Benett.kennedy-Quanto With Copula

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Quanto pricing with Copulas

Michael N. Bennett, Joanne E. Kennedy

University of Warwick

May 29, 2003

Abstract

We study the practical problem of pricing a particular multi-assetoption, a quanto FX option. The Black model, which corresponds to a

jointly lognormal distribution of asset prices at expiry, is inconsistent withthe implied volatility smile for each of the three relevant currency pairs.We demonstrate a practical methodology for constructing a model forthe joint distribution that is calibrated to all relevant implied volatilities.The margins of this distribution are determined separately in an initialstage. To calibrate the joint distribution to the implied volatility smile onthe remaining FX rate, we perturb the dependence structure associatedwith the Black model (the Normal copula) in order to influence the taildependence characteristics of the resulting joint distribution.

We calibrate our model to a number of real-life scenarios correspondingto several maturities and currency set-ups. We find that a well-known ad-hoc adjustment to the Black pricing formula often gives lower quanto call

prices than those calculated under our transformed copula model. Therelative difference in quanto prices with strikes furthest away from at-the-money is occasionally large (10-15%).

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1 Introduction

The price of any European multi-asset option may be written as an integral

involving the joint density of the asset prices at expiry. Information regarding

this joint density is given by the market prices of financial options written

on the relevant underlying assets. Under the Black model, asset prices are

jointly lognormal (under the appropriate equivalent martingale measure) and

analytical formulae are available. However, both the marginal distributions and

the dependence structure associated with this model may not be consistent with

information given by the prices of vanilla options, since implied volatilities are

strike-dependent. Constructing an alternative model with realistic properties

that may be calibrated to all relevant market data is a very general problem

in finance. Quanto FX options, spread options, basket options etc. are allexamples of such multi-asset pricing problems. It is extremely important to

practitioners to find an effective solution.

In our pricing methodology, we employ a copula function as a model for

the dependence structure. A copula provides the link between the multivariate

joint distribution and the univariate marginals. This allows us to separate the

modelling of the marginal distributions from the modelling of the dependence

structure, permitting a two-stage calibration. The concept of obtaining implied

marginal distributions from market prices of vanilla options is certainly not new

(see, for example, [Dupire, 1994]) and there is a growing literature in this area.

However, there currently exists no literature concerning the practical extraction

of the entire joint distribution of asset prices from implied volatilities.The application of copulas in finance has recently attracted a great deal of

interest. [Embrechts et al, 2001] provides a survey of copulas and dependence

concepts in relation to finance, with particular focus on elliptical, Archimedean

and Marshall-Olkin copulas and applications in insurance risk and market risk

(VaR). In relation to option pricing, practical interest has recently focused on

credit derivatives, where simple parametric copulas are used to capture the de-

pendence between default and asset prices (see, for example,

[Schonbucher at al, 2001]). However, in this case the dependence structure is

given exogenously and is not implied from prices of vanilla options. [Rosenberg, 2003]

applies a similar methodology to the FX problem we consider, estimating the

dependence function via a non-parametric method based on historical return

data (see Section 3.2).

Considering the standard quanto FX option, market prices of relevant vanilla

options provide both information regarding the marginal distributions of the two

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relevant FX rates at expiry and also partial information on the dependence struc-

ture (see Section 3.1). Calibration of marginal distributions is accomplished inan initial stage. Given these marginal distributions, we find that the depen-

dence structure associated with the Black model (the bivariate Normal copula)

is insufficiently flexible to permit calibration to the final set of available im-

plied volatilities. We accomplish this final calibration step by modifying the tail

dependence structure of the Normal copula using a suitable parametric trans-

formation. This transformation is based on a generic parametric transformation

of a bivariate copula first suggested in [Genest, 2000]. The resulting model is

calibrated to the prices of vanilla options for various maturities and currency

set-ups.

We have only looked at one multi-asset option in this paper. As mentioned

above, this methodology is clearly applicable to other multi-asset options suchas spread options. Also, this methodology may have more general application in

many real-world statistical problems where the Normal copula does not provide

a sufficiently accurate model for the dependence structure under consideration,

for example in scenario analysis or stress-testing.

The rest of this paper is organised as follows: In Section 2, we describe the

problem of pricing a standard quanto FX call option and outline current market

practice. In Section 3, we give an overview of our proposed pricing methodology,

introduce the notion of dependence modelling with copulas and describe how an

appropriate transformation of the Normal copula provides a suitable model for

the dependence structure, which may be calibrated to market prices of vanilla

options. We present the results of calibration to actual market data for several

currency set-ups and maturities in Section 4. In Section 5, the prices of a quanto

FX option under our proposed model are compared against prices calculated

using the Black model and those given by an alternative ad-hoc approximation.

Our conclusions are presented in Section 6.

2 Motivation

2.1 Quanto Call Option Pricing Problem

Consider the standard quanto FX call option pricing problem. Let Dit,T denote

the time-t value in currency i of the zero coupon discount bond with maturity

T. Let Qi be the equivalent martingale measure associated with this numeraire.

Also let Xi,jt , (i = j) denote the value in currency i of one unit of currencyj at time t. For an arbitrage-free economy we must have Xi,jt = (X

j,it )

1 and

Xi,jt = Xk,jt /X

k,it .

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For the quanto pricing problem we only need to consider three currencies.

A standard quanto FX call option is a contract which pays the holder a totalof (Xi,jT K)+ in the quanto currency k, where is a known constant,the conversion factor. Note that is simply a scaling factor. Currency i is

commonly referred to as the money currency and currency j as the dealt

currency. Since we may renumber currencies at our convenience, without loss of

generality we choose currency one to be the quanto currency, currency two to be

the money currency and currency three to be the dealt currency. By standard

arbitrage pricing theory, the value of this quanto call option at time zero is given

by

Cquanto0 = D10,TEQ1 [(X

2,3T K)+], (1)

where Q1 is the equivalent martingale measure associated with the numeraire

D1t,T. Therefore the value of the quanto option is completely determined if the

distribution of X2,3T under Q1 is known.

This quanto FX option must be priced consistently with the prices of vanilla

options written on the three relevant underlying FX rates, X1,2T , X1,3T and X

2,3T .

The prices of vanilla options on each of these FX rates are determined by the

market for a finite set of strikes.

The prices of vanilla options on X1,jT (j = 2, 3) allow us to recover the

marginal distributions of X1,jT under Q1, under suitable assumptions regarding

the shape of the distribution. This subproblem has received a great deal of

attention recently in the literature and we proceed by choosing a simple param-

eterisation of the distribution, described in Section 3.1.1.Prices of vanilla options on X2,3T give information about the distribution

of this FX rate under Q2. A change of measure shows these options provide

partial information regarding the joint distribution of (X1,2T , X1,3T ) under Q

1,

since X2,3T = X1,3T /X

1,2T .

Under a suitable model for the joint density, we may calculate the distri-

bution of X2,3T under Q1 and hence the price of the quanto FX option. The

problem is therefore to formulate a model such that we capture the dependence

structure implied by the third set of option prices (on X2,3T ) while incorporating

the information provided about the marginal distributions given by the first two

sets of prices (on X1,2T and X1,3T respectively).

In the following, we choose to separate the modeling of the marginal distri-

butions and the dependence structure by specifying the dependence structure

in terms of a copula function, described in Section 3.2. Note that this allows us

to interchange the method we have used in Section 3.1.1 to obtain the marginal

distributions referred to above with other more sophisticated methods. Once we

have calibrated the marginal distributions to the first two sets of option prices,

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the problem remains to choose a suitable parameterisation of the dependence

structure such that the associated joint density is well calibrated to the thirdset of vanilla option prices.

2.2 Current market practice

In the market, the standard approach to pricing quanto FX options is based

on a Black-type model where we assume correlated lognormal dynamics for the

forward FX rates. Let Mi,jt,T denote the forward FX rate at time t with maturity

T quoted in units of currency i per unit of currency j. Then by no arbitrage,

Mi,jt,T = Djt,TX

i,jt /D

it,T

andMi,jt,T = (M

i,jt,T)

1, Mi,jt,T = Mk,jt,T/M

k,it,T.

Under the Black model, we assume the forward rates M1,jt,T follow the process

dM1,2t,T = (1) M1,2t,T dW

1t , dM

1,3t,T =

(2) M1,3t,T dW2t

under the equivalent martingale measure Q1 corresponding to the numeraire

D1t,T, where (1) and (2) are (strike-independent) volatilities of vanilla options

on X1,2T and X1,3T respectively and W

1t and W

2t are standard Brownian motions

under Q1 such that

dW1t dW2t = dt.

It is straightforward to show that under Q2

,

dM2,3t,T = (3) M2,3t,T dW

3t

where

((3))2 = ((1))2 + ((2))2 2(1)(2) (2)and W3t is a standard Brownian motion under Q

2.

Consider a contract which pays VT = (X2,3T K) at time T in currency

one. Taking the expectation under Q1, the usual pricing formula yields

V0 = (M2,30,Te

(((1))2(1)(2))(T) K),

which is zero if the strike K is given by the quanto forward rate

M2,30,T

= M2,30,Te(((1))2(1)(2))T.

In the Black model, X2,3T is lognormal under Q1, thus evaluating the expec-

tation (1), we find the price of the quanto option is given by

Cquanto0 = D10,T[M

2,30,T

(d1) K(d2)] (3)

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where

d1 =log(M2,3

0,T

/K)

(3)

T+ 12

(3)T , d2 = d1 (3)T

and () is the standard cumulative Normal distribution function. Note thatunder the Black model, (3) is the volatility of vanilla options on X2,3T , given

by Equation (2). Thus given the (strike-independent) volatilities (1), (2) and

(3), we may infer the implied correlation under the Black model.

The above model is a good benchmark against which to compare more so-

phisticated models. To understand the deficiencies of this model, it is necessary

to look at the implied volatilities corresponding to the vanilla options written

on the relevant underlying assets. Recall that it is standard market convention

to quote a vanilla FX option price in units of its implied volatility, that is, the

corresponding value of the volatility parameter which gives the correct market

price when entered into the Black formula. Thus the Black formula is simply

used as a 1-1 mapping of market prices to Black volatilities. 1

In practice the Black volatility corresponding to a quoted vanilla option

price is dependent on the strike of the option, indicating that the assumptions

underlying the Black model do not hold (this well-known feature is termed

the volatility smile). Therefore the assumption that X1,2T , X1,3T are jointly

lognormal under Q1 may b e inappropriate. In general, it is not possible to

obtain an analytical form for the distribution of X2,3T . To incorporate the effect

of the volatility smile, practitioners often adopt an ad-hoc approach and modify

the Black formula (3) as follows. The value of may be determined as usual from

the at-the-money volatilities of options on X1,2T , X1,3T and X

2,3T using Equation

(2), then an ad-hoc approximation for the price of the quanto option may be

found by substituting the actual strike-dependent volatility (3)(K) for (3) in

Equation (3), where (3)(K) is the Black volatility corresponding to the price of

the vanilla option X2,3T with the same strike K as the quanto option in question.

In Section 5.2 we compare these Black-adjusted prices with those obtained

by the copula approach, described below. Note that this ad-hoc modification

of the Black formula does not provide a model for the joint distribution of X1,2Tand X1,3T . Therefore the only comparison with the copula approach available to

us is a simple comparison of calculated quanto prices for a range of strikes.

1Recall that the at-the-money FX option is that which has strike given by the currentforward FX rate. Options with higher (lower) instrinsic value are known as in-the-money(out-of-the-money) options. Market convention also dictates that the strike of a vanilla FXoption is quoted in terms of the corresponding delta, i.e. the value of the delta parameterwhich gives the correct value of the strike of the option under the Black model, given theimplied volatility of the option. By convention, the at-the-money volatility quote is usedin this calculation. For a more detailed description of market practice, see, for example,[Malz, 1997].

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3 Methodology

3.1 Overview of pricing methodology

The price of a standard quanto option may be evaluated directly given the dis-

tribution of X2,3T under Q1, which in turn may be computed from the joint

distribution function of X1,2T and X1,3T under Q

1. We model this joint distribu-

tion function such that it is well calibrated to all relevant vanilla FX options as

follows.

In our modeling framework we choose to specify the dependence structure

between X1,2T and X1,3T under Q

1 using a function known as a copula (described

in Section 3.2). This allows us to separate the modeling of the implied marginal

distributions (incorporating information given by market quotes for the prices

of vanilla options on X1,2T and X1,3T ), from the modeling of the dependence

structure (for which we have only partial information given by the prices of

vanilla options on X2,3T ).

In the first stage of this methodology, we choose to model the implied

marginal distributions by parameterising each distribution as a mixture of log-

normals (outlined in Section 3.1.1). Since we subsequently model the depen-

dence structure via a copula function, this method may be interchanged with

more sophisticated methods without affecting the remainder of the methodology.

Once the marginal distributions have been determined, the joint distribution

of X1,2T and X1,3T under Q

1 may be computed under the assumption that the

dependence structure is modeled appropriately using a given copula. Underthis model for the joint distribution, prices of vanilla options on X2,3T may be

calculated by numerical integration. The problem remains to choose a suitable

copula such that that the corresponding joint distribution can be calibrated to

market quotes for the prices of vanilla options on X2,3T .

There are infinitely many models for the joint distribution which are well-

calibrated to all relevant market prices since the available prices of vanilla op-

tions on X2,3T only provide partial information regarding the dependence struc-

ture. We proceed to model the dependence structure by perturbing the Normal

copula, which is the model for the dependence structure associated with the

Black pricing formula.

Under the Black pricing model, implied volatilities are independent of thestrike. The correlation parameter controls the overall level of association

between X1,2T and X1,3T under Q

1 and therefore the level of (flat) volatilities

of vanilla options on X2,3T . In the general case implied volatilities are strike-

dependent, so the Black model can only be calibrated to at-the-money volatili-

ties (by choosing the value of to satisfy (2)). We shall see that if we replace

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the lognormal marginals by those actually implied by vanilla options on X1,2T

and X1,3

T , we find that the parameter of the Normal copula still controls theoverall level of association and consequently the general level of volatilities of

options on X2,3T (see Figure 3, Section 4.2).

Given the implied marginal distributions, we shall find that typically it is

necessary to alter the tail dependence characteristics of the Normal copula in

order to match the volatility smile of vanilla options on X2,3T (see Section 3.3.3).

We choose to accomplish this by modeling the dependence structure using a

specific parametric transformation of the Normal copula (based on the generic

copula transformation suggested in [Genest, 2000]). We shall see that the pa-

rameter of the original Normal copula still controls the overall level of volatil-

ities of options on X2,3T under this new model for the joint distribution provided

the transformation function exhibits certain qualitative features.Calibration to implied volatilities of options on X2,3T is performed by a non-

linear weighted least squares optimisation procedure. The resulting joint distri-

bution is consistent with distributional information provided by vanilla options

on all three FX rates.

3.1.1 Implied marginal distributions

We now consider the problem of recovering the implied marginal distributions

of the FX rates X1,jT (j = 2, 3) from market quotes of prices of vanilla options

written on these FX rates, which are available for a finite set of strikes. 2 One

suitable method for constructing the marginal distributions such that they areconsistent with this market data is to assume a suitable parametric form such

as a mixture of two lognormal distributions.

This method is computationally simple, efficient and the optimisation ap-

pears to be relatively stable (see [Bahra, 1997]). It is often used as a bench-

mark with which to compare more sophisticated models (see, for example,

[Bliss et al, 2000], [Campa et al, 1997] and [Coutant et al, 1998])). The mix-

ture of lognormals model clearly reduces to the special case of a single lognor-

mal distribution (Black model) in the case that the mixture parameter is one

or zero; this is extremely useful in testing the numerical implementation of the

following copula model. Both density and distribution functions are analytical

functions of the cumulative Normal distribution function.The mixture of lognormals distribution is calibrated to market data as fol-

lows. A vanilla call option on the FX rate X1,jt , (j = 2, 3) with strike K gives

2Of course, if we did know prices of vanilla options for all strikes this would completelyspecify the marginal distributions ofX

1,jT (j = 2, 3) (see [Dupire, 1994]).

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the holder a payoff of (X1,jT K)+ at time T, hence by the usual arbitragepricing argument the value of this option at time zero is given by

C1,j0 = D10,TEQ1 [(X

1,jT K)+]. (4)

This expectation may be written as an integral involving the probability density

function of X1,jT under Q1.

If the distribution of X1,jT under Q1 is assumed to be a mixture of lognor-

mals, analytical formulae are obtained for prices of vanilla options in terms of

the standard cumulative Normal distribution function (see Appendix A for de-

tails). The density may be fitted to market quotes by performing a suitable

optimisation to find the density parameters which minimise the squared error

between model and market prices of the relevant vanilla options. Notice that

the forward FX rate M1,j0,T is the mean of the marginal distribution ofX1,jT under

Q1. This is fitted exactly, reducing the number of free distribution parameters

by one.

Note that this method always results in proper distribution and density

functions which are both smooth and differentiable; this is important in the

following derivations and is not necessarily apparent in some methods.3 A

number of other methods may have been suggested with superior numerical

properties at the expense of more complex implementation.4 However, as noted

earlier it is possible to substitute an alternative to this method of obtaining the

marginal distributions without affecting the remainder of the methodology.

3.1.2 Partial Information on Joint Distribution

We now consider the partial information provided about the joint distribution

of X1,2T and X1,3T under Q

1 which is given by the prices of vanilla options on

X2,3T . The value in currency two of a vanilla call option on X2,3T with strike K

at time zero is given by

C2,30 (K) = D20,TEQ2 [(X

2,3T K)+],

3We require that the density f1,j() is both continuous and differentiable, with f1,j(0) = 0,

0 f1,j(u) du = 1 and f1,j(u) > 0 for all u R. In some smoothing-type methods where

call volatilities are interpolated and differentiated with respect to the strike to obtain thedistribution (following [Dupire, 1994]), lognormal tails are pasted onto the central part of the

distribution and it is not always straightforward to smooth the density function across the join whilst ensuring

0 f1,j(u) du = 1 .

4The mixture of lognormals model has the disadvantage that the single optimal fit availableis not guaranteed to match market quotes to sufficient accuracy in all cases. More sophisti-cated smoothing-type methods usually have a goodness-of-fit parameter allowing a closerfit to market quotes at the cost of reduced smoothness. [Bliss et al, 2000] indicate that thesemethods may offer a more stable fit by observing the effects of perturbing market quotes. Alsosee [Malz, 1997], [Sherrick at al, 1996] and [Ait-Sahalia at al, 1998].

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where Q2 is the equivalent martingale measure corresponding to the numeraire

D2t,T. We show that prices of such options provide information on the values of

integrals over specific regions of the joint distribution of (X1,2T , X1,3T ) under Q

1,

via a change of measure.

The Radon-Nokodym derivative ofQj relative to Qi is given by

dQj

dQi

Ft

=Mi,jt,T

Mi,j0,T,

so as a consequence of the Radon-Nikodym theorem

EQ2 [(X2,3T K)+] = EQ1

dQ2

dQ1

Ft

(X2,3T K)+

= X2,10D10,TD20,T

EQ1 [X1,2T (X

2,3T K)+].

Therefore

C2,30 (K) = X2,10 D

10,TEQ1 [(X

1,3T KX1,2T )+]

= X2,10 D10,T

0

Kv

(u Kv)fQ1X1,2

T,X1,3

T

(u, v) du dv, (5)

where fQ1

X1,2T

,X1,3T

(, ) is the joint density of X1,2T and X1,3T under Q1. If this jointdensity is known, the integral in Equation (5) may be evaluated numerically and

hence vanilla call option prices C2,30 (K) may be calculated for various strikes Kand compared against market quotes.

If the marginal distributions of X1,3T and X1,2T are already determined (as

described in Section 3.1.1), the model of the joint distribution is completed by

specifying a suitable model for the dependence structure of (X1,3T , X1,2T ) under

Q1. The remaining practical problem is to choose the copula to match the partial

information given by the integrals (5) corresponding to the market prices of

vanilla options on X2,3T for different strikes K.

3.2 Dependence Modeling with Copulas

A bivariate copula is a function which allows us to investigate the scale-invariantdependence structure of two variables. It provides the link between the marginal

distribution functions and the joint distribution function. By changing the

copula it is possible to alter the dependence structure of the joint distribution

without altering the marginal distributions.

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Here it is convenient to define a bivariate copula as a bivariate distribution

function whose one-dimensional margins are uniform on the interval [0,1]. As-suming we know the margins, the Theorem below shows that we can obtain the

copula distribution from the joint distribution and vice versa.

There is a growing literature on copulas and their application in finance. For

formal definitions, theoretical properties and comparisons of common paramet-

ric copulas we refer the reader to [Nelson, 1998] and [Joe, 1997]. [Embrechts et al, 2001]

provides a survey of copulas and dependence concepts in relation to finance, with

particular focus on Marshall-Olkin, elliptical and Archimedean copulas and ap-

plications in insurance risk and market risk (VaR).

Recent work has recognised the potential application of copulas in derivative

pricing, especially in the field of credit derivatives. However, the copula is gener-

ally specified exogeneously. Usually either the copula is assumed to take a par-ticular parametric form, which may have more desirable dependence character-

istics than the Normal copula for the particular application (such as the Student

t copula), or the copula is estimated using historical data. [Rosenberg, 2003]

performs the latter, applying non-parametric techniques to estimate the depen-

dence copula between two FX rates, while using implied volatilities to construct

the margins. However, the implied volatilities on the cross-FX rate X2,3T do not

match market quotes. Practitioners usually need a model that is well calibrated

to current market prices. There is presently no literature regarding the practical

application of copulas to option pricing where the copula is calibrated to the

prices of vanilla options.

3.2.1 Sklars Theorem

Sklars theorem explains the role of copulas in describing the relationship be-

tween a multivariate joint distribution function and its univariate marginal dis-

tributions.

Theorem (Existence) Let H be a standard joint distribution function and let

F and G be the margins of H. Then there exists a copula C such that for all

x, y R,H(x, y) = C(F(x), G(x)). (6)

Corollary If the marginal distribution functions F and G are strictly increasing,

then

C(u, v) = H(F1(u), G1(v)).

Since it is a prerequisite that the marginal densities of X1,2T and X1,3T under

Q1 are continuous, if we have in mind a model for the continuous joint density

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we may calculate the unique copula associated with this joint density and vice

versa.

3.2.2 Parametric copulas

The most common practical example of a copula is probably the Normal copula,

which has distribution function

CN(u, v | ) = (1(u), 1(v)),

where is the standard bivariate Normal distribution function with correla-

tion and is the standard univariate Normal distribution function. By the

Corollary above, this is the copula implied by a standard bivariate Normal joint

distribution function with standard univariate Normal marginal distributions.It is clear that this elliptical copula exhibits both positive and negative depen-

dence (for positive and negative values of respectively).

In the following we shall refer informally to the behaviour of the copula in

the extremes of the upper-right and lower-left quadrants of the domain as upper

and lower tail dependence respectively. Thus if we perturb the Normal copula to

increase upper tail dependence, we will observe that the probability of observing

very high values in both variables is higher than under the Normal copula.

The Normal copula defined above describes the dependence structure asso-

ciated with the Black model. We shall see below that in order to calibrate our

joint distribution to vanilla options on X2,3T , we require a dependence model

with much greater flexibility. Specifically, the copula must exhibit both posi-

tive and negative overall dependence structures and it must be possible to alter

simultaneously the level of upper-tail and lower-tail dependence (see Section

3.3.3).

For most common parametric copulas in the literature it is not possible to

alter the overall level of dependence and the upper and lower tail dependence

characteristics simultaneously. This is a feature of all single parameter copulas.

A few two-parameter copulas have been studied in the literature, but these are

generally quite inflexible and it is difficult to construct a multi-parameter copula

directly which has the desired properties. However, we shall see that a copula

with all the required characteristics may be constructed by a transformation of

the Normal copula (see Section 3.3.3).

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3.3 Fitting a copula using the information available on the

joint distribution3.3.1 Calculation of vanilla call prices under a given copula model

Suppose we have in mind a suitable model for the dependence structure of X1,2Tand X1,3T under Q

1 in terms of a specific copula. In order to assess the fit of this

copula, we need to be able to calculate the prices of vanilla options on X2,3T .

Using the marginal distributions which have already been determined (as

described in Section 3.1.1), it is possible to recover the associated joint density

function as discussed below. From this joint density, prices of vanilla options

on X2,3T may be calculated by numerical integration (see Section 3.3.2).

Denote the joint distribution function ofX1,2T and X1,3T underQ

1 by FQ1

X1,2T

,X1,3T

(, )and the associated marginal distribution functions by FQ1

X1,2T

() and FQ1X1,3

T

() re-spectively. If the dependence structure between X1,2T and X

1,2T under Q

1 may

be modeled appropriately by a known copula with density function c(, ), thenusing Sklars theorem (6), the joint density is given by

fQ1

X1,2T

,X1,3T

(u, v) = fQ1

X1,2T

(u)fQ1

X1,3T

(v)c(FQ1

X1,2T

(u), FQ1

X1,3T

(v)).

If the copula density is known, we may now calculate the price of a vanilla

option with a given strike by evaluating the integral in Equation (5) numerically.

These model prices may then be compared against market quotes.

3.3.2 Reformulating Integration Region

We now consider the integration regions corresponding to the integral in Equa-

tion (5) for different strikes. These integrals correspond to the prices of vanilla

options on X2,3T . We need to understand how the Normal copula must be modi-

fied such that the resulting joint distribution is well calibrated to market quotes

for a given set of strikes. We find that contour plots of copulas vs. Normal

marginals provide a very helpful clue as to how to accomplish this.

Suppose for the moment that the marginal distributions FQ1

X1,jT

() are bothlognormal (j = 1, 2). Then for a given strike K, the integration region associated

with the integral in Equation (5) corresponds to the region of the copula density

below the line log(v) = log(u) + K when plotted against Normal marginals (seeFig. 1). In the general case, of course, the integration region will not take such

a simple form. However, since the observed implied marginals will not be too

far removed from the single lognormal model, any intuition we can gain in this

simple setting should carry over to a more realistic model for the marginals.

13


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log(u)

log

(v)

-2 -1 0 1 2

-2

-1

0

1

20.02

0.02

0.04

0.060.08

0.10.12

0.140.16

0.18

0.2

0.22

u=v

log(v)=log(u)+K

Figure 1: Where the joint distribution ofX1,2T and X1,2T under Q

1 has lognormal mar-gins (i.e. flat volatilities), the integration region associated with integral in Equation(5) is the shaded region below the line log(v) = log(u)+ K. For high strikes, integrationregion is moved further to the bottom right. Contours shown correspond to a Normalcopula with positive correlation.

log(u)

log

(v)

-2 -1 0 1 2

-2

-1

0

1

2

0.02

0.02

0.04

0.06

0.080.1

0.120.14

0.160.18

0.2

0.2

0.20.2

0.2

0.22

u=v

log(u)+log(v)=K

Figure 2: The reformulated integration region associated with integral in Equation(7) is that above the line log(u) + log(v) = K. For high strikes, this region is movedfurther to the upper right. Contours shown correspond to the same Normal copulashown in Fig. 1.

14


15/44

Suppose now that we wish to modify the Normal copula to fit the strike-

dependent volatilities of vanilla call options on X2,3

T , but still retain a symmetricstructure, i.e. c(u, v) = c(v, u) for all u, v R. If the implied volatility of avanilla call with a high strike is higher than expected under the Normal copula

model, then if we wish our copula model to correctly price this option we must

increase the value of the integral (5). This may be accomplished by increasing

the mass of the copula density corresponding to high values of u and low values

of v (see Fig. 1). However, due to the symmetry of the copula this would also

increase the mass of the copula density corresponding to low values of u and

high values of v, thus also increasing call prices with low strikes simultaneously.

A symmetric structure is a feature of most copulas that are suitable for

practical applications. The application of non-symmetric copulas is a more

complex problem requiring much greater computational complexity. We notethat there is very little literature regarding the practical application f non-

symmetric copulas. It is therefore desirable to reformulate the model so that

symmetric copulas may be used.

If we specify the joint density in terms of 1/X1,2T = X2,1T and X

1,3T under Q

1,

C2,30 (K) = X2,10 D

10,T

0

K/v

(u K/v)fQ1X2,1

T,X1,3

T

(u, v) du dv, (7)

where

fQ1

X2,1T

,X1,3T

(u, v) = fQ1

X2,1T

(u)fQ1

X1,3T

(v)c(FQ1

X2,1T

(u), FQ1

X1,3T

(v)) (8)

and both fQ1X

2,1T

() and FQ1X

2,1T

() may be found from fQ1X

1,2T

() and FQ1X

1,2T

() directly(see Appendix A). If the margins are lognormal, the region of the copula cor-

responding to the integration region of Equation (7) is illustrated in Fig. 2 for

a given strike. The values of integrals (7) for different values of K may now

be altered independently using a symmetric copula by shifting the mass of the

copula density along the line u = v. Thus relatively high call prices for high

strikes may be obtained by increasing upper tail dependence and so forth. By

controlling the upper and lower tail dependence of the copula, it is possible to

alter model call prices for different strikes such that they match market quotes.

Equations (7) and (8) allow us to calculate prices of vanilla options un-

der our copula model for any given strike. Of course, under the Black model(corresponding to lognormal margins and a Normal copula with correlation ),

Equation (7) may be evaluated analytically. The implied volatility associated

with these call prices is given by

((3))2 = ((1))2 + ((2))2 + 2(1)(2), (9)

15


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where (1) and (2) are the (flat) implied volatilities of call options on X1,2T and

X1,3

T respectively.5

3.3.3 Transformation of Normal Copula

The copula which we calibrate to the prices of vanilla options on X2,3T will

not be unique because these prices only give partial information regarding the

dependence structure. However, we have greater confidence in our inferences if

our chosen copula model is close to the Normal copula, which is the dependence

model associated with the Black pricing formula. Therefore the copula we choose

to fit to market data is a perturbation of the Normal copula.

We construct a new symmetric copula by a suitable transformation of the

Normal copula, modifying the tail dependence structure such that the resulting

joint density is well calibrated to prices of vanilla options on X2,3T . The following

Theorem gives sufficient conditions under which the transformed copula is itself

a copula (for proof see Appendix B).

Theorem [Genest, 2000] Let : [0, 1] [0, 1] be a continuous, twice differen-tiable concave function such that (0) = 0 and (1) = 1. Then

C(u, v) := 1(C((u), (v))) (10)

is a copula if C(, ) is a copula.

The density of the transformed copula C is obtained by taking the partial

derivative of (10) with respect to both arguments;

c(u, v) =(u)(v)

(C(u, v))

c((u), (v))

(C(u, v))

[(C(u, v))]2C

u((u), (v))

C

v((u), (v))

, (11)

where c(, ) is the density of the original copula C.This transformation allows us to construct infinitely many new copulas from

any given copula. If our starting point is the Normal copula with parameter

, then we may use the transformation function to modify the tail dependencecharacteristics of the copula as required. Note that the identity transformation

(x) = x does not alter the original copula.

Observing the form of Equation (10), notice that the behaviour of (x) near

x = 1 controls the upper tail dependence of the transformed copula. To see

5Note has changed sign, c.f. Equation (2).

16


17/44

this, note that for u, v both near one, C((u), (v)) will be near one, hence it

is the behaviour of (x) and 1

(x) for x close to one which controls the levelof C(u, v). Similarly, the behaviour of (x) near x = 0 controls the lower tail

dependence of the transformed copula.

Henceforth suppose the original copula C in (10) is the Normal copula. We

require that the second derivative (x) of the transformation function be near

zero around x = 12

or else the transformation will significantly alter the overall

dependence characteristics of the copula. Provided this is the case, experimen-

tation shows that the parameter still controls the overall dependence charac-

teristics of the copula. This allows us to use the endpoints of the transformation

function to alter the tail dependence characteristics. Increasing the size of the

second derivative of the transformation near zero (one) is observed to increase

the lower (upper) tail dependence of the transformed copula.Heuristically, how can we see this from Equation (11)? We are performing

a slight modification of the Normal copula, hence the optimal transformation

function should be close to (x) = x. For transformation functions with char-

acteristics described above, increasing the size of () < 0 near zero or onewill increase the second term of Equation (11), which will eventually dominate

since () will be close to one and the tails of cN((u), (v)) decay exponen-tially. This follows because the quantities (), () and CNu (, ) do not varyas fast as (). The mass of the copula density is shifted toward those regionscorresponding to lower or upper tail dependence.

If the copula C is a Normal copula then the transformed density function

(11) may be evaluated directly because the normal copula density cN and the

partial derivative CNv are analytical (only requiring an approximation to the

standard Normal cumulative distribution, see Appendix B) and the distribution

function of the bivariate Normal copula CN(, ) may be approximated to highaccuracy (see, for example, [Genz, 2002]).

It remains to parameterise the transformation function such that its second

derivative may be modified in order to alter the upper and lower tail dependence

as required.

3.3.4 Calibration of transformation function

Given the correlation of the Normal copula which defines the dependencestructure of X2,1T and X

1,3T under Q

1 and assuming the transformation function

has been parameterised and satisfies the conditions of the above Theorem,

the density of the transformed Normal copula may be computed, hence the joint

density of X2,1T and X1,3T under Q

1 may be found using Equation (8). Vanilla

call option prices C2,30 (K) are computed under this model by evaluating the

17


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integral in Equation (7). These prices can then be compared against market

quotes.A non-linear optimisation may be performed to find the values of the trans-

formation parameters which give the closest match of model and market vanilla

option prices. A standard non-linear least squares algorithm will find the opti-

mal solution z Rn which minimises the objective function

r(z)2 =mi=1

ri(z)2,

where n m.Let the residuals ri be given by

ri(z) = wi(observedi

modeli (z)), i = 1, . . . , m

where observedi are the available market implied volatility quotes associated

with the prices of vanilla options on X2,3T with strikes Ki, modeli (z) are the cor-

responding implied volatilities calculated under the transformed copula model

with parameter vector z Rn and wi are given weights.In order to modify the tail dependence structure of the Normal copula, it

is necessary to construct a transformation function with some flexibility, incor-

porating our prior intuition regarding the shape of a suitable transformation

function as discussed in Section 3.3.3. Therefore we choose to specify as a

cubic spline with k + 1 suitable predefined knot points pi [0, 1], such that theconstraints on in the Theorem of Section 3.3.3 are satisfied.

Since the size of the second derivative gives a good indicator of the increase

in tail dependence, it makes sense to parameterise this directly. Hence is

found by specifying the second derivative of the spline at each knot point, then

recovering the coefficients of the spline by making use of the requirement that

(0) = 0 and (1) = 1.

At each optimisation step, the first component of z = (z1, . . . , zn)T may be

used to determine the correlation of the Normal copula (1, 1) (for example,via := sin(z1)).

Suppose we choose knot points p = (p0, p1, . . . , pk)T with p0 = 0 and pk = 1.

Define the cubic spline between the jth and (j + 1)th nodes by

Sj(x) :=3i=0

ai,j(x pj)i, (12)

for x [pj , pj+1] and coefficients ai,j to be determined.For a well-defined optimisation we have n m, so may be constructed

using at most m1 parameters. Since 12

Sj (xpj) = a2,j +3a3,j(xpj), setting

18


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a2,j := z2j+1 < 0 (j = 1, . . . , k 1) leads to a negative second derivative for allx [0, 1] if the endpoints of the cubic spline

(0) = S0 (0) := 2a2,0 < 0, (1) = Sk (0) := 2a2,k < 0

are given. Solving for the other coefficients by incorporating the endpoints

(0) = 0 and (1) = 1 results in a cubic spline with negative curvature (see

Appendix C). However, the size of the predefined values of (0) and (1)

will have a material impact on the shape of .

In the current pricing problem we calibrate to prices of vanilla options for

m = 5 strikes. Since the objective of the transformation is to alter tail depen-

dence and not the general dependence structure (which we control by altering ),

we expect the second derivative of the optimal solution to be near zero around

x = 12

(see discussion in Section 3.3.3). Therefore we introduce the condition

( 12 ) = 0, reducing the number of free parameters to be fitted by one. Let knot

points be given by p = (0, p1,12

, p2, 1)T where p1, p2 are given. Then using n = 5

parameters, the transformed copula is fully specified by setting := sin(z1) and

constructing as a cubic spline with these knot points satisfying

a2,0 := z22 , a2,1 := z23 , a2,2 := 0, a2,3 := z24 , a2,4 := z25with endpoints (0) = 0 and (1) = 1. For details see Appendix C.

At each optimisation step, we propose a parameter vector z which corre-

sponds to a value of and a candidate transformation function given by the

cubic spline . We require that satisfy all of the conditions of the Theorem of

Section 3.3.3. Since the construction above gives rise to a cubic spline that takes

positive values and has negative curvature, it remains to check that (x) 1for all x [0, 1]. But is concave and continuous therefore this condition musthold unless (1) < 0. This is trivial to check. If this is the case, a large value

of the objective function may be returned to the optimisation loop so that the

current trial solution z will be discarded. In practice we find this situation does

not arise as long we choose the starting point of the optimisation z(0) such that

this condition holds.6

6For instance, one suitable starting point is the vector z = (c, 0, . . . ,0)T for some constantc R. This corresponds to a transformation function given by the identity transformation,resulting in a Normal copula with correlation determined by the value of c (e.g. = sin(c)).

19


20/44

4 Results of Calibration to European Call Op-

tions

4.1 Description of Dataset

Our model for the joint distribution is fitted to actual market quotes for the

implied volatilities of vanilla options written on exchange rates between the three

currencies JPY, EUR and USD, with strikes corresponding to standard values

of the option delta given by 10%, 25% and 50% and standard maturities 1M,

3M, 6M, 1Y and 2Y (see Appendix F). All option price data and corresponding

FX rates and discount factors were obtained at the close of 7 July 2001.

4.2 Comparison of implied volatilities of model prices of

vanilla options on X2,3

T against market quotes

Before presenting our findings on the calibration of our model to market data, it

is instructive to first consider the joint distribution model given by the Normal

copula with marginal distributions given by market data. Consider as exam-

ple the dataset corresponding to maturity 1M and the currency set-up where

(CCY1,CCY2,CCY3)=(USD,EUR,JPY). Implied volatilities of vanilla call op-

tions C2,30 (K) under the Normal copula model with various values of correlation

are displayed in Fig. 3. As expected, it is observed that changing alters the

overall dependence structure (as with the Black model) and therefore results

in a parallel shift in model implied volatilities with negligible change in slope

or curvature. Therefore it is not possible to find a close match to the market

quotes simply by varying .

By contrast, Fig. 4 indicates that in this case the transformed Normal copula

offers essentially an exact fit to market quotes (with an objective function of

zero to 14 s.f. for this dataset), with transformation function () shown in Fig.5. This transformation function is constructed by cubic splines as described in

Section 3.3.4, with knot points p = (0, 0.1, 0.5, 0.9, 1)T.

To see directly the effect of the transformation of the Normal copula, com-

pare the contour plot of the optimal transformed copula (Fig. 7) against that

of the optimal Normal copula (Fig. 8), which is calculated by finding the value

of the parameter that minimises the same objective function described in Sec-tion 3.3.4. Notice that the optimal transformed copula exhibits much higher

upper tail dependence as well as higher lower tail dependence than the Normal

copula. As discussed in Section 3.3.3, the resulting upper and lower tail depen-

dence characteristics of the transformed copula correspond to the graph of ()near one and zero respectively (Fig. 6). Experimentation shows that in order

20


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Strike

Volatility

0.0094 0.0096 0.0098 0.0100 0.0102 0.0104

0.1

2

0.1

4

0.1

6

0.1

8

Actual Volatilityrho=0.2rho=0.1rho=0rho=-0.1rho=-0.2rho=-0.3

Figure 3: Implied volatility of options on X2,3T under the Normal copula model forvarious values of correlation . Here (CCY1,CCY2,CCY3)=(USD,EUR,JPY) and therelevant marginal distributions have been calibrated to option prices with maturity1M. Also shown are the actual market quotes for these implied volatilities.

Strike

Volatility

0.0094 0.0096 0.0098 0.0100 0.0102 0.0104

0.1

4

0.1

5

0.1

6

0.1

7

0.1

8

Actual Volatility

Transformed CopulaOptimal Normal Copula

Figure 4: Implied volatility of options on X2,3

T corresponding to the currency set-up (USD,EUR,JPY) and maturity 1M, under the transformed Normal copula modelwith optimal parameters. Also shown are the actual market quotes for these impliedvolatilities and the implied volatilities under the optimal Normal copula model (whichminimises the same objective function as that used to assess the fit of the optimaltransformed Normal copula).

21


22/44

to attain a good fit to market quotes, a great deal of curvature is required near

zero and one, whilst close to zero near1

2 .We now consider the results of the calibration to all currency set-ups and

all five maturities for which we have market implied volatility quotes. We find

that the transformed copula model offers a better fit to market data than the

Normal copula in all cases.

For the general case where the smile in implied volatilities is symmetric, the

calibration to market data is particularly good (see Appendix D). This is also

the case when implied volatilities of options on X2,3T are skewed but increasing as

a function of strike. We find that for a given currency set-up, the behaviour of

is usually stable across different maturities (see Fig. 13). This is to be expected,

since for each currency pair in our dataset the implied volatility smile has the

same qualitative features across maturities, hence the overall correction ofthe Normal copula achieved by the calibrated transformation function should

be the same. In general the dependence structure implied by market quotes of

vanilla options on X2,3T has greater upper and lower tail dependence than that

exhibited by a Normal copula.

For the scenarios where the implied volatilities of options on X2,3T are heavily

skewed and decreasing, the fit is occasionally not as good. For the currency set-

up (EUR,USD,JPY) and maturities 1M and 3M, the market volatility quotes

that we calibrate to are almost flat for strikes above ATM rather than the usual

more symmetric shape and it is difficult to fit this skew. The fit of the model may

be improved by shifting the knot points appropriately; experimentation shows

this works extremely well for the 3M maturity. Alternatively, observe that the

ideal transformation function exhibits a great deal of curvature in its second

derivative near zero and one in all cases and a piecewise linear construction

is a poor approximation to this. The parameterisation of the transformation

function may be altered in order to allow such curvature and thus afford a

better fit in these cases (see Appendix C).

However, the fit to the currency set-up (EUR,JPY,USD) is found to be

extremely good for all maturities. Interchanging currencies two and three also

results in a model which may be used to price any quanto FX option with quanto

currency EUR. Therefore for any currency set-up and meturity we are able to

find a well-calibrated model for either the joint density fX2,1T ,X1,3T or fX3,1T ,X1,2Tand we are in a position to value a quanto FX option.

22


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x

phi(x)

0.0 0.2 0.4 0.6 0.8 1.00

.0

0.2

0.4

0.6

0.8

1.0

Figure 5: Optimal transformation function () used to match volatilies in datasetcorresponding to the currency triple (USD,EUR,JPY) and maturity 1M (Fig. 4).

phi(x)

0.0 0.2 0.4 0.6 0.8 1.0

-4

-3

-2

-1

0

x

Figure 6: Second derivative () of transformation function used to match volatiliesin dataset corresponding to the currency triple (USD,EUR,JPY) and maturity 1M(Fig. 4).

23


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log(u)

log

(v)

-2 -1 0 1 2

-2

-1

0

1

2

0.02

0.02

0.02

0.02

0.02

0.02

0.04

0.06

0.080.10.120.14

0.16

Figure 7: Contour plot of optimal Normal copula corresponding to dataset corre-sponding to maturity 1M and the currency set-up (USD,EUR,JPY), plotted againstNormal margins. Optimal value of is found to be -0.1299.

log(u)

log

(v)

-2 -1 0 1 2

-2

-1

0

1

2

0.02 0.02

0.040.06

0.08

0.10.120.14

0.16

Figure 8:Contour plot of optimal transformed Normal copula fitted to dataset cor-

responding to maturity 1M and the currency set-up (USD,EUR,JPY), plotted againstNormal margins. Although the overall level of association is similar to the optimalNormal copula (see Fig. 7), much greater upper tail dependence is required in order tofit the implied volatilities of vanilla options on X2,3T with high strikes. The transformedcopula also exhibits increased lower tail dependence and a much higher central peak.

24


25/44

5 Quanto Option Pricing

5.1 Valuation of Quanto FX Options under a given Copula

Model

The value of a standard quanto FX call option paying out in currency one is

computed by evaluating the expectation

EQ1 [(X2,3T K))+],

as described in Section 2.1. This is straightforward once the distribution of X2,3Tunder Q1 is known.

Recall X2,3T = X2,1T X

1,3T and so

fQ1

X2,3T

(z) =

0

1

xfQ

1

X2,1T

,X1,3T

z

x, x

dx.

Therefore the price of a standard quanto call option may be evaluated directly

by numerical integration:

Cquanto0 = D10,TEQ1 [(X

2,3T K)+]

= D10,T

K

0

1

xfQ

1

X2,1T

,X1,3T

zx

, x

(z K) dx dz. (13)

It is useful to note that the double integral in (13) corresponds to a region

of the joint density fQ1

X2,1

T,X1,3

T

(p,q) bounded below by the line p = K/q. If the

margins are lognormal, this corresponds to the region of the contour plot of the

copula density above the line log(p) + log(q) = K when plotted against Normal

margins (c.f. Fig. 2). This is the same integration region as a vanilla call option

on X2,3T with strike K. Once the implied marginals have been determined, the

value of (13) depends critically on the distribution of the copula mass along the

line u = v.

In our model we perturb the Normal copula to match information on the

dependence structure contained in vanilla option prices on X2,3T . Given the im-

plied marginal distributions, the price of a quanto option is likely to be highly

dependent on the fit of the copula to these vanilla option prices. Any distortion

of the copula density other than along the line u = v (for instance any asym-metry) is unlikely to have a material effect on the price of the quanto option.

Thus we have more confidence that although our fitted joint distribution model

is not the only copula that could be calibrated to market prices, any other cop-

ula calibrated to market data is unlikely to result in markedly different quanto

prices.

25


26/44

Prices of standard quanto call and put options with the same strike are

linked by the put-call parity relationship

Cquanto0 (K) Pquanto0 (K) + D10,TK = D10,TEQ1 [X2,3T ],

where the right-hand side is independent of the strike K (but is still dependent

on the particular model of the joint distribution). It is a useful numerical

check to verify that put-call parity holds for calculated quanto prices under the

transformed copula model.

5.2 Comparison of Quanto Prices under Copula model vs.

Black approximation

We now compare the prices of quanto options calculated under the transformedcopula model (evaluating Equation (13)) with both quanto prices calculated

under the standard Black model (calibrated to at-the-money volatilities) and

prices calculated using the adjusted Black formula (defined in Section 2.2).

With the adjustment of the Black formula we attempt to allow for the smile in

implied volatilities on X2,3T , but we ignore the effect of any smile behaviour in

X1,2T and X1,3T . This ad-hoc formula does not correspond to any joint distribution

of X1,2T and X1,3T , so the only comparison of these approaches available to us is

a simple comparison of prices. Note that all prices correspond to given values

of the conversion factor , but prices for other values of are easily obtained

since this is just a scaling factor.

From the results given in Appendix E it is clear that both the adjusted Black

formula and the transformed copula model give much higher quanto prices than

the simple Black model. This is to be expected. The adjusted Black formula

allows the practitioner to compensate somewhat for the increase in implied

volatility of X2,3T for strikes away from at-the-money (assuming the lowest im-

plied volatility corresponds to the at-the-money strike, which is usually the

case). The transformed copula model allows for the smile in implied volatilities

of options on all three relevant currency pairs.

We now examine the difference between the prices of standard quanto call (or

put) options calculated via the adjusted Black formula and the corresponding

prices calculated under the transformed copula model. Although call and putquanto prices are generally close, there are sometimes significant differences,

usually with strikes furthest from at-the-money. This difference can be as great

as 15% of the option value for standard strikes. In order to assist practitioners

familiar with the adjusted Black formula, we have also included in our results

the value of the adjusted implied volatility parameter (3) that must be entered

26


27/44

into the adjusted Black formula to obtain the model quanto price (calculated

under our calibrated transformed Copula model). The adjustment required inthe size of this parameter is usually in the region of 0.1-0.3% (in standard units of

implied volatility) but is occasionally in the region of 1-1.3% for strikes furthest

away from at-the-money.

For the datasets used in this exercise, the transformed copula model usually

gives higher quanto call prices than the adjusted Black formula. The relative

difference commonly increases with the level of the strike of the quanto option.

Often the absolute difference in prices is of similar magnitude across strikes.

The only scenario where model call prices are slightly below adjusted Black

prices corresponds to the currency set-up (JPY,EUR,USD) with maturity 2Y.

There is no significant qualitative difference between the implied volatilities

of the relevant vanilla options corresponding to this maturity and the impliedvolatilities for other maturities. In general, the implied volatilities corresponding

to the marginals of this dataset appear slightly more symmetric than those

corresponding to shorter maturities.

The transformed copula model usually leads to lower quanto put prices than

the adjusted Black formula. The greatest differences often occur for strikes away

from at-the-money, though there is not such a clear trend as that observed for

call prices. In a few scenarios, put prices under the transformed copula model

are slightly above those calculated using the adjusted Black formula for some

strikes, but on the whole these relative differences are small (see all results

corresponding to maturity 1M and also (JPY,USD,EUR) 2Y).

Using the results of these scenarios alone, it is not straightforward to deter-

mine exactly under what conditions the difference in model and adjusted Black

prices will be greatest because the adjusted Black formula does not correspond to

a valid model for the joint distribution ofX1,2T and X1,3T . The transformed copula

model provides quanto prices as a result of a calibration to implied volatilities

on the three relevant FX rates and it is not possible to alter the implied volatil-

ities on one FX rate without adjusting the entire calibration. Further scenario

analysis is required to understand the exact relationship between adjusted Black

prices and model prices for different patterns of the relevant implied volatilities,

however this is beyond the scope of this paper.

6 Conclusion

This paper presents a new methodology for the valuation of quanto FX options,

in which we separate the modelling of the dependence structure of the relevant

FX rates from the modelling of the implied marginal distributions. Our model

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for the dependence structure is a copula obtained by a suitable perturbation of

the Normal copula. This allows us to modify the upper and lower tail depen-dence characteristics appropriately to allow calibration to the smile in implied

volatilities on all three relevant FX rates. Most ad-hoc approaches to this prob-

lem do not capture all the relevant information contained in market quotes,

indeed many do not correspond to a proper model for the joint distribution of

the relevant FX rates.

Although our model for the dependence structure (the transformed Normal

copula) is not the only model which may be calibrated to market prices, it

is generally close to the dependence model corresponding to the Black pricing

formula. We adjust the dependence structure in such a way that even though

the calibrated dependence model is not unique, we have a degree of confidence

that any other model calibrated to all market data will give similar values ofthe quanto option.

The resulting quanto prices evaluated under a number of real scenarios are

generally close to prices calculated using the ad-hoc adjusted Black formula,

which is certainly a much better proxy for the model price than the standard

Black formula. However, we find the adjusted Black model often gives lower

quanto call prices and higher quanto put prices than those calculated under the

transformed copula model. The relative difference in quanto prices is occasion-

ally large (10-15%) for standard strikes furthest away from at-the-money.

The quanto FX option pricing problem we have focused on in this paper is

a particular application of what is a general method for perturbing a Normal

copula (the dependence model associated with the Black model) to modify the

tail dependence characteristics of the overall dependence structure. A similar

methodology may be applied to any two-asset or hybrid option pricing problem

if vanilla option quotes exist to inform the implied dependence structure. Where

such option quotes are unavailable (as is the case with equity spread options),

historical data may be used to estimate the copula directly (see for example

[Rosenberg, 2003]). However the method outlined in this paper could easily be

adapted so that the transformed Normal copula is estimated from historical

data, allowing construction of a smoother copula that places less reliance on the

quality of the historical data and the estimation methods.

This methodology may also have other applications in the statistical mod-elling of real-world dependence where the Normal copula does not provide a

sufficiently realistic model for the dependence structure, for example in scenario

analysis or stress-testing.

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References

[Ait-Sahalia at al, 1998] Ait-Sahalia, Y. & Lo, A.W., 1998, Nonparametric Estima-tion of State-Price Densities Implicit in Financial Asset Prices,Journal of Finance.

[Bahra, 1997] Bahra, B., 1997, Implied Risk-Neutral Probability Density Func-tions from Option Prices: Theory and Application, Bank of Eng-land Working Paper.

[Bliss et al, 2000] Bliss, R.R. & Panigirtzoglou, N., 2000, Testing the stability ofimplied probability density functions, Bank of England WorkingPaper.

[Burden et al, 1997] Burden, R.L. & Faires, J.L., 1997, Numerical Analysis, SixthEdition, Brooks/Cole.

[Campa et al, 1997] Campa, J.M., Chang, P.H.K. & Reider, R.L., 1997, Implied Ex-change Rate Distributions: Evidence from OTC Option Markets,Working Paper.

[Coutant et al, 1998] Coutant, S., Jondeau, E. & Rockinger, M., 1998, Reading In-terest Rate and Bond Futures Options Smiles: How PIBOR andNotional Operators Appreciated the 1997 French snap election,Banque de France Working Paper.

[Dupire, 1994] Dupire, B., 1994, Pricing with a Smile, Risk Magazine 7 (Jan-uary).

[Embrechts et al, 2001] Embrechts, P., Lindsog, F. & McNeil, A., 2001, ModellingDependence with Copulas and Applications to Risk Management,ETHZ Working Paper.

[Genest, 2000] Genest, 2000, Distributions with Given Marginals and StatisticalModelling, Conference Proceedings, Barcelona, July 17-20, 2000.

[Genz, 2002] Genz, A., 2002, Function for computing bivariate normal proba-bilities, Washington State University Working Paper.

[Joe, 1997] Joe, 1997, Multivariate Models and Dependence Concepts, Chap-man & Hall.

[Malz, 1997] Malz, M., 1997, Estimating the Probability Distribution of theFuture Exchange Rate from Option Prices, Journal of Derivatives.

[More, 1977] More, J.J., 1977, The Levenberg-Marquardt algorithm: imple-mentation and theory, Numerical Analysis, pp. 105-116, LectureNotes in Mathematics 630, Springer-Verlag.

[Nelson, 1998] Nelson, R.B., 1998, An Introduction to Copulas, Lecture Notes InStatistics, Springer-Verlag.

[Rosenberg, 2003] Rosenberg, J.V., 2003, Non-parametric Pricing of MultivariateContingent Claims, Journal of Derivatives.

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[Schonbucher at al, 2001] Schonbucher, P.J. & Schubert, D., 2001, Copula-Dependent Default Risk in Intensity Models, Working Paper,University of Bonn.

[Sherrick at al, 1996] Sherrick, B.J., Garcia, P. & Tirupattur, V., 1996, RecoveringProbabalistic Information From Option Markets: Tests of Distri-butional Assumptions, Journal of Futures Markets.

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A Calculation of Implied Marginal Density and

Distribution FunctionsAs outlined in Section 3.1.1, initial calibration of marginal distributions to impliedvolatility quotes is by a weighted non-linear least squares optimisation, similarly to[Bahra, 1997].

Fix the maturity T. Suppose the probability density function fQi

Xi,jT

() of the FXrate Xi,jT under the measure Q

i is given by a mixture of lognormal distributions

fQi

Xi,jT

(x) = i,jL(i,j1 , i,j1 | x) + (1 i,j)L(i,j2 , i,j2 | x), (14)

where i,jk = i,jk (i,jk )2T /2, i,jk = i,jk

T , L() is the standard lognormal density

function

L(i,jk , i,jk | x) = 1

xi,jk

2exp

(log x i,jk )22(i,jk )

2

, k = 1, 2

and i,j , i,j1 , i,j2 ,

i,j1 and

i,j2 are parameters to be determined from market quotes

of vanilla option prices as follows.The price of a vanilla FX call option on Xi,jT is given by direct evaluation of the

expectation given in Equation (4):

Ci,j0 = Di0,T[

i,j{ei,j1 +(i,j1 )2/2(di,j1,1) K(di,j2,1}+(1 i,j){ei,j2 +(i,j2 )2/2(di,j1,2) K(di,j2,2}], (15)

where

d

i,j

1,k = log K + i,jk + (

i,jk )

2

i,jk , d

i,j

2,k = d

i,j

1,k i,j

k , k = 1, 2

and () is the standard Normal cumulative distribution function. Put prices areobtained similarly.

Values of parameters i,j , i,j1 , i,j2 ,

i,j1 and

i,j2 are found by a standard least-

squares optimisation procedure that minimises the weighted squared error betweenmarket quotes for implied volatilities of vanilla options and the corresponding valuesunder this model. Since we know the FX forward Mi,j0,T is the mean of the distribution

of the FX rate Xi,jT under the measure Qi,

Mi,j0,T = i,je

i,j1 +(

i,j1 )

2/2 + (1 i,j)ei,j2 +(i,j2 )2/2, (16)

therefore i,j2 may be found immediately given the other parameters. A computationalrequirement of subsequent joint density calculations is that the forward rate implied by

the marginal distribution is exact (at least to around 8 s.f.); the following optimisationprocedure is also found to be far more stable if the forward is not allowed to be a freeparameter (c.f. [Bahra, 1997]).

The distribution function ofXi,jT under Qi may be found from the density function

(14) by direct calculation:

FQi

Xi,jT

(x) = i,j

log x i,j1

i,j1

+ (1 i,j)

log x i,j2

i,j2

.

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Given the distribution of Xi,jT under Qi it is possible to calculate the distribution

and density functions of the inverse FX rate Xj,i

Tunder the corresponding measure

Qj , so it is only necessary to perform a single calibration of model parameters for eachcurrency pair Xi,jT , i > j under Q

i.Let us assume the density of Xi,jT under Q

i is given by a mixture of lognormaldistributions as outlined above. By no arbitrage

Pj,i0 (K) = KXj,i0 C

i,j0 (1/K),

hence by differentiating both sides with respect to K we see that

FQj

Xj,iT

(K) =i,je

i,j1 +(

i,j1 )

2/2(di,j1,1) + (1 i,j)ei,j2 +(

i,j2 )

2/2(di,j1,2)

i,jei,j1 +(

i,j1 )

2/2 + (1 i,j)ei,j2 +(i,j2 )2/2,

which is the distribution function of a mixture of two lognormal distributions.

B Copula calculations

The Normal copula has distribution function

CN(u, v | ) = (1(u), 1(v)),

where is the standard bivariate Normal distribution function with correlation and is the standard univariate Normal distribution function (for which standardapproximations are available).

Let x = 1(u) and y = 1(v). By standard transformation of variables,

cN(u, v | ) = (1 2)1/2 exp

x2 + y2

2 1

2(1

2)

(x2 + y2 2xy)

. (17)

The partial derivative of the Normal copula with respect to one of its arguments isrequired in evaluation of the transformed Normal copula density (11). By integrationof (17) we obtain

C

q(p,q) =

1(p) 1(q)

1 2

.

We transform the Normal copula CN in Section 3.3.3 via

C(u, v) := 1(C((u), (v))), (18)

where : [0, 1] [0, 1] is a continuous, twice differentiable concave function such that(0) = 0 and (1) = 1. This has density given by the partial derivative of (18) withrespect to both its arguments,

c(u, v) = (u)(v)

(C(u, v))

c((u), (v))

(C(u, v))

[(C(u, v))]2C

u((u), (v))

C

v((u), (v))

, (19)

where c(, ) is the density of the original copula C.

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Following [Genest, 2000], to show the transformation (18) results in a copula, notethat if is concave then

(1(C(u, v)))

[(1(C(u, v)))]2C

u(u, v)

C

v(u, v) 0,

hence

c(u, v) (1(C(u, v)))

[(1(C(u, v)))]2C

u(u, v)

C

v(u, v) 0. (20)

Integrating the transformed density (19) between 0 < u1 u2 < 1 and 0 < v1 v2

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Benett.kennedy-Quanto With Copula