�i�j is a single�shout �oor with an initial �oor

struck at S�uj� This is equal to an at�the�money single�shout �oor F�i��

scaled to the appropriate index level S�uj instead of S�� The initial shout

�oor price is given by F�� For a twice shout option with an initial strike

equation �� has to be adjusted as follows

�F�n�j � K � S�uj� if S�u

j � K� ��

� �� otherwise� � ��

Extension to a k�times shout �oor is straightforward� with k being any

positive integer� In fact the value can be calculated by recursion on the num�

ber of shout possibilities� where the recursion is based on the k�shout option

with initial at�the�money strike� Hence� the recursion is on �Fk rather than

the shout �oor value Fk� However the shout �oor values can be determined

from the lattice for �F �k � ��

The computation time for a twice�shout �oor is twice as long as that

for a single�shout �oor� because both the lattices for �F�i�j and F�i�j must

be constructed� For the case of a k�shout �oor� after the �rst shout� the

value of the derivatives is equal to a �k� ��shout �oor with an initial strike

determined at the time of the �rst shout� After the second shout� the value

becomes equal to a �k � ��shout �oor with an initial strike higher than the

previous one� Hence� the computation time for a k�shout �oor is k�times as

long as that for a single�shout �oor� since all the lattices for �F�i�j� �F�i�j� � � � �

and Fki�j are needed to determine the initial price of a k�shout �oor� The

computation time for the single shout �oor has the order O�n�� with n the

��


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n No� of shout possibilities��

��

Table �� Multiple Shout Floor Prices without initial strike

S� � �� T � �� r � �� q � �� and � � ��

n No� of shout possibilities��

��

Table �� Multiple�Shout Floor Prices with initial strike

S� � �� K � �� T � �� r � �� q � �� and � � ��

number of time steps� Hence� the computation time for a k�shout �oor has

the order O�kn��

Values for multiple�shout �oors without and with initial strike are given in

Tables and �� respectively� We see that the convergence as a function of the

number of steps n in the tree is quite good when only a small number of shouts

is allowed� Convergence is slower when there are more shouts involved� as

the uncertainty compounds with each additional possibility to shout� Indeed

the values for shout �oors with initial strikes are always higher than the

values without initial strike� However� the values without initial strike but

with one more shout possibility are again higher than those with initial strike

�Compare e�g� column of Table with column � of Table ��

��


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As remarked earlier� a shout �oor with an in�nite number of shout possi�

bilities is worth the same as a lookback put� A lookback put option under the

same parameters as in Table is worth �� which is substantially higher

than a shout �oor with a small number of shout possibilities� To analyze the

large di�erence we calculated the value of shout��oor with �� shout possi�

bilities with a tree of �� steps� In that case it is optimal to shout at every

point in time� even if the price is below an earlier set strike price� Nothing is

lost by shouting� since the number of times one is allowed to shout is equal

to the number of possible times to shout� The value in this case is ��

This seems still far away from �� with even one hundred shout possibili�

ties� However� Cheuk and Vorst �� have documented that for lookback

options there can be large price di�erences for lookbacks where the under�

lying asset is observed continuously and lookbacks where only discrete time

observations are used� Since� the option has a maturity of � years� �� ob�

servation points amounts to observing the underlying asset value only every

�� days� Using the lookback model of Cheuk and Vorst with �� observa�

tion points and �� time steps leads to a value of �� Hence� the value

is indeed close to �� the value of a �� time shout option� The small

di�erence might be due to the fact that the lookback option is valued by a

binomial tree� while the shout option is valued by a trinomial tree� Given the

long maturity of the option� one should allow for considerable more shout

opportunities than �� to come close to the analytic lookback value�

��


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� Conclusion

In this paper� we discuss shout �oors� which can be packaged with an index to

create an interesting structure for investment� The shout �oors give investors

the �exibility to in�uence the outcome of their investments� As such� this

product has a certain appeal to those who feel the need for protection� but

�nd it inappropriate to set a �oor immediately� For a single�shout �oor

without initial strike� a semi�analytic pricing formula is derived� If the shout

�oor comes with an initial strike or if the holder is allowed to shout more

than once� a numerical method based on the trinomial lattice has to be

applied� We assume the holder can shout at any given time during the

trading hours� In the rare event that the holder is restricted to shout only

after the closing of the market at the closing price� the model would have

to be modi�ed to make distinctions between time points one is allowed to

shout and time points that do not allow one to shout� This is similar to

the distinction between observation and non�observation points applied for

barrier and lookback options in Cheuk and Vorst ��a� ��b and ��

��


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References

Boyle� P� P� �� A lattice framework for option pricing with two state

variables� Journal of Financial and Quantitative Analysis � � ��

Cheuk� T� H� F� and T� C� F� Vorst ��a�� Breaking down barriers�

RISK April ��

Cheuk� T� H� F� and T� C� F� Vorst ��b�� Complex Barrier Options�

Journal of Derivatives Fall ��

Cheuk� T� H� F� and T� C� F� Vorst �� Currency lookback options and

observation frequency� A binomial approach� Journal of International

Money and Finance� forthcoming�

Omberg� E� �� E�cient discrete time jump process models in option

pricing� Journal of Financial and Quantitative Analysis � � ��

Thomas� B� �� Something to shout about� RISK May ��

Van Moerbeke� P� �� Optimal stopping and free boundary problems�

Rocky Mountain Journal of Mathematics ��

��


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Generating Correlated Random Variables:Simple Techniques and Implementation

Henry Partier∗

Abstract

Simulation analysis is a commonly used numerical technique for financial riskmeasurement and the valuation of exotic derivatives and structured products.An important element of this analysis is the generation of valid correlatedrandom variables when dealing with higher dimensional valuation problems ormultifactor financial risk quantification. This article aims to illustrate someof the commonly adopted approaches as well as their associated practicalimplementation.

Keywords: Monte Carlo simulation, Matrix decompositions, Principal Com-ponent Analysis, VBA(Excel).

1 Introduction

The benefit of simulation methods is best appreciated when dealing with higher di-mensional problems that may not have an analytically tractable solution. In financethese higher dimensional problems include valuation of exotic derivatives or structuredproducts with numerous contingent payoffs, or the risk quantification of a portfolio ofsecurities that are sensitive to a number of risk factors.

A key step in implementing simulations for higher dimensional problems is the gener-ation of correlated random variables. The generation of correlated random variables isa means by which the average linear association between risk factors or valuation in-puts can be represented. Usually, this is performed by reconstructing a valid correlationstructure through a linear combination of the n series of iid standard normal randomvariables W = (w1, . . . ,wn).

∗The author is a Senior Associate with PricewaterhouseCoopers (PwC) Global Risk ManagementSolutions in Hong Kong. The views expressed in this article do not represent the views of PwC and arethe sole responsibility of the author. Specific comments or inquiries regarding the contents of this articlemay be directed to the author on Tel: (852) 2289 1963 or email: [email protected].


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It is important to note that one must exercise caution when using linear correlation asa measure of association between random variables. Financial risk managers should beaware of situations in which linear correlation fails to be a valid and coherent measure ofassociation. Disregarding these issues will potentially lead to incorrect inferences drawnfrom violations of the necessary and sufficient conditions that validate the reliance onlinear correlation. For an accessible exposition into the uses and abuses of correlationin finance, see [1] and [2].

This article will go through two particular aproaches to generating correlated randomvariables, namely the Cholesky Decomposition and Principal Component Analysis of theunderlying correlation structure. Techniques regarding estimation and implementationin VBA(Excel) will also be covered.

2 Cholesky Decomposition

For any positive semi-definite (psd) symmetric matrix1, the Cholesky decomposition isperhaps the easiest to implement and represents one of the quickest and most commonlyused approaches to generating correlated random variables. It has a similar represen-tation to that of the LU-decomposition, with the upper triangular matrix U being thetranspose of the lower triangular matrix L. For a valid correlation matrix R, such de-compostion is of the form R = LLT .

The elements of the lower triangular matrix L is commonly referred to as Choleskyfactors. These Cholesky factors form the coefficients of the linear combination of thegenerated iid random variable series. The resulting transformed series should have thesame correlation structure R imposed on it through the Cholesky factors in R ≡ LLT .

This decomposition is conceptually akin to deriving the “square root” of each element ofR given that the outer product of the decomposition is a symmetric sum of squares andcross product(SSCP) matrix. The SSCP martix is then equated with R, which allows usto recursively solve for each element of L in column (j ) by row (i) order. Following [4],for any n-dimensional psd symmetric matrix, we have the following recursive equationfor i = 1, . . . , n.

lii =

rii −i−1∑j=1

l2ij

12

(1)

For j = 1, . . . , n and i = (j + 1) , . . . , n, we have

lij =1

ljj

rij −j∑

k=1

likljk

(2)

Equations (1) and (2) above provide the estimates for the main diagonal and non-zerooff-diagonal elements of L respectively, with r representing the elements of R.

1These include valid covariance and correlation matrices denoted henceforth by S and R respectively


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Having estimated the Cholesky factors based on the above procedure, reconstructing thevalid correlation structure on the iid random variables W involves the straightforwardlinear transformation Z = WL. The transformed random variable series Z should onaverage have a correlation structure equivalent to that given by R. Sample source codeis presented in Appendix 1 using the procedures described above.

The main drawback of this approach is that a solution can only be obtained for psdsymmetric matrices. Therefore, in the event that the resulting correlation (or covari-ance) matrix is non-psd (and hence invalid), more involved techniques are required togenerate valid correlated random variables.

3 Principal Component Analysis (PCA)

In the event that R (or sample covariance matrix S) of the jointly distributed randomvariables X becomes non-psd and hence invalid, certain adjustments may be followed toarrive at a valid correlation structure. By definition, a symmetric matrix, say R, is psdif the associated quadratic form satisfies the following inequality aTRa ≥ 0, for a 6= 0.A common way to identify whether this is satistified involves analysing the character-istic roots (eigenvalues) of R. Given some a 6= 0, the preceeding inequality will not besatisfied if R has at least one negative eigenvalue.

By the spectral decomposition (or alternatively the more general singular value de-composition) of R, we can factor out its eigensystem in the form R = PΛPT , withΛ = diag(λ1, . . . , λn) representing the ordered eigenvalues with associated eigenvectorsP = (p1, . . . ,pn). There are a number of ways to estimate Λ and P as described andimplemented in [4]. However, for much simpler packages such as VBA(Excel), a morepractical approach needs to be used. One such technique involves an indirect way of es-timating the eigenvalue and associated eigenvectors of R through Principal ComponentAnalysis (PCA).

PCA and its associated methodology is commonly used in dealing with multi-dimensionaldata reduction, multicollinearity, factor estimation as well as cannonical correlationstudies. Principal components are formed from linear combinations of the underlyingrandom variables y = Xb under the condition that V ar(y) is maximised subject tothe normalising orthogonal constraints BTB = I. The matrix B represents coefficientvectors bi, which for i, j = 1, . . . , n has2.

bTi bj =

{10

for i = jfor i 6= j

(3)

The solution to the coefficient vectors bi, is obtained by maximising the following objec-tive function for the correlation of the linear transformations subject to the constraints

2Note that this technique can be similarly applied to correlation matrices (as the covariance matrixof standardised variables) in which the objective is still to maximise a quadratic form relative to certainequality constraints given that S = V1/2RV1/2 with V1/2 = diag(σ1, . . . , σn). However, its shouldbe pointed out that the resulting principal components (or associated eigensystem) is not the same asthose derived using S (i.e. the covariance matrix of the non-standardised data).

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Financial Engineering Review38

in (3).

maxbi⊥bj

bTi Rbj

bTi bj

(4)

The resulting maximum values of the objective function in (4) and the associated coeffi-cient vectors bi for i = 1, . . . , n are the ordered eigenvalues and associated eigenvectorsof the correlation matrix R, with the constraints in (3) also allowing for the alternativeequality condition

Corr(yi,yj) =

{10

for i = jfor i 6= j

(5)

Having estimated the eigenvalue and eigenvectors of R, we can proceed to check thenecessary condition for a correlation matrix to be valid. The implementation of theabove procedure is presented in Appendix 2. The routine is used in conjunction witha transformation procedure to recover a valid correlation matrix discussed in the nextsection.

4 Transformations

Given the existence of some λ < 0, a number of techniques may be employed to recovera valid correlation matrix3. The objective is to recover a correlation matrix that is validand comes close to the desired correlation structure as measured by an appropriate errorestimator.

Instances in which a desired correlation structure may prove invalid is usually found instress testing or scenario analysis. For instance, a financial risk manager may wish toinvestigate the effect of a change in the correlation structure on the risks of a portfolio.A short-cut approach often employed is to shock the correlation coefficients arbitrarily(rather than adjusting the underlying data directly), which as a consequence can leadto invalid correlation matrices.

Among the different methods for recovering valid correlation matrices, perhaps one ofthe easiet to implement computationally is the approach proposed by the authors in[3] and [5], which involves the transformation of the diagonal matrix Λ and so-calledpseudo-square root spectral decomposition. The transformed diagonal matrix enforcesthe condition that λi ≥ 0 for all i = 1, . . . , n as follows,

Λ = diag

{λi

0if λi ≥ 0if λi < 0

(6)

The transformation also involves the diagonal scaling matrix T, which is an inverse

3For examples, see [5] or [6]


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eigenvalue weighted sum of the squared row elements of the eigenvector matrix as fol-lows,

T = diag{ti = 1/Σn

j=1(b2ijλj)

}(7)

The finalised adjustment to recover a valid correlation matrix involves the followingdecomposition,

R =(T1/2BΛ1/2

) (T1/2BΛ1/2

)T(8)

Tests performed in [3] and [5] show that the estimated total difference between R andR, as measured by the sum of squared element differences4,

ε =n∑

i=1

n∑j=1

(rij − rij)2 (9)

is relatively “small” and at least produces correlation matrices that are in line withother methods for recovering valid correlation matrices. The algorithm for this transfor-mation is presented in Appendix 3 in conjunction with the eigenvalue and eigenvectorscomputed through PCA in the preceeding section.

Having transformed the structure into a valid correlation matrix, reconstructing a validcorrelation structure through W may be facilitated using the procedures described insection 2. Alternatively, the eigenvalues and associated eigenvectors of the valid cor-relation matrix can be used in a linear transformation with the coefficients being theproduct of the square root of the eigenvalues and the eigenvectors as follows5,

Z = (PΛ1/2

)WT (10)

The transformation in (10) has P and Λ1/2 = diag(λ1/21 , . . . , λ1/2

n ) as the eigenvectorsand square root of the eigenvalues of (8), with ZT = (z1, . . . , zn) being the n columnsof (kx1) series of correlated random variables.

We will investigate the performance of the techniques in sections 3 and 4, together withthe developed algorithm presented in the appendices. The code that is presented in thisarticle can be constructed and implemented quite readily. Note however that it is just

4The authors in [3] and [5] define this estimate as the χ2elements = Σi Σj(rij − rij)2, which seems to

suggest that it behaves as such. However, it should be pointed out that perhaps due to factors such asthe degree of non-psd as well as dimension of the correlation structure, errors may persist that couldlead to violations of the necessary and sufficient conditions for this statement to be true.

5Note that the eigenvectors are orthogonal such that P−1 = PT . Infact we actually make use of theinverse in constructing the linear combinations of independent random variables in W to reproduce thecorrelated series. The row elements of P (which are equal to the column elements of its inverse), formthe weights in the resulting (nxk) linear combinations, where k represents the number of iterations inthe simulation analysis.


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one of numerous alternatives to the possible structure of the program such that the pro-gram runs more efficiently. The trade off as per usual in most commercial applicationsis speed of implementation versus efficient code. The reader is encouraged to modifythe coding to render it more efficient.

5 Performance

In order to test the approach discussed in Section 3 and 4, we construct a valid n = 20dimensional correlation matrix, and arbitrarily shock the n = 1, 2, 10 random variableseries by ±0.50, subject to the elements being −1 ≤ r ≤ 1. The resulting correlationmatrix is denoted by R. We then run the PCA and Transformation code in Appendices2 and 3 respectively on a number of increasing dimensions n = 3, 5, 8, 10, 12, 15, 20 ofR. The results are presented in Table 1 below.

CPU Timen (secs) Order Degree ε RMSE3 0.2813 0 0 0 05 0.9531 1 -0.0489 0.0812 0.05708 3.8438 1 -0.1237 1.2589 0.140210 7.8516 1 -0.1189 1.8310 0.135312 15.9788 2 -0.1149 2.2377 0.124715 34.1484 2 -0.1201 3.0854 0.117120 135.7930 2 -0.1408 6.7830 0.1302

Table 1: Simulation results of PCA and transformations

Table 1 includes details of the processing time to complete both the PCA and transfor-mation to a valid correlation matrix in seconds. It also includes information regardingthe Order and Degree of non-psd of R defined as the number of λi < 0 and 1

n

∑(λi < 0)

respectively. In terms of error, it includes data on ε as defined in equation (9), as wellas an element-wise average departure from validity in the form of a root mean squarederror RMSE = 1

n

√ε.

Besides the first run on n = 3 dimensions, which yielded a valid correlation matrix,all other trials resulted in an invalid correlation structure that required the necessarytransformation. As expected, the degree of computation time and sum of squared dif-ferences increases with the number of dimensions. However, the average element-wisedeparture from validity denoted by the RMSE appears to have a tentative relation tothe degree or order of non-psd as well as the dimensions. This observation may be dueto the structure of the data and the arbitrary manipulation of 3 out of the 20 variables.The degree of non-psd when spread over increasing dimensions appears to change onlyslightly in terms of element-wise differences. However, it would be interesting for thesake of completeness to investigate the specific cause of this result in more detail.

Other noticeable results include the processing time. The CPU time for n = 20 appearsto be quite long (greater than 2 minutes). Furthermore, it appears to have some form of


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positive higher orderrelationship with the number of dimensions. The results indicatethat this relationship may eventually become unmanageable for more extensive applica-tions (e.g. risk quantification of a portfolio that has exposure to more than 50 marketrisk factors).

The seemingly significant CPU time is not surprising given that VBA (Excel) was notprimarily designed to cater for heavy duty optimisation. Furthermore, the processingspeed is directly affected by the imposed precision level, which if lowered could perhapslead to faster CPU times. But the trade off in doing so, especially for higher dimensionsand departure from validity, is the potentially larger sum of squared differences. Thus,it seems that this implementation may prove to be inappropriate for significantly higherdimensionsal problems.

6 Conclusion

This article illustrates some of the common approaches to generating correlated randomvariables and the associated practical implementation using a widely available softwarepackage. Performance results show that with the number of dimensions n ≤ 20, theimplementation provides a reasonable alternative for recovering valid correlation matri-ces for simulation procedures in financial risk quantification and derivative pricing. Forhigher dimensions however, the implementation in this article is not suitable given therelatively lenghty processing time. This processing time tends to become unmanageableas the number of dimensions increase. Nevertheless, the process has provided some prac-tical insight into the methodology and highlighted alternative ways of implementationthat may be useful for practitioners.


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References

[1] Embrechts, P., McNeil, A. J., and Straumann, D., “Correlation and Dependencein Risk Management: Properties and Pitfalls”, in Dempster, M. (ed), (2002), RiskManagement: Value at risk and beyond, CambridgeUniversity Press.

[2] Embrechts P., McNeil A.J., and Straumann, D.,“Correlation: Pitfalls and Alterna-tives”, RISK, May, pages 69-71.

[3] Jackel, P., Monte Carlo Methods in Finance, (2001), John Wiley & Sons.

[4] Press, W. H., Teulovsky, S. A., Vetterling, W. T., and Flannery, B. P., NumericalRecipes in C, The Art of Scientific Computing, Second Edition, (1992), CambridgeUniversity Press.

[5] Rebonato, R. and Jackel, P., (1999), “The most general methodology for creatingvalid correlation matrices for risk management and option pricing purposes”, WorkingPaper, Quantitative Research Centre, NatWest Group.

[6] Zhang, Z. and Wu, L., (2000), “Optimal low-rank approximation to a correlationmatrix”, Working Paper, Department of Mathematics, The Hong Kong University ofScience and Technology, forthcoming in Linear Algebra and its applications.


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Appendix 1 - Cholesky Decomposition

The following algoritm is an implementation of the Cholesky decomposition with thefirst element of the correlation matrix r1,1 in Sheet1.Range(“B2”). This procedure pro-vides a decomposition only for valid correlation (or covariance) matrices. As such, anerror command has been incorporated to alert of potential problems in the correlationmatrix to be decomposed.

Option Base 1

Sub Cholesky()

Dim corr As Range, i as Integer, j as Integer, k as Integer, m as Integer

Dim Sum as Double, Prod as Double

With Sheet1.Range("B2")

Set corr = Range(.End(xlDown), .End(xlToRight))

End With

Row = corr.Rows.Count

Col = corr.Columns.Count

OnError Resume Next

MsgBox ("Cholesky procedure failed, check data")

ReDim Factor(Row, Col) As Variant

For j = 1 To Col

For i = j To Row

Factor(j, i) = 0

If i = j Then

For k = 1 To j - 1

Sum = (Factor(i, k) ^ 2) + Sum

Next k

Factor(i, j) = Sqr(corr(i, j) - Sum)

Sum = 0

Else:

For m = 1 To j - 1

Prod = (Factor(i, m) * Factor(j, m)) + Prod

Next m

Factor(i, j) = (corr(i, j) - Prod) / Factor(j, j)

Prod = 0

End If

Sheet2.Cells(1 + i, 1 + j) = Factor(i, j)

Next i

Next j

End Sub


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Appendix 2 - PCA

The following sample code may be used to solve for the eigenvalues and eigenvectors ofthe entered correlation matrix by invoking the VBA(Excel) reference to Solver. Solveroptions are kept as default (Newton method for optimistion), with iteration and precisionincreased to 1000 and 10−9 respectively for convergence. Additionally, it is assumed thatthe first element of the correlation matrix r1,1 is in Sheet1.Range(“B2”).

Sub PCA()

Application.ScreenUpdating = False

Dim corr As Range, Row As Integer, Col As Integer, i as integer

With Sheet1.Range("B2")

Set corr = Range(.End(xlDown), .End(xlToRight))

End With

Row = corr.Rows.Count

Col = corr.Columns.Count

Sheet1.Select

Sheet1.Range(Cells(2, 2), Cells(Row + 1, Col + 1)).Name = "Corr"

Sheet2.Select

Range(Cells(2, 2), Cells(Row + 1, Col + 1)).Name = "Eigenvect"

Range(Cells(2, Col + 3), Cells(Row + 1, (2 * Col) + 2)).Name = "Eigenval"

Range(Cells(Row + 4, 2), Cells((2 * Row) + 3, Col + 1)).Name = "Orthogonal"

For j = 1 To Col

For i = 1 To Row

Range("Eigenvect").Cells(i, j) = 0

Next i

Next j

Range("Orthogonal").Select

Selection.FormulaArray = "=MMULT(TRANSPOSE(Eigenvect),Eigenvect)"

Range("Eigenval").Select

Selection.FormulaArray = "=MMULT(MMULT(TRANSPOSE(Eigenvect),Corr),Eigenvect)"

For i = 1 To Col

SolverReset

Solveroptions iterations:=1000, precision:=0.00000001

solverok _

SetCell:=Range("Eigenval").Cells(i, i), MaxMinVal:=1, _

ByChange:=Range(Range("Eigenvect").Cells(1, i), _


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Range("Eigenvect").Cells(Row, i))

SolverAdd CellRef:=Range("Orthogonal").Cells(i, i), _

Relation:=2, FormulaText:=1

If i <> 1 Then

SolverAdd CellRef:=Range(Range("Orthogonal").Cells(i - 1, i), _

Range("Orthogonal").Cells(1, i)), Relation:=2, FormulaText:=0

End If

SolverSolve (True)

Next i

Call Transform(Row, Col)

End Sub


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Appendix 3 - Transformation

The transformation algorithm presented below takes the input from the PCA routinein terms of the underlying correlation structure and its dimensions, as well as the de-rived eigenvalue and associated eigenvectors. The corresponding error estimators ε andRMSE, as well as degree of non-psd of the original correlation struture are also com-puted based on the estimated (transformed) valid correlation structure and the derivedeigenvalues of the original correlation structure respectively.

Option Base 1

Sub Transform(Row As Integer, Col As Integer)

Application.ScreenUpdating = False

Dim Corr_prime, B_prime, epsilon As Double, rmse As Double, degree as Single

Dim i as Intger, j as Integer

ReDim Lambda(Row, Col) As Double, rootLambda(Row, Col) As Double

ReDim RootInvT(Row, Col) As Double

For j = 1 To Col

For i = 1 To Row

If i = j And Range("Eigenval").Cells(i, j) >= 0 Then

Lambda(i, j) = Range("Eigenval").Cells(i, j)

degree = Lambda(i, j) + degree

ElseIf Range("Eigenval").Cells(i, j) < 0 Then

Lambda(i, j) = 0

End If

Next i

Next j

For i = 1 To Row

SumT = 0

For j = 1 To Col

RootInvT(i, j) = 0

rootLambda(i, j) = Sqr(Lambda(i, j))

SumT = (Range("Eigenvect").Cells(i, j) ^ 2) * Lambda(j, j) + SumT

Next j

RootInvT(i, i) = Sqr(1 / SumT)

Next i

B_prime = WorksheetFunction.MMult(RootInvT, WorksheetFunction.MMult _

(Range("Eigenvect"), rootLambda))

Corr_prime = WorksheetFunction.MMult(B_prime, WorksheetFunction.Transpose _

(B_prime))

epsilon = WorksheetFunction.SumXMY2(Corr_prime, Range("Corr"))

rmse = Sqr(epsilon / (Row * Col))

degree = (Col-degree)/Col

End Sub


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Binomial Approach for Lookback Options

Mark Ioffe∗

Abstract

In applying the Binomial method for calculating the price of path-dependentoptions, in particular lookback options, it is necessary to solve two equallydifficult problems - the problem of defining the pay-off function at the expira-tion moment, as well as capturing the effect of time on this pay-off function.If there is no known analytical solution or benchmark, the results of the Bi-nomial method are doubtful.

Keywords: Binomial approximation, Finite difference approximation, Look-back options.

1 Introduction

In spite of all my respect towards the importance, usefulness, relative simplicity andpopularity of the CRR Binomial method as well as towards its authors, I cant helppoint out that from a mathematical point of view, it is only the one of the count-less numerical methods of solving partial differential equations, which although is easyto implement, is unfortunately not the best one under certain situations (for exam-ple, see 1). In particular, using this method to solve Cauchy problems (problems withgiven initial conditions) or problems with boundary and initial conditions for parabolicequation of the second order, requires the use of specific (oblique) explicit finite differ-ence schemes. Therefore, to use the Binomial method we need to have both a parabolicpartial differential equation of the second order and the initial and boundary conditions.

We will investigate the performance of the Binomial method in pricing fixed and floatingstrike path dependent options. The results will highlight the differences in option valuesattributable to the Binomial method compared to an analytic benchmark.

∗The author is Head of Financial Engineering at EGAR Technology Inc., N.Y.. Specific com-ments or inquiries regarding the contents of this article may be directed to the author on email:[email protected].


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2 Pricing Framework

First, we shall investigate how the mentioned equation and initial conditions appear inthe Black-Sholes model while calculating premium of options, with pay-offs defined onlyby stock price at expiration moment. In this model the increase of stock price S(t) isdefined for 0 ≤ t ≤ T by the equation,

dS(t) = S(t)rdt + S(t)σdW (t) (1)

The pay-off at expiration T is equal to,

C = φ(S(T ))e−rT (2)

Function φ(S(T )) is supposed to be given and defined by the kind of option. As will beshowed below, since we need to use the numerical method, this function has to satisfysome specific smooth conditions. Let us make the following transformation of variableS(t),

x(t) =1

σln

(S(t)

S(0)

)(3)

It follows from equations (1) and (3) that the change in x(t) can be expressed as follows,

dx(t) = γdt + dW (t) (4)

with γ = (r − 12σ2)/σ. The pay-off at expiry T is now defined as,

C = φ(S(0)eσx)e−rT (5)

It is obvious that process x(t) is a Markov stochastic process depending only on theone variable γ. To avoid arbitrage, the option price has to be equal to the average ofdistribution of x(t) defined by equation (5). In order to simplify the notation, we canomit the discount in the following formulas Without loss of generality, in which theoption premium P can be expressed as the following integral,

P =

∞∫−∞

1√2πt

e−12t

(x−γt)2C(x)dx (6)

Let us now introduce a new variable z and consider the following function Ψ(z, t).

Ψ(z, t) =

∞∫−∞

1√2πt

e−12t

(z−x+γt)2C(x)dx (7)


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Obviously, function Ψ(z, t) satisfies the following parabolic differential equation,

∂Ψ(z, t)

∂t= γ

∂Ψ(z, t)

∂z+

1

2

∂2Ψ(z, t)

∂z2(8)

Because of the well-known properties of the density function of the Gaussian distribu-tion, it is Dirac Delta function at t=0, and we have,

Ψ(z, t = 0) = C(z) (9)

At z = 0 function Ψ(z, t) is equal to the option premium. Therefore, any option withpay-off defined only by the stock price at the expiration moment satisfies the parabolicdifferential equation (8). Hence, to calculate the premium or to calculate integral (6)we can use any corresponding numerical method, such as the Binomial method.

3 Lookback Options

For path-dependent options, in particular lookback options, it is much more difficultboth to find the corresponding differential equation and initial conditions. Becausewe know an explicit analytical solutions for calculating lookback option premium, in-ferences of which are based on the known formulas for density of distribution of themaximum and minimum of a given stochastic process, we can analyze the “correctness”of premiums calculated using the Binomial model. We can write the known formula forcalculating premium of a floating strike call option P(CFloat) as the following,

P (CFloat) =

∞∫−∞

0∫−∞

(S(0)eσz − S(0)eσy)∂

∂y

1√2πt

e2γy− 12t

(z−γt−2y)2χ(y, z)dzdy (10)

with the condition,

χ(y, z) =

{10

if y < 0 and z > yif y < 0 and z < y

(11)

Let us write formula (10) as the average of a function of Gaussian variables, or as in(7).

P (CFloat) =

∞∫−∞

1√2πt

e−12t

(z−γt)2Θ(z, t)dx (12)

with,

Θ(z, t) =√

2πte−12t

(z−γt)20∫

−∞

(S(0)eσz − S(0)eσy)∂

∂y

1√2πt

e2γy− 12t

(z−γt−2y)2χ(y, z)dy (13)


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It follows from (12) and (13) that the pay-off of path-dependent options depends notonly on stock price but also on the period of time from today to expiration. Hence, ifwe introduce function Ψ1(z, t) analogically to (7), we have,

Ψ1(z, t) =

∞∫−∞

1√2πt

e−12t

(z−x+γt)2Θ(x, t)dx (14)

Equation (14) satisfies the following differential equation,

∂Ψ(z, t)

∂t= γ

∂Ψ(z, t)

∂z+

1

2

∂2Ψ(z, t)

∂z2+

∞∫−∞

1√2πt

e−12t

(z−x+γt)2 ∂Θ(x, t)

∂tdx (15)

Therefore, when we use the binomial method for premium calculation, there is calcu-lation error that is connected with dependence of pay-off on time. If the function isgiven, as it is in the considered example when we know the exact analytical solution,then calculating the integral in the right part of (15) will help us take into account thiserror. For options with little time before expiration we can simplify the calculation asfollows,

∞∫−∞

1√2πt

e−12t

(z−x+γt)2 ∂Θ(x, t)

∂tdx ∼=

∂Θ(z + γt, t)

∂t=

Θ(γt, t + ∆t)−Θ(γt, t)

∆t(16)

We can use the same approach for the premium of Fixed Strike (K) Lookback call

options (CFixed) with function Θ1(z, t) and yk = 1σ

ln(

KS(0)

), which can be described

analogically to (12) and (13) as,

P (CFixed) =

∞∫−∞

1√2πt

e−12t

(z−γt)2Θ1(z, t)dx (17)

with,

Θ1(z, t) =√

2πte−12t

(z−γt)2∞∫

yk

(S(0)eσy −K)∂

∂y

−1√2πt

e2γy− 12t

(z−γt−2y)2χ1(y, z)dy (18)

and the condition,

χ1(y, z) =

{10

if y > yk and z < yif y > yk and z > y

(19)


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4 Pricing Analysis

It is to be mentioned that the integrals in formulas (12,13) and (17,18) are explicitlycalculated by means of using the Gaussian distribution function. Based on the abovemethod for parabolic nonhomogeneous equation (15) with corresponding initial con-ditions, a comparison is performed using a finite difference scheme corresponding toBinomial method. Premiums for the following options were calculated (see 2),

1. European floating strike lookback call(S(0)=100, rd=0.04, rf=0.07, T=0.5)

2. European fixed strike lookback call(S(0)=100, K=100, rd=0.04, rf=0.07, T=0.5)

In both cases the decision has converged if we have nstep=50. All results are representedin Table 1 below. (nstep=50).

Lookback Option PriceOption σ = 0.1 σ = 0.2 σ = 0.3

Binomial Float 4.6810 9.7645 14.6919Analytic Float 4.6799 9.7915 14.6875Binomial Fixed 6.7066 13.8602 21.3750Analytic Fixed 4.9231 10.7644 16.8762

Table 1: Lookback option pricing results

As it can be seen in Table 1, the errors in the estimated premium for European floatingstrike lookback call options are insignificant; meanwhile, those for European fixed strikelookback call options reach 40%. There are three kinds of errors in calculating premiumof path-dependent options with finite difference method,

1. Error in definition of pay-off function

2. Error brought about by dependence of pay-off function on time

3. Error of numerical method brought about by representation of solution of differ-ential equation as the solution of finite difference equations. Error in arithmeticalcalculations also belongs to this kind of error

In our example, there are no errors of the first kind, because we use the known ana-lytical equation for definition of pay-off function of lookback options. Increasing thenumber of steps in the difference scheme, we can make the errors of the third kind verysmall. In our example, as there is conversion on the first 50 steps, we can suppose thatthere are almost no errors of the third kind. Hence, the main source of errors in ourapproximation is the dependence of pay-off function on time.

If we know the exact analytical solution, as in our example, we can diminish this error bymeans of using more complex approximation of dependence of pay-off function on time.In general case, not using the known analytical solution, to realize Binomial methodit is necessary to define pay-off function at the expiration moment. It is supposed in


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[2] that for European floating strike lookback call option this formula is equal to (see

formual (7)) S(tn)(1− u−k), where u = eσ√

δt, N=nstep, k=the number of steps up.

In classic binomial nodes, the values of stock prices are equals to, for 0 ≤ i ≤ nstep,0 ≤ k ≤ i,

S(i, k) = S(0)u2k−i (20)

Because of our assumption, we can consider that the pay-off function in nodes F0(k) isequal to, for 0 ≤ k ≤ nstep,

F0(k) = S(0)(u(2k−nstep) − u(k−nstep)) (21)

It follows from formula (3) and (20) that the corresponding values of variable z(k) areequal to,

z(k) = (2k − nstep)√

δt (22)

Using the formulas (12,13) and (18) we can calculate the real value of pay-off function innode F1(k). The results of calculations are presented in Figure 1. Series 1 correspondsto the pay-off function in knot F0(k) from [2]; Series 2 corresponds to the real valuesthat can be calculated if we know option price.

Figure 1: Payoff function different approaches

As it can be seen in Fig.1, the supposed function is significantly different from the real.To correct this difference in initial values in [2] it is used a non-classical scheme of Bino-mial method (formulas (10) and (11)) in [2]. In the classical scheme of Binomial method,the price of the option in k-knot on i-step is calculated with the option price in k+1 knotand k knot on I+1 step. In [2] the authors suppose to calculate option price in k knot on


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i step with the option price in k+1 and k-1 knot on i+1 step. If we use this schema forcalculating the price of european call with parameters. S(0)=100, Strike=100, rd=0.04,rf=0.07, σ = 0.1, T=0.5, then after 50 steps we will have the value 2.2427 while the realvalue, calculated from the Black-Sholes formula or calculated with classical Binomialmethod with nstep=50, is 2.0758. But even with these changes of Binomial method wecan rely on the results only if we simulate the significant number of steps (∼5000).

It is supposed in [2] that for European fixed strike lookback call pay-off function is equalto zero. The real value of this function that is calculated with formulas (17,18) and (22)is shown on Fig.2.

Figure 2: Payoff function using eqn (17,18) and (22)

To have the reliable result in this case, it is supposed in [2] to solve a non-homogeneousequation with the right part.

5 Conclusion

Based on the results above, we can infer the following points regarding the use of theBinomial method in pricing path dependent options, such as Lookback options.

1. To apply Binomial method for the calculating of path-dependent options it isinitially necessary to solve equally difficult problem, the problem of definition ofpay-off function at the expiration moment

2. It is necessary to take into account in Binomial schema that pay-off functiondepends on time.

3. If there is no known analytical solution, or benchmark, the results of Binomialmethod are doubtful.


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References

[1] Ioffe, G. and Ioffe, M.,“Application of the finite differences method for pricing barrieroptions”.

[2] Cheuk, T. H. P. and Vorst A. C. P. (1997), “Currency lookback options and observa-tion frequency: A binomial approach”, Journal of International Money and Finance,Vol.16 No.2, pages 173-187.


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