currency-protected swaps and swaptions with nonzero...

Currency-Protected Swaps

and Swaptions with

Nonzero Spreads in a

Multicurrency LMM

JUI-JANE CHANGSON-NAN CHENTING-PIN WU∗

Despite the fact that currency-protected swaps and swaptions are widely traded inthe marketplace, pricing models for zero-spread swaps, and swaptions have rarelybeen examined in the extant literature. This study presents a multicurrency LIBORmarket model and uses it to derive pricing formulas for currency-protected swapsand swaptions with nonzero spreads. The resulting pricing formulas are shown tobe feasible and tractable for practical implementation and their hedging strategiesare also provided. Our pricing formulas provide prices close to those computedfrom Monte Carlo simulation, but involve far less computation time, and therebyoffering almost instant price quotes to clients and daily marking-to-market tradingbooks, and facilitating efficient risk management of trading positions. C© 2012Wiley Periodicals, Inc. Jrl Fut Mark

∗Correspondence author, Department of Finance, National Central University, 300 Jhongda Road, JhongliCity, Taoyuan County 32001, Taiwan. Tel: 886-3-4227151 ext. 66252, Fax: 886-3-4252961, e-mail:[email protected]

Received July 2011; Accepted April 2012

� Jui-Jane Chang is an assistant professor at the Department of Financial Engineeringand Actuarial Mathematics, Soochow University, Taiwan.

� Son-Nan Chen is a professor at the Shanghai Advanced Institute of Finance (SAIF),Shanghai Jiao Tong University, China.

� Ting-Pin Wu is an associate professor at the Department of Finance, National CentralUniversity, Taiwan.

The Journal of Futures Markets, Vol. 00, No. 0, 1–41 (2012)C© 2012 Wiley Periodicals, Inc.Published online in Wiley Online Library (wileyonlinelibrary.com).DOI: 10.1002/fut.21567

The Journal of Futures Markets, Vol. 33, No. 9, 827-867 (2013)The Journal of Futures Markets, Vol. 33, No. 9, 827-867 (2013)

© 2012 Wiley Periodicals, Inc.© 2012 Wiley Periodicals, Inc.

Published online 11 June 2012 in Wiley Online Library (wileyonlinelibrary.com).Published online 11 June 2012 in Wiley Online Library (wileyonlinelibrary.com).

DOI: 10.1002/fut.21567DOI: 10.1002/fut.21567

*

828828 Chang, Chen, and Wu Chang, Chen, and Wu

1. INTRODUCTION

A currency-protected swap (CPS), also called a cross-currency basis swap, (LI-BOR) differential swap or interest rate index swap, is a variation of a plain-vanilla interest rate swap. A CPS involves the exchange of two series of floatingpayments, both denominated in the same currency, while reference rates areLIBOR rates in two different currencies. A nonzero spread is often contracted ina CPS to compensate for the difference between two interest rates and the rel-ative credit ratings of two counterparties. Just like ordinary interest rate swaps,CPSs are normally settled in arrears, and there is no exchange of principal.

The first CPS was traded in late 1990 between Credit Suisse First Bank anda Japanese insurance company, and the instrument has been widely traded sincethen.1 The main impetus to the growth of the CPS market lies in a differencein the relation between global term structures of interest rates; the differencesin long-term interest rates among a number of currencies fail to reflect thedifferentials among their short-term rates. For example, a considerable numberof CPSs were traded in the period between the second half of 1991 and early1992 due to the fact that some interest rate term structures in the globalmarket were upward sloping (e.g., in the United States), while others were flator downward sloping (primary in Europe), which led to the accelerating growthof the CPS market.

The major use of CPSs is to exploit the floating rate difference or themoney market interest rate difference between two currencies without incurringdirect exchange rate risk. For example, a portfolio manager can employ CPSsto enhance current yield if he can take a correct view on the movement ofinterest rate differentials between two currencies. In currencies with relativelyhigh interest rates, borrowers may enter into CPSs to reduce interest costs bycapitalizing on the interest rate differential between two currencies. In addition,corporate treasurers may use CPSs to transform their yields (from assets) orcosts (from liabilities) to link interest rates in one currency with those in anotherfavorable currency, while payments are still made in domestic currency.

Currency-protected swaptions (CPSOs), which provide their owners theright to enter or terminate CPSs, have also been traded in over-the-countermarkets. An investor who wants to take a position in a CPS in the future andfinds that the current CPS price is attractive, can lock in the CPS price via aCPSO.

To present, few research papers have examined the pricing and hedging ofCPSs and CPSOs. Litzenberger (1992) first discussed CPSs and proposed someanalytical pricing methods in his presidential address. Turnbull (1993) adopts

1According to the data from BIS, the outstanding notional principal of CPSs is amounted to about $37 billionin 2007.

Journal of Futures Markets DOI: 10.1002/fut

Currency-Protected Swaps and Swaptions 829 Currency-Protected Swaps and Swaptions 829

the cross-currency Gaussian Heath, Jarrow, and Morton (1992, HJM) modelto price and hedge CPSs. Jamshidian (1994) examines the pricing and hedgingof CPSs in a replication approach by using domestic and foreign zero-couponbonds (ZCB), but only a general expression of the solution is provided, whichcannot be used without further analysis. Both Wei (1994) and Chang, Chung,and Yu (2002) employ an Ornstein–Uhlenbeck process to specify the behaviorof domestic and foreign risk-free interest rates and used the resulting model toprice CPSs. The studies cited above provide theoretically sound pricing modelsfor CPSs. Unfortunately, they are intractable in parameter calibration and thusnot well suited for practical implementation. Moreover, the rates specified intraditional interest rate models, such as Vasicek (1977) and Gaussian HJM(1992), can yield negative values with positive probability, which may result inpricing errors in many cases.2

To overcome the obstacles in parameter calibration of traditional interestrate models, the LIBOR market model (LMM) is thus developed by Brace,Gatarek, and Musiela (1997), Musiela and Rutkowski (1997), and Miltersen,Sandmann, and Sondermann (1997). The rates specified in the LMM aremarket-observable LIBOR rates, which have been used as underlying inter-est rates in many popular financial instruments. The LIBOR rate modeled inthe LMM is lognormally distributed, thereby avoiding the pricing errors result-ing from possible negative interest rates. The LMM can be used to price mostinterest rate derivatives, and thus the interest rate risks can be consistentlymanaged.3 In addition, the most important merits of the LMM are its feasi-bility and tractability in parameter calibration. The LMM can simultaneouslycalibrate market-quoted cap volatilities and the correlation matrix of underlyingforward LIBOR rates.

Due to the advantages presented above, Schlogl (2002) adopted a cross-currency LMM to price CPSs and CPSOs. However, the spread of CPSs andCPSOs in Schlogl (2002) is set to zero, and thus their pricing formulas cannotbe used to price nonzero-spreads CPSs and CPSOs that are more popularlytraded in the marketplace. In addition, by following interest rate parity, thereis a concrete relation among domestic and foreign forward LIBOR rates andthe forward exchange rate, so are their volatilities. Therefore, Schlogl (2002)imposes a restriction on the choice of lognormally distributed underlying vari-ables. He then derives a CPSO pricing formula under the assumption that thedomestic forward LIBOR rate and forward exchange rate volatilities are deter-ministic.4 However, under the model setting of Schlogl (2002), the parameter

2Please refer to Rogers (1996) for the negative interest rate problem.3The LMM are also adopted to price exotic interest rate swaps and options. Please refer to Wu and Chen(2009a, 2010).4Refer to Schlogl (2002) for a detailed discussion.



calibration of the resulting CPSO pricing formula is not easy to carry out inpractice.

Based on the arbitrage-free conditions in Amin and Jarrow (1991), thepresent study extends the LMM to a multicurrency LIBOR market model(MLMM), and then adopts it to examine the pricing and hedging of CPSsand CPSOs.5 It assumes the domestic and foreign forward LIBOR rate volatil-ities are deterministic, which makes parameter calibration more tractable.6 Byextending the pricing formulas in Schlogl (2002), our pricing formulas can beused to price nonzero-spread CPSs and CPSOs that are frequently traded inthe marketplace. In addition, we further examine the pricing and hedging of avariant CPS, which exchanges interest rates in one foreign currency to anotherforeign currency, both denominated in domestic currency. Variant CPSs arealso popular in the marketplace due to its great versatility, and its options arealso examined in this analysis.

The remainder of this study is organized as follows. In Section 2, we in-troduce the MLMM and its associated implementation techniques in prac-tice. A generalized lognormal distribution is also introduced in this section. InSection 3, we develop the pricing and hedging of CPSs. The pricing and hedgingof CPSOs are presented in Section 4. In Section 5, we provide numerical stud-ies to examine the accuracy of our pricing formulas. In Section 6, we presentour conclusions.

2. THE MODEL

In Section 2.1, we briefly review the MLMM and various approximate tech-niques that are widely used in practical operation. In Section 2.2, we presentthe approximate dynamics of swap rates within the MLMM framework. In Sec-tion 2.3, we introduce a generalized lognormal distribution, which is used toapproximate the distribution of the difference between domestic and foreignswap rates.

2.1. Review of the MLMM and ApproximateTechniques

Assume that trading takes place continuously in time over an interval [0, τ ],0 < τ < ∞. The uncertainty is described by the filtered probability space(�,F,Q, {F}t∈[0,T ]), where the filtration is generated by an independent

5Our MLMM involves interest rates in the environment with the domestic and two various foreign currencies,while Schlogl (2002) specifies interest rates in the domestic and only one foreign currency.6Both domestic and foreign caps (floors) are the most liquid source of interest rate volatility information, andtheir maturity range extends to as far as 20 years.



m-dimensional standard Brownian motion W(t) = (W1(t), W2(t), . . . , Wm(t))and Q represents the domestic spot martingale probability measure. We listthe following notations with the subscript “d” representing domestic country;“ f ” and “g” two other foreign countries:

Lk(t, T) = the kth country’s forward LIBOR rate contracted at time t forborrowing and lending during the time period [T, T + δ], where0 ≤ t ≤ T < T + δ ≤ τ and k ∈ {d, f, g }.

Bk(t, T) = the time t price of the kth country’s ZCB expiring and paying onedollar at time T .

rk(t) = the kth country’s risk-free short rate at time t.Xk(t) = the spot exchange rate at time t for one unit of the kth country’s

currency in terms of domestic currency.QT = the domestic forward martingale measure with respect to the

numeraire Bd (·, T).

The relationship between LIBOR rates and ZCB prices can be specified asfollows:

1 + δLk(t, T) = Bk(t, T)Bk(t, T + δ)

, k ∈ {d, f, g }. (1)

Based on the arbitrage-free conditions in Amin and Jarrow (1991), weextend the LMM to the MLMM. The result is presented in the following propo-sition.7

Proposition 1: MLMM under domestic spot martingale measure Q. Under thedomestic spot martingale measure Q, the processes of the forward LIBOR rates,the ZCB prices and the spot exchange rates are given as follows:

d Ld (t, T)Ld (t, T)

= γd (t, T) · σBd (t, T + δ) dt + γd (t, T) · dW(t), (2)

d Bd (t, T)Bd (t, T)

= rd (t) dt − σBd (t, T) · dW(t), (3)

d Lk(t, T)Lk(t, T)

= γk(t, T) · (σBk(t, T + δ) − σX k(t))

dt + γk(t, T) · dW(t), k ∈ { f, g },

(4)

7The derivation procedure can be obtained from Wu and Chen (2007a).



d Bk(t, T)Bk(t, T)

= (rk(t) + σX k(t) · σBk(t, T)

)dt − σBk(t, T) · dW(t), k ∈ { f, g }, (5)

d Xk(t)Xk(t)

= (rd (t) − rk(t)

)dt + σX k(t) · dW(t), k ∈ { f, g }, (6)

where · stands for the inner product of vectors. m-dimensional volatility vectorfunctions, γk(t, T), k ∈ {d, f, g }, and σX k(t), k ∈ { f, g }, are deterministic andsatisfy the standard regular conditions. The bond volatility vector function,σBk(t, T), k ∈ {d, f, g }, is defined as follows:

σBk(t, T) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩�δ−1(T−t)�∑

j=1

δLk(t, T − jδ)1 + δLk(t, T − jδ)

γk(t, T − jδ) t ∈ [0, T − δ]

& T − δ > 0,

0 otherwise,

(7)

where �δ−1(T − t)� denotes the greatest integer that is less than δ−1(T − t) andt ∈ [0, T].

Benner, Zyapkov, and Jortzik (2009, BZJ) also propose another approachto specify the stochastic processes, whose result is equivalent to Proposition1 and is presented in Proposition 2. Proposition 1 carries simpler notations,and thereby greatly facilitating the pricing process and achieving more elegantpricing formulas. Therefore, the following pricing models are derived based onProposition 1.

Proposition 2: BZJ (2009) Approach for the MLMM under Q. Under Q, theprocesses of the forward LIBOR rates, the ZCB prices, and the spot exchangerates are given as follows:


= ρLd(T),Bd(T)VLd (t, T)VBd (t, T)dt + VLd (t, T)d ZLd(T)(t),

d Bd (t, T)Bd (t, T)

= rd (t)dt − VBd (t, T)d Z Bd(T)(t),

d Lk(t, T)Lk(t, T)

= (ρLk(T),Bk(T)VLk(t, T)VBk(t, T) − ρLk(T),X kVLk(t, T)VX k(t, T)

)dt

+ VLk(t, T)d ZLk(T)(t),

d Bk(t, T)Bk(t, T)

= (rk(t) + ρBk(T),X kVBk(t, T)VX k(t, T)

)dt − VBk(t, T)d Z Bk(T)(t),



d Xk(t)Xk(t)

= (rd (t) − rk(t)

)dt + VX k(t)d Z X k(t),

where

d ZLd(T)(t) = γd (t, T) · dW(t)VLd (t, T)

, d Z Bd(T)(t) = σBd (t, T) · dW(t)VBd (t, T)

,

d ZLk(T)(t) = γk(t, T) · dW(t)VLk(t, T)

, d Z Bk(T)(t) = σBk(t, T) · dW(t)VBk(t, T)

,

d Z X k(t) = σX k(t) · dW(t)VX k(t)

,

and

ρLd(T),Bd(T)dt = d ZLd(T)(t)d Z Bd(T)(t)VLd (t, T)VBd (t, T)

, ρLk(T),Bk(T)dt = d ZLk(T)(t)d Z Bk(T)(t)VLk(t, T)VBk(t, T)

,

ρLk(T),X kdt = d ZLk(T)(t)d Z X k(T)(t)VLk(t, T)VX k(t, T)

, ρBk(T),X kdt = d Z Bk(T)(t)d Z X k(T)(t)VBk(t, T)VX k(t, T)

,

and VL∗(t, T) = ‖γ∗(t, T)‖, VB∗(t, T) = ‖σB∗(t, T)‖ and VX∗(t, T) = ‖σX∗(t, T)‖,∗ ∈ {d, f, g }.

The interest rates specified in the MLMM are market-observable LIBORrates. Moreover, the cap and floor pricing formulas within the MLMM frame-work are the Black’s formulas, which are widely used for price quotations inthe marketplace. Therefore, the model parameters, including initial LIBORrates, Lk(0, ·), and the volatilities, γk(t, T), can be easily extracted from marketdata, and σBk(t, T) can be calculated from (7). The volatilities of exchangerates can be obtained either from historical data of exchange rates or im-plied volatilities of currency options. Thus, the parameters in the MLMMcan be easily calibrated using market data. This feature makes pricing for-mulas derived within the MLMM framework more tractable and feasible inpractice.

Equation (7) indicates that σBk(t, T + δ) is composed of the LIBOR rateprocesses, and thus it is stochastic rather than deterministic, thereby causingEquations (2) and (4) to be insolvable and the distribution of Lk(T, T) is un-known. However, we can approximate σBk(t, T) by σ 0

Bk(t, T), which is defined



by

σ 0Bk(t, T) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩�δ−1(T−t)�∑

j=1

δLk(0, T − jδ)1 + δLk(0, T − jδ)

γk(t, T − jδ), t ∈ [0, T − δ]

& T − δ > 0,

0 otherwise,

(8)

where the calendar time of the process Lk(t, T − jδ) in (7) is frozen at its initialtime 0 and thus the process σ 0

Bk(t, T) becomes deterministic. By substitutingσ 0

Bk(t, T + δ) for σBk(t, T + δ) in the drift terms of (2) and (4), we can computeLk(T, T), which follows approximately a lognormal distribution.

This approximate technique is also used in BGM (1997), Schlogl (2002),and Wu and Chen (2007), and has been widely employed in practice due toits accuracy. The result of the lognormalization is concluded in the followingproposition.

Proposition 3: Approximate LIBOR rate dynamics under Q. The approximatedynamics of forward LIBOR rates under Q are given as follows:


= γd (t, T) · σ 0Bd (t, T + δ)dt + γd (t, T) · dW(t), (9)

d Lk(t, T)Lk(t, T)

= γk(t, T) · (σ 0Bk(t, T + δ) − σX k(t)

)dt + γk(t, T) · dW(t), (10)

where k ∈ { f, g } and 0 ≤ t ≤ T ≤ τ .

Occasionally, it is more convenient to compute expected cash flows indifferent forward martingale measures. The following proposition specifies thegeneral rule under which the LIBOR rate dynamics change following a changein the underlying measure. This rule is useful for deriving the pricing formulasof interest rate derivatives.8

Proposition 4: Drift adjustment technique in different measures. The dynam-ics of forward LIBOR rates under an arbitrary domestic forward martingalemeasure QT ′

is given as follows:


= γd (t, T) · (σ 0Bd (t, T + δ) − σ 0

Bd (t, T ′))dt + γd (t, T) · dW(t), (11)

8For the pricing technique associated with changing measure, please refer to Sundaram (1997) and Benninga,Bjork, and Wiener (2002).



d Lk(t, T)Lk(t, T)

= γk(t, T) · (σ 0Bk(t, T + δ) − σ 0

Bd (t, T ′) − σX k(t))dt + γk(t, T) · dW(t),

(12)

where k ∈ { f, g } and 0 ≤ t ≤ min(T, T ′).9

2.2. Approximate Dynamics of Swap RatesWithin the MLMM Framework

The swap market model (SMM) and the LMM are popular in practice dueto their agreement with Black’s formulas for caps and swaptions.10 However,it is well known that the LMM and the SMM are incompatible, and thuschoosing either one of the two models as a pricing foundation is problematic.We adopt the LMM framework as the central model due to its mathematicaltractability and versatility for pricing a variety of instruments with the sameanalytical framework. In this subsection, we introduce an approach to findingan approximate dynamics of swap rates within the MLMM framework.

Consider the tenor structure {T0, T1, . . . , Tn} with δ = Tj − Tj−1, j =1, 2, . . . , n.11 Define the N-year domestic forward swap rate, observed at time tand matured at time T0 with 0 ≤ t ≤ T0, as follows:

Sd (t, T0) =n∑

j=1

w j (t)Ld (t, Tj−1), (13)

where N = nδ and w j (t) is the weight of the jth LIBOR rate and defined by

w j (t) = δBd (t, Tj )C(t; T1, Tn)

, (14)

and

C(t; T1, Tn) = δ

n∑k=1

Bd (t, Tk) (15)

9We employ W(t) to denote an independent d-dimensional standard Brownian motion under an arbitrarymeasure.10The SMM is presented in Jamshidian (1997), under which the forward swap rate is a martingale andswaptions can be priced using the Black’s formula.11For simplicity, we set δ as a constant. Its value may depend on the number of days in each accrual period,most often equal to three or six months.



denotes the present value of a basis point. According to (1), Equation (13) canbe rewritten as

Sd (t, T0) = Bd (t, T0) − Bd (t, Tn)C(t; T1, Tn)

, (16)

which can be regarded as a tradable asset expressed in units of C(t; T1, Tn).Assume that the market is arbitrage-free and complete and there exists an

equivalent martingale measure with respect to the numeraire C(t; T1, Tn) that isdenoted by QC. Under QC, any tradable asset expressed in terms of C(t; T1, Tn)must evolve as a martingale process, and thus the dynamics of Sd (t, T0) shouldhave the following form12:

d Sd (t, T0)Sd (t, T0)

= σSd (t, T0) · dW(t), (17)

where σSd (t, T0), defined in (A1), denotes the volatility of a domestic forwardswap rate within the MLMM framework. Because σSd (t, T0) is a stochasticprocess, Sd (T0, T0) is incomputable and its distribution is unknown. In AppendixA, we approximate σSd (t, T0) by a deterministic process σ 0

Sd (t, T0) defined by

σ 0Sd (t, T0) =

n∑k=1

ζ 0k (0)γd (t, tk−1) +

n∑k=1

φ0k (0)σ 0

Pd(t, tk) −

n∑k=1

ζ 0k (0)σ 0

Pd(t, tk), (18)

where φ0k (0) and ζ 0

k (0) are defined in (A4) and (A5), respectively. In this way,Sd (T0, T0) can be computed by

Sd (T0, T0) = Sd (0, T0) exp(

−12

∫ T0

0‖σ 0

Sd (t, T0)‖2dt +∫ T0

0σ 0

Sd (t, T0) · dW(t))

,

(19)

and ln Sd (T0, T0) is normally distributed with mean and variance, defined, re-spectively, by

Md = ln Sd (0, T0) − 12

∫ T0

0‖σ 0

Sd (t, T0)‖2dt,

V 2d =

∫ T0

0‖σ 0

Sd (t, T0)‖2dt.

12For details regarding this statement, refer to Harrison and Kreps (1979), Harrison and Pliska (1981, 1983),and Jamshidian (1997).



2.3. The Generalized Lognormal Distribution

As we will see later, a CPSO can be regarded as an option on the spreadbetween the domestic and the foreign swap rates, both of which are lognormallydistributed. To price the CPSO, the distribution of the spread must be specified.To simplify the analysis, we reduce the problem to a simple structure. Assumethat Sd and Sf are two lognormal random variables and Y = Sd − Sf . The goalis to compute the following expectation:

E[(Y − K )+], (20)

where K is a constant and (A)+ = Max(A, 0). Although the distribution of Y isunknown, its first three moments, E[Y ], E[Y 2], and E[Y 3], can be computed.In this section, we introduce a generalized family of lognormal distributions toapproximate the distribution of Y .

The generalized family of lognormal distributions includes four types: regu-lar, shifted, negative, and negative-shifted distributions. Their probability den-sity functions are defined as follows.13

Definition 1: The probability density functions of the generalized lognormaldistributions are defined as follows:

(a) Regular lognormal distribution:

f (x) = 1

βx√

2πexp

(− 1

2β2(log x − α)2

), x > 0. (21)

(b) Shifted lognormal distribution:

f (x) = 1

β(x − γ )√

2πexp

(− 1

2β2

(log(x − γ ) − α

)2), x > γ. (22)

(c) Negative lognormal distribution:

f (x) = −1

βx√

2πexp

(− 1

2β2

(log(−x) − α

)2), x < 0. (23)

13For details regarding the generalized family of lognormal distributions, please refer to Johnson, Kotz, andBalakrishnan (1994) and Li, Deng, and Zhou (2008).



(d) Negative-shifted lognormal distribution:

f (x) = 1

β(−x − γ )√

2πexp(

− 12β2

(log(−x − γ ) − α)2), x < −γ, (24)

where α, β, and γ denote, respectively, the scale, shape, and locationparameters.

The relation among the above four distributions is given as follows: if a ran-dom variable X has a regular lognormal distribution, then the random variableX + γ has a shifted lognormal distribution; −X is a negative lognormal distri-bution; −(X + γ ) is a negative-shifted lognormal distribution. We denote thefirst three moments of X by M1(α, β, γ ), M2(α, β, γ ), and M3(α, β, γ ), whichcan be computed for each of the four types of distributions. The result is givenin the following proposition.

Proposition 5: For each distribution, the first three moments in terms of pa-rameters α, β, and γ are presented as follows:

(a) If X has a regular lognormal distribution, then its first three moments arecomputed as follows:

M1(α, β, γ ) = exp{α + 1

2β2}

,

M2(α, β, γ ) = exp{2α + 2β2},

M3(α, β, γ ) = exp{

3α + 92β2}

.

(b) If X has a shifted lognormal distribution, then its first three moments arecomputed as follows:

M1(α, β, γ ) = γ + exp{α + 1

2β2}

,

M2(α, β, γ ) = γ 2 + 2τ exp{α + 1

2β2}

+ exp{2α + 2β2},

M3(α, β, γ ) = γ 3 + 3γ 2 exp{α + 1

2β2}

+ 3γ exp{2α + 2β2}



+ exp{

3α + 92β2}

.

(c) If X has a negative lognormal distribution, then its first three moments aregiven by

M1(α, β, γ ) = − exp{α + 1

2β2}

,

M2(α, β, γ ) = exp{2α + 2β2},

M3(α, β, γ ) = − exp{

3α + 92β2}

.

(d) If X has a negative-shifted lognormal distribution, then its first three mo-ments are given by

M1(α, β, γ ) = −γ − exp{α + 1

2β2}

,

M2(α, β, γ ) = γ 2 + 2γ exp{α + 1

2β2}

+ exp{2α + 2β2},

M3(α, β, γ ) = −γ 3 − 3γ 2 exp{α + 1

2β2}

− 3γ exp{2α + 2β2}

− exp{

3α + 92β2}

.

The parameters of a generalized lognormal random variable X (which willbe used to approximate Y ) can be obtained by the moment matching method,and specifically, solving the following system of equations14:

M1(α, β, γ ) = E[Y ]

M2(α, β, γ ) = E[Y 2] ⇒ α, β, γ . (25)

M3(α, β, γ ) = E[Y 3]

The remaining problem is choosing an appropriate distribution X from thegeneralized family of lognormal distributions to approximate Y . The determinant

14The analytical solution of this system of equation can be obtained in Chang, Chen, and Wu (2012).



criterion of the approximate distribution X depends on both γ , computed from(25), and the skewness of Y , which is defined as follows:

SKY = E[Y − EY

]3(E[Y − EY

]2)1.5 . (26)

The determinant procedure is specified as follows. If Y has a positive skew-ness (SKY > 0), then the shifted lognormal random variable X is used tocompute (25) and obtain γ . Next, if γ ≥ 0, then X follows the regular log-normal distribution; otherwise, X follows the shifted distribution. On theother hand, if Y has a negative skewness (SKX < 0), then the negative-shiftedX is used to compute (25) and obtain γ . Next, if γ ≥ 0, then X followsthe negative lognormal distribution; otherwise, X follows the negative-shifteddistribution.

Once the approximate distribution X is determined (one of the four types:(21)–(24)) and its parameters computed (from (25)), Equation (20) can beapproximately computed by

E[(Y − K )+

] ≈ E[(X − K )+

]. (27)

This approximate technique will be used to price CPSOs in Section 4.

3. PRICING AND HEDGING CPSs

In this section, we employ the MLMM to price and hedge two popular CPSs,both of which have domestic-currency notional principals. The first type isa plain-vanilla CPS, denoted by CPS1, which exchanges domestic-currencyinterest for foreign-currency interest. The second type is a variant CPS, denotedby CPS2, which swaps foreign-currency interest for another foreign-currencyinterest.

3.1. Pricing and Hedging CPS1

A CPS1 is a contract that starts at time T0 with reset dates T0 < T1 < · · · < Tn−1

and payment dates T1 < T2 < · · · < Tn. For simplicity, we define δ = Tj+1 − Tj ,j = 0, 1, . . . , n − 1. During the tenor of a swap, notional principal, denoted byVd , is fixed and denominated in domestic currency. At each payment date Tj

for j = 1, . . . , n, a domestic counterparty pays a foreign counterparty a do-mestic floating rate payment, δ(Ld (Tj−1, Tj−1) + K ), and receives from theforeign counterparty a foreign floating rate payment, δL f (Tj−1, Tj−1). Both



payments are made in domestic currency and K denotes a spread in basispoints.

To the domestic counterparty that pays the domestic floating rate paymentsand receives the foreign floating rate payment, the cash flow stream is given asfollows:

At time T1 : VdδL f (T0, T0) − Vdδ[Ld (T0, T0) + K1]

At time T2 : VdδL f (T1, T1) − Vdδ[Ld (T1, T1) + K1]...

......

At time Tn : VdδL f (Tn−1, Tn−1) − Vdδ[Ld (Tn−1, Tn−1) + K1].

The pricing formula of the CPS1 is presented in the following theorem. Theproof is given in Appendix B.

Theorem 1: The pricing formula of a CPS1. Within the MLMM framework,the price of a CPS1 at time T0 is given by

CPS1(T0) = Vdδ

n∑j=1

Bd (T0, Tj ){L f (T0, Tj−1)ηT0f (T0, Tj−1) − Ld (T0, Tj−1) − K1},

(28)

where

ηT0f (T0, Tj−1) = exp

(∫ Tj−1

T0

γ f (t, Tj−1) · (σ T0B f (t, Tj ) − σ

T0Bd (t, Tj ) − σX f (t)

)dt)

.

(29)

Our CPS1 pricing formula within the MLMM framework has some practicaladvantages over the CPS1 pricing formulas in Turnbull (1993), Wei (1994),and Chang et al. (2002). First, the model parameters, including the initialLIBOR rates, exchange rate and volatilities, can be easily extracted from marketdata, thereby making our CPS1 pricing formula more feasible and tractable inpractice. Second, the forward LIBOR rates within the MLMM framework arepositive, thus avoiding pricing errors due to possible negative rates. In addition,most actively traded interest rate derivatives can be priced within the MLMMframework so that efficient risk management can be carried out consistently.

According to Equation (1), Equation (28) can be rewritten as follows:

Vd

n∑j=1

{QA f (T0, Tj )

[B f (T0, Tj−1) − B f (T0, Tj )

]− δK1 Bd (T0, Tj )}



+ Vd(Bd (T0, Tn) − 1

), (30)

where

QA f (t, Tj ) = Bd (t, Tj )B f (t, Tj )

ηT0f (T0, Tj−1), t ∈ [T0, Tj ].

QA f (T0, Tj ) denotes the quanto adjustment arising from a time-Tj payoff thatis measured in the f th country’s currency and made in domestic currency attime T0. Therefore, QA f (T0, Tj )B f (T0, Tj−1) can be viewed as the price of aquanto bond (a foreign bond denominated in domestic currency).

CPS1s are developed to allow investors to capitalize on the expected dif-ferentials between the domestic and the foreign money market rates withoutincurring any direct exchange rate risk exposure. However, the exchange ratestill affects the CPS1 price. Specifically, the CPS1 price is affected by both theexchange rate volatility and the correlation between the exchange rate and theforeign LIBOR rates. Both affect the CPS1 via the term QA f (T0, ·).

Expression (30) provides an explicit construction of a hedging (replicating)portfolio for a CPS1. Because a CPS1 involves three different variates, includingthe domestic and the foreign interest rates and the exchange rate, it requiresthree financial instruments to construct a hedging portfolio, namely one do-mestic ZCB and two foreign ZCBs. To replicate a CPS1, an issuer must firstborrow Vd units of domestic currency and then use the fund to construct thefollowing hedging portfolio, whose jth component, j = 1, 2, . . . , n, is presentedas follows:

P (1)j = �

(1)1 j B f (T0, Tj−1) + �

(1)2 j B f (T0, Tj ) + �

(1)3 j Bd (T0, Tj ),

where

�(1)1 j = +Vd QA f (T0, Tj ),

�(1)2 j = −Vd QA f (T0, Tj ),

�(1)3 j =

{−VdδK1, j = 1, 2, . . . , n − 1,

+Vd (1 − δK1), j = n,

where “+” denotes a long position and “−” denotes a short position.



Pricing a newly issued CPS1 is equivalent to computing a fair spread K1,which is given as follows:

K1 =n∑

j=1

w j (T0)ηT0f (T0, Tj−1) −

n∑j=1

w j (T0)Ld (T0, Tj−1), (31)

where w j (T0) is defined in (14). By setting the spread of the CPS1 as K1,the CPS1 becomes a fair contract. The fair spread K1 bears a resemblance tothe difference between two weighted averages of the foreign and the domesticLIBOR rates. The spread K1 functions as a compensation for the counter-party that pays higher (domestic or foreign) interest rates and receives lower(foreign or domestic) interest rates. Besides, K1 is affected by the volatilityof the foreign exchange rate and the correlation between the term structuresof interest rates and the foreign exchange rate. Both affect K1 through theterm η

T0f (T0, Tj−1).

3.2. Pricing and Hedging CPS2

CPS2 contracts, which exchange the f th country’s floating rate payments forthe g th country’s floating rate payments, both denominated in domestic cur-rency, are also popular in the over-the-counter market. CPS2s allow investorsto capitalize the expected differences between two foreign money market rateswithout incurring any direct exchange rate risk. International companies mayuse CPS2s to transform their yields (assets) or costs (liabilities) from linkingwith the f th country’s interest rates to the g th country’s interest rates. Portfoliomanagers can employ CPS2s to enhance current yields, and foreign currencyborrowers can use them to lower their foreign interest rate costs, if they cantake an accurate view on the relative movement of two foreign-currency interestrates.

Consider the same tenor structure and notations used in Section 3.1. To adomestic counterparty that pays the f th country’s floating rate payments andreceives the g th country’s floating rate payments, the cash flow stream of aCPS2 is given as follows:

At time T1 : VdδLg (T0, T0) − Vdδ[L f (T0, T0) + K2]

At time T2 : VdδLg (T1, T1) − Vdδ[L f (T1, T1) + K2]...

......

At time Tn : VdδLg (Tn−1, Tn−1) − Vdδ[L f (Tn−1, Tn−1) + K2].



The pricing formula of a CPS2 is presented in Theorem 2, whose proof isgiven in Appendix B.

Theorem 2: Pricing formula of a CPS2. Within the MLMM framework, theprice of a CPS2 at time T0 is given by

CPS2(T0) = Vdδ

n∑j=1

Bd (T0, Tj ){

Lg (T0, Tj−1)ηT0g (T0, Tj−1)

− L f (T0, Tj−1)ηT0f (T0, Tj−1) − K2

}, (32)

where

ηT0g (T0, Tj−1) = exp

(∫ Tj−1

T0

γg (t, Tj−1) · (σ T0Bg (t, Tj ) − σ

T0Bd (t, Tj ) − σX g (t)

)dt)

.

(33)

Similar to (30), Equation (32) can be rewritten as follows:

Vd

n∑j=1

{QAg (T0, Tj )

[Bg (T0, Tj−1) − Bg (T0, Tj )

]− QA f (T0, Tj )

[B f (T0, Tj−1) − B f (T0, Tj )

]− δK2 Bd (T0, Tj )}, (34)

where

QAg (t, Tj ) = Bd (t, Tj )Bg (t, Tj )

ηT0g (T0, Tj−1), t ∈ [T0, Tj ].

QAg (T0, ·) denotes the quanto adjustment that occurs when a payoff is mea-sured in the g th country’s currency and made in domestic currency. Expression(34) provides an explicit way of constructing a hedging portfolio for a CPS2.Because a CPS2 involves five different terms (two foreign interest rates, onedomestic interest rate and two exchange rates), it needs five financial instru-ments to construct a hedging portfolio (one domestic ZCB, two f th country’sZCBs and two g th country’s ZCBs). To hedge a CPS2, an issuer must con-struct the following hedging portfolio, whose jth component, j = 1, 2, . . . , n,is represented as follows:

P (2)j = �

(2)1 j Bg (T0, Tj−1) + �

(2)2 j Bg (T0, Tj ) + �

(2)3 j B f (T0, Tj−1)

+ �(2)4 j B f (T0, Tj ) + �

(2)5 j Bd (T0, Tj ),



where

�(2)1 j = +Vd QAg (T0, Tj ),

�(2)2 j = −Vd QAg (T0, Tj ),

�(2)3 j = −Vd QA f (T0, Tj ),

�(2)4 j = +Vd QA f (T0, Tj ),

�(2)5 j = −Vd δK2.

At an inception date T0, pricing a newly issued CPS2 is to compute a fairspread K2 such that the CPS2 becomes a fair contract. The fair spread K2 canbe computed as follows:

K2 =n∑

j=1

w j (T0)Lg (T0, Tj−1)ηT0g (T0, Tj−1)

−n∑

j=1

w j (T0)L f (T0, Tj−1)ηT0f (T0, Tj−1), (35)

where w j (T0) is defined in (14). This bears a resemblance to the differencebetween two weighted averages of the g th country and the f th country LIBORrates. The spread K2 serves as a compensation for the counterparty that payshigher interest rates and receives lower interest rates. In addition, K2 is affectedby both the foreign exchange rate volatilities and the correlation between theterm structures of interest rates and the foreign exchange rates, both of whichare impounded in the terms ηT0

g (T0, Tj−1) and ηT0f (T0, Tj−1).

4. PRICING AND HEDGING CPSOs

Consider the tenor structure of an underlying CPS defined in Section 3. A CPSOis an option on a CPS, and its cash flow at maturity date T0 (0 < T0 < T1) isdefined as follows:

CPSOk(T0) = Max(CPSk(T0), 0), k = 1 and 2. (36)

The pricing formula of a CPSO1 with a zero spread (K1 = 0) is examined inSchlogl (2002), whereas the pricing formula of a frequently traded CPSO1

with nonzero-spread (K1 ≥ 0) has (to the best of our knowledge) not yet been



studied. In this section we intend to derive, respectively, the pricing formulasof a CPSO1 and a CPSO2 with nonzero spreads.

4.1. Pricing and Hedging a CPSO1 withSpread K1 ≥ 0

In this subsection, we derive the pricing formula of a CPSO1 with a spreadK1 ≥ 0. The resulting formula is more general and useful in practice than thecase with K1 = 0 derived in Schlogl (2002). According to (28), we can rewrite(36) as follows:

CPSO1(T0) = VdC(T0; T1, Tn)Max(Y1 − K1, 0), (37)

where

Y1 = Sf (T0, T0) − Sd (T0, T0), (38)

C(t; T1, Tn) = δ

n∑k=1

Bd (t, Tk), (39)

Sf (t, T0) =n∑

j=1

w j (t)L f (t, Tj−1)η0f (T0, Tj−1), (40)

Sd (t, T0) =n∑

j=1

w j (t)Ld (t, Tj−1), (41)

w j (t) = δBd (t, Tj )/C(t; T1, Tn), t ∈ [0, T0]. (42)

Sd (·, T0) and Sf (·, T0) represent a domestic and a quanto foreign forwardswap rate, respectively, both of which are lognormally distributed accordingto the approximation technique presented in Section 2.2. In light of (37) and(38), a CPSO1 can be regarded as a spread option on the difference betweenforeign and domestic swap rates. Based on the martingale pricing method andthe approximation technique presented in Section 2.3, the pricing formulaof a CPSO1 can be derived under the martingale probability measure QC

with respect to the numeraire C(t; T1, Tn). The resulting formula is given inTheorem 3. Its proof is presented in Appendix C.



Theorem 3: Pricing formula of a CPSO1 with Spread K1 ≥ 0. Based onthe martingale pricing method and the approximation technique presented inSection 2.3, the pricing formula of a CPSO1 with K1 ≥ 0 is computed withinthe MLMM framework as follows:

(a) If the approximate distribution for Y1 is a regular lognormal distribution,

CPSO1(0) = VdC(0; T1, Tn)((Sf (0, T0) − Sd (0, T0))N(a11) − K1 × N(a12)),

(43)

where a11 =ln(

Sf (0, T0) − Sd (0, T0)K1

)+ 1

2β2

1

β1and a12 = a11 − β1.

(b) If the approximate distribution for Y1 is a shifted-regular lognormaldistribution,

CPSO1(0) = VdC(0; T1, Tn)((Sf (0, T0) − Sd (0, T0) − γ1)N(b11)

− (K1 − γ1)N(b12)), (44)

where b11 =ln(

Sf (0, T0) − Sd (0, T0) − γ1

K1 − γ1

)+ 1

2β2

1

β1and b12 = b11 − β1.

(c) If the approximate distribution for Y1 is a negative lognormal distribution,

CPSO1(0) = VdC(0; T1, Tn)((Sf (0, T0) − Sd (0, T0))N(c11) − K1 × N(c12)),

(45)

where c11 = −ln(

Sf (0, T0) − Sd (0, T0)K1

)+ 1

2β2

1

β1and c12 = c11 + β1.

(d) If the approximate distribution for Y1 is a negative-shifted lognormaldistribution,

CPSO1(0) = VdC(0; T1, Tn)((Sf (0, T0) − Sd (0, T0) + γ1)N(d11)

− (K1 + γ1)N(d12)), (46)

where d11 = −ln(

Sf (0, T0) − Sd (0, T0) + γ1

K1 + γ1

)+ 1

2β2

1

β1and d12 = d11 + β1.



Using the system of equations in (25), the parameters, α1, β1, and γ1, canbe computed in terms of EQC

[Y1], EQC[Y 2

1 ], and EQC[Y 3

1 ], which are definedin (C4)–(C6), respectively, in Appendix C.

The CPSO1 pricing formulas shown in (43)–(46) resemble the swaptionpricing formula presented in Jamshidian (1997), except that the underlyingswap rate is replaced by a difference between foreign and domestic swap rates.Compared to the formula (K1 = 0) presented in Schlogl (2002), our pricingformulas are more capable of pricing a CPSO1 with spread K1 > 0, which ismore frequently traded than that with zero spread (K1 = 0) in the marketplace.In addition, the parameters required to compute the CPSO1 price can be easilyextracted from market data, thereby making the pricing formulas more feasibleand tractable for market practitioners.

Because Equations (43)–(46) are derived under the MLMM framework,they provides us a hedging strategy that is consistent with caps, floors, and otherexotic interest rate derivatives. For further examination regarding an explicithedging strategy, we take (43) as a demonstration and other related cases canbe performed accordingly. Equation (43) can be rewritten as follows:

CPSO1(0) = Vd

n∑j=1

{QA f (0, Tj )N(a11)[B f (0, Tj−1) − B f (0, Tj )]

− δK1 N(a12)Bd (0, Tj )}

+ Vd N(a11)(Bd (0, Tn) − Bd (0, T0)). (47)

If a financial institution issues a CPSO1, then Equation (47) reveals a clue forconstructing a hedging portfolio composed of various long and short positionsin Bd (0, ·) and B f (0, ·), whose jth component, j = 1, 2, . . . , n, is established by

P (3)j = �

(3)1 j B f (0, Tj−1) + �

(3)2 j B f (0, Tj ) + �

(3)3 j Bd (0, Tj ) + �

(3)4 j Bd (0, Tj−1),

where

�(3)1 j = +Vd QA f (0, Tj )N(a11),

�(3)2 j = −Vd QA f (0, Tj )N(a11),

�(3)3 j =

{−VdδK1N(a12), j = 1, 2, . . . , n − 1,

+Vd N(a11) − VdδK1N(a12), j = n,

�(3)4 j =

{−Vd N(a11), j = 1,

0, j = 2, 3, . . . , n.



4.2. Pricing and Hedging a CPSO2 withSpread K2 ≥ 0

In this subsection, we present the pricing formula of a CPSO2 with a spreadK2 ≥ 0. According to (32) and (36), we may rewrite the final payoff of a CPSO2

as follows:

CPSO2(T0) = VdC(T0; T1, Tn)Max(Y2 − K2, 0), (48)

where

Y2 = Sg (T0, T0) − Sf (T0, T0), (49)

Sg (t, T0) =n∑

j=1

w j (t)(Lg (t, Tj−1)ηT0g (T0, Tj−1)), t ∈ [0, T0]. (50)

Sg (·, T0) and Sf (·, T0) represent, respectively, the g th country’s and the f thcountry’s quanto forward swap rates, both of which are lognormally distributedaccording to the approximation technique described in Section 2.2. Therefore,a CPSO2 can be considered a spread option on the difference between twoquanto foreign swap rates. Based on the martingale pricing method and theapproximation technique presented in Section 2.3, the pricing formula of aCPSO2 can be derived, and is given in Theorem 4. Its proof is omitted for thesake of brevity.

Theorem 4: Pricing formula of a CPSO2 with spread K2 ≥ 0. Based on themartingale pricing method and the approximation technique presented in Sec-tion 2.3, the pricing formula of a CPSO2 with K2 ≥ 0 is computed within theMLMM framework as follows:

(a) If the approximate distribution for Y2 is a regular lognormal distribution,

CPSO2(0) = VdC(0; T1, Tn)((Sg (0, T0) − Sf (0, T0))N(a21) − K2 × N(a22)),

(51)

where a21 =ln(

Sg (0, T0) − Sf (0, T0)K2

)+ 1

2β2

2

β2and a22 = a21 − β2.



(b) If the approximate distribution for Y2 is a shifted-regular lognormaldistribution,

CPSO2(0) = VdC(0; T1, Tn)((Sg (0, T0) − Sf (0, T0) − γ2)N(b21)

− (K2 − γ2)N(b22)), (52)

where b21 =ln(

Sg (0, T0) − Sf (0, T0) − γ2

K2 − γ2

)+ 1

2β2

2

β2and b22 = b21 − β2.

(c) If the approximate distribution for Y2 is a negative lognormal distribution,

CPSO2(0) = VdC(0; T1, Tn)((Sg (0, T0) − Sf (0, T0))N(c21) − K2 × N(c22)),

(53)

where c21 = −ln(

Sg (0, T0) − Sf (0, T0)K2

)+ 1

2β2

2

β2and c22 = c21 + β2.

(d) If the approximate distribution for Y2 is a negative-shifted lognormaldistribution,

CPSO2(0) = VdC(0; T1, Tn)((Sg (0, T0) − Sf (0, T0) + γ2)N(d21)

− (K2 + γ2)N(d22)), (54)

where d21 = −ln(

Sg (0, T0) − Sf (0, T0) + γ2

K2 + γ2

)+ 1

2β2

2

β2and d22 = d21 + β2.

Using the system of equations in (25), the parameters, α2, β2, and γ2, can becomputed in terms of EQC

[Y2], EQC[Y 2

2 ], and EQC[Y 3

2 ], which are presentedin (C15)–(C17), respectively.

The parameters defined in Equations (51)–(54) can be easily extractedfrom market data. Due to the versatility of the MLMM, the CPSO2 pricingformulas lead us to a hedging strategy that is consistent with other relatedexotic interest rate derivatives. We take (52) as a demonstration and otherrelated cases can be inferred accordingly. Equation (52) can be rewritten asfollows:



CPSO2(0) = Vd

n∑j=1

{QAg (0, Tj )N(b21)

[Bg (0, Tj−1) − Bg (0, Tj )

]− QA f (0, Tj )N(b21)

[B f (0, Tj−1) − B f (0, Tj )

]− [

δγ2 N(b21) + δ(K2 − γ2)N(b22)]Bd (0, Tj )

}(55)

If a financial institution issues a CPSO2, then the below hedging portfolio canbe constructed, and its interpretation is similar to (47):

P (4)j = �

(4)1 j Bg (0, Tj−1) + �

(4)2 j Bg (0, Tj ) + �

(4)3 j B f (0, Tj−1)

+ �(4)4 j B f (0, Tj ) + �

(4)5 j Bd (0, Tj ),

where

�(4)1 j = +Vd QAg (0, Tj )N(b21),

�(4)2 j = −Vd QAg (0, Tj )N(b21),

�(4)3 j = −Vd QA f (0, Tj )N(b21),

�(4)4 j = +Vd QA f (0, Tj )N(b21),

�(4)5 j = −Vd

[δγ2N(b21) + δ(K2 − γ2)N(b22)

].

5. NUMERICAL STUDIES

In this section, we provide numerical examples for analyzing the accuracy of thepricing formulas developed here by comparing the results to those of a MonteCarlo (MC) simulation. We only present the results for Theorems 2 and 4,while those for Theorems 1 and 3 are similar, and are thus omitted to conservespace. The results show that our pricing formulas are robustly and sufficientlyaccurate. We introduce the implementation of the numerical studies in detailas follows.15

First, we consider CPS2s that pay Japanese LIBOR rates plus a spreadand receive U.K. LIBOR rates, both denominated in U.S. dollar. The CPS2sconsidered have maturities of three, five, and seven years, and the spread is150 basis points. Second, we consider CPSO2s on the five-year CPS2 thatpay Japanese LIBOR rates plus a spread and receive U.K. LIBOR rates, both

15For the parameter calibration of the MLMM, please refer to Wu and Chen (2007a, 2007b).



TABLE I

Numerical Examples of CPS2 in Basis Points

Three-year CPS2 Five-year CPS2 Seven-year CPS2

Date Theorem 2 MC Theorem 2 MC Theorem 2 MC

March 1, 2007 739.8 739.8 1,074.0 1,074.1 1,231.1 1,230.8June 1, 2007 815.6 815.6 1,203.2 1,203.2 1,397.5 1,397.3September 3, 2007 840.7 840.8 1,239.5 1,239.5 1,456.5 1,456.3December 3, 2007 693.6 693.5 1,036.9 1,036.8 1,233.8 1,232.9March 3, 2008 602.5 602.5 956.1 956.1 1,175.4 1,175.2June 2, 2008 726.5 726.5 1,068.2 1,069.1 1,234.1 1,234.5September 1, 2008 653.9 653.9 981.9 981.8 1,152.7 1,152.5December 1, 2008 144.1 144.1 305.5 306.1 423.0 423.2

aThe three-, five-, and seven-year CPS2s that pay Japanese LIBOR rates plus 150 basis points and receive U.K. LIBORrates, both denominated in U.S. dollars, are seasonally priced based on the market data in 2007 and 2008. The notionalvalue is assumed to be $1. The simulation is implemented based on 30,000 paths. MC stands for the result of the MonteCarlo simulation. The standard error of each simulation is about 2 basis points, on average.

denominated in U.S. dollar. The maturity periods of CPSO2s are one, three,and five years, and the spreads of their underlying CPS2 are 50, 100, 150, 200,and 250 basis points.

To examine the accuracy and robustness of the derived formula underdifferent market scenarios, both CPS2s and CPSO2s are priced using marketdata chosen quarterly in 2007 and 2008.16 The notional principal for each caseis assumed to be $1 and each simulation is implemented based on 30,000 paths.The results are listed in Tables I and II, and show that the approximate formulasare sufficiently accurate as compared to the MC simulation and robust underdifferent market scenarios over the past two years.

6. CONCLUSION

Within the framework of a multifactor MLMM, we derive the approximate pric-ing formulas for CPS1s, CPS2s, CPSO1s, and CPSO2s with nonzero spreads.As compared with traditional interest rate models, the LIBOR rates specifiedin the MLMM are market-observable and the model parameters can be eas-ily extracted from market data. Therefore, our pricing models are feasible andtractable for practical implementation.

Without our pricing formulas, CPSs and CPSOs are generally computedin practice using inefficient and time-consuming MC simulation. As shownin Section 5, our pricing formulas yield prices close to those computed inMC simulations, but in a fraction of the time. Our formulas provide market

16The market data used for numerical studies are available upon request from the authors.



TABLE II

Numerical Examples of CPSO2s in Basis Points

One-year CPSO2 Three-year CPSO2 Five-year CPSO2

Date Spread Theorem 4 MC Theorem 4 MC Theorem 4 MC

March 1, 2007 50 1,319.7 1,320.0 927.9 927.9 676.4 673.9100 1,112.6 1,112.6 742.8 743.6 521.9 523.0150 905.3 905.3 561.7 561.9 374.4 371.8200 698.8 698.0 384.9 385.4 238.3 236.0250 489.5 490.8 217.9 219.4 121.5 123.8

June 1, 2007 50 1,439.9 1,439.6 1,019.1 1,019.4 755.9 758.2100 1,235.5 1,235.4 837.7 837.8 602.6 605.1150 1,031.3 1,031.2 658.1 657.4 454.4 451.0200 828.3 827.8 479.8 479.8 314.7 311.2250 620.5 621.0 305.2 307.1 189.1 187.5

September 3, 2007 50 1,485.5 1,486.2 1,085.0 1,085.2 816.0 813.6100 1,278.1 1,278.4 900.8 900.1 660.3 658.9150 1,070.4 1,070.6 716.7 716.7 509.3 506.9200 862.6 862.8 534.9 536.5 365.8 367.1250 654.0 654.5 359.8 361.2 233.9 231.1

December 3, 2007 50 1,332.3 1,331.8 1,029.2 1,028.7 803.0 800.6100 1,116.5 1,116.3 834.2 833.8 639.7 638.8150 901.3 901.4 641.3 640.8 482.4 485.5200 685.7 686.4 449.3 451.3 334.3 331.8250 473.7 472.8 274.5 276.8 201.2 200.2

March 3, 2008 50 1,330.1 1,330.6 1,116.4 1,117.0 900.5 903.9100 1,112.2 1,111.9 917.0 915.4 733.5 730.9150 890.0 889.8 715.8 715.8 572.3 570.1200 667.5 668.4 521.2 523.3 419.7 416.4250 448.6 449.0 346.8 348.5 279.9 276.4

June 2, 2008 50 1,356.2 1,355.8 993.5 990.4 762.0 765.1100 1,138.9 1,138.3 796.2 794.9 602.4 604.4150 923.7 922.8 602.2 601.6 450.8 452.2200 705.2 706.4 416.0 416.9 311.3 314.5250 489.9 490.7 250.9 253.6 189.8 186.3

September 1, 2008 50 1,302.8 1,301.6 989.4 989.4 765.8 762.9100 1,085.7 1,084.3 794.5 792.8 605.6 602.2150 867.0 867.9 597.6 598.2 453.8 450.4200 651.8 651.5 412.4 414.3 314.3 311.2250 437.7 437.8 256.4 257.1 193.0 191.1

December 1, 2008 50 884.5 883.8 857.6 856.1 800.9 803.2100 662.6 661.1 653.3 651.2 634.6 631.1150 461.9 460.7 478.0 476.0 488.8 485.5200 304.3 305.6 336.8 338.4 366.4 363.3250 199.2 198.2 236.4 238.6 267.7 264.4

The one-, two-, and three-year CPSO2s on a CPS2 that pay Japanese LIBOR rates plus a spread and receive U.K.LIBOR rates, both denominated in U.S. dollars, are quarterly priced based on the market data in 2007 and 2008. Thespreads considered are 50, 100, 150, 200, and 250 basis points. The notional value is assumed to be $1. The simulationis implemented based on 30,000 paths. MC stands for the result of the Monte Carlo simulation. The standard error of eachsimulation is about 2 basis points, on average.



practitioners with a new, efficient and time-saving approach for offering al-most instantly quoted prices to clients and the daily marking-to-market tradingbooks, and facilitating efficient risk management of trading positions. Giventhese advantages, the presented formulas are worth recommending to marketpractitioners.

APPENDIX A: DERIVATION OF FORWARDSWAP RATE VOLATILITIES

This appendix intends to derive forward swap rate volatilities, σ 0Sd (t, T0),

σ 0Sf (t, T0) and σ 0

Sg (t, T0), within the MLMM framework. We first derive thedomestic swap rate volatility, σ 0

Sd (t, T0). Taking logarithm to (13), we have

ln(Sd (t, T0)

) = ln

(n∑

i=1

Bd (t, Ti )Ld (t, Ti−1)

)− ln

(n∑

i=1

Bd (t, Ti )

),

and

∂ ln(Sd (t, T0)

)∂ Bd (t, Tk)

= Ld (t, Tk−1)n∑

i=1Bd (t, Ti )Ld (t, Ti−1)

− 1n∑

i=1Bd (t, Ti )

,

∂ ln(Sd (t)

)∂Ld (t, Tk−1)

= Bd (t, Tk)n∑


, for k = 1, 2, . . . , n.

By employing Ito’s lemma, σSd (t, T0) is derived as follows:

σSd (t, T0) =n∑

k=1

[∂ ln Sd (t, T0)∂ Bd (t, Tk)

(−σ 0Bd (t, Tk)Bd (t, Tk)

)+ ∂ ln Sd (t, T0)

∂Ld (t, Tk−1)γd (t, Tk−1)Ld (t, Tk−1)

]

=n∑

k=1

ζk(t)γd (t, Tk−1) +n∑

k=1

φk(t)σ 0Bd (t, Tk) −

n∑k=1

ζk(t)σ 0Bd (t, tk), (A1)

where

φk(t) = Bd (t, Tk)n∑

i=1Bd (t, Ti )

, (A2)



ζk(t) = Bd (t, Tk)Ld (t, Tk−1)n∑


. (A3)

Both φk(t) and ζk(t) look like weights, and thus σSd (t, T0) can be regardedas a composite of three weighted averages. Because the weights φk(t) and ζk(t),for k = 1, 2, . . . , n, are stochastic, so is σSd (t, T0). Although the variability ofφk(t) and ζk(t) are small (to be shown later), we can freeze the calendar time ofboth Bd (t, ·) and Ld (t, ·) in (A2) and (A3) at its initial time 0, and the resultingprocesses become deterministic and are defined by

φ0k (0) = Bd (0, Tk)

n∑i=1

Bd (0, Ti ), (A4)

ζ 0k (0) = Bd (0, Tk)Ld (0, Tk−1)

n∑i=1

Bd (0, Ti )Ld (0, Ti−1). (A5)

By replacing φk(t) and ζk(t), respectively, by φ0k (0) and ζ 0

k (0), σ 0Sd (t, T0)

becomes deterministic and given by

σ 0Sd (t, T0) =

n∑k=1

ζ 0k (0)γd (t, Tk−1) +

n∑k=1

φ0k (0)σ 0

Bd(t, Tk) −

n∑k=1

ζ 0k (0)σ 0

Bd(t, Tk).

(A6)

This approximation is widely employed in practice to price swap-type in-struments within the MLMM framework and examined to be robustly accuratein the numerical examples given in Section 5. For the rest of this appendix,we provide the main theoretical foundation to show that the variability of theweights, φk(t) and ζk(t), is small, and thus it is reasonable to freeze these pro-cesses at time 0. For the sake of brevity, we only show the weight ζk(t) for ademonstration. Rewrite ζk(t) as follows:

ζk(t) = Y (t, Tk)n∑

i=1Y (t, Ti )

,

where Y (t, Ti ) = Bd (t, Ti )Ld (t, Ti−1) for i = 1, 2, . . . , n.



Based on Ito’s lemma, the dynamics of ζk(t) can be derived and its volatilityterm, σζk (t), is presented as follows:

σζk (t) =n∑

i=1

⎡⎢⎢⎢⎢⎢⎣1

n∑j=1

Y (t, Tj )

(σBd (t, Ti ) − γd (t, Ti−1)

)⎤⎥⎥⎥⎥⎥⎦

+ 1Y (t, Tk)

(γd (t, Tk−1) − σBd (t, Tk)

)= 1

Y (t, Tk)

[n∑

i=1

ζk(t)(σBd (t, Ti ) − γd (t, Ti−1)

)+ (γd (t, Tk−1) − σBd (t, Tk)

)].

Because Y (t, Ti ) ≈ Y (t, Tk), σζk (t) can be approximately rewritten as follows:

σζk (t) = 1Y (t, Tk)

×

⎡⎢⎢⎢⎢⎢⎣n∑

i=1

ζi (t)σBd (t, Ti ) − σBd (t, Tk)︸︷︷︸(A7)

+ γd (t, Tk−1) −n∑

i=1

ζi (t)γd (t, Ti−1)︸︷︷︸(A8)

⎤⎥⎥⎥⎥⎥⎦ .

Because the plus and minus terms in (A7) and (A8) are roughly equal, thevolatility of ζk(t) tends to be small. Therefore, this fact justifies that ζ 0

k (0) is agood approximation to ζk(t). Similarly, we can also approximate φk(t) by φ0

k (0).Similarly, the swap volatilities of the f th and g th country’s swap rates,

σ 0Sf (t, T0) and σ 0

Sg (t, T0), can be derived as follows:

σ 0Sf (t, T0) =

n∑k=1

θ0k (0)γ f (t, Tk−1) +

n∑k=1

φ0k (0)σ 0

Bd (t, Tk) −n∑

k=1

θ0k (0)σ 0

Bd (t, Tk),

(A9)

σ 0Sg (t, T0) =

n∑k=1

ψ0k (0)γg (t, Tk−1) +

n∑k=1

φ0k (0)σ 0

Bd (t, Tk) −n∑

k=1

ψ0k (0)σ 0

Bd (t, Tk),

(A10)



where

θ0k (0) = Bd (0, Tk)L f (0, Tk−1)η0

f (T0, Tk−1)n∑

i=1Bd (0, Ti )L f (0, Ti−1)η0

f (T0, Ti−1), (A11)

ψ0k (0) = Bd (0, Tk)Lg (0, Tk−1)η0

g (T0, Tk−1)n∑

i=1Bd (0, Ti )Lg (0, Ti−1)η0

g (T0, Ti−1), (A12)

where η0f (T0, Ti−1) and η0

g (T0, Ti−1) are defined in (B3).

APPENDIX B: PROOF OF THEOREMS 1AND 2

Based on the changing-measure mechanism in Proposition 4, the dynamicsof Ld (·, Tj−1) and Lk(·, Tj−1), k ∈ { f, g }, under QTj are given as follows:

d Ld (t, Tj−1)Ld (t, Tj−1)

= γd (t, Tj−1) · dW(t), (B1)

d Lk(t, Tj−1)Lk(t, Tj−1)

= γk(t, Tj−1) · (σ T0Bk(t, Tj ) − σ

T0Bd (t, Tj ) − σX k(t)

)dt

+ γk(t, Tj−1) · dW(t), (B2)

and

Ld (Tj−1, Tj−1) = Ld (T0, Tj−1) exp(

−12

∫ Tj−1

T0

‖γd (t, Tj−1)‖2dt

+∫ Tj−1

T0

γd (t, Tj−1) · dW(t))

,

Lk(Tj−1, Tj−1) = Lk(T0, Tj−1)ηT0k (T0, Tj−1) exp

(−1

2

∫ Tj−1

T0

‖γd (t, Tj−1)‖2dt

+∫ Tj−1

T0

γd (t, Tj−1) · dW(t))

,

where

ηT0k (T0, Tj−1)=exp

(∫ Tj−1

T0

γk(t, Tj−1) · (σ T0Bk(t, Tj )− σ

T0Bd (t, Tj )−σX k(t)

)dt)

.

(B3)



Based on the martingale pricing method, the price of a CPS1 can be com-puted as follows:

CPS1(T0) = δ

n∑j=1

Bd (T0, Tj )EQTj [L f (Tj−1, Tj−1) − Ld (Tj−1, Tj−1) − K1|FT0

]= δ

n∑j=1

Bd (T0, Tj ){EQTj [

L f (Tj−1, Tj−1)|FT0

]− Ld (T0, Tj−1) − K1

}

= δ

n∑j=1

Bd (T0, Tj ){L f (T0, Tj−1)ηT0f (T0, Tj−1) − Ld (T0, Tj−1) − K1}.

Similarly, the price of a CPS2 can be computed as follows:

CPS2(T0) = δ

n∑j=1

Bd (T0, Tj ){

Lg (T0, Tj−1)ηT0g (T0, Tj−1)

− L f (T0, Tj−1)ηT0f (T0, Tj−1) − K2

}.

APPENDIX C: PROOF OF THEOREM 3

Based on the martingale pricing method, the price of a CPSO1 at time 0 iscomputed as follows:

CPSO1(0)

= Bd (0, T0)EQT0{Max

(CPS1(T0), 0

)∣∣∣F0

}= Vd Bd (0, T0)EQT0

{C(T0; T1, Tn)Max

(Sf (T0, T0) − Sd (T0, T0) − K1, 0

)∣∣∣F0

}= VdC(0; T1, Tn)EQT0

{dQC

dQT0Max

(Sf (T0, T0) − Sd (T0, T0) − K1, 0

)∣∣∣F0

}= VdC(0; T1, Tn)EQC

{Max

(Sf (T0, T0) − Sd (T0, T0) − K1, 0

)∣∣∣F0

}, (C1)

where

dQC

dQT0= C(T0; T1, Tn)/C(0; T1, Tn)

Bd (T0, T0)/Bd (0, T0)



denotes the Randon–Nikodym derivative that defines a martingale probabilitymeasure with respect to the numeraire C(t; T1, Tn) which is given by

QC(A) =∫

ω∈A

dQC

dQT0dQT0, ∀ A ∈ F .

According to (40) and (41), both Sd (t, T0) and Sf (t, T0) can be regardedas tradable assets expressed in C(t; T1, Tn) units, and thus must evolve as mar-tingale processes under the measure QC. Based on the approximate techniquepresented in Section 2.2, the dynamics of Sd (t, T0) and Sf (t, T0) should havethe following form under the measure QC:

d Sk(t, T0)Sk(t, T0)

= σ 0Sk(t) · dW(t),

and

Sk(T0, T0)=Sk(0, T0) exp(−1

2

∫ T0

0‖σ 0

Sk(t)‖2dt +∫ T0

0σ 0

Sk(t) · dW(t))

,

k ∈ {d, f }, (C2)

where σ 0Sd (t) and σ 0

Sf (t) are defined in (A6) and (A9), respectively. Therefore,Sk(T0, T0) has a lognormal distribution with ln Sk(T0, T0) ∼ N(Mk, V 2

k ), where

Mk = ln Sk(0, T0) − 12

∫ T0

0‖σ 0

Sk(t)‖2dt, (C3a)

V 2k =

∫ T0

0‖σ 0

Sk(t)‖2dt. (C3b)

To compute (C1), we can use the approximation technique presented inSection 2.3. Let Y1 = Sf (T0, T0) − Sd (T0, T0), and its first three moments arecomputed as follows:

EQC[Y1] = Sf (0, T0) − Sd (0, T0), (C4)

EQC[Y 2

1 ] = Sf (0, T0)2 exp(∫ T0

0‖σ 0

Sf (t)‖2dt)

+ Sd (0, T0)2 exp(∫ T0

0‖σ 0

Sd (t)‖2dt)

− 2Sf (0, T0)Sd (0, T0) exp(∫ T0

0σ 0

Sd (t) · σ 0Sf (t)dt

), (C5)



EQC[Y 3

1 ] = Sf (0, T0)3 exp(

3∫ T0

0‖σ 0

Sf (t)‖2dt)

− Sd (0, T0)3 exp(

3∫ T0

0‖σ 0

Sd (t)‖2dt)

− 3Sf (0, T0)2Sd (0, T0) exp(∫ T0

0‖σ 0

Sf (t)‖2 + 2σ 0Sd (t) · σ 0

Sf (t)dt)

+ 3Sf (0, T0)Sd (0, T0)2 exp(∫ T0

0‖σ 0

Sd (t)‖2 + 2σ 0Sd (t) · σ 0

Sf (t)dt)

.

(C6)

Solving the system of equations in (25) for α1, β1, and γ1 and computingthe skewness of Y1 in (26), we can find a generalized lognormal distribution X1

to approximate Y1, and the price of a CPSO1 can be computed for each type ofX1 as follows:

CPSO1(0) = VdC(0; T1, Tn)EQC{max(X1 − K1, 0)|F0} (C7)

= VdC(0; T1, Tn)(EQC

[X1 I{X1≥K1}] − K1EQC[I{X1≥K1}]

), (C8)

where I{·} is an indicator function and X1 follows one of the four types ofgeneralized lognormal distributions. We first consider that X1 follows a regularlognormal distribution.

Assume that X1 has a regular lognormal distribution, namely ln(X1) ∼N(α1, β

21 ), where α1 and β2

1 can be computed via (25) in terms of EQC[Y1] and

EQC[Y 2

1 ] as follows:

α1 = ln(EQC

[Y1])

− 12β2

1 ,

β21 = ln

(EQC

[Y 21 ])

− 2 ln(EQC

[Y1])

.

The second expectation of (C8) is computed as follows:

EQC[I{X1≥K1}

] = P(

ln(X1) ≥ ln(K1))

= N

⎛⎜⎝ ln(EQC

[Y1]/K1

)− 1

2β2

1

β1

⎞⎟⎠ . (C9)



The first expectation of (C8) is computed as follows:

EQC[

X1 I{X1≥K1}]

= EQC[eln(X1) I{ln(X1)≥ln(K1)}

]= EQC

[eβ1 Z+α1 I{β1 Z+α1≥ln(K1)}

](where Z ∼ N(0, 1)),

= eα1

∫ ∞

ln(K1)−α1β1

eβ1z 1√2π

e−12 z2

dz

= eα1+ 12 β2

1

∫ ∞

ln(K1)−α1β1

1√2π

e−12 (z−β1)2

dz

= eα1+ 12 β2

1

∫ ∞ln(K1)−α1−β2

1β1

1√2π

e−12 u2

du (let U = Z − β1)

= eα1+ 12 β2

1 N

(α1 + β2

1 − ln(K1)

β1

)

= EQC[Y1]N

⎛⎜⎝ ln(EQC

[Y1]/K1

)+ 1

2β2

1

β1

⎞⎟⎠ . (C10)

Equations (C8)–(C10) lead to (43).Second, consider a shifted lognormal distribution X1 = V + γ1, where γ1

is a constant and V is a regular lognormal distribution with ln(V ) ∼N(α1, β21 ).

Equation (C7) can be rewritten as follows:

CPSO1(0) = VdC(0; T1, Tn)EQC[(X1 − K1)+

],

= VdC(0; T1, Tn)EQC[(V − (K1 − γ1))+

]. (C11)

By replacing K1 in both (C9) and (C10) by K1 + γ1, the CPSO1 pricing formulacan be derived and is given in (44).

Third, let X1 be a negative lognormal distribution, namely X1 = −V , whereV is a regular lognormal distribution. Equation (C7) can be rewritten as follows:

CPSO1(0) = VdδC(0; T1, Tn)EQC[(− V − K1)+]

, (C12)

= VdδC(0; T1, Tn)(

− EQC[V I{V≤−K1}] − K1EQC

[I{V≤−K1}]). (C13)



Equation (C13) can be computed similarly to the first and second expectationin (C8), which leads to (45).

Fourth, let X1 be a negative-shifted lognormal distribution, namely X1 =−(V + γ1), where V is a regular lognormal distribution and γ1 is a constant.Equation (C7) can be rewritten as follows:

CPSO1(0) = VdC(0; T1, Tn)EQT [(−V − (K1 + γ ))+

], (C14)

By replacing K1 in (C12) by K1 + γ1, the CPSO1 pricing formula can be derivedand is given in (46).

The pricing formulas of a CPSO2 can be derived similarly and the derivationis omitted for the sake of brevity. We only provide the formulas for EQC

[Y2],EQC

[Y 22 ], and EQC

[Y 32 ] as follows:

EQC[Y2] = Sg (0, T0) − Sf (0, T0), (C15)

EQC[Y 2

2 ] = Sg (0, T0)2 exp(∫ T0

0‖σ 0

Sg (t)‖2dt)

+ Sf (0, T0)2 exp(∫ T0

0‖σ 0

Sf (t)‖2dt)

− 2Sg (0, T0)Sf (0, T0) exp(∫ T0

0σ 0

Sg (t) · σ 0Sf (t)dt

), (C16)

EQC[Y 3

2 ] = Sg (0, T0)3 exp(

3∫ T0

0‖σ 0

Sg (t)‖2dt)

− Sf (0, T0)3 exp(

3∫ T0

0‖σ 0

Sf (t)‖2dt)

− 3Sg (0, T0)2Sf (0, T0) exp(∫ T0

0‖σ 0

Sg (t)‖2 + 2σ 0Sf (t) · σ 0

Sg (t)dt)

+ 3Sg (0, T0)Sf (0, T0)2 exp(∫ T0

0‖σ 0

Sf (t)‖2 + 2σ 0Sf (t) · σ 0

Sg (t)dt)

.

(C17)

APPENDIX D: EXAMINATION OF THEACCURACY OF THE TWOAPPROXIMATIONS

Because the resulting pricing formulas are derived based on two approxi-mations for the LIBOR rate and the swap rate dynamics, this appendix intendsto examine their accuracy. These two approximations are extensively used inpractice and can also be found in Brace et al. (1997), Schlogl (2002), Wu and



Chen (2008, 2009b), and Brigo and Mercurio (2006). To conserve space inpresentation, we only examine the domestic LIBOR rates and swap rates, andthe results of the foreign rates are similar.

In Proposition 1, the domestic LIBOR rate under Q is derived and specifiedby the following process:


= γd (t, T) · σBd (t, T + δ)dt + γd (t, T) · dW(t), (D1)

where σBd (t, T + δ) is defined in (7). To make (D1) solvable, we approximate(D1) by the following process:


= γd (t, T) · σ 0Bd (t, T + δ)dt + γd (t, T) · dW(t), (D2)

where σ 0Bd (t, T + δ) is defined in (8).

To examine the accuracy of the approximation, we employ Monte Carlo(MC) simulation to simulate 30,000 samples of the LIBOR rates for each of twocases: the first case is implemented according to (D1), denoted by L(i)

d1(T, T), andthe second is based on (D2), denoted by L(i)

d2(T, T), where the index i stands forthe ith sample. The simulated LIBOR rates have ten different times to maturity,which are generated under eight different market scenarios, namely, using themarket data on March 1, 2007, June 1, 2007, September 3, 2007, December3, 2007, March 3, 2008, June 2, 2008, September 1, 2008, and December 1,2008.17

We compute the root-mean-square relative error (RMSRE) by

RMSRE =

√√√√√ n∑i=1

e2i

n,

where ei = (L(i)d1(T, T) − L(i)

d2(T, T))/L(i)d1(T, T), and the result is presented in

Table III. Based on the simulation result, the approximation technique areaccurate and robust over the past two years. Even on the most volatile dateof the LIBOR rates, December 1, 2008, the largest percentage relative errorof L(10, 10) associated with the longer maturity date 10 is about 4%, whichshould still be acceptable in practice.

17The market data can be obtained upon request from the authors, and the calibration procedure can referto Wu and Chen (2011).



TABLE III

The Root-mean-square Relative Error Between Actual and Approximate LIBOR Rates

RMSRE

March1, 2007

(%)

June 1,2007(%)

September3, 2007

(%)

December3, 2007

(%)

March3, 2008

(%)

June 2,2008(%)

September1, 2008

(%)

December1, 2008

(%)

1 0.011 0.012 0.033 0.052 0.112 0.092 0.133 0.8492 0.027 0.029 0.071 0.112 0.242 0.195 0.294 1.3903 0.055 0.058 0.124 0.194 0.408 0.323 0.484 1.8924 0.096 0.101 0.195 0.298 0.602 0.474 0.696 2.3195 0.153 0.161 0.285 0.426 0.826 0.652 0.935 2.7126 0.230 0.241 0.399 0.582 1.084 0.863 1.210 3.1107 0.327 0.343 0.534 0.760 1.364 1.095 1.503 3.4538 0.449 0.469 0.693 0.965 1.676 1.353 1.824 3.7719 0.593 0.619 0.876 1.194 2.017 1.647 2.174 4.00110 0.767 0.802 1.088 1.451 2.386 1.973 2.551 4.013

aThe accuracy of the LIBOR rate approximation is examined in this table. The simulated LIBOR rates have ten differenttimes to maturity, and they are generated under eight different market scenarios, namely using the market data on March1, 2007, June 1, 2007, September 3, 2007, December 3, 2007, March 3, 2008, June 2, 2008, September 1, 2008, andDecember 1, 2008. Each result is based on 30,000 sample paths.

Next, we test the accuracy of the swap rate approximation. We have shownthat the swap rate dynamics follows the following process:


= σSd (t, T0) · dW(t), (D3)

where σSd (t, T0), defined in (A1). To make Sd (T0, T0) computable, we approxi-mate (D3) by the following process:


= σ 0Sd (t, T0) · dW(t), (D4)

where σ 0Sd (t, T0) is defined in (18).

To examine the accuracy of the approximation, we employ MC simulationto simulate 30,000 samples of the five-year swap rates for each of the two cases:the first case is implemented according to (D3), denoted by S(i)

d1(T, T), and thesecond is based on (D4), denoted by S(i)

d2(T, T), where the index i stands forthe ith sample and T = 5. The simulated five-year swap rates are generated byusing the same market data as in Table III.


The Root-Mean-Square Relative Error Between Actualand Approximate LIBOR RatesThe Root-Mean-Square Relative Error Between Actualand Approximate LIBOR Rates


TABLE IV

The Root-mean-square Relative Error Between Actual and Approximate Swap Rates

RMSRE

March1, 2007

(%)

June 1,2007(%)

September3, 2007

(%)

December3, 2007

(%)

March3, 2008

(%)

June 2,2008(%)

September1, 2008

(%)

December1, 2008

(%)

1 0.192 0.189 0.243 0.293 0.453 0.377 0.461 0.7382 0.188 0.183 0.248 0.310 0.430 0.387 0.466 0.7303 0.184 0.184 0.239 0.295 0.471 0.393 0.464 0.7694 0.187 0.186 0.248 0.298 0.452 0.383 0.460 0.7825 0.193 0.188 0.245 0.308 0.448 0.387 0.468 0.7446 0.187 0.186 0.242 0.313 0.450 0.387 0.454 0.7457 0.186 0.189 0.247 0.299 0.452 0.396 0.452 0.7528 0.189 0.189 0.242 0.318 0.450 0.384 0.459 0.7689 0.191 0.190 0.252 0.301 0.442 0.386 0.472 0.75110 0.186 0.186 0.243 0.299 0.459 0.386 0.462 0.750

aThe accuracy of the swap rate approximation is examined in this table. The simulated five-year swap rates are generatedunder eight different market scenarios, namely using the market data on March 1, 2007, June 1, 2007, September 3,2007, December 3, 2007, March 3, 2008, June 2, 2008, September 1, 2008, and December 1, 2008. Each result is basedon 30,000 sample paths.

We compute the RMSRE by

RMSRE =

√√√√√ n∑i=1

ε2i

n,

where εi = (S(i)d1(T, T) − S(i)

d2(T, T))/S(i)d1(T, T), and the result is presented in

Table IV. According to the simulation result, the approximation technique areaccurate and robust over the past two years. In addition, because the swap ratesare the weighted averages of LIBOR rates, the average effect makes the swaprate approximation more stable among different lengths of times to maturity.

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