on cross-currency models with stochastic volatility and correlated-fx

8/6/2019 On Cross-Currency Models With Stochastic Volatility and Correlated-FX

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On Cross-Currency Models with

Stochastic Volatility and Correlated

Interest Rates

Lech A. Grzelak, Cornelis W. Oosterlee

February 6, 2011

Abstract

We construct multi-currency models with stochastic volatility and correlatedstochastic interest rates with a full matrix of correlations. We first dealwith a foreign exchange (FX) model of Heston-type, in which the domesticand foreign interest rates are generated by the short-rate process of Hull-White [Hull and White, 1990]. We then extend the framework by modelingthe interest rate by a stochastic volatility displaced-diffusion Libor MarketModel [Andersen and Andreasen, 2002], which can model an interest rate smile.We provide semi-closed form approximations which lead to efficient calibration ofthe multi-currency models. Finally, we add a correlated stock to the frameworkand discuss the construction, model calibration and pricing of equity-FX-interestrate hybrid payoffs.

Keywords: Foreign exchange (FX); Stochastic Volatility; Heston Model; Stochastic Interest Rates;

Interest Rate Smile; Forward Characteristic Function; Hybrids; Affine Diffusion; Efficient Calibration.

1 Introduction

Since the financial crisis, investors tend to look for products with a long time horizon,that are less sensitive to short-term market fluctuations. When pricing these exoticcontracts it is desirable to incorporate in a mathematical model the patterns present inthe market that are relevant to the product.

Due to the existence of complex FX products, like the Power-Reverse Dual-Currency [Sippel and Ohkoshi, 2002], the Equity-CMS Chameleon or the Equity-LinkedRange Accrual TRAN swaps [Caps, 2007], that all have a long lifetime and are sensitiveto smiles or skews in the market, improved models with stochastic interest rates needto be developed.

The literature on modeling foreign exchange (FX) rates is rich and many stochasticmodels are available. An industrial standard is a model from [Frey and Sommer, 1996;Sippel and Ohkoshi, 2002], where log-normally distributed FX dynamics are assumed

and Gaussian, one-factor, interest rates are used. This model gives analytic expressionsfor the prices of basic products for at-the-money options. Extensions on the interest rateside were presented in [Schlogl(b), 2002; Mikkelsen, 2001], where the short-rate modelwas replaced by a Libor Market Model framework.

A Gaussian interest rate model was also used in [Piterbarg, 2006], in which alocal volatility model was applied for generating the skews present in the FX market.In another paper, [Kawai and Jackel, 2007], a displaced-diffusion model for FX wascombined with the interest rate Libor Market Model.

Delft University of Technology, Delft Institute of Applied Mathematics, Mekelweg 4, 2628 CD,Delft, The Netherlands, and Rabobank International, Jaarbeursplein 22, 3521 AP, Utrecht. E-mailaddress: [email protected].

CWI Centrum Wiskunde & Informatica, Amsterdam, the Netherlands, and Delft University ofTechnology, Delft Institute of Applied Mathematics. E-mail address: [email protected].

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Stochastic volatility FX models have also been investigated. For example,in [Haastrecht and Pelsser, 2011] the Schobel-Zhu model was applied for pricing FXin combination with short-rate processes. This model leads to a semi-closed formfor the characteristic function. However, for a normally distributed volatility processit is difficult to outperform the Heston model with independent stochastic interestrates [Haastrecht and Pelsser, 2011].

Research on the Heston dynamics in combination with correlated interest rateshas led to some interesting models. In [Andreasen, 2006] and [Giese, 2006] anindirectly imposed correlation structure between Gaussian short-rates and FX waspresented. The model is intuitively appealing, but it may give rise to very largemodel parameters [Antonov et al., 2008]. An alternative model was presentedin [Antonov and Misirpashaev, 2006; Antonov et al., 2008], in which calibrationformulas were developed by means of Markov projection techniques.

In this article we present an FX Heston-type model in which the interest rates arestochastic processes, correlated with the governing FX processes. We first discuss the He-ston FX model with Gaussian interest rate (Hull-White model [Hull and White, 1990])short-rate processes. In this model a full matrix of correlations is used.

This model, denoted by FX-HHW here, is a generalization of our workin [Grzelak and Oosterlee, 2011], where we dealt with the problem of finding an

affine approximation of the Heston equity model with a correlated stochastic interestrate. In this paper, we apply this technique in the world of foreign exchange.

Secondly, we extend the framework by modeling the interest rates by amarket model, i.e., by the stochastic volatility displaced-diffusion Libor MarketModel [Andersen and Andreasen, 2002; Piterbarg, 2005]. In this hybrid model, calledFX-HLMM here, we incorporate a non-zero correlation between the FX and theinterest rates and between the rates from different currencies. Because it is not possibleto obtain closed-form formulas for the associated characteristic function, we use alinearization approximation, developed earlier, in [Grzelak and Oosterlee, 2010].

For both models we provide details on how to include a foreign stock in the multi-currency pricing framework.

Fast model evaluation is highly desirable for FX options in practice,

especially during the calibration of the hybrid model. This is themain motivation for the generalization of the linearization techniquesin [Grzelak and Oosterlee, 2011; Grzelak and Oosterlee, 2010] to the world of foreignexchange. We will see that the resulting approximations can be used very well in theFX context.

The present article is organized as follows. In Section 2 we discuss the extension ofthe Heston model by stochastic interest rates, described by short-rate processes. Weprovide details about some approximations in the model, and then derive the relatedforward characteristic function. We also discuss the models accuracy and calibrationresults. Section 3 gives the details for the cross-currency model with interest rates drivenby the market model and Section 4 concludes.

2 Multi-Currency Model with Short-Rate InterestRates

Here, we derive the model for the spot FX, (t), expressed in units of domesticcurrency, per unit of a foreign currency.

We start the analysis with the specification of the underlying interest rate processes,rd(t) and rf(t). At this stage we assume that the interest rate dynamics are defined viashort-rate processes, which under their spot measures, i.e., Qdomestic and Zforeign,are driven by the Hull-White [Hull and White, 1990] one-factor model:

drd(t) = d(d(t) rd(t))dt + ddWQd (t), (2.1)drf(t) = f(f(t)

rf(t))dt + fdW

Zf (t), (2.2)

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where WQd (t) and WZf(t) are Brownian motions under Q and Z, respectively. Parameters

d, f determine the speed of mean reversion to the time-dependent term structurefunctions d(t), f(t), and parameters d, f are the volatility coefficients.

These processes, under the appropriate measures, are linear in their state variables,so that for a given maturity T (0 < t < T) the zero-coupon bonds (ZCB) are known tobe of the following form:

Pd(t, T) = exp (Ad(t, T) + Bd(t, T)rd(t)) ,

Pf(t, T) = exp (Af(t, T) + Bf(t, T)rf(t)) ,(2.3)

with Ad(t, T), Af(t, T) and Bd(t, T), Bf(t, T) analytically known quantities (see forexample [Brigo and Mercurio, 2007]). In the model the money market accounts aregiven by:

dMd(t) = rd(t)Md(t)dt, and dMf(t) = rf(t)Mf(t)dt. (2.4)

By using the Heath-Jarrow-Morton arbitrage-free argument, [Heath et al., 1992], thedynamics for the ZCBs, under their own measures generated by the money savingsaccounts, are known and given by the following result:

Result 2.1 (ZCB dynamics under the risk-free measure). The risk-free dynamics of thezero-coupon bonds, Pd(t, T) and Pf(t, T), with maturity T are given by:

dPd(t, T)

Pd(t, T)= rd(t)dt

Tt

d(t, s)ds

dWQd (t), (2.5)

dPf(t, T)

Pf(t, T)= rf(t)dt

Tt

f(t, s)ds

dWZf (t), (2.6)

where d(t, T), f(t, T) are the volatility functions of the instantaneous forward ratesfd(t, T), ff(t, T), respectively, that are given by:

dfd(t, T) = d(t, T) Tt

d(t, s)ds + d(t, T)dWQd (t), (2.7)

dff(t, T) = f(t, T)

Tt

f(t, s)ds + f(t, T)dWZf(t). (2.8)

Proof. For the proof see [Musiela and Rutkowski, 1997].

The spot-rates at time t are defined by rd(t) fd(t, t), rf(t) ff(t, t).By means of the volatility structures, d(t, T), f(t, T), one can define a number

of short-rate processes. In our framework the volatility functions are chosen to bed(t, T) = d exp(d(T t)) and f(t, T) = f exp(f(T t)). The Hull-Whiteshort-rate processes, rd(t) and rf(t) as in (2.1), (2.2), are then obtained and the termstructures, d(t), f(t), expressed in terms of instantaneous forward rates, are alsoknown. The choice of specific volatility determines the dynamics of the ZCBs:

dPd(t, T)

Pd(t, T)= rd(t)dt + dBd(t, T)dW

Q

d (t),

dPf(t, T)

Pf(t, T)= rf(t)dt + fBf(t, T)dW

Zf (t), (2.9)

with Bd(t, T) and Bf(t, T) as in (2.3), given by:

Bd(t, T) =1

d

ed(Tt) 1

, Bf(t, T) =

1

f

ef (Tt) 1

. (2.10)

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For a detailed discussion on short-rate processes, we refer to the analysis of Musiela andRutkowski in [Musiela and Rutkowski, 1997]. In the next subsection we define the FXhybrid model.

2.1 The Model with Correlated, Gaussian Interest Rates

The FX-HHW model, with all processes defined under the domestic risk-neutralmeasure, Q, is of the following form:

d(t)/(t) = (rd(t) rf(t)) dt+

v(t)dWQ (t), (0) > 0,

dv(t) = (v v(t))dt+

v(t)dWQv (t), v(0) > 0,

drd(t) = d(d(t) rd(t))dt+ ddWQd (t), rd(0) > 0,drf(t) =

f(f(t) rf(t)) f,f

v(t)

dt+ fdWQf (t), rf(0) > 0.

(2.11)Here, the parameters , d, and f determine the speed of mean reversion of the latterthree processes, their long term mean is given by v, d(t), f(t), respectively. Thevolatility coefficients for the processes rd(t) and rf(t) are given by d and f and the

volatility-of-volatility parameter for process v(t) is .In the model we assume a full matrix of correlations between the Brownian motionsW(t) =

WQ (t), W

Qv (t), W

Qd (t), W

Qf (t)T

:

dW(t)(dW(t))T =

1 ,v ,d ,f

,v 1 v,d v,f,d v,d 1 d,f,f v,f d,f 1

dt. (2.12)Under the domestic-spot measure the drift in the short-rate process, rf(t), gives rise to

an additional term, f,f

v(t). This term ensures the existence of martingales,under the domestic spot measure, for the following prices (for more discussion,see [Shreve, 2004]):

1(t) := (t)Mf(t)

Md(t)and 2(t) := (t)

Pf(t, T)

Md(t),

where Pf(t, T) is the price foreign zero-coupon bond (2.9), respectively, and the moneysavings accounts Md(t) and Mf(t) are from (2.4).

To see that the processes 1(t) and 2(t) are martingales, one can apply the Itoproduct rule, which gives:

d1(t)

1(t)=

v(t)dWQ (t), (2.13)

d2(t)

2(t)

= v(t)dWQ (t) + fBf(t, T)dW

Qf (t). (2.14)

The change of dynamics of the underlying processes, from the foreign-spot to thedomestic-spot measure, also influences the dynamics for the associated bonds, which,under the domestic risk-neutral measure, Q, with the money savings account consideredas a numeraire, have the following representations

dPd(t, T)

Pd(t, T)= rd(t)dt + dBd(t, T)dW

Qd (t), (2.15)

dPf(t, T)

Pf(t, T)=

rf(t) ,ffBf(t, T)

v(t)

dt + fBf(t, T)dWQ

f (t), (2.16)

with Bd(t, T) and Bf(t, T) as in (2.10).

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2.2 Pricing of FX Options

In order to perform efficient calibration of the model we need to be able toprice basic options on the FX rate, V(t, X(t)), for a given state vector, X(t) =[(t), v(t), rd(t), rf(t)]T:

V(t, X(t)) = EQ Md(t)Md(T) max((T) K, 0)F(t) ,with

Md(t) = exp

t0

rd(s)ds

.

Now, we consider a forward price, (t), such that:

EQ

max((T) K, 0)Md(T)

F(t) = V(t, X(t))Md(t)

=: (t).

By Itos lemma we have:

d(t) =

1

Md(t) dV(t) rd(t)V(t)

Md(t) dt, (2.17)

with V(t) := V(t, X(t)). We know that (t) must be a martingale, i.e.: E(d(t)) = 0.Including this in (2.17) gives the following Fokker-Planck forward equation for V:

rdV =1

22f

2V

r2f+ d,fdf

2V

rdrf+

1

22d

2V

r2d+ v,ff

v

2V

vrf

+v,dd

v2V

vrd+

1

22v

2V

v2+ ,ff

v

2V

rf+ x,dd

v

2V

rd

+,v v2V

v+

1

22v

2V

2+

f(f(t) rf) ,ff

v V

rf

+d(d(t) rd)V

rd + (v v)V

v + (rd rf)

V

+V

t .

This 4D PDE contains non-affine terms, like square-roots and products. It is thereforedifficult to solve it analytically and a numerical PDE discretization, like finite differences,needs to be employed. Finding a numerical solution for this PDE is therefore ratherexpensive and not easily applicable for model calibration. In the next subsection wepropose an approximation of the model, which is useful for calibration.

2.2.1 The FX Model under the Forward Domestic Measure

To reduce the complexity of the pricing problem, we move from the spot measure,generated by the money savings account in the domestic market, Md(t), to the forward

FX measure where the numeraire is the domestic zero-coupon bond, Pd(t, T). Asindicated in [Musiela and Rutkowski, 1997; Piterbarg, 2006], the forward is given by:

FXT(t) = (t)Pf(t, T)

Pd(t, T), (2.18)

where FXT(t) represents the forward exchange rate under the T-forward measure, and(t) stands for foreign exchange rate under the domestic spot measure. The superscriptshould not be confused with the transpose notation used at other places in the text.

By switching from the domestic risk-neutral measure, Q, to the domestic T-forwardmeasure, QT, the discounting will be decoupled from taking the expectation, i.e.:

(t) = Pd(t, T)ET

max

FXT(T) K, 0

|F(t)

. (2.19)

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In order to determine the dynamics for FXT(t) in (2.18), we apply Itos formula:

dFXT(t) =Pf(t, T)

Pd(t, T)d(t) +

(t)

Pd(t, T)dPf(t, T) (t) Pf(t, T)

P2d (t, T)dPd(t, T)

+(t)Pf(t, T)

P3d (t, T)(dPd(t, T))

2 +1

Pd(t, T)(d(t)dPf(t, T))

Pf(t, T)P2f(t, T)

(dPd(t, T)d(t)) (t)P2d (t, T)

dPd(t, T)dPf(t, T). (2.20)

After substitution of SDEs (2.11), (2.15) and (2.16) into (2.20), we arrive at the followingFX forward dynamics:

dFXT(t)

FXT(t)= dBd(t, T)

dBd(t, T) ,d

v(t) d,ffBf(t, T)

dt

+

v(t)dWQ (t) dBd(t, T)dWQd (t) + fBf(t, T)dWQf (t). (2.21)

Since FXT(t) is a martingale under the T-forward domestic measure, i.e.,Pd(t, T)E

T(FXT(T)

|F(t)) = Pd(t, T)FX

T(t) =: Pf(t, T)(t), the appropriate Brownian

motions under the Tforward domestic measure, dWT (t), dWTv (t), dWTd (t) anddWTf (t), need to be determined.

A change of measure from domestic-spot to domestic T-forward measure requires achange of numeraire from money savings account, Md(t), to zero-coupon bond Pd(t, T).In the model we incorporate a full matrix of correlations, which implies that all processeswill change their dynamics by changing the measure from spot to forward. Lemma 2.2provides the model dynamics under the domestic T-forward measure, QT.

Lemma 2.2 (The FX-HHW model dynamics under the QT measure). Under the T- forward domestic measure, the model in (2.11) is governed by the following dynamics:

dFXT(t)

FX

T

(t)

=v(t)dWT (t) dBd(t, T)dWTd (t) + fBf(t, T)dWTf (t), (2.22)

where

dv(t) =

(v v(t)) + v,ddBd(t, T)

v(t)

dt +

v(t)dWTv (t), (2.23)

drd(t) =

d(d(t) rd(t)) + 2dBd(t, T)

dt + ddWTd (t), (2.24)

drf(t) =

f(f(t) rf(t)) f,f

v(t) + dfd,fBd(t, T)

dt + fdWTf (t),

(2.25)

with a full matrix of correlations given in (2.12), and with Bd(t, T), Bf(t, T) givenby (2.10).The proof can be found in Appendix A.

From the system in Lemma 2.2 we see that after moving from the domestic-spotQ-measure to the domestic T-forward QT measure, the forward exchange rate FXT(t)does not depend explicitly on the short-rate processes rd(t) or rf(t). It does not containa drift term and only depends on dWTd (t), dW

Tf (t), see (2.22).

Remark. Since the sum of three correlated, normally distributed random variables,Q = X+ Y + Z, remains normal with the mean equal to the sum of the individual meansand the variance equal to

v2Q = v2X + v

2Y + v

2Z + 2X,YvXvY + 2X,ZvXvZ + 2Y,Z vYvZ ,

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we can represent the forward (2.22) as:

dFXT/FXT =

v + 2dB2d +

2fB

2f 2,ddBd

v

+2,ffBf

v 2d,fdfBdBf 12 dWTF . (2.26)

Although the representation in (2.26) reduces the number of Brownian motions in

the dynamics for the FX, one still needs to find the appropriate cross-terms, likedWTF (t)dW

Tv (t), in order to obtain the covariance terms. For clarity we therefore prefer

to stay with the standard notation.

Remark. The dynamics of the forwards, FXT(t) in (2.22) or in (2.26), do not dependexplicitly on the interest rate processes, rd(t) and rf(t), and are completely described bythe appropriate diffusion coefficients. This suggests that the short-rate variables will notenter the pricing PDE. Note, that this is only the case for models in which the diffusioncoefficient for the interest rate does not depend on the level of the interest rate.

In next section we derive the corresponding pricing PDE and provide modelapproximations.

2.3 Approximations and the Forward Characteristic Function

As the dynamics of the forward foreign exchange, FXT(t), under the domestic forwardmeasure involve only the interest rate diffusions dWTd (t) and dW

Tf (t), a significant

reduction of the pricing problem is achieved.In order to find the forward ChF we take, as usual, the log-transform of the forward

rate FXT(t), i.e.: xT(t) := log FXT(t), for which we obtain the following dynamics:

dxT(t) =

(t,

v(t)) 12

v(t)

dt+

v(t)dWT (t)dBddWTd (t)+fBfdWTf (t), (2.27)

with the variance process, v(t), given by:

dv(t) = (v v(t)) + v,ddBdv(t) dt + v(t)dWTv (t), (2.28)where we used the notation Bd := Bd(t, T) and Bf := Bf(t, T), and

(t,

v(t)) = (x,ddBd x,ffBf)

v(t) + d,fdfBdBf 12

2dB

2d +

2fB

2f

.

By applying the Feynman-Kac theorem we can obtain the characteristic function of theforward FX rate dynamics. The forward characteristic function:

T := T(u, X(t), t , T ) = ET

eiuxT(T)F(t) ,

with final condition, T(u, X(T), T , T ) = eiuxT(T), is the solution of the following

Kolmogorov backward partial differential equation:

T

t=

(v v) + v,dd

vBd(t, T) T

v+

1

2v (t, v)

2T

x2

T

x

+

x,vv v,dd

vBd(t, T) + v,ff

vBf(t, T) 2T

xv+

1

22v

2T

v2.

This PDE contains however non-affine

v-terms so that it is nontrivial to findthe solution. Recently, in [Grzelak and Oosterlee, 2011], we have proposed twomethods for linearization of these non-affine 1 square-roots of the square root pro-

1According to [Duffie et al., 2000] the n-dimensional system of SDEs:

dX(t) = (X(t))dt + v(X(t))dW(t),

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cess [Cox et al., 1985]. The first method is to project the non-affine square-root termson their first moments. This is also the approach followed here 2.

The approximation of the non-affine terms in the corresponding PDE is then doneas follows. We assume:

v(t) E

v(t)

=: (t), (2.29)

with the expectation of the square root of v(t) given by:

E

v(t)

=

2c(t)e(t)/2

k=0

1

k!((t)/2)k

1+2 + k

(

2+ k)

, (2.30)

and

c(t) =1

42(1 et), = 4v

2, (t) =

4v(0)et

2(1 et) . (2.31)

(k) is the gamma function defined by:

(k) =

0

tk1etdt.

Although the expectation in (2.30) is a closed form expression, its evaluation is ratherexpensive. One may prefer to use a proxy, for example,

E(

v(t)) 1 + 2e3t, (2.32)

in which the constant coefficients 1, 2 and 3 can be determined by asymptotic equalitywith (2.30) (see [Grzelak and Oosterlee, 2011] for details).

Projection of the non-affine terms on their first moments allows us to derive thecorresponding forward characteristic function, T, which is then of the following form:

T(u, X(t), t , T ) = exp(A(u, ) + B(u, )xT(t) + C(u, )v(t)),

where = Tt, and the functions A() := A(u, ), B() := B(u, ) and C() := C(u, )are given by:

B() = 0,

C() = C() + (B2() B())/2 + x,vB()C() + 2C2()/2,A() = vC() + v,dd()Bd()C() (, ())

B2() B()

+ (v,dd()Bd() + v,ff()Bf()) B()C(),

with (t) = E(

v(t)), and Bi() = 1i

ei 1 for i = {d, f}. The initial conditions

are: B(0) = iu, C(0) = 0 and A(0) = 0.With B() = iu, the complex-valued function C() is of the Heston-

type, [Heston, 1993], and its solution reads:

C() =1 ed

2(1 ged) ( x,viu d) , (2.33)

with d =

(x,viu )2 2iu(iu 1), g = x,viu d x,viu + d .

is of the affine form if:

(X(t)) = a0 + a1X(t), for any (a0, a1) Rn Rnn,

v(X(t))v(X(t))T = (c0)ij + (c1)TijX(t), for arbitrary (c0, c1) R

nn Rnnn,

r(X(t)) = r0 + rT1 X(t), for (r0, r1) R R

n,

for i, j = 1, . . . , n, with r(X(t)) being an interest rate component.2Since the moments of the square-root process under the T-forward measure are difficult to determine

for v(t) we have set v,d = 0 or, in other words, the expectation is calculated under measure Q.8
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The parameters , , x,v are given in (2.11).Function A() is given by:

A() =

0

v + v,dd(s)Bd(s) v,dd(s)Bd(s)iu

+v,ff(s)Bf(s)iuC(s)ds + (u2 + iu)

0

(s, (s))ds, (2.34)

with C(s) in (2.33). It is most convenient to solve A() numerically with, for example,Simpsons quadrature rule. With correlations v,d, v,f equal to zero, a closed-formexpression for A() would be available [Grzelak and Oosterlee, 2011].

We denote the approximation, by means of linearization, of the full-scale FX-HHWmodel by FX-HHW1. It is clear that efficient pricing with Fourier-based methods canbe done with FX-HHW1, and not with FX-HHW.

By the projection of

v(t) on its first moment in (2.29) the corresponding PDE isaffine in its coefficients, and reads:

T

t= ((v v) + 1)

T

v+

1

2v (t, (t))

2T

x2

T

x

+ (x,vv 2) 2T

xv+ 1

22v

2T

v2, with: (2.35)

T(u, X(T), T , T ) = ET

eiuxT(T)F(T) = eiuxT(T),

and (t, (t)) = 3 + d,fdfBd(t, T)Bf(t, T) 12

2dB2d(t, T) +

2fB

2f(t, T)

.

The three terms, 1, 2, and 3, in the PDE (2.35) contain the function (t):

1 := v,ddBd(t, T)(t),

2 := (v,ddBd(t, T) v,ffBf(t, T)) (t),3 := (x,ddBd(t, T) x,ffBf(t, T)) (t).

When solving the pricing PDE for t T, the terms Bd(t, T) and Bf(t, T) tend to zero,and all terms that contain the approximation vanish. The case t 0 is furthermoretrivial, since

v(t)

t0 E(

v(0)).Under the T-forward domestic FX measure, the projection of the non-affine terms

on their first moments is expected to provide high accuracy. In Section 2.5 we performa numerical experiment to validate this.

It is worth mentioning that also an alternative approximation for the non-affineterms

v(t) is available, see [Grzelak and Oosterlee, 2011]. This alternative approach

guarantees that the first two moments are exact. In this article we stay, however, withthe first representation.

2.4 Pricing a Foreign Stock in the FX-HHW Model

Here, we focus our attention on pricing a foreign stock, Sf(t), in a domestic market.With this extension we can in principle price equity-FX-interest rate hybrid products.

With an equity smile/skew present in the market, we assume that Sf(t) is given bythe Heston stochastic volatility model:

dSf(t)/Sf(t) = rf(t)dt+

(t)dWZSf (t),

d(t) = f( (t))dt+ f

(t)dWZ (t),

drf(t) = f(f(t) rf(t)))dt+ fdWZf (t),(2.36)

where Z indicates the foreign-spot measure and the model parameters, f, f, f, f(t)and f, are as before.

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Before deriving the stock dynamics in domestic currency, the model has to becalibrated in the foreign market to plain vanilla options. This can be efficiently donewith the help of a fast pricing formula.

With the foreign short-rate process, rf(t), established in (2.11) we need to determinethe drifts for Sf(t) and its variance process, (t), under the domestic spot measure. Theforeign stock, Sf(t), can be expressed in domestic currency by multiplication with the

FX, (t), and by discounting with the domestic money savings account, Md(t). Sucha stock is a tradable asset, so the price (t)Sf(t)/Md(t) (with (t) in (2.11), Sf(t)from (2.36) and the domestic money-saving account Md(t) i n (2.4)) needs to be amartingale.

By applying Itos lemma to (t)Sf(t)/Md(t), we find

d

(t)Sf (t)Md(t)

(t) Sf(t)

Md(t)

= ,Sf

v(t)

(t)dt +

v(t)dWQ (t) +

(t)dWZSf , (2.37)

where Q and Z indicate the domestic-spot and foreign-spot measures, respectively. Tomake process (t)Sf(t)/Md(t) a martingale we set:

dWZ

Sf (t) = dWQ

Sf ,Sfv(t)dt,where v(t) is the variance process of FX defined in (2.11).

Under the change of measure, from foreign to domestic-spot, Sf(t) has the followingdynamics:

dSf(t)/Sf(t) = rf(t)dt +

(t)dWZSf (t)

=

rf(t) ,Sf

v(t)

(t)

dt +

(t)dWQSf (t). (2.38)

The variance process is correlated with the stock and by the Cholesky decompositionwe find:

d(t) = f( (t))dt + f(t)Sf ,dWZSf (t) +1 2Sf ,dWZ (t)=

f( (t)) Sf ,Sf ,f

(t)

v(t)

dt + f

(t)dWQ (t). (2.39)

Sf(t) in (2.38) and (t) in (2.39) are governed by several non-affine terms. Assumingthat the foreign stock, Sf(t), is already calibrated to market data, we only need tosimulate the foreign stock dynamics in the domestic market. Monte Carlo simulationof the foreign stock under domestic measure can be done as, for example, presentedin [Andersen, 2007]. The outstanding property of Andersens QE Monte Carlo scheme isthat the Heston model can be accurately simulated when the Feller condition is satisfiedas well as when this condition is violated.

2.5 Numerical Experiment for the FX-HHW Model

In this section we check the errors resulting from the various approximations of theFX-HHW1 model. We use the set-up from [Piterbarg, 2006], which means that theinterest rate curves are modeled by ZCBs defined by Pd(t = 0, T) = exp(0.02T) andPf(t = 0, T) = exp(0.05T). Furthermore,

d = 0.7%, f = 1.2%, d = 1%, f = 5%.

We choose 3:

= 0.5, = 0.3, v = 0.1, v(0) = 0.1. (2.40)

3The model parameters do not satisfy the Feller condition, 2 > 2v.

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1.1 1.2 1.3 1.4 1.5

8

9

10

11

12

13

Strike

imp

liedBlackVolatility

Calibration results

1Y Market

1Y FXHHW

0.6 0.8 1 1.2 1.4 1.69

10

11

12

13

14

15

16

17

18

Strike

imp

liedBlackVolatility

Calibration results

10Y Market

10Y FXHHW

0.4 0.6 0.8 1 1.2 1.4

14

15

16

17

18

19

20

21

22

23

24

Strike

imp

liedBlackVolatility

Calibration results

20Y Market

20Y FXHHW

Figure 2.1: Comparison of implied volatilities from the market and the FX-HHW1model for FX European call options for maturities of 1, 10 and 20 years. The strikesare provided in Table B.1 in Appendix B. (0) = 1.35.

Our experiments show that the model can be well calibrated to the market data. For

long maturities and for deep-in-the money options some discrepancy is present. This ishowever typical when dealing with the Heston model (not related to our approximation),since the skew/smile pattern in FX does not flatten for long maturities. This wassometimes improved by adding jumps to the model (Bates model). In Appendix B inTable B.4 the detailed calibration results are tabulated.

Short-rate interest rate models can typically provide a satisfactory fit to at-the-money interest rate products. They are therefore not used for pricing derivatives thatare sensitive to the interest rate skew. This is a drawback of the short-rate interest ratemodels. In the next section an extension of the framework, so that interest rate smilesand skews can be modeled as well, is presented.

3 Multi-Currency Model with Interest Rate Smile

In this section we discuss a second extension of the multi-currency model, in whichan interest rate smile is incorporated. This hybrid model models two types of smiles, thesmile for the FX rate and the smiles in the domestic and foreign fixed income markets.We abbreviate the model by FX-HLMM. It is especially interesting for FX productsthat are exposed to interest rate smiles. A description of such FX hybrid products canbe found in the handbook by Hunter [Hunter and Picot, 2005].

A first attempt to model the FX by stochastic volatility and interest rates driven by amarket model was proposed in [Takahashi and Takehara, 2008], assuming independencebetween log-normal-Libor rates and FX. In our approach we define a model with non-zero correlation between FX and interest rate processes.

As in the previous sections, the stochastic volatility FX is of the Heston type, whichunder domestic risk-neutral measure, Q, follows the following dynamics:

d(t)/(t) = (. . . )dt+

v(t)dWQ (t), S(0) > 0,

dv(t) = (v v(t))dt+

v(t)dWQv (t), v(0) > 0,(3.1)

with the parameters as in (2.11). Since we consider the model under the forward measurethe drift in the first SDE does not need to be specified (the dynamics of domestic-forwardFX (t)Pf(t, T)/Pd(t, T) do not contain a drift term).

In the model we assume that the domestic and foreign currencies are independentlycalibrated to interest rate products available in their own markets. For simplicity, wealso assume that the tenor structure for both currencies is the same, i.e., Td Tf ={T0, T1, . . . , T N T} and k = Tk Tk1 for k = 1 . . . N . For t < Tk1 we define the

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forward Libor rates Ld,k(t) := Ld(t, Tk1, Tk) and Lf,k(t) := Lf(t, Tk1, Tk) as

Ld,k(t) :=1

k

Pd(t, Tk1)

Pd(t, Tk) 1

, Lf,k(t) :=1

k

Pf(t, Tk1)

Pf(t, Tk) 1

. (3.2)

For each currency we choose the DD-SV Libor Market Model from [Andersen and Andreasen, 2002]for the interest rates, under the T-forward measure generated by the numeraires Pd(t, T)

and Pf(t, T), given by:

dLd,k(t) = vd,kd,k(t)

vd(t)

d(t)

vd(t)dt + dWd,Tk (t)

,

dvd(t) = d(vd(0) vd(t))dt + d

vd(t)dWd,Tv (t),

(3.3)

and

dLf,k(t) = vf,kf,k(t)

vf(t)

f(t)

vf(t)dt + dWf,Tk (t) ,

dvf(t) = f(vf(0) vf(t))dt + f

vf(t)dWf,Tv (t), (3.4)with

d(t) = N

j=k+1

jd,j(t)vd,j1 + jLd,j(t)

dk,j , f(t) = N

j=k+1

jf,j (t)vf,j1 + jLf,j(t)

fk,j , (3.5)

where

d,k = d,kLd,k(t) + (1 d,k)Ld,k(0),f,k = f,kLf,k(t) + (1 f,k)Lf,k(0).

The Brownian motion, dWd,Tk , corresponds to the k-th domestic Libor rate, Ld,k(t),

under the T-forward domestic measure, and the Brownian motion, d

Wf,Tk relates to the

k-th foreign market Libor rate, Lf,k(t) under the terminal foreign measure T.In the model v

d,k(t) and v

f,k(t) determine the level of the interest rate volatility

smile, the parameters d,k(t) and f,k(t) control the slope of the volatility smile,and d, f determine the speed of mean-reversion for the variance and influencethe speed at which the interest rate volatility smile flattens as the swaption expiryincreases [Piterbarg, 2005]. Parameters d, f determine the curvature of the interestrate smile.

The following correlation structure is imposed 4, between

FX and its variance process, v(t): dWT (t)dWTv (t) = ,vdt,

FX and domestic Libors, Ld,j(t): dWT (t)dW

d,Tj (t) =

d,jdt,

FX and foreign Libors, Lf,j(t): dWT (t)d

Wf,Tj (t) =

f,jdt,

Libors in domestic market: dWd,T

k

(t)dWd,T

j

(t) = dk,jdt,

Libors in foreign market: dWf,Tk (t)dWf,Tj (t) = fk,jdt,Libors in domestic and foreign markets: dWd,Tk (t)d

Wf,Tj (t) = d,fk,j dt.(3.6)

We prescribe a zero correlation between the remaining processes, i.e., between

Libors and their variance process,

dWd,Tk (t)dWd,Tv (t) = 0, dWf,Tk (t)dWf,Tv (t) = 0,

4As it is insightful to relate the covariance matrix with the necessary model approximations, thecorrelation structure is introduced here by means of instantaneous correlation of the scalar diffusions.

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Libors and the FX variance process,

dWd,Tk (t)dWTv (t) = 0, dWf,Tk (t)dWTv (t) = 0,

all variance processes,

dWT

v

(t)dWd,T

v

(t) = 0, dWT

v

(t)dWf,Tv (t) = 0, dWd,Tv (t)dWf,Tv (t) = 0,FX and the Libor variance processes,

dWT (t)dWd,Tv (t) = 0, dW

T (t)dWf,Tv (t) = 0.

The correlation structure is graphically displayed in Figure 3.1.

Figure 3.1: The correlation structure for the FX-HLMM model. Arrows indicate non-zero correlations. SV is Stochastic Volatility.

Throughout this article we assume that the DD-SV model in (3.3) and (3.4) is alreadyin the effective parameter framework as developed in [Piterbarg, 2005]. This meansthat approximate time-homogeneous parameters are used instead of the time-dependentparameters, i.e., k(t) k and vk(t) vk.

With this correlation structure, we derive the dynamics for the forward FX, givenby:

FXT(t) = (t)Pf(t, T)

Pd(t, T), (3.7)

(see also (2.18)) with (t) the spot exchange rate and Pd(t, T) and Pf(t, T) zero-couponbonds. Note that the bonds are not yet specified.

When deriving the dynamics for (3.7), we need expressions for the zero-coupon bonds,

Pd(t, T) and Pf(t, T). With Equation (3.2) the following expression for the final bondcan be obtained:

1

Pi(t, T)=

1

Pi(t, Tm(t))

Nj=m(t)+1

(1 + jLi,j(t)) , for i = {d, f}, (3.8)

with T = TN and m(t) = min(k : t Tk) (empty products in (3.8) are defined tobe equal to 1). The bond Pi(t, TN) in (3.8) is fully determined by the Libor ratesLi,k(t), k = 1, . . . , N and the bond Pi(t, Tm(t)). Whereas the Libors Li,k(t) are definedby System (3.3) and (3.4), the bond Pi(t, Tm(t)) is not yet well-defined in the currentframework.

To define continuous time dynamics for a zero-coupon bond, interpolation

techniques are available (see, for example, [Schlogl(a), 2002; Piterbarg, 2004;

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Davis and Mataix-Pastor, 2009; Beveridge and Joshi, 2009]). We consider herethe linear interpolation scheme, proposed in [Schlogl(a), 2002], which reads:

1

Pi(t, Tm(t))= 1 + (Tm(t) t)Li,m(t)(Tm(t)1), for Tm(t)1 < t < Tm(t). (3.9)

In our previous work, [Grzelak and Oosterlee, 2010], this basic interpolation technique

was very satisfactory for the calibration. By combining (3.9) with (3.8), we find for thedomestic and foreign bonds:

1

Pd(t, T)=

1 + (Tm(t) t)Ld,m(t)(Tm(t)1) N

j=m(t)+1

(1 + jLd(t, Tj1, Tj)) ,

1

Pf(t, T)=

1 + (Tm(t) t)Lf,m(t)(Tm(t)1) N

j=m(t)+1

(1 + jLf(t, Tj1, Tj)) .

When deriving the dynamics for FXT(t) in (3.7) we will not encounter any dt-terms (asFXT(t) has to be a martingale under the numeraire Pd(t, T)).

For each zero-coupon bond, Pd(t, T) or Pf(t, T), the dynamics are determined underthe appropriate T-forward measures (for Pd(t, T) the domestic T-forward measure, andfor Pf(t, T) the foreign T-forward measure). The dynamics for the zero-coupon bonds,driven by the Libor dynamics in (3.3) and (3.4), are given by:

dPd(t, T)

Pd(t, T)= (. . . )dt

vd(t)

Nj=m(t)+1

jvd,jd,j(t)

1 + jLd,j(t)dWd,Tj (t), (3.10)

dPf(t, T)

Pf(t, T)= (. . . )dt

vf(t)

Nj=m(t)+1

jvf,jf,j (t)

1 + jLf,j (t)dWf,Tj (t), (3.11)

and the coefficients were defined in (3.3) and (3.4).

By changing the numeraire from Pf(t, T) to Pd(t, T) for the foreign bond, only thedrift terms will change. Since FXT(t) in (3.7) is a martingale under the Pd(t, T) measure,it is not necessary to determine the appropriate drift correction.

By taking Equation (2.20) for the general dynamics of (3.7) and neglecting all thedt-terms we get

dFXT(t)

FXT(t)=

v(t)dWT (t) +

vd(t)N

j=m(t)+1

jvd,jd,j(t)

1 + jLd,j(t)dWd,Tj (t)

vf(t)N

j=m(t)+1

jvf,jf,j (t)

1 + jLf,j(t)dWf,Tj (t). (3.12)

Note that the hatinW, disappeared from the Brownian motion dWf,Tj (t) in (3.12) whichis an indication for the change of measure from the foreign to the domestic measure forthe foreign Libors.

Since the stochastic volatility process, v(t), for FX is independent of the domesticand foreign Libors, Ld,k(t) and Lf,k(t), the dynamics under the Pd(t, T)-measure do notchange 5 and are given by:

dv(t) = (v v(t))dt +

v(t)dWTv (t). (3.13)

5In [Grzelak and Oosterlee, 2010] the proof for this statement is given when a single yield curve isconsidered.

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The model given in (3.12) with the stochastic variance in (3.13) and the correlationsbetween the main underlying processes is not affine. In the next section we discuss alinearization.

3.1 Linearization and Forward Characteristic Function

The model in (3.12) is not of the affine form, as it contains terms likei,j(t)/(1 + i,jLi,j(t)) with i,j = i,jLi,j(t) + (1 i,j)Li,j(0) for i = {d, f}.In order to derive a characteristic function, we freeze the Libor rates, which isstandard practice (see for example [Glasserman and Zhao, 1999; Hull and White, 2000;Jackel and Rebonato, 2000]), i.e.:

Ld,j(t) Ld,j(0) d,j Ld,j(0),Lf,j(t) Lf,j(0) f,j Lf,j(0). (3.14)

This approximation gives the following FXT(t)-dynamics:

dFXT(t)

FXT(t)

v(t)dWT (t) +

vd(t)

jAd,jdW

d,Tj (t)

vf(t)

jAf,jdW

f,Tj (t),

dv(t) = (v v(t))dt + v(t)dWTv (t),dvi(t) = i(vi(0) vi(t))dt + i

vi(t)dW

i,Tv (t),

with i = {d, f}, A = {m(t) + 1, . . . N }, the correlations are given in (3.6) and

d,j :=jvd,jLd,j(0)

1 + jLd,j(0), f,j :=

jvf,jLf,j (0)

1 + jLf,j(0). (3.15)

We derive the dynamics for the logarithmic transformation of FXT(t), xT(t) =log FXT(t), for which we need to calculate the square of the diffusion coefficients 6.

With the notation,

a := v(t)dWT (t), b :=vd(t)jA

d,jdWd,Tj (t), c :=

vf(t)jA

f,jdWf,Tj (t),

(3.16)

we find, for the square diffusion coefficient (a + b c)2 = a2 + b2 + c2 + 2ab 2ac 2bc.So, the dynamics for the log-forward, xT(t) = log FXT(t), can be expressed as:

dxT(t) 12

(a + b c)2 +

v(t)dWT (t) +

vd(t)A

d,jdWd,Tj (t)

vf(t)A

f,jdWf,Tj (t), (3.17)

with the coefficients a, b and c given in (3.16). Since

Nj=1

xj2

=N

j=1

x2j +

i,j=1,...,Ni=j

xixj , for N > 0,

6As in the standard Black-Scholes analysis for dS(t) = vS(t)dW(t), the log-transform givesdlog S(t) = 1

2v2dt + vdW(t).

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with the final condition T(u, X(T), T) = eiuxT(T). Since all coefficients in this PDE are

linear, the solution is of the following form:

T(u, X(t), t , T ) = exp

A(u, ) + B(u, )xT(t) + C(u, )v(t)

+Dd(u, )vd(t) + Df(u, )vf(t)

, (3.22)

with := T t. Substitution of (3.22) in (3.21) gives us the following system of ODEsfor the functions A() := A(u, ), B() := B(u, ), C() := C(u, ), Dd() := Dd(u, )and Df() := Df(u, ):

A() = f(t)(B2() B())/2 + dvd(0)D1() + fvf(0)D2() + vC(),B() = 0,

C() = (B2() B())/2 + (x,vB() ) C() + 2C2()/2,Dd() = Ad(t)(B

2() B())/2 dDd() + 2dD2d()/2,Df() = Af(t)(B

2() B())/2 fDf() + 2fD2f()/2,

with initial conditions A(0) = 0, B(0) = iu, C(0) = 0, Dd(0) = 0, Df(0) = 0 with Ad(t)and Af(t) from (3.18), (3.19), respectively, and f(t) as in (3.20).

With B() = iu, the solution for C() is analogous to the solution for the ODE for theFX-HHW1 model in Equation (2.33). As the remaining ODEs involve the piecewise con-stant functions Ad(t), Af(t) the solution must be determined iteratively, like for the pureHeston model with piecewise constant parameters in [Andersen and Andreasen, 2000].For a given grid 0 = 0 < 1 < < N = , the functions Dd(u, ), Df(u, ) andA(u, ) can be expressed as:

Dd(u, j) = Dd(u, j1) + d(u, j),

Df(u, j) = Df(u, j1) + f(u, j),

for j = 1, . . . , N , and

A(u, j) = A(u, j1) + A(u, j) 1

2 (u

2

+ u) j

j1 f(s)ds,

with f(s) i n (3.20) and analytically known functions k(u, j), for k = {d, f} andA(u, j):

k(u, j) :=

k k,j 2kDk(u, j1)

(1 ek,jsj )(2k(1 k,jek,jsj )),and

A(u, j) =v

2

( x,viu dj)sj 2log

(1 gjedjsj )

(1 gj)

+vd(0)

d

2d (d d,j)sj 2log (1 d,jed,jsj ) (1 d,j)+vf(0)

f2f

(f f,j )sj 2log

(1 f,jef,jsj )

(1 f,j )

,

where

dj =

(x,viu )2 + 2(iu + u2), gj = ( x,viu) dj 2C(u, j1)

( x,viu) + dj 2C(u, j1) ,

k,j =

2k + 2kAk(t)(u

2 + iu), k,j =k k,j 2kDk(u, j1)k + k,j 2kDk(u, j1)

,

with sj = j j1, j = 1, . . . , N , Ad(t) and Af(t) are from (3.18) and (3.19).

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The resulting approximation of the full-scale FX-HLMM model is called FX-LMM1here.

3.2 Foreign Stock in the FX-HLMM Framework

We also consider a foreign stock, Sf(t), driven by the Heston stochastic volatilitymodel, with the interest rates driven by the market model. The stochastic processesof the stock model are assumed to be of the same form as the FX (with one, foreign,interest rate curve) with the dynamics, under the forward foreign measure, given by:

dSTf (t)

STf (t)=

(t)dWf,TSf (t) +

vf(t)N

j=m(t)+1

jvf,jf,j (t)

1 + jLf,j(t)dWf,Tj (t),

d(t) = f( (t))dt + f

(t)dWf,T (t). (3.23)

Variance process, (t), is correlated with forward stock ST(t).We move to the domestic-forward measure. The forward stock, STf , and forward

foreign exchange rate, FXT(t), are defined by

STf (t) = Sf(t)

Pf(t, T), FXT(t) = (t) Pf(t, T)

Pd(t, T). (3.24)

The quantity

STf (t)FXT(t) =

Sf(t)

Pf(t, T)(t)

Pf(t, T)

Pd(t, T)=

Sf(t)

Pd(t, T)(t), (3.25)

is therefore a tradable asset. So, foreign stock exchanged by a foreign exchange rate anddenominated in the domestic zero-coupon bond is a tradable quantity, which impliesthat STf (t)FX

T(t) is a martingale. By Itos lemma, one finds:

dSTf (t)FX

T(t)STf (t)FXT(t) = dFXT(t)

FXT(t) +

dSTf (t)

STf (t)+dFXT(t)FXT(t) dS

Tf (t)

STf (t) . (3.26)

The two first terms at the RHS of (3.26) do not contribute to the drift. The last terminvolves all dt-terms, that, by a change of measure, will enter the drift of the varianceprocess d(t) in (3.23).

3.3 Numerical Experiments with the FX-HLMM Model

We here focus on the FX-HLMM model covered in Section 3 and consider the errorsgenerated by the various approximations that led to the model FX-HLMM1. We haveperformed basically two linearization steps to define FX-HLMM1: We have frozenthe Libors at their initial values and projected the non-affine covariance terms on a

deterministic function. We check, by a numerical experiment, the size of the errors ofthese approximations.

We have chosen the following interest rate curves Pd(t = 0, T) =exp(0.02T), Pf(t = 0, T) = exp(0.05T), and, as before, for the FX stochasticvolatility model we set:

= 0.5, = 0.3, v = 0.1, v(0) = 0.1. (3.27)

In the simulation we have chosen the following parameters for the domestic and foreignmarkets:

d,k = 95%, vd,k = 15%, d = 100%, d = 10%, (3.28)

f,k = 50%, vf,k = 25%, f = 70%, f = 20%. (3.29)

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pricing of FX-interest rate hybrid products that are not exposed to the smile in the fixedincome markets.

For hybrid products sensitive to the interest rate skew a second model was presentedin which the interest rates were driven by the stochastic volatility Libor Market Model.

For both hybrid models we have developed approximate models for the pricing ofEuropean options on the FX. These pricing formulas form the basis for highly efficient

model calibration strategies.The approximate models are based on the linearization of the non-affine termsin the corresponding pricing PDE, in a very similar way as in our previous arti-cle [Grzelak and Oosterlee, 2011] on equity-interest rate options. The approximatemodels perform very well in the world of foreign exchange.

These models can also be used to obtain an initial guess when the full-scale modelsare used.

Acknowledgments

The authors would like to thank to Sacha van Weeren and Natalia Borovykh fromRabobank International for fruitful discussions and helpful comments.

References[Andersen, 2007] L.B.G. Andersen, Simple and efficient simulation of the Heston stochastic

volatility model. J. of Comp. Finance, 11:142, 2007.[Andersen and Andreasen, 2000] L.B.G. Andersen and J. Andreasen, Volatility Skews and

Extensions of the Libor Market Model. Appl. Math. Finance, 1(7): 132, 2000.[Andersen and Andreasen, 2002] L.B.G. Andersen and J. Andreasen, Volatile Volatilities.

Risk, 15(12): 163168, 2002.[Andreasen, 2006] J. Andreasen, Closed Form Pricing of FX Options Under Stochastic Rates

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A Proof of Lemma 2.2

Since the domestic short rate process, rd(t), is driven by one source of uncertainty(only one Brownian motion dWQd (t)), it is convenient to change the order of the

state variables, from dX(t) = [dFXT(t)/FXT(t), dv(t), drd(t), drf(t)]T to dX(t) =

[drd(t), drf(t), dv(t), dFXT(t)/FXT(t)]T and express the model in terms of the inde-

pendent Brownian motions dWQ(t) = [dWd(t), dWf(t), dWv(t), dW(t)]T, i.e.:drddrfdv

dFXT/FXT

= (X)dt +

d 0 0 00 f 0 00 0

v 0

dBd fBf 0

v

H

dWQddWQfdWQvdWQ

, (A.1)which, equivalently, can be written as:

dX(t) = (X)dt + AHdWQ(t), (A.2)where (X) represents the drift for system dX(t) and H is the Cholesky lower-triangular matrix of the following form:

H =

1 0 0 0

H2,1 H2,2 0 0H3,1 H3,2 H3,3 0H4,1 H4,2 H4,3 H4,4

=

1 0 0 0f,d H2,2 0 0v,d H3,2 H3,3 0,d H4,2 H4,3 H4,4

. (A.3)The representation presented above seems to be favorable, since the short-rate processrd(t) can be considered independently of the other processes.

The matrix model representation in terms of orthogonal Brownian motions resultsin the following dynamics for the domestic short rate rd(t) under measure Q:

drd(t) = d(d(t) rd(t))dt + 1(t)dWQ(t),and for the domestic ZCB:

dPd(t, T)

Pd(t, T)= rd(t)dt + Bd(t, T)1(t)dWQ(t),

with k(t) being the kth row vector resulting from multiplying the matrices A and H.Note, that for the 1D Hull-White short rate processes 1(t) =

d, 0, 0, 0

.

Now, we derive the Radon-Nikodym derivative [Geman et al., 1996], TQ(t),:

TQ(t) =dQT

dQ=

Pd(t, T)

Pd(0, T)Md(t). (A.4)

By calculating the Ito derivative of Equation (A.4) we get:

dTQ

TQ= Bd(t, T)1(t)dWQ(t), (A.5)

which implies that the Girsanov kernel for the transition from Q to QT is given byBd(t, T)1(t) which is the T-bond volatility given by dBd(t, T), i.e.:

TQ = exp

1

2

T0

B2r (s, T)21 (s)ds +

T0

Br(s, T)1(s)dWQ(s) . (A.6)So,

d

WT(t) = Bd(t, T)T1 (t)dt + d

WQ(t).

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on cross-currency models with stochastic volatility and correlated-fx

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