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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 dependen