“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 © The author(s) 2021. This publication is an open access article. 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 follow- ing a square-root process and is designed to be flexible enough to allow for the incorporation of as much market infor- mation 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 ana- lytical 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 ff Lt() Lttt (, , ) L ()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.