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An asymptotic FX option formula in the cross currency Libor market model Atsushi Kawai∗ Peter J¨ ckel† a First version: 25th October 2006 This version: 3rd February 2007 Abstract In this article, we introduce analytic approximation formulae for FX options in the Libor market model (LMM). The method to derive the formulae is an asymptotic expansion technique introduced in Kawai [Kaw03]. We ﬁrst apply the method to the lognormal LMM and lognormal FX model. Then, the method is applied to the displaced diffusion LMM and the displaced diffusion FX model. Some numerical examples show that the derived formulae are sufﬁciently accurate for practical applications. 1 Introduction The Libor market model developed by Brace, Gatarek, and Musiela [BGM97], Jamshidian [Jam97], Mil- tersen, Sandmann, and Sondermann [MSS97] is one of the most popular interest rate models among both academics and practitioners. It is interesting to use the model not only for pure interest rate products but also for long-dated equity/FX products or hybrid products. In this article, we focus on modeling cross cur- rency FX markets using the Libor market model, that is, the cross currency hybrid LMM. The dynamics of the model are a straightforward extension of the standard LMM formulation, and considerations regarding the speciﬁc choice of FX state variable (spot, forward, rolling spot) are discussed in the literature [Sch02]. Whilst a variety of accurate and efﬁcient approximations for vanilla swaption pricing, and thus model calibration are available [JR00, HW00, Kaw02, Kaw03], very little has been published with respect to vanilla FX option approximations in cross-currency FX/interest rate models. The most notable exceptions are the very recent works by Osajima [Osa06] for a Gaussian forward rate stochastic volatility setup, and [AM06a, AM06b] for a cross-currency Libor market model without explicit skew on FX and interest rates. Since European FX options are the most important hedge instruments for the cross-currency exposure of FX/interest rate con- tracts, and since the pricing of vanilla options using Monte Carlo simulations for calibration purposes can be rather cumbersome, analytical approximations for FX plain vanilla option prices are highly desirable. Using an asymptotic expansion method introduced in Kawai [Kaw03], we derive an analytic approximation formula for European FX options in the lognormal1 cross currency hybrid LMM. Then, we extend the method to the displaced diffusion cross currency hybrid LMM. In the next section, we brieﬂy review our model setup. In section 3, we present the FX option formula directly in terms of a strike-dependent equivalent Black implied volatility for the lognormal LMM. Then, in section 4, we extend our results to the extended cross-currency LMM in which the domestic rates, the foreign rates, and the FX spot process all are governed by a skew-generating local volatility process known as displaced diffusion [Rub83]. Following that, we show some numerical results for comparison of analytics and Monte Carlo simulations. Finally, we conclude. ∗ Merrill Lynch † Global head of credit, hybrid, inﬂation, and commodity derivative analytics, ABN AMRO, 250 Bishopsgate, London EC2M 4AA, UK 1 Note that we use the term lognormal often in an approximate sense. Strictly speaking, due to stochastic drift terms, marginal distributions of forward interest and FX rates are not exactly lognormal. 1 2 The cross-currency hybrid LMM In the lognormal multi-currency LMM, the spot FX rate Q, the domestic forward Libor rates fiD and the foreign forward Libor rates fiF evolve lognormally according to the following stochastic differential equations in the domestic TN forward measure. • The spot FX rate N −1 D D dQ D F fk τk = µQ dt + σQ dWQ , with µQ = r − r − σQ σD D D k D Q,fk . (2.1) Q k=0 1 + fk τk Note that the domestic and foreign short rates rD and rF are herein only auxiliary concepts which we can convert into discrete forward rates once we integrate over time. Alternatively, we can consider forward FX rates in their own natural measure which removes any drift terms altogether, and this is indeed the approach we take in the derivation of our formulae in the appendix. • The domestic forward Libor rates N −1 dfiD D D fk τk D D = µD dt + σi dWiD , i with µD i = D −σi σD D D k D D fi ,fk . (2.2) fi k=i+1 1 + fk τk • The foreign forward Libor rates dfiF = µF dt + σi dWiF , i F (2.3) fiF with i F F N −1 D D fk τk fk τk µF i = F σi −σQ F Q,fi + σF F F fi ,fk − σD D F fi ,fk . (2.4) k=0 1 + fk τk k F F k=0 1 + fk τk k D D Correlations are incorporated by the fact that the individual standard Wiener processes in equation (2.1), (2.2) and (2.3) satisfy E dWQ dWiD = Q,fi dt D , E dWQ dWiF = Q,fi dt F , E dWiD dWjD = fi ,fj dt D D , (2.5) E dWiF dWjF = fi ,fj dt F F , E dWiD dWjF = fi ,fj dt D F . 3 The asymptotic FX option formula Based on the dynamics of the FX rate and interest rates described above, we can obtain an analytic approximate European FX option formula using the asymptotic expansion method. The detailed derivation of the formula can be found in appendix A. Indeed, the formula is in the form of the European Black option formula and the resulting Black volatility contains all the information of stochasticity of interest rates. Let QTN (0) be the forward FX rate for maturity TN , that is, F PTN (0) QTN (0) = Q(0) · D , (3.1) PTN (0) D F where PTN (0) and PTN (0) are domestic and foreign TN discount factors respectively as seen at time 0, i.e. today, though, we shall in the following often omit the explicit mentioning “(0)” for the sake of brevity. Then, approximate European FX option prices with strike K and maturity TN are given by the following formulae. • Call option D PTN · QTN · Φ(h) − K · Φ(h − σBlack TN ) , (3.2) 2 • Put option D PTN · K · Φ(−h + σBlack TN ) − QTN · Φ(−h) , (3.3) where 1 2 ln (QTN /K) + 2 σBlack TN h= √ , (3.4) σBlack TN Φ(·) is the cumulative standard normal distribution function, and 2 v1 1 11 2c1 2 1 σBlack = 1+ − 2c1 g0 + c2 + 2 − g0 + c3 + + 2c4 Q2 N v1 T . (3.5) TN QTN 12QTN QTN 12Q2 N T Also, g0 = QTN − K . (3.6) Before we provide the details of the constants v1 , c1 , c2 , c3 and c4 , we need to make some auxiliary deﬁnitions. First, consider the integrated covariances between the processes: TN TN D cQ,Q = σQ (u)σQ (u) du cQ,fk = D σQ (u)σk (u) Q,fk (u) du D 0 0 TN TN F D D cQ,fk = F σQ (u)σk (u) Q,fk (u) du F cfk ,fjD = D σk (u)σj (u) fk ,fj (u) du D D (3.7) 0 0 TN TN F F D F cfk ,fjF = F σk (u)σj (u) fk ,fj (u) du F F cfk ,fjF = D σk (u)σj (u) fk ,fj (u) du D F 0 0 Second, deﬁne the weights of initial forward Libor rates as follows. D D F F D fk (0)τk F fk (0)τk wk = D D , wk = F F . (3.8) 1 + fk (0)τk 1 + fk (0)τk D D F F D fk (0)τk F fk (0)τk yk = 2 , yk = 2 . (3.9) D D F F 1 + fk (0)τk 1 + fk (0)τk D F D F Then deﬁne constant v1 , v2,k , v2,k , v3,k and v3,k as follows. N −1 N −1 D D F F D F D F v1 = cQ,Q + wk wj cfk ,fjD D + wk wj cfk ,fjF F − 2wk wj cfk ,fjF D +2 wk cQ,fk − wk cQ,fk D F (3.10) j,k=0 k=0 N −1 N −1 D D F F D F v2,k = cQ,fk + D wj cfk ,fjD D − wj cfk ,fjF D v2,k = cQ,fk + F wj cfk ,fjD − wj cfk ,fjF F F (3.11) j=0 j=0 N −1 k N −1 D D F F D v3,k = − wj cfk ,fjD D v3,k = −cQ,fk + F wj cfk ,fjF F − wj cfk ,fjD F (3.12) j=k+1 j=0 j=0 Now, the constants c1 , c2 , c3 , and c4 are given as follows. N −1 1 2 D D 2 F F 2 c1 = 2 v1 + yk v2,k − yk v2,k , (3.13) 2QTN v1 k=0 c2 = − 5c2 + 2g1 + 2g3 + d1 , 1 (3.14) 3 c3 = 3c2 − 2g1 + 2g2 − 2g3 − d1 + d2 , 1 (3.15) N −1 1 D D D F F F c4 = yk v2,k v3,k − yk v2,k v3,k , (3.16) 2 2Q2 N v1 T k=0 with 1 g1 = , (3.17) 6Q2 N T N −1 1 D D 2 F F 2 g2 = yk v2,k − yk v2,k , (3.18) 2 2Q2 N v1 k=0 T N −1 1 D D 2 D F F F 2 g3 = yk v2,k 3v1 + v2,k − yk v2,k 3v1 + v2,k , (3.19) 3 6Q2 N v1 k=0 T N −1 1 3 D D 2 F F 2 d1 = v1 + 2v1 yk v2,k − yk v2,k (3.20) 3 Q 2 N v1 T k=0 N −1 D D D D F F F F D F D F + yk yj v2,k v2,j cfk ,fjD + yk yj v2,k v2,j cfk ,fjF − 2yk yj v2,k v2,j cfk ,fjF D F D , j,k=0 and N −1 N −1 1 2 D D2 F F2 d2 = v1 + yk v2,k + yk v2,k + D D yk yj c2 D ,f D + yk yj c2 F ,f F − 2yk yj c2 D ,f F F F f f D F f . (3.21) 2 2Q2 N v1 T k=0 j,k=0 k j k j k j 4 The FX option formula with skew This section extends the FX option formula when the spot FX and interest rates follow displaced diffusion processes. We assume the following dynamics for the spot FX rate, the domestic forward Libor rates and the foreign forward Libor rates. • The spot FX rate N −1 d (Q + sQ ) Q (fk +sD )τk D D = µQ dt + σQ dWQ , with µQ = rD − rF − σQ k D D 1+fk τk D σk D Q,fk . (4.1) Q + sQ Q + sQ k=0 • The domestic forward Libor rates N −1 d fiD + sD i (fk +sD )τk D D D D D = µD dt + σi dWi , i with µD i = D −σi k D D 1+fk τk D σk D D fi ,fk . (4.2) fi + si k=i+1 • The foreign forward Libor rates d fiF + sFi F = µF dt + σi dWiF , i (4.3) fiF + sF i with i N −1 Q + sQ (fk +sF )τk F F (fk +sD )τk D D µF i = F σi − σQ F Q,fi + k F F 1+fk τk F σk fiF ,fk F − k D D 1+fk τk D σk D F fi ,fk . (4.4) Q k=0 k=0 4 Then, in this extended framework the approximate European FX option prices with strike K and maturity TN are given by the following formulae. • Call option D PTN · QDisp · Φ(h) − K Disp · Φ(h − σDisp TN TN ) , (4.5) • Put option D PTN · K Disp · Φ(−h + σDisp TN ) − QDisp · Φ(−h) TN , (4.6) where 1 2 ln QDisp /K Disp + 2 σDisp TN TN h= √ , (4.7) σDisp TN and v1 1 11 2c1 1 2 σDisp = β 2 1+ − 2c1 g0 + c2 + 2 − 2 g0 + c3 + Disp 2 + 2c4 Q 2 N v1 . T TN QDisp TN 12 QDisp TN QDisp TN 12 QTN (4.8) Here g0 = QTN − K , QDisp = QTN + sQ , TN K Disp = K + sQ , and β = QTN /QDisp . TN (4.9) Furthermore, we have new constants v1 , c1 , c2 , c3 and c4 . Let us ﬁrst deﬁne the displacement correction factor as 1 1 1 γ = 1 + sQ + . (4.10) 2 QTN Q Second, deﬁne the weights of initial forward Libor rates as follows. D D fk (0) + sD τk k D F fk (0) + sF τk F k F wk = D D wk = F F (4.11) 1 + fk (0)τk 1 + fk (0)τk 2 2 D fk (0) + sD k D τk − sD τk k D F fk (0) + sF k F τk − sF τk k F D F yk = yk = (4.12) D D 2 F F 2 1+ fk (0)τk 1 + fk (0)τk D F D F Then, deﬁne the constants v1 , v2,k , v2,k , v3,k , v3,k , v4 and v5 as follows. N −1 N −1 2 D D F F D F D F v1 = γ cQ,Q + wk wj cfk ,fjD D + wk wj cfk ,fjF F − 2wk wj cfk ,fjF D + 2γ wk cQ,fk − wk cQ,fk D F (4.13) j,k=0 k=0 N −1 N −1 D D F F D F v2,k = γcQ,fk + D wj cfk ,fjD − wj cfk ,fjF D D v2,k = γcQ,fk + F wj cfk ,fjD − wj cfk ,fjF F F (4.14) j=0 j=0 N −1 k N −1 D D F F D v3,k = − wj cfk ,fjD D v3,k = −γcQ,fk + F wj cfk ,fjF − F wj cfk ,fjD F (4.15) j=k+1 j=0 j=0 N −1 N −1 D F D v4 = γcQ,Q + wj cQ,fjD − wj cQ,fjF v5 = − wj cQ,fjD . (4.16) j=0 j=0 Now, the constants c1 , c2 , c3 and c4 are given by: N −1 1 2 2 D D 2F F 2 c1 = 2 v1 − γ (γ − 1) v4 + yk v2,k − yk v2,k , (4.17) 2QTN v1 k=0 c2 = − 5c2 + 2g1 + 2g3 + d1 , 1 (4.18) 5 c3 = 3c2 − 2g1 + 2g2 − 2g3 − d1 + d2 , 1 (4.19) N −1 1 D D D F F F c4 = γ (γ − 1) v4 v5 + yk v2,k v3,k − yk v2,k v3,k , (4.20) 2 2Q2 N v1 T k=0 with 1 g1 = , (4.21) 6Q2 N T N −1 1 2 2 D D 2F F 2 g2 = − γ (γ − 1) v4 + γ (γ − 1) v4 cQ,Q + yk v2,k − yk v2,k , (4.22) 2 2Q2 N v1 T k=0 1 2 3 g3 = − 3γ (γ − 1) v1 v4 + γ (2γ − 1) (γ − 1) v4 (4.23) 3 6Q2 N v1 T N −1 D D 2 D F F F 2 + yk v2,k 3v1 + v2,k − yk v2,k 3v1 + v2,k , k=0 1 d1 = v1 + γ 2 (γ − 1)2 v4 cQ,Q − 2γ (γ − 1) v1 v4 3 2 2 3 Q 2 N v1 T N −1 D D 2F F 2 +2 yk v1 v2,k − yk v1 v2,k (4.24) k=0 N −1 D D F F − 2γ (γ − 1) yk v2,k v4 cQ,fk − yk v2,k v4 cQ,fk D F k=0 N −1 D D D D F F F F D F D F + yk yj v2,k v2,j cfk ,fjD + yk yj v2,k v2,j cfk ,fjF − 2yk yj v2,k v2,j cfk ,fjF D F D , (4.25) j,k=0 and 1 d2 = v1 + γ 2 (γ − 1)2 c2 − 2γ (γ − 1) v4 2 Q,Q 2 2 2Q2 N v1 T N −1 N −1 D D 2 F F 2 + yk v2,k − yk v2,k − γ (γ − 1) yk c2 D − yk c2 F D F Q,f Q,f (4.26) k k k=0 k=0 N −1 + D D F F D F yk yj c2 D ,f D + yk yj c2 F ,f F − 2yk yj c2 D ,f F f f f . k j k j k j j,k=0 5 Numerical results In this section, we present some numerical results showing the accuracy of our analytic approximations by comparing with Monte Carlo valuations. In ﬁgure 1, we show the FX implied volatility proﬁle for different maturities from three months to ﬁfteen years. Both the domestic and the foreign currency’s interest rates were set to be ﬂat at 5%, and the yield curves were individually driven by a single factor, with equal levels of volatility. All displacement coefﬁcients s(·) were set to zero. As we can see, this kind of symmetric setup gives rise to a moderate symmetric smile that is generated solely by the fact that the FX rate’s instantaneous dynamics have a drift component that is stochastic in its own right in a non-Gaussian fashion. In ﬁgure 2, we repeated the same experiment with 6 13.0% 27% 26% 12.5% 25% 3m Analytic 3m Monte Carlo 6m Analytic 6m Monte Carlo 24% 5y Analytic 5y Monte Carlo 7y Analytic 7y Monte Carlo 1y Analytic 1y Monte Carlo 3y Analytic 3y Monte Carlo 10y Analytic 10y Monte Carlo 15y Analytic 15y Monte Carlo 23% 12.0% 22% 21% 11.5% 20% 19% 11.0% 18% 17% 10.5% 16% 15% 10.0% 14% 75% 80% 85% 90% 95% 100% 105% 110% 115% 120% 125% 75% 80% 85% 90% 95% 100% 105% 110% 115% 120% 125% Figure 1: Numerical and analytical implied volatilities for different maturities T as a function of K/QT , i.e. strike divided by forward FX rate. fiD = fiF = 5%, sQ = sD = sF = 0, σQ = 10%, σi = σi = 40%, ρfiD fj = ρfiF fj = 1, ρQ,fj = 0.3, ρQ,fj = −0.3, i j D F D F D F ρfj ,fj = 0.3 D F 10.6% 18% 17% 10.5% 16% 10.4% 5y Analytic 5y Monte Carlo 7y Analytic 7y Monte Carlo 10y Analytic 10y Monte Carlo 15y Analytic 15y Monte Carlo 3m Analytic 3m Monte Carlo 6m Analytic 6m Monte Carlo 1y Analytic 1y Monte Carlo 3y Analytic 3y Monte Carlo 15% 10.3% 14% 10.2% 13% 10.1% 12% 10.0% 11% 75% 80% 85% 90% 95% 100% 105% 110% 115% 120% 125% 75% 80% 85% 90% 95% 100% 105% 110% 115% 120% 125% Figure 2: Numerical and analytical implied volatilities for different maturities T as a function of K/QT , i.e. strike divided by forward FX rate. fiD = 6%, fiF = 2%, sQ = sD = sF = 0, σQ = 10%, σi = 20%, σi = 60%, ρfiD fj = ρfiF fj = e− |ti −tj |/10 , i j D F D F ρfj ,fj = ρQ,fj = ρQ,fj = 0, D F D F different interest rate and volatility levels with fully factorised (i.e. decorrelated) interest curve dynamics. Note that the difference in interest rates and volatilities gives rise to a skew for FX implied volatilities, despite the fact that absolute interest rate volatility levels in the two currencies are approximately equal (∼ 1.2%). In ﬁgures 3 to 5, we show the analytical results in comparison to simulation data for the same overall scenario as in ﬁgure 2, but for a range of skew parameters sQ and different maturities. The FX displaced diffusion parameter σQ was rescaled for different sQ according to σQ = 10% · Q(0)/(Q(0) + sQ ) which gives 7 rise to the appearance that the implied volatility curves, with varying sQ , pivot about the point where the FX spot is in relation to the respective FX forward. The last set of results shown in ﬁgure 6 is for a market- 11.5% 11.5% 11.0% 11.0% 10.5% 10.5% 10.0% 10.0% 9.5% 9.5% 9.0% 9.0% 8.5% 8.5% (a) Analytic (b) Analytic (c) Analytic (d) Analytic (a) Analytic (b) Analytic (c) Analytic (d) Analytic (a) Monte Carlo (b) Monte Carlo (c) Monte Carlo (d) Monte Carlo (a) Monte Carlo (b) Monte Carlo (c) Monte Carlo (d) Monte Carlo 8.0% 8.0% 75% 80% 85% 90% 95% 100% 105% 110% 115% 120% 125% 75% 80% 85% 90% 95% 100% 105% 110% 115% 120% 125% Figure 3: Numerical and analytical 3 month (left) and 6 month (right) implied volatilities with different FX skew settings: (a) sQ = 8 log2 (10) · Q (almost normal), (b) sQ = Q (similar to square root distribution), (c) sQ = 0 (almost lognormal), (d) Q sQ = − log2 ( 3/2) · Q (positive skew). fiD = 6%, fiF = 2%, sD = sF = 0, σQ = 10% · Q+sQ , σi = 20%, σi = 60%, i j D F ρfiD fj = ρfiF fj = e− |ti −tj |/10 , ρfj ,fj = ρQ,fj = ρQ,fj = 0 D F D F D F 11.5% 12.5% 12.0% 11.0% 11.5% 10.5% 11.0% 10.0% 10.5% 9.5% 10.0% 9.0% 9.5% 8.5% 9.0% (a) Analytic (b) Analytic (c) Analytic (d) Analytic (a) Analytic (b) Analytic (c) Analytic (d) Analytic (a) Monte Carlo (b) Monte Carlo (c) Monte Carlo (d) Monte Carlo (a) Monte Carlo (b) Monte Carlo (c) Monte Carlo (d) Monte Carlo 8.0% 8.5% 75% 80% 85% 90% 95% 100% 105% 110% 115% 120% 125% 75% 80% 85% 90% 95% 100% 105% 110% 115% 120% 125% Figure 4: Numerical and analytical 1 year (left) and 3 year (right) implied volatilities with different FX skew settings: (a) sQ = 8 log2 (10)·Q (almost normal), (b) sQ = Q (similar to square root distribution), (c) sQ = 0 (almost lognormal), (d) sQ = − log2 ( 3/2)· Q Q (positive skew). fiD = 6%, fiF = 2%, sD = sF = 0, σQ = 10% · Q+sQ , σi = 20%, σi = 60%, ρfiD fj = ρfiF fj = e− |ti −tj |/10 , i j D F D F ρfj ,fj = ρQ,fj = ρQ,fj = 0 D F D F given USD (domestic) and EUR (foreign) interest rate scenario as seen in the market for Friday October 13, 2006. Forward rate volatility term structures were calibrated to caplet prices. Speciﬁcally, term structures of 8 13.5% 14.5% 13.0% 14.0% 12.5% 13.5% 12.0% 13.0% 11.5% 12.5% 11.0% 12.0% 10.5% 11.5% 10.0% 11.0% 9.5% 10.5% (a) Analytic (b) Analytic (c) Analytic (d) Analytic (a) Monte Carlo (b) Monte Carlo (c) Monte Carlo (d) Monte Carlo (a) Analytic (b) Analytic (c) Analytic (d) Analytic (a) Monte Carlo (b) Monte Carlo (c) Monte Carlo (d) Monte Carlo 9.0% 10.0% 75% 80% 85% 90% 95% 100% 105% 110% 115% 120% 125% 75% 80% 85% 90% 95% 100% 105% 110% 115% 120% 125% Figure 5: Numerical and analytical 5 year (left) and 7 year (right) implied volatilities with different FX skew settings: (a) sQ = 8 log2 (10)·Q (almost normal), (b) sQ = Q (similar to square root distribution), (c) sQ = 0 (almost lognormal), (d) sQ = − log2 ( 3/2)· Q Q (positive skew). fiD = 6%, fiF = 2%, sD = sF = 0, σQ = 10% · Q+sQ , σi = 20%, σi = 60%, ρfiD fj = ρfiF fj = e− |ti −tj |/10 , i j D F D F ρfj ,fj = ρQ,fj = ρQ,fj = 0 D F D F 8.20% 9.80% 9.60% 8.00% 9.40% 7.80% 9.20% 5y Analytic 7y Analytic 10y Analytic 15y Analytic 5y Monte Carlo 7y Monte Carlo 10y Monte Carlo 15y Monte Carlo 9.00% 7.60% 8.80% 7.40% 8.60% 3m Analytic 6m Analytic 1y Analytic 3y Analytic 3m Monte Carlo 6m Monte Carlo 1y Monte Carlo 3y Monte Carlo 8.40% 7.20% 8.20% 7.00% 8.00% 75% 80% 85% 90% 95% 100% 105% 110% 115% 120% 125% 75% 80% 85% 90% 95% 100% 105% 110% 115% 120% 125% Figure 6: Numerical and analytical implied volatilities for market-calibrated USD (domestic) / EUR (foreign) rates and volatilities on Friday October 13, 2006. instantaneous volatility of individual forward rates were deﬁned by the parametric Nelson-Siegel form σ(t, T ) = kT · (a + b · (T − t)) · e−c·(T −t) + d (5.1) with a = −0.074514253 b = 0.208715347 c = 0.606615724 d = 0.107550229 for USD (domestic) (5.2) a = −0.071209658 b = 0.196349282 c = 0.632737991 d = 0.100540646 for EUR (foreign) 9 for the instantaneous volatility of a forward rate expiring at T . This leaves a scaling constant kT for each forward rate expiring at T permitting calibration to market observable caplet prices. The forward rates and kTi scaling numbers are shown in ﬁgure 7. Note that the Nelson-Siegel parametrisation of instantaneous 6.00% 1.12 1.1 5.50% 1.08 1.06 5.00% 1.04 4.50% 1.02 1 4.00% 0.98 forward rates (USD) 0.96 3.50% forward rates (EUR) k (USD) 0.94 k (EUR) 3.00% 0.92 2006-10-13 2009-07-09 2012-04-04 2014-12-30 2017-09-25 2020-06-21 Figure 7: Forward rates (left axis) and kTi volatility scaling factors (right axis) from calibration to market on Friday October 13, 2006, as used for the results shown in ﬁgure 6. 9.5% 9.0% 8.5% 8.0% 7.5% 7.0% 6.5% 6.0% 5.5% 2006-10-13 2009-10-12 2012-10-12 2015-10-13 2018-10-13 2021-10-13 Figure 8: Instantaneous FX volatility function σQ (t) calibrated to market observable plain vanilla option prices at the money on Friday October 13, 2006, as used for the results shown in ﬁgure 6. volatility (5.1) with scaling constants kTi means that perfect time homogeneity of volatility is given when all of the kTi are identical. This is almost achieved for EUR, as can be seen in ﬁgure 7, and a reasonably high degree of time homogeneity is also given for USD since all the kTi values are near unity. Correlations between interest rates within each yield curve were given by 1 √ √ ρfiD fjD (t) = ρfiF fjF (t) = e− 5 | ti −t− tj −t| . (5.3) There was no cross-currency interest rate correlation, i.e. ρfiD fjF = 0. Correlations between domestic interest 1√ rates and the spot FX rate was ρQ,fiD (t) = − 1 e− 5 4 ti −t and the correlation of the FX spot rate with foreign 1 −5 1√ ti −t forward rates was ρQ,fiF (t) = 4 . Displacements of forward rates were individually set to sD = fiD e i and sF = fjF (similar to square root distribution). The spot FX rate was undisplaced. The instantaneous FX j driver volatility shown in ﬁgure 8 was calibrated as a piecewise constant function to match market observable 10 plain vanilla option prices at the money out to ten years, and extrapolated ﬂat beyond that. As can be seen in ﬁgure 6, for a market-realistic scenario, the proposed approximations are of superb quality even for long dated FX options. 6 Conclusion In this article, we derived analytic approximation formulae for FX options in Libor market models. It turns out that the derived formulae are accurate enough for use in practical applications. The asymptotic expansion method was straightforwardly extended to cross currency FX markets form single currency interest markets, which was analyzed previously by Kawai [Kaw03]. The method has proved to be very powerful and ﬂexible, whence it can be applied to other stochastic processes such as other interest rate and stochastic volatility models. A Derivation of the FX option formula The formula is obtained using the asymptotic expansion method. The method is fully explained in Kawai [Kaw03] by applying the method to a European swaption pricing in the LMM. To derive the formula, it is more convenient to express the stochastic differential equation (2.1), (2.2) and (2.3) as being driven by 2N + 1 independent standard Wiener processes W by decomposing the covariance structure into orthogonal components. • The spot FX rate N −1 D D dQ fk τk ˜· = µQ dt + σQ dW , with µQ = rD − rF − σQ σD D D k D Q,fk . (A.1) Q k=0 1 + fk τk • The domestic forward rates N −1 dfiD D D fk τk D = µD dt + σi · dW , i ˜D with µD = −σi i D σD D D k D D fi ,fk . (A.2) fi k=i+1 1 + fk τk • The foreign forward rates dfiF = µF dt + σi · dW , i ˜F (A.3) fiF with i F F N −1 D D fk τk fk τk µF i = F σi −σQ F Q,fi + σF F F fi ,fk − σD D F fi ,fk , (A.4) k=0 1 + fk τk k F F k=0 1 + fk τk k D D ˜ ˜D ˜F where σQ , σi and σi are 2N + 1 dimensional vectors satisfying 2 σQ · σQ = σQ , ˜ ˜ ˜ ˜D D σQ · σi = σQ σi D Q,fi , ˜ ˜F F σQ · σi = σQ σi F Q,fi , (A.5) ˜D ˜D D D σi · σj = σi σj D D fi ,fj , ˜F ˜F F F σi · σj = σi σj F F fi ,fj , ˜D ˜F D F σi · σj = σi σj D F fi ,fj . (A.6) From equations (A.1), (A.2) and (A.3), the TN -forward FX rate as deﬁned in (3.1) follows N −1 D D N −1 F F dQTN (t) fk τk fk τk = (˜Q (t) + η(t)) · dWt , σ with η= σD − ˜ σF . ˜ (A.7) QTN (t) k=0 1 + fk τk k D D k=0 1 + fk τk k F F 11 Naturally, it is a martingale in the TN measure which obviates any considerations regarding the domestic and foreign short rates we may have had with respect to (2.1) and similar. Now, as is usual with asymptotic expansions, we insert a smallness parameter ε into equation (A.7) to obtain TN (ε) (ε) QTN (TN ) = QTN (0) + ε QTN (t) σQ (t) + η (ε) (t) · dWt , ˜ (A.8) 0 with N −1 D (ε) D N −1 F (ε) F (ε) fk τk fk τk η = ˜D σk − ˜F σk . (A.9) D (ε) D F (ε) F k=0 1+ fk τk k=0 1+ fk τk Note that the superscript (ε) is not meant to indicate the ε-th derivative but instead denotes dependence on ε. Here, perturbed interest rates follow TN TN (ε) (0) (ε) fiD = fiD (0) + ε fiD (0) µD i du + ε fiD ˜D σi · dW , (A.10) 0 0 with N −1 D D (0) fk (0)τk µD i D = −σi σD fi ,fk . (A.11) k=i+1 1 + fk (0)τk k D D and TN TN (ε) (0) (ε) fiF = fiF (0) +ε fiF (0) µF i du + ε fiF ˜F σi · dW , (A.12) 0 0 as well as i F F N −1 D D (0) fk (0)τk fk (0)τk µF i F = σi F σF F k F F fi ,fk − σQ F fi ,Q − D σD D k F fi ,fk . (A.13) k=0 1 + fk (0)τk k=0 1 + fk (0)τk Notice that the drift terms of the interest rates are approximated deterministically using initial interest rates as a consequence of the Itˆ -Taylor expansions in (A.8), (A.10), and (A.12). This makes both the derivation o and the resulting formula simpler, yet it remains accurate. By applying a Taylor series expansion in ε to the forward FX rate (A.8), we obtain a third-order asymptotic expansion. (ε) ∂Q(ε) 1 ∂ 2 Q(ε) 1 ∂ 3 Q(ε) QTN (TN ) = QTN (0) + ε + ε2 + ε3 + O ε4 , (A.14) ∂ε ε=0 2 ∂ε2 ε=0 6 ∂ε3 ε=0 where TN ∂Q(ε) = QTN (0) σQ + η (0) · dWu , ˜ (A.15) ∂ε ε=0 0 TN u1 ∂ 2 Q(ε) = 2QTN (0) σQ + η (0) · dWu2 σQ + η (0) · dWu1 ˜ ˜ (A.16) ∂ε2 ε=0 0 0 T u1 TN u1 N −1 N D D fk (0)τk (0) + 2QTN (0) D 2 µD k ˜D du2 σk · dWu1 + ˜D ˜D σk · dWu2 σk · dWu1 ( D 1+fk (0)τk ) k=0 0 0 0 0 T u1 TN u1 N −1 N F F fk (0)τk (0) − 2QTN (0) F 2 µF k ˜F du2 σk · dWu1 + σk F · dWu2 σk F · dWu1 . ˜ ˜ ( F 1+fk (0)τk ) k=0 0 0 0 0 12 and TN u1 u2 ∂ 3 Q(ε) = 6QTN (0) σQ + η (0) · dWu3 σQ + η (0) · dWu2 σQ + η (0) · dWu1 ˜ ˜ ˜ ∂ε3 ε=0 0 0 0 TN u1 u2 N −1 D D fk (0)τk + 6QTN (0) 2 ˜D ˜D σk · dWu3 σk · dWu2 σQ + η (0) · dWu1 ˜ (1+fk (0)τk ) D D k=0 0 0 0 TN u1 u2 N −1 F F fk (0)τk − 6QTN (0) 2 ˜F ˜F σk · dWu3 σk · dWu2 σQ + η (0) · dWu1 ˜ (1+fk (0)τk ) F F k=0 0 0 0 TN u1 u2 N −1 D D fk (0)τk + 6QTN (0) 2 ˜D σk · dWu3 σQ + η (0) · dWu2 σk · dWu1 ˜ ˜D (1+fk (0)τk ) D D k=0 0 0 0 TN u1 u2 N −1 F F fk (0)τk − 6QTN (0) 2 ˜F σk · dWu3 σQ + η (0) · dWu2 σk · dWu1 ˜ ˜F (1+fk (0)τk ) F F k=0 0 0 0 TN u1 u2 N −1 D D fk (0)τk + 6QTN (0) 2 ˜D ˜D σQ + η (0) · dWu3 σk · dWu2 σk · dWu1 ˜ (1+fk (0)τk ) D D k=0 0 0 0 TN u1 u2 N −1 F F fk (0)τk − 6QTN (0) 2 σQ + η (0) · dWu3 σk · dWu2 σk · dWu1 ˜ ˜F ˜F (1+fk (0)τk ) F F k=0 0 0 0 TN u1 N −1 D D fk (0)τk + 6QTN (0) 2 ˜D ˜ σk · σQ + η (0) du2 σk · dWu1 ˜D (1+fk (0)τk ) D D k=0 0 0 TN u1 N −1 F F fk (0)τk − 6QTN (0) 2 ˜F ˜ σk · σQ + η (0) du2 σk · dWu1 ˜F (1+fk (0)τk ) F F k=0 0 0 TN u1 u2 N −1 D D fk (0)τk + 6QTN (0) 2 ˜D ˜D ˜D σk · dWu3 σk · dWu2 σk · dWu1 (1+fk (0)τk ) D D k=0 0 0 0 TN u1 u2 N −1 F F fk (0)τk − 6QTN (0) 2 ˜F ˜F ˜F σk · dWu3 σk · dWu2 σk · dWu1 + O (f τ )2 . (A.17) (1+fk (0)τk ) F F k=0 0 0 0 Note that in the above expressions many dependencies on u3 , u2 , or u1 are not mentioned explicitly for the sake of legibility. Next, deﬁne X (ε) as 1 (ε) X (ε) = QTN (TN ) − QTN (0) . (A.18) ε Then we can ﬁnd an asymptotic expansion for the density function of X (ε) as (ε) c1 3 fX (x) = ϕv0 (x) + ε x − 3v0 x ϕv0 (x) + εc4 x2 − v0 ϕv0 (x) v0 g1 + g3 4 + ε2 2 x − 6v0 x2 + 3v0 ϕv0 (x) + ε2 g2 x2 − v0 ϕv0 (x) v0 1 c2 1 d1 4 + ε2 1 x6 − 15v0 x4 + 45v0 x2 − 15v0 ϕv0 (x) + ε2 2 2 3 2 x − 6v0 x2 + 3v0 ϕv0 (x) 2 v0 2 v0 13 1 + ε2 d2 x2 − v0 ϕv0 (x) + O ε3 , x , (A.19) 2 where ϕv0 (x) is the Gaussian density function with mean 0 and variance v0 = Q 2 N v1 T (A.20) and v1 deﬁned in (3.10), and c1 , c2 , c3 and c4 are constants deﬁned in (3.13), (3.14), (3.15) and (3.16) respec- tively. As a result, letting ε = 1, it follows that an asymptotic FX call option price is D 2 2 PTN · G (v0 ) − 2c1 v0 g0 G (v0 ) + 2c2 v0 g0 G (v0 ) + v0 c2 g0 + c3 v0 + 2c3 v0 G (v0 ) , 1 2 (A.21) where the function G (x) is deﬁned as 2 g e−g0 /2x G (x) = g0 Φ √0 +x √ , with g0 = QTN − K . (A.22) x 2πx Since the dynamics of the forward FX rate is close to lognormal, a further procedure2 that matches coefﬁcients to an analogous expansion for the standard Black model within the same order in ε improves the accuracy. Finally, we obtain the FX option formulae (3.2) and (3.3). B Derivation of the FX option formula with skew From the SDE (4.1), (4.2) and (4.3), the forward FX rate follows N −1 N −1 dQTN sQ (fk +sD )τk D D (fk +sF )τk F F = 1+ ˜ σQ + η · dW , with η = k D D 1+fk τk ˜D σk − k F F 1+fk τk ˜F σk . (B.1) QTN Q k=0 k=0 Now, by allowing a small perturbation ε, we can rewrite the SDE (B.1) as TN (ε) (ε) sQ QTN (TN ) = QTN (0) + ε QTN 1+ σQ + η (ε) ˜ · dW , (B.2) Q 0 with D (ε) (ε) N −1 fk D + sD τk k N −1 F fk + sF τk k F (ε) η = ˜D σk − ˜F σk . (B.3) D (ε) D F (ε) F k=0 1+ fk τk k=0 1+ fk τk The perturbed spot FX rate follows TN TN (ε) Q(ε) = Q(0) + µQ du + ε σ (˜Q ) dW , (B.4) 0 0 with N −1 (ε) D F D D fk (0) + sD τk D k µQ =Q r −r − εσQ D D σk D Q,fk , (B.5) k=0 1 + fk (0)τk and D PTN (0) Q(0) = Q(0) F . (B.6) PTN (0) 2 whose details are explained in [Kaw03] 14 Here, perturbed interest rates follow TN TN (ε) (0) (ε) fiD = fiD (0) +ε fiD (0) + sD i µD i du + ε fiD + sD i ˜D σi dW , (B.7) 0 0 with N −1 (0) D D fk (0) + sD τk D k µD i = D −σi D D σk fi ,fk . 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