A Multifactoral Cross-Currency LIBOR Market Model with a FX Volatility Skew”

“A multifactoral cross-currency LIBOR market model with a FX volatility skew”

AUTHORS Wolfgang Benner Lyudmil Zyapkov

ARTICLE INFO Wolfgang Benner and Lyudmil Zyapkov (2008). A multifactoral cross-currency LIBOR market model with a FX volatility skew. Banks and Bank Systems, 3(4)

RELEASED ON Friday, 06 February 2009

JOURNAL "Banks and Bank Systems"

FOUNDER LLC “Consulting Publishing Company “Business Perspectives”

NUMBER OF REFERENCES NUMBER OF FIGURES NUMBER OF TABLES 0 0 0

businessperspectives.org Banks and Bank Systems, Volume 3, Issue 4, 2008

Wolfgang Benner (Germany), Lyudmil Zyapkov (Germany) A multifactoral Cross-Currency LIBOR Market Model with an FX volatility skew Abstract Based on LIBOR Market Models, we develop a rigorous pricing framework for cross-currency exotic interest rate instruments under a uniform probability measure and in a multifactoral environment that accounts for the empirically observed foreign exchange skew. The model resorts to a stochastic volatility approach with volatility dynamics following a square-root process and is designed to be flexible enough to allow for the incorporation of as much market information as possible. Using the Fourier transform, we produce closed-form valuation formulas for FX options by obtain- ing an explicit expression for the characteristic function, though in a mildly approximate fashion for the sake of analytical tractability. The main focus is placed on FX markets, in terms of which the calibration of model parameters can be performed on a wide range of FX options expiries and strikes. Keywords: Cross-Currency LIBOR Market Model, stochastic volatility, Fourier transform, foreign exchange skew, forward probability measure. JEL Classification: G13, E43, F31. Introductioni natural to resort to an extension of the lognormal- type dynamics of the forward forex rate that is based The origins of the proposed Cross-Currency LIBOR on stochastic volatility. Market Model (CCLMM) can be traced back to the need of developing a unified pricing framework for This paper proposes an integrated CCLMM under a a number of cross-currency exotics. Initially con- uniform pricing measure in a multifactoral environ- fronted with a hybrid structure that required the ment that allows for as much flexibility as possible simultaneous description of highly correlated inter- in calibrating model parameters to market data. The est rate markets, foreign exchange (forex) rate and pricing measure will be uniform as it will be appli- hazard rate dynamics, the present work gained im- cable to (i) simple financial instruments that are petus from the necessity to determine the value of a affected only by the domestic interest rate market or cross-currency swap, which was to serve as an un- the foreign interest rate market but not both, as well derlying of various derivative products, at an arbi- as to (ii) complex financial instruments that are af- trary future date. Typically, FX options exhibit a fected by both the domestic and foreign interest rate significant volatility skew that manifests itself in the markets linked by the forex market. With the inten- at-the-money (ATM) implied volatility’s underesti- tion to derive valuation formulas, we deflate all mation of in-the-money (ITM) option prices and stochastic price processes using a single numeraire overestimation of out-of-the-money (OTM) ones, regardless of the market the price process belongs to whereby the ATM implied volatility has been ob- or is affected by, thus ensuring pricing consistency tained by inversion of an ATM option pricing for- between the markets and allowing the evaluation of mula based on a lognormal stochastic evolution of complex financial structures within a LIBOR Mar- the forward forex rate. Moreover, it seems impossi- ket Model setup. The model design must be capable ble for the most cross-currency derivatives to of reflecting market implied volatilities and exoge- choose a particular strike, or a specific maturity of nously assigned correlation structures between the an FX option since they usually represent long- interest rates and FX dynamics. However, the main dated exotic structures that either cannot be decom- focus will be placed on the calibration to FX options posed into plain-vanilla FX options, or at best de- for various maturities and strikes simultaneously, pend on FX options for a wide range of strikes and while retaining one-factor assumptions for both maturities. Aggravating matters even further, exotic interest rate markets. Though somewhat restrictive cross-currency interest rate derivatives are rarely at first glance, this choice keeps the number of structured to depend on ATM volatilities. They are model parameters to be calibrated low affecting usually designed with strikes far away from at-the- high speed of calibration without sacrificing accu- money. Hence, the volatility function needs to be racy of valuation. In addition, the model developed calibrated to prices of FX options across all avail- here can easily be used as a stepping stone to incor- able maturities and strikes as suggested by Piterbarg porating interest rate volatility smiles on a multi- (2006). He asserts that a model similar to that of currency basis, which remains a subject of future Schloegl (2002) based on LIBOR Market Models, research. The various extensions of the forward yet accounting for forex smiles in a proper manner, LIBOR models could serve as a starting point of this and a good FX option calibration algorithm still effort. One possibility would be the postulation of awaits development. For this purpose, it appears alternate interest rate dynamics such as local volatil- 73 Banks and Bank Systems, Volume 3, Issue 4, 2008 ity type of extensions based on constant elasticity of measure variance (CEV) processes pioneered by Andersen et 1.1. Definitions. Given a filtered probability space al. (2000), or the adoption of a displaced-diffusion tN approach as elaborated, for example, by Benner et :,,^`Ft P ttt[,0 N ] al. (2007). Jump-diffusions are treated in Glasser- satisfying the usual condi- d 1,2,3,4 man et al. (2003a, 2003b), but have not gained much Wt()d () ^ tN ` acceptance due to their producing of non-time- tions, let the tuple ttt[,0 N ] denote a d- homogeneous volatility term structures and some dimensional Brownian motion that introduces the other calibration complications. Finally as the mod- source of uncertainty to the correlated dynamics of elling technique with probably the greatest explana- the foreign exchange market, both the domestic and tory power, the inclusion of stochastic volatility in foreign interest rate markets and to the mean- the LMM is considered by three main research reverting square root process of the common volatil- streams: Andersen et al. (2005) and Andersen et al. Vt() (2002), on which Piterbarg (2003) builds using the ity shared equally by forward forex rates of method of calibration by parameter averaging as any maturity. In addition, we assume that the filtra- described in Piterbarg (2005a, 2005b) and providing ^Ft ` tN tion ttt[,0 N ] is the usual P -augmentation of formulas that relate market and model skews and d 1,2,3 volatilities directly without the need to develop Wt()d () ^ tN ` closed-form solutions of European option valuation the natural filtration generated by ttt[,0 N ], problems. Joshi et al. (2003) choose a distinctly so that it is right-continuous and complete. A com- different way of analyzing the evolution of the 0 tt ... t mon set of LIBOR maturities 01 N is swaption volatility matrix over time by assuming a defined for both the domestic and the foreign cur- specific time-homogeneous instantaneous volatility rency markets. May the following symbols function whose parameters are allowed to vary sto- f chastically. LstT(,, ) and LstT(,, ) indicate domestic and s The paper is organized as follows. Section 1 devel- foreign (forward) LIBORs as of time , starting at ops a unified pricing framework under a uniform time t and maturing at time T respectively with domestic forward measure. It determines both the stTdd . For the sake of simplicity, dynamics of the domestic/foreign LIBORs and the Lt() Lttt (, , ) Lff()tLttt (, , ) forward forex rate with stochastic volatility. The tii i1 and tii i1 are reason why forward forex rates are being modelled designated to represent one-year forward rates start- directly is that, by definition, they represent price ti ttii1 1 processes of tradable securities as opposed to spot ing at and ending at with Lttt(,, ) Ltttf (,, ) forex rates. In fact, each forward forex rate follows iN 0,..., 1. Moreover, i and i a martingale under its natural forward measure, so Qt() that its dynamics are, under such a measure, fully specify spot rates, whereas stands for the spot specified by its volatility process. Section 2 derives forex rate as units of domestic currency per unit of an FX option pricing formula with stochastic forex foreign currency. The forward forex rate, at which volatilities based on mildly approximate assumptions investors can buy or sell foreign currency for set- in order to preserve the analytical tractability of the tlement at a future date, is determined by model. It also offers an elaborate overview of the FX() t Bf (, t t ) Q () t B (, t t ) iii. implemented calibration procedure. The last section concludes with a brief summary of the main results In terms of zero-coupon bonds, B(,tti ) and and some suggestions for future work. For the sake of f denote the domestic and foreign bond lucid presentation, all purely technical details are B (,tti ) reserved to the appendices at the end of the paper. respectively, while the LIBORs are given by: 1. Cross-Currency LMM under uniform probability §·Btt(, ) § Bf (, tt ) · L() t DDD11ii 1 , Lf () t 1 with t t 1. tiii¨¸ ti ¨f ¸ iii1 ©¹Btt(,ii11 ) © B (, tt ) ¹

In particular:

11§·11f § · Lttt(,,10 ) DD¨¸ 1 , L (,, ttt 10 ) ¨f 1 ¸ withD 01 t td 1. ©¹Btt(,11 ) © B (, tt ) ¹ 74 Banks and Bank Systems, Volume 3, Issue 4, 2008

Their stochastic evolution is characterized solely by this place since we primarily concentrate on retain- f ing sufficient control over forex smiles and develop- the respective volatility functions J i ()t and J i ()t , ing a practicable FX option calibration algorithm for which are assumed deterministic within the main a wide range of maturities and across a variety of framework and can be calibrated independently for strikes. For convenience, the terminal forward prob- the domestic and foreign LIBORs to single-currency ability measure P associated with the domestic caps and swaptions using the well-known tech- tN niques for the single-currency LMM suggested by bond maturing at the terminal date tN is chosen to Rebonato (2002) and Brigo et al. (2002). To capture be the uniform martingale measure throughout the the implied volatility’s functional dependence on the paper. Though of marginal importance to the analy- corresponding interest rate option’s strike, a dis- sis, the structure of both bond volatilities placed-diffusion approach according to Rubinstein ()f (1983) can be adopted, as shown by Benner et al. V (,ttN ) is nonetheless needed and is determined (2007). However, it will not be further pursued in according to Benner et al. (2007): N 1 DDLttt()ff(, , ) Lttt() (,, ) VJJ()fff(,tt ) iii10() (, tt , t ) 1() (,, ttt ). Ni¦ ()ffi11() i 1 1(DDiiLttt,,i10) 1( Lttt,,1)

1.2. Modeling forward forex rates with stochastic volatility. We begin by assuming the following general dynamics for the CCLMM:

f PP dLii() ttN dL () t fftN PJii()t dt () t dW12 (), tf P i () t dt J i () t dW (), t Ltii() L () t

PP dQ() t qqttNN PV()tdt () tdWQQ (), t W1,2, o PtBROWNian motions. Qt() N (1)

dW12() t dW () t UU 12 dt , dW 1 () t dWQQ () t 1 dt , dW 2 () t dW Q () t U 2 Q dt

The last forward forex rate within the exemplary tenor structure represents a martingale under P : tN

PP P P dFXN () t fqttNN t N fx t N VV(,t tNQ ) dW21 () t () t dW () tVV (, t tN ) dW () t N () t dW3 (), t where FXN () t fx2222 f q f q q f VVVVVNN()t () t () t N () t 2 N () ttVN ()UV1,2 2 () ttV NQNQ ()UV 1, 2 () ttV ()U 2, (2)

dW13() t dW () t UU 13 dt , dW 23 () t dW () t 23dt

fx by switching from the natural to the terminal meas- Strictly speaking, V N ()t is a stochastic quantity ure, as will be shown shortly. through its dependence on the realization of the LIBOR rates3. Though, it could be made condition- Such a model of the forward forex rate can safely be ally deterministic to a high degree of accuracy by used to price FX options with different maturities some quite sophisticated and extremely precise ap- but the same strike. When simultaneously pricing proximation methods, or simply by classical “drift- options with various strikes, however, the natural freezing” techniques4. Yet another approach of di- question arises as to how to account for the usually rectly calibrating V fx ()t as a (deterministically) observed smile effect. For this reason, we resort to an extension of the lognormal-type dynamics of the variable model parameter is being pursued hence- forward forex rate beyond the geometric BROWNian forth since the forward forex rate represents the motion and postulate a stochastic volatility evolu- price process of a tradable asset denominated by the tion in conformity with Heston (1993) based on a corresponding numeraire, hence an observable security, as it has already been mentioned above. The common volatility Vt() that follows a mean- decomposition of the volatility in (2) serves solely reverting square-root process under the physical for the purpose of enabling us to determine the drift probability measure: of any forward forex rate prior to the terminal one

75 Banks and Bank Systems, Volume 3, Issue 4, 2008

P dFXN () t fx tN VDN ()tVtdWt ()34 (), dVt () T Vtdt ()[ VtdWt () () FXN () t (3) dW1() t dW 4 () t UUU 14 dt , dW 2 () t dW 4 () t 24 dt , dW 3 () t dW 4 () t 34 dt Any previous forward forex rate no longer follows a martingale, but its dynamics under P can be deter- tN mined according to: f P BttQt(,ii ) () Btt (, ) dFXi () t fx fx tN With FXi () t PVii()t dt () t V () t dW3 (), t Btt(,NN ) Btt (, ) FXi () t PP fx ttNN wherePVii() tdt () tdW11 () t V N () tdW () t ฀ PP PP P fqttNN tt NNtN VVVVViQ()tdW21 () t () tdW () tN () tdW () ti () tdW1 () t N (tdW)(1 t) N 1 PPD jjjLtt(, , t1 ) VJfx ()t VtdW ()ttNN () t () tdW () t it31¦ j ji 1(D jjjLtt,,) t1 N 1 D Ltt(, , t ) PJfx ()tt jjj1 ()Vfx ()tV ()tU , it¦ j i13 ji 1(D jjjLtt,,) t1 while the volatility process evolves under the uniform martingale measure as follows:

PP ttNNdB(, t tN ) WithdWtdWt44() () dWtdWt 44 () ()VU (, ttN ) 14 dt Btt(,N ) P dV() t ªºDT V () t [ V () t V (, t t ) U dt [ V () t dWtN (). t ¬¼ N 14 4 Finally, the drift functions of both term structures of the foreign term structure at last attainable. Since any traded asset scaled by the respective numeraire interest rates under Pt remain to be computed. It is N will be a martingale under the corresponding meas- well-understood that the drift of the domestic LI- ff BOR takes on the expression: ure, we infer that LtBttQtNN1() (, ) (), which N 1 DJLtt(, , t ) () tJ () t represents a portfolio of foreign bonds in domestic jjjtt1 ji Pi ()ti ¦ , N 1. currency, will be driftless when divided by the nu- ji 1 1(D jjjLtt,,) t1 meraire B(,ttN ): A sequential procedure starting with the terminal foreign LIBOR, and moving backwards until the spot LIBOR rate is reached, renders the evolution of LtBttQtff() (, ) () LtBttQtff () (, ) () dttNN 1 ªºPJVfff() ()tx ()tV ()td UtNN1 t ¬¼NNN11 23 Btt(,N ) Btt(,)N ff ff LtBttQt() (, ) ()PP LtBttQt () (, ) () tNN 1 fftN tNN 1 xtN JVNN12()tdWt () () tVtd ()Wt3 () Btt(,NN ) Btt (, ) fffx PJNN11()tt ()VUN ()tV ()t 23 . Applying the same reasoning to an arbitrary foreign LIBOR prior to the terminal one, we obtain: LtBttQtff() (, ) () Btt(, ) Withtii 1 Lf () t FX () ti1 and i N 1 tii 1 Btt(,NN ) Btt (, ) ff N 1 dLtBªº() () tQtB ()/ () t §·DJjtLt() t () t ¬¼tii 1 N ¨¸PJff()ttV V fx () ()t UJ f () t jj Udt ff itii i123 t¦ 12 LtBtQtBt() () ()/ () ¨¸ji 1 1D Lt () tiN1 1 N ©¹jtj N 1 PPDJjtLt() t () t P JVff()tdWttNN () tx () t VtdW () () tjj dW t N () t tii 21 3¦ 1(D Lt) 1 ji 1 jtj

N 1 DJjtLt() t () t PJVfff ()ttVx () ()t UJ f () t jj U . itiii123 t¦ 1(D Lt) 12 ji 1 jtj 76 Banks and Bank Systems, Volume 3, Issue 4, 2008

Summarizing the results so far, the proposed Cross-Currency LIBOR Market Model with an incorporated forex smile is fully described by the following system of stochastic differential equations: N 1 dLt () t DJjjjtLtt(, , t1 ) () t P i JJ()tj dt () t dWtN () t ttii¦ 1 Lt()ji1 1D Lttt (, , ) tji jj1

f §·N 1 dLt () t DJjtLt() t () t P i JVff()tt¨¸x ()V ()t U jj U dtt Jf ()dWtN ()t f tii 123¦ 12ti 2 Lt()¨¸ji 1 1D Lt () tji ©¹tj N 1 () DJjjjtLtt(, , t1 ) () t P dFXi t j fx fx tN ¦ VUii()t V () t13 dt V () t V () t dW 3 () t (4) FXi ( t)1(ji D jjjLtt,, t1) P dV() t ªºDT V () t [ V () t V (, t t ) U dt [ V () t dWtN () t ¬¼ N 14 4

dWml() t dW () t U ml dt , where m , l ^` 1,2,3,4 and mz l . the CCLMM with a forex smile in (4), consists in The correlation coefficients Uml , ml,1 ,2,3, can the development of an effective and fast calibration be chosen either by historical estimation, or by par- algorithm, at the core of which a closed-form FX simonious parameterization of the correlation func- option valuation formula stands. tion1 and subsequent calibration to the information extracted from occasionally observed prices of 2. Option pricing formulas and calibration quanto interest rate contracts. Assuming that both routines the domestic and the foreign LIBOR volatilities 2.1. FX option valuation by Fourier transform. have previously been calibrated independently to Regardless of the model chosen for the evolution of single-currency caps and swaptions as indicated t th above, the model still needs to be calibrated to the forward forex rate, the price at 0 of the i FX available FX options if it is intended to be used as a P option under the natural measure ti and under the pricing tool for cross-currency exotics. Therefore, P the main purpose of this article, aside from deriving equivalent measure tN is given by:

PPªºBtt(, ) FXoptt( ) BttE ( , )ti ªº FXt () K |FF BttE ( , )tN FXt () K i | . iiitNi00 ¬¼0 0 «» t0 ¬¼Btt(,N ) 5 instruments, which is at the same time flexible The driftless dynamics of FXi () t under their own natural probability measure will produce the correct enough to allow for the incorporation of the entire option price only if the discounting of the payoff is market information, as of FX markets essentially carried out using the appropriate numeraire – in this meaning calibration to the whole range of FX options prices across all available maturities and case the bond maturing at t . The use of any other i strikes. As a step prior to the actual pricing of de- measure will introduce a covariance between the rivative structures, the calibration of the model can discounting and the payoff itself, for example when be carried out under any probability measure since the pricing of plain-vanilla options on the whole we calibrate to plain-vanilla FX options, whose spectrum of forward forex rates FXi () t , payoffs involve only a single forward forex rate at a iN 1,..., , is accomplished under a single measure particular point of time. It is acceptable to use a model separately calibrated for each option expiry like the terminal one. In order to recover the same since vanillas depend on the terminal distribution of option value, this fact has to be compensated for by the underlying only as opposed to exotics whose altering the drift of the forward forex rate as shown values usually depend on the full dynamics through previously. However, facing a complex pricing time of a whole range of FX rates. Consequently, we problem, which entails simultaneously several FX revert every time to the natural measure P , effec- rates, a particular measure has to be specified and ti once it has been chosen, the presence of non-zero tively using for calibration always the same model drifts is unavoidable and the need to formulate a though under different probability measure, to cir- model for the stochastic evolution of the underly- cumvent the unpleasant dependence alluded to above, ing(s) like (4) is inevitable. This has been the main which will admittedly complicate the producing of a purpose of our work – the development of a viable closed-form solution to the option pricing problem. pricing model for exotic cross-currency interest rate 77 Banks and Bank Systems, Volume 3, Issue 4, 2008

The value of the FX option can be rewritten in terms i teristic function It () of ln FXtii ( ) Yt ( ) of the real part of its Fourier transform, where the 0 P only unknown parameter is the conditional charac- I i ()uEe tiiiuY() t |F , eventually arriving at: tt00

§·ffiuln K i iuln K i FXopt() t 11ªºeuIItt() i§· 11 ªº eu () i 0 FX( t )¨¸ Re«»0 du K¨¸ Re«»0 du . (5) Bt(,) ti 0 ¨¸ 2SI³³ iui ( i )¨¸ 2 S iu 0 it©¹00¬¼«»0 ©¹¬¼«» Similar descriptions of the option price in a different for the Fourier transform designed to use the FFT to form have already been derived by numerous au- price options efficiently. In the appendix we propose thors, e.g., Bakshi et al. (2000) and Scott (1997), a different approach, which draws directly upon and numerically determined on the assumption that Lévy’s inversion theorem, and avail ourselves of the the characteristic function is known analytically. Gauss-Laguerre Quadrature to obtain the best nu- One disadvantage of the formula above is the singu- merical estimate of the Fourier integrals in (5). larity of the integrand at the required evaluation Therefore, the option pricing reduces to the calcula- point u 0 , which ultimately precludes the application of the unknown conditional characteristic function of the Fast Fourier Transform (FFT). Therefore, tion. The dynamics directly relevant to valuing the Carr et al. (1999) develop a new analytic expression FX option are: i1 D jtLt() D Lt() WithVJ(, t t ) j () t 0 t JU (); t dW () t dW () t dt ; m , l 1,3,4 itt¦ j mlml^` j 1 1(DDLt)1( Lt) jtj 0 t

1 P dY() t VVfx () t2 V () t dt fx () t V () t dWti () t ii2 i 3 (6) P dV() t ªºDT V () t [ V () t V (, t t ) U dt [ V () t dWti () t ¬¼ i 14 4 i1 dLt () t DJkkktLtt(, , t1 ) () t P j JJ()tdk t ()tdWti ().t L ()tLttjj¦ 1D (,tt ,t ) 1 t j kj 1 kkk1 It is well-known that according to the Markov property the characteristic function:

P I i (uyvl , , , ,..., l ) Etii eiuY() t | Yt ( ) yVt , ( ) vL , ( t ) l ,..., L ( t ) l ti0 01 i 0 0 tt01 00i 01i is determined as the solution of a partial differential function of Vt() and by eliminating the stochastic equation (PDE) that can be found through Feynman- dependence on the LIBORs via the bond volatilities Kac’s theorem. To provide a closed-form solution in V (,tt ) since the asymptotic form of the drift in the the spirit of Heston (1993), however, we need to i ensure the linearity of the coefficients in the related dynamics of the LIBORs rules out linearity with PDE. This property is obviously destroyed by the respect to lk , ki 0,..., 1. The classical approach presence of the drift correction term to handling the LIBORs is by freezing them at their initial value. In addition, we need to approximate [ Vt()VU (, tti ) 14 in the dynamics of the volatility the square root of the volatility process within the process due to the change of measure. It becomes immediately apparent that the only way to explicitly drift function. Consequently, the dynamics of Vt() calculate the wanted characteristic function is by become approximately of a square-root type and making the drift of the volatility process an affine after redefining the system of SDEs (6):

1(§·Vt)PP With Vt()|¨¸ Vt ( ) , dWttii () tdW () t U dtand 2 ¨¸03434 ©¹Vt()0 i1 D jtLt()0 D Lt() VV(,)tt| (,) t t j J () t 00t0 J () t ii00¦1(DDLt) tj 1( Lt) t0 0 (7) j 1 jtj 00 t0 0

1 P dY() t VVfx () t2 V () t dt fx () t V () t dWti () t ii2 i 3

ªº1(§·Vt) P dV() t «»DT V ( t)[V¨¸Vt ( ) ( t , t )U dt [ VtdW ()ti (), t 2 ¨¸00Vt() i 14 4 ¬¼«»©¹0 we come by the following PDE in the backward variables and the respective boundary condition:

78 Banks and Bank Systems, Volume 3, Issue 4, 2008 wwIIªº111V (,)tt wI «»DT vV [()(,)t Vtt U [0 i U v Vfx () t2 v wwtv22200ii 14 14wy 0 0 ¬¼«»Vt()0 11ww22IIw2I V[fx ()tv22 vVfx () t[U v 0 (8) 22wwyv22ii00wvwy34 iiuy It ()ue . Suggested by the linearity of the PDE’s coefficients in v , we propose a solution like: I i (u , y , v ) eCt()() t00 Dt t v iuy , where C (0) 0 and D (0) 0. (9) t0 By plugging this ansatz into (8), we obtain two ordinary differential equations (ODEs): wD 11ªºV (,)tt 1 1 [[U22Di«»0 i uVfx ()t[U DVVDu22fx ()t iu fx ()t 2 wt 22Vt() 14ii 0 34 2 0 2i 0 0 ¬¼«»0 (10) wC ªº1 DV«»DT [()(,)t00 Vtti U 14 . wt0 ¬¼2 As shown in the appendix, the first one is a Riccati In the first place, we can check whether the FX op- equation (see Oksendal (2000) Chapter 6) which can tion market is flat-smiled or not. The standard pro- be solved by reducing it to a second-order linear cedure would be to take the ATM strike, which hap- ODE, whereas the second one is solved by direct pens to be that of the third FX option in the table integration. By means of their explicit solutions, as above since G ()j 0 and consequently given by (B16) and (B18) respectively, we obtain an Ki() FXt ( ), and to compute the ATM implied analytical expression for the characteristic function ji0 (9) that enables us to numerically determine the volatility by inversion of the lognormal option option price (5). valuation formula. The same volatility is then util- ized to price options with different strikes given by 2.2. The calibration algorithm. For illustration (11). The results, presented in Table 2, underline purposes, we now consider a fictitious example of once again the fact that FX options exhibit a pro- fitting the model to FX options across five different nounced volatility skew with ITM options being expiries and strikes. It is unnecessary to accentuate underestimated whereas OTM ones are being sys- that the procedure can theoretically be extended to an arbitrarily wide range of maturities and strikes at tematically overestimated. From there the need the expense of rising computational time since the stems to go beyond the standard geometric number of model parameters to match increases ac- BROWNian motion and to resort to an extension of cordingly. Market prices of FX options in basis points, the lognormal-type dynamics of the forward forex as displayed in Table 1, for strikes generated by rate that is based on stochastic volatility. Table 2. FX call prices (in bp) computed with the K i FX t e 1.0 iuu G j , j i 0 (11) ATM implied volatility where i,j 5,...1 and G j 1,5.0,0,5.0,1 Expiry FXopt 1 FXopt 2 FXopt 3 FXopt 4 FXopt 5 have been taken as a basis for the ensuing calibra- 1y tion routine. 71.97134 46.15283 25.64710 11.96459 4.55733

2y Table 1. Market prices of FX calls (in bp) for differ- 88.39820 52.03995 23.56308 7.48320 1.54316 ent strikes and maturities 3y Expiry FXopt 1 FXopt 2 FXopt 3 FXopt 4 FXopt 5 97.65900 55.09986 21.40343 4.78883 0.53401

1y 74.31572 47.68933 25.64710 11.21897 3.99027 4y 102.85492 57.42792 20.92330 3.94375 0.31927 5y 2y 90.20941 53.70243 23.56308 6.77430 1.26618 105.22879 58.83967 21.14829 3.80896 0.28176

3y 98.96450 56.78179 21.40343 4.11798 0.38920 The calibration is performed on the FX call prices 4y 103.73044 58.84490 20.92330 3.29027 0.20156 from Table 1, where the model prices are determined by (5). To obtain the best numerical estimate 5y 105.93154 60.10471 21.14829 3.11624 0.15548 of both Fourier integrals, we employ the Gauss- Laguerre Quadrature, which is a Gaussian Quadra- 79 Banks and Bank Systems, Volume 3, Issue 4, 2008 ture over the interval [0,f ) with a weighting func- In order to capture certain features of a given exotic x instrument, one could alternatively try to achieve the tion wx() e (see Abramowitz et al. (1972)). best fit to only a specific portion of the matrix of The model parameters to be calibrated are: (a) the market prices sacrificing the remaining part of it. parameters of the volatility process, initially set to The quality of the partial calibration will be gov- [ 0.01, D 0.02 , T 0.001 and Vt(0 ) 0.01; erned by the, in this case, non-constant weights Z (b) the FX volatility coefficients, initially set to assigned to the elements of the '2 distance func- fx V i []i 1 exp( 0.05(i 0.2)) , i 1,...,5; (c) tion. The choice of the weights will mostly depend the correlation coefficients, set to on the particular pricing problem.

U14 U 34 0.2 . Aiming to reproduce FX option Conclusions values for a wide range of strikes and expiries, we We proposed an integrated Cross-Currency LIBOR solve the calibration problem by simultaneously Market Model under a uniform probability measure varying the model parameters (a), (b) and (c) until in a multifactoral environment. The chief purpose of the sum of squared basis point differences between our paper has been the development of a viable pric- model and market prices has been minimized, ing framework for exotic cross-currency interest rate which, for the sake of completeness, is reported here instruments that is at the same time flexible enough to have been achieved at 8.56059 bp. We use a fast unconstrained non-linear minimization algorithm, to allow for the incorporation of as much available the Davidon-Fletcher-Powell (DFP) conjugate gra- market information as possible. In terms of FX mar- dient method as described in Press et al. (1996), to kets, on which the main focus of this work has been 2 placed, fulfilling this purpose in a satisfying manner make the distance ' between the model and mar- required the calibration to the whole range of FX ket matrices as small as possible: options prices across all available maturities and 5 2 2mmarketod el (12) strikes. This line of modelling has been reinforced ' ¦ Zij FXopt ij FXopt ij o Min! ij,1 by the significant volatility skew typically observed The calibration has been carried out with identical with FX options and the fact that it seems impossi- constant weights Z , which essentially corresponds ble for the most cross-currency derivatives to to an attempt to obtain the best global fit to the choose a particular strike, or a specific expiry of an “complete” market information, as represented by FX option since they usually depend on a variety of our market prices matrix. The resulting optimal strikes and maturities. The procedure eventually solution is shown in Table 3 below. culminated in an extension of the lognormal-type dynamics of the forward forex rate beyond the geo- Table 3. The best overall fit to the matrix of market metric BROWNian motion and the postulation of a call prices in Table 1 Expiry FXopt 1 FXopt 2 FXopt 3 FXopt 4 FXopt 5 stochastic volatility evolution in conformity with Heston (1993) based on a unique volatility, common 1y to all forward FX rates, that follows a mean- 73.44406 47.58264 26.25862 11.46718 3.54044 reverting square-root process. After determining an

2y analytical expression for the conditional characteris- 89.40880 53.47140 24.14783 6.74194 0.80012 tic function in a mildly approximate fashion for the sake of mathematical tractability, closed-form for- 3y 98.25958 56.37568 21.92059 3.97862 0.14489 mulas for FX option prices have been obtained and numerically estimated with the aid of the Gauss- 4y 103.30069 58.59928 21.35842 3.08668 0.05267 Laguerre Quadrature. The model prices computed in 5y 105.40284 59.02163 19.80163 2.06586 0.00988 this manner served as the backbone of the ensuing calibration routine, by means of which we matched The stochastic volatility model brings about a sig- the model parameters so that the distance between nificant improvement over the deterministic volatil- the model and market prices matrices has been ity one in any case. Prices of mid-maturity FX calls minimized. We have seen that introducing stochas- (i.e., 3y and 4y) are reproduced with a very good tic volatility led to a substantial improvement over precision. For very short- and long-maturity options the deterministic model in any case, although the (i.e., 5y), however, the goodness of fitting worsens fitting quality slightly worsened for very short- and suggesting that the assumption of a unique volatility especially long-dated options as compared to mid- process Vt() common to all forward forex rates maturity ones. This feature has been ascribed to the might be too restrictive, especially when calibrating possibility of our unique volatility process common to a very wide range of maturities. to all forward FX rates being too restrictive, especially when faced with a wide maturity spectrum to

80 Banks and Bank Systems, Volume 3, Issue 4, 2008 calibrate to. More flexibility could be introduced by parameters. Above all, the pricing of cross-currency considering a different stochastic volatility process exotic interest rate products would become a very for the dynamics of each forward FX rate, however, difficult task since the drift functions within the inevitably making the calibration more cumbersome dynamics of both the foreign LIBOR and the for- because of the additional volatility and correlation ward forex rate would, aside from the unpleasant parameters and raising the potential problem of stochastic dependence on LIBOR rates, involve overfitting the model due to the increased number of extra intra- and intercorrelated volatility processes. References 1. Abramowitz, M. and I. A. Stegun (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, pp. 890 and 923. 2. Andersen, L., and J. Andreasen (2000). “Volatility Skews and Extensions of the LIBOR Market Model”, Applied Mathematical Finance, 7, 1-32. 3. Andersen, L., and J. Andreasen (2002). “Volatile Volatilities”, RISK December, 163-168. 4. Andersen, L. and R. Brothertone-Ratcliffe (2005). “Extended LIBOR Market Models with Stochastic Volatility”, The Journal of Computational Finance, 9, 1. 5. Bakshi, G., and D. Madan (2000). “Spanning and Derivative Security Valuation”, Journal of Financial Economics, 55, 205-238. 6. Benner, W., Zyapkov, L., and S. Jortzik (2007). “A Multifactoral Cross-Currency LIBOR Market Model”, SSRN working paper. 7. Brigo, D., and F. Mercurio (2002). “Calibrating Libor,” RISK January, 117-121. 8. Carr, P., and D. Madan (1999). “Option Valuation using the Fast Fourier Transform”, Journal of Computational Finance, 2, 61-73. 9. Daniluk, A., and D. Gatarek (2005). “A fully lognormal LIBOR Market Model”, RISK September, 115-119. 10. Heston, S. (1993). “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options”, The Review of Financial Studies, 6, 2, 327-343. 11. Gil-Pelaez, J. (1951). “Note on the Inversion Theorem”, Biometrika, 38, 3/4, 481-482. 12. Glasserman, P., and N. Merener, 2003a, “Numerical Solution of Jump-Diffusion LIBOR Market Models”, Finance and Stochastics, 7, 1, 1-27. 13. Glasserman, P. and S.G. Kou (2003b). “The Term Structure of Simple Forward Rates with Jump Risk”, Mathe- matical Finance, 13, 3, 383-410. 14. Joshi, M., and R. Rebonato (2003). “A Displaced-Diffusion Stochastic Volatility LIBOR Market Model: Motiva- tion, Definition and Implementation”, Quantitative Finance, 3, 6, 458-469. 15. Oksendal, B. (2000). Stochastic Differential Equations. An Introduction with Applications, Springer Verlag, Berlin Heidelberg New York. 16. Piterbarg, V. (2003). “A Stochastic Volatility Forward LIBOR Model with a Term Structure of Volatility Smiles”, SSRN Working Paper. 17. Piterbarg, V. (2005a). “Stochastic Volatility Model with Time-Dependent Skew”, Applied Mathematical Finance, 12, 2, 147-185. 18. Piterbarg, V. (2005b). “Time to Smile”, RISK May, 71-75. 19. Piterbarg, V. (2006). “Smiling Hybrids”, RISK May, 66-71. 20. Press, W.H., Teukolsky, S.A., Vetterling, W.T., and B.P. Flannery (1996). Numerical Recipes in C. The Art of Scientific Computing, Cambridge University Press, Cambridge, pp. 425-430. 21. Rebonato, R. (2002). Modern Pricing of Interest Rate Derivatives. The LIBOR Market Model and Beyond, Prince- ton University Press, New Jersey. 22. Rubinstein, M. (1983). “Displaced Diffusion Option Pricing”, Journal of Finance, 38, 213-217. 23. Schloegl, E. (2002). “A Multicurrency Extension of the Lognormal Interest Rate Market Models”, Finance and Stochastics, 6, 173-196. 24. Schoenmakers, J., and B. Coffey, 2003, “Systematic Generation of Correlation Structures for the LIBOR Market Model”, International Journal of Theoretical and Applied Finance, 6, 4, 1-13. 25. Scott, L. (1997). “Pricing Stock Options in a Jump-Diffusion Model with Stochastic Volatility and Interest Rates: Application of Fourier Inversion Methods”, Mathematical Finance, 7, 413-426. Appendix A. FX option pricing formula Based on a derivation of the inversion theorem by Gil-Pelaez (1951), we determine the probability of finishing in-the- money, where ln FXii () t Y () t and ln(Kk ) , as follows: ff WithI ii() u euY dPY () cos() uY i sin() uY dPY (), t0 ³³ f f

81 Banks and Bank Systems, Volume 3, Issue 4, 2008

ff 2i sin( uk ) cos( uYdPY ) ( ) 2cos( uk ) i sin( uYdPY ) ( ) euiukII i()eu iuk i () ³³ tt00 f f iu iu ff cos(uk ) i sin( uY ) dP ( Y ) i sin( uk ) cos( uY ) dP ( Y ) iuk i ³³ ªºeuIt () and Re «»0 f f ¬¼«»iu iu ffiuk i iuk i iuk i 11euIItt()eu ()(11 ªºeuIt ) P()^`Yt k 00du Re «»0 du i 22S ³³iu2 S iu 00¬¼«» f iuk i 11 ªºeuIt () P()^`^`Yt! k 1P() Yt k Re«»0 du (A12) ii2 S ³ iu 0 ¬¼«» The delta of the option is somewhat convoluted, though it can be determined by similar techniques. For any positive numbers O and H , we have:

OOiuk i ffiuk i() u i Y OY iu() Y k 11ªºeuIt () i ªºªºee dPY()1 ee dPY() Re«»0 du Re«»«» du Re du SS³³iu³ iu S ³³ iu HH¬¼«»f ¬¼¬¼ Hf f O f O 1Y ªº cosuY ( k ) i sin uY ( k ) 1Y sin uY ( k ) ³³edPY()Re«» du ³³ edPY() du SSf H ¬¼iuf H u Proceeding from this expression now, we obtain the step function by simply letting H tend to zero and O tend to in- finity as shown in Gil-Pelaez (1951). Thereafter, we conveniently arrive at the wanted conditional expectation in the following manner: 1sffO inuY( k)1 11 edPYYY()lim du signY ( kedPY ) () edPYY () edPYY () ³³Oof ³ ³ ³ S f H o0 H u 22f Yk!2 Yk

iif uki IItt()ie1 ªº (ui ) With eYi dP() Y I ( i ) e Y dP () Y 00 Re«» du , ³³³t0 2 S iu Yk! 0 ¬¼«» essentially meaning that the delta of the option is defined by:

f iuk i 11 ªºeuIt () i Re «»0 du (A13) 2(SI³ iui i) 0 ¬¼«»t0 P Thus, additionally observing that I i ()iEe tiiiiYt()() |F FXt (), the option price is readily computed in tt00 i0 terms of the conditional characteristic function like:

FXopt() t P i 0 EFti ªº XtK() |F ¬¼it0 Bt(,)0 ti

iff iuk i iuk i FXopt() t §·IItt()ie11ªº (ui )§1 ªº e I t ()u· i 0 ¨¸00Re«»duK ¨ Re «»0 du¸(A14) Bt(,) t ¨¸22SS³³iu¨ iu ¸ 0 i ©¹00¬¼«»© ¬¼«»¹ §·ffiuk i iuk i FXopt() t 11 ªºeuIt ( i)(§·11 ªºeuIt ) i 0 FX() t ¨¸ Re«»0 du K¨¸Re«»0 du . Bt(, t) i 0 ¨¸2 S ³³iuISi () i¨¸ 2 iu 0 i ©¹00¬¼«»t0 ©¹¬¼«»

82 Banks and Bank Systems, Volume 3, Issue 4, 2008

Appendix B. Solution of the ordinary differential equations Starting with the Riccati equation in (10): wD 11V (,)tt aD22 bD c,, with a [ b [UV0 i iufx ( t )[U D, wt 22Vt() 14i 0 34 0 0 (B15) 11 cuti 22VVfx() ut fx (), 2 22ii00 it has been led back to a second-order linear differential equation by substitution: Put aD v v'' aD v '2 v bv ca uu''' Substitute v andsin ce v'2' v u' bu' acu . uu The solution ansatz is of an exponential type and is plugged into the ODE to be solved:

bbr2 4 ac ezt() t0 z2 bz ac 0, z 1,2 2 ultimately arriving at the following general solution along with the respective boundary condition:

b b2244 ac b b ac tt tt Ae2200 Be ' u ' From D(0) 0 and D u ( t t0 ) 0 au tt 0 bb2244 ac bb ac AB 0. 22 However, we are not aimed at finding an explicit solution of the second-order linear differential equation. We do not need to compute A and B separately, which is by the way based on a single boundary condition impossible, the ratio

A/ B would suffice since we actually seek to determine Dt() t0 . Having made this crucial observation, we obtain:

Abba2 4 c With B bb2 4 ac

22bb2244 ac bb ac tt tt b b44 ac2200 b b ac ' AeBeB : u 22 Dt() t0 b b2244 ac b b ac au tt tt aAe2200 aBe: B

22b b2244 ac b b ac b b24 ac b b 44 ac tt00 b b ac tt tt0 ee22 : e2 22 Dt() t0 2 b b222444 ac b b ac b b ac bb 4 ac tt000 tt tt aea222 ee: bb2 4 ac 2 2 b ba 44cbba c2 e bactt4 0 Dt() t 22 0 2 bb 4 ac2 aa eba4 ct t0 bb 2 4 ac §· 2 ¨¸bactt2 4 ba41cb¨¸ e 0 Dt() t . (B16) 0 ¨¸2 2a bb4 ac2 ¨¸1 e bactt4 0 ¨¸2 ©¹bb4 ac 83 Banks and Bank Systems, Volume 3, Issue 4, 2008

The second ODE: wC 1 DT_.,D with DT _ DT[ V ()(,) t00 V t ti U 14 (B17) wt0 2 is solved by direct integration:

t With C(0) 0 and j b2 4 ac C (0) C ( t t ) DT _ D ( t u )du 0 ³ t0

tt00 tt ju jb1 e b jju Ct()_ t0 DT Dudu ()_ DT du . Put 1 e y ³³2abbj ju j 001 e bj

bj jtt() (1yb )( j ) 1 e 0 1 jb bj bj Ct() t DT _ dy 0 ³ 2(ajbj y1 y) 1 bj

bjjtt()bjjtt() 11ee00 jbbj b j jb bj 1 C()t t D __TDdy T dy 0 ³³ 2(ajbj y b j)2( ajbj y1 y) 11 bjbj

jtt() §·bj jt() t bj 0 1 e 0 1 e jbbj¨¸bj jb bj §·11 Ct() t DT_ln¨¸ DT _ dy 0 bj ³ ¨¸ 22aj b j¨¸ ajbj ©¹ y1 y ¨¸1 1 ©¹bj bj

§·bjjt( t)(§· bjjt t) 11ee00 DT_(bj )¨¸b jb DT _( jb ) ¨¸j Ct() t0 ln¨¸ ln ¨¸ 2aj¨¸bj 2 aj ¨¸bj ¨¸11 ¨¸ ©¹bj ©¹bj bj jtt() 1 e 0 jb bj 1 Ct() t DT_(dy1) 0 ³ 21ajbj y 1 bj

§·bj jt() t 1 e 0 DT_ ¨¸bj jb Ct() t0 ln¨¸ DT _ ()t t0 . (B18) aa¨¸bj 2 j ¨¸1 ©¹bj In conclusion, we hold (B16) and (B18) to be the solutions of the ordinary differential equations in (10) being integral part of the characteristic function (9), whereby a , b , c and D _T are the substitutes defined previously by (B15) and (B17) respectively.