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An asymptotic FX option formula in the cross currency Libor market

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An asymptotic FX option formula in the cross currency Libor market Powered By Docstoc
					 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   .                      (B.8)
                                                                  k=i+1
                                                                             1 + fk (0)τk
and
                                                  TN                                            TN
                         (ε)                                                  (0)                          (ε)
                     fiF       = fiF (0) + ε           fiF (0) + sF
                                                                  i        µF
                                                                            i       du + ε           fiF         + sF
                                                                                                                    i    ˜F
                                                                                                                         σi   dW ,          (B.9)
                                                  0                                            0

as well as
                               i                                                              N −1
          (0)
                                      F           F
                                    fk (0) + sF τk F
                                              k
                                                                                                      D
                                                                                                     fk (0) + sD τk D
                                                                                                               k
                                                                                                                   D
       µF
        i       =    F
                    σi                     F    F
                                                    σk            F F
                                                                 fi ,fk   − σQ     F
                                                                                  fi ,Q   −                 D    D
                                                                                                                     σk        F
                                                                                                                              fi ,fk   .   (B.10)
                          k=0
                                     1 + fk (0)τk                                             k=0
                                                                                                      1 + fk (0)τk

Applying an asymptotic method in this setting, we obtain the FX option formula (4.5) and (4.6).


References
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[AM06b] A. Antonov and T. Misirpashaev. Markovian Projection onto a Displaced Diffusion: Generic For-
        mulas with Applications. Technical report, Numerix, October 2006. ssrn.com/abstract=
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[Kaw03] A. Kawai. A new approximate swaption formula in the LIBOR market model: an asymptotic
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[KT01]       N. Kunitomo and A. Takahashi. The asymptotic expansion approach to the valuation of interest
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[KT03]    N. Kunitomo and A. Takahashi. Applications of the Asymptotic Expansion Approach based on
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[Osa06]   Y. Osajima. The asymptotic expansion formula of implied volatility for dynamic SABR model
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