implied multivariate value-at-risk

1

IMPLIED MULTIVARIATE VALUE-AT-RISK

byJosé M. Campa*

IESE Business SchoolNBER and CEPR

andP.H. Kevin Chang**

Credit Suisse First Boston

revised: October 2001

ABSTRACT

This paper proposes methods for computing multivariate risk-neutral distributionsfrom option prices. We illustrate the differences between the risk-neutral pdfs obtainedfrom multivariate estimation and those from the use of univariate techniques. Theestimation of multivariate distributions has important useful applications for financialhedging, derivative pricing, and the evaluation of correlation, contagion, and othermeasurements of comovement among the underlying securitires. We use an examplewith options on the dollar-mark, dollar-streling and mark-sterling exchange rates tohighlight the differences in the estimated measures of comovement from the multivariatedistribution and those obtained from historical estimation and univariate techniques. Ingeneral, the risk-neutral measures imply fatter distribution tails and higher correlations oflarge movements of the underlying securities than those suggested by historical prices.

____________* 44 West 4th Street, New York, NY 10012. (212) 998-0429. [email protected]

For helpful comments, we are grateful to Jens Carsten Jackwerth, Joshua Rosenberg, Charlie Thomas,and conference participants at the Wharton Risk Management Workshop, and the BIS conference on“Estimating and Interpreting Probability Density Functions”. We also thank seminar participants at theBank of England, Bank of Spain, MEFF, New York University and Federal Reserve Bank of New York. For data, we thank Richard Holmes of Citibank and John Slee and John Quayle of NatWest Markets.

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I. Introduction

Options, as forward-looking financial indicators, have gained increasing attention in recent years

as informative predictors of future asset price behavior. Several recent papers have documented that

implied volatility contains information not found in the underlying time series that is useful for predicting

future volatility [Day and Lewis (1992); Scott (1992); Kroner, Kneafsey, and Claessens (1993);

Lamoureux and Lastrapes (1993); Jorion (1995), Figlewski (1997)]. In a different approach to

characterizing future asset price behavior, resent research [Shimko (1993), Rubinstein (1994), Abken

(1995), Lo and Ait-Sahalia (1995), Jackwerth and Rubinstein (1996), Malz (1996a), McCauley and Melick

(1996a,b), Mizrach (1996), Melick and Thomas (1997)] has used option prices to derive the implied risk-

neutral density function of the underlying variable, based on the methods of Breeden and Litzenberger

(1978). Other papers have shown that options anticipated the 1987 stock market crash (Bates 1991) or

the 1992 ERM crisis [Campa and Chang (1996), Malz (1996b), Mizrach (1996)]---often better than did

time series data or, in the case of the ERM, interest rate differentials. In brief, the recent literature has

exhibited an increased appreciation as well as greater exploitation of the information content of options.

The information content in options evaluated in the literature has generally focused on options on a

single asset. However, there are many applications in which the user is not only interested in forecasting

the behaviour of a single asset but in the joint bejaviour of more that one asset. For instance, an European

oil importer might be interested in the behaviour of the dollar price of oil jointly with the expected

movement of the Euro/USD exchange rate. An international fund manager might be interested in the joint

behaviour of its home currency exchange rate relative to other currencies in which it has an investment

interest. A U.S. investor evaluating its expected return on the Japanese stock index would be interested in

the joint behaviour of the yen/dollar exchange rate as well as the expected movement in the nikkei index.

In all these circumstances an investor is interested in the expected correlation among two financial assets.

In recent work, documented the ability of foreign exchange options to predict correlations among

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currency returns in different markets [Campa and Chang (1998), Bodurtha and Shen (1995) and Lopez

and (2000)] use implied volatilities from foreign currency options on three currency-pairs to compute an

"implied correlation" among the expected returns on two currencies. They find that correlations estimated

this way can help forecast future correlations and in some cases implied correlation is a better forecast

than measures based on historical data.1

The purpose of this paper is to estimate multivariate distributions based on options data and to

evaluate the information that can be extracted from these distributions to inform investment decision

making. We extend three of the most widely used methods to calculate univariate risk-neutral distributions

to the multivariate case and evaluate the advantages and problems in estimation faced by each of these

methods. We also propose several measures derived from this multivariate distributions that might help

asset managers to conform a better picture of the underlying expectations for risk management.

The estimation and use of the full implied multivariate distribution rather than the first and second

moments of the distribution can be particularly useful in certain contents in which we are really interested

primarily on the shape of certain domain of the distribution, not in its full domain. For instance, in the

context of risk-management the interest usually lies in estimating the tails of low returns of the distribution

of a certain portfolio that the institution is holding. There has also recently been increased interest in

international finance in the issue of contagion and the behaviour of currency exchange rates in crisis

situations in which one or several currencies suffer large devaluations, such as the events during the Asian

crisis of 1998. In order to provide an answer to these situations, an analysts needs to observe the

complete multivariate distribution, and evaluate its shape in the particular domains of interest. In an

applied context, correlation is important in areas such as asset management, risk management, and the

pricing and hedging of certain derivative instruments dependent on correlation.2 Accurate prediction of

1 Technically, implied correlations also exist in various "quanto" options in which the underlying securityis quoted in one currency while the payoff is made in another, as in the case of the Nikkei options traded inChicago. The liquidity of such markets is much smaller than that of the mark-yen options market.

2 Derivative instruments dependent on correlation include "best-of-two" asset options, outperformance

4

correlation of key international financial variables---such as interest rates, exchange rates, equity markets,

commodity prices---clearly would have important benefits for portfolio managers, risk managers, and

financial regulators alike. More generally, correlation will be important in any asset-pricing environment in

which investors are assumed to be risk-averse. With risk-aversion, second moments necessarily play a

key role, and when the universe consists of multiple risky assets, covariance terms can easily outnumber

variance terms.

We will illustrate the methodology with an example using option data on the mark-dollar, mark-

sterling and sterling-dollar exchange rates to compute the bivariate distribution on the dollar-mark and

dollar-sterling returns. We will compare the measures of correlation, Value-at-Risk (VaRs), and

probability measures of certain comovements. We will also look at the differences in pricing behavior of

certain derivative products whose price depends on the co-movement of both securities. Finally, we will

look at the conditional distribution of returns on each asset given a certain return on the other asset. Our

results provide strong evidence that the implied multivariate distribution predicts much larger probabilities

of large joint depreciations of the mark and sterling relative to the dollar, than those computed from

historical returns, or from the information obtained using the information from only at-the-money options.

Recent academic work on correlation, dealing mainly with equity markets, has focused on whether

correlation is constant over time (Kaplanis (1988), Longin and Solnik (1995) Forbers and Rigobon (1999))

or linked to the business cycle (Erb, Harvey, and Viskanta (1995)). In general, however, relative to the

extensive literature on the predictability of returns and volatility, comparatively little has been written on

the predictability of co-movements in two or more assets.

The remainder of this paper is organized as follows. Section II describes how to extend existing

methods to estimate univariate risk-neutral distributions to the multivariate case.. Section III presents our

options, basket options, quanto options, swaptions, quanto swaps, diff swaps. Correlation enters directly("first order effects") into the option payoffs in spread options, outperformance options, yield curve options,cross-currency caps. Correlation results in the modification of payoffs ("second order effects") in diff swapsand quanto options.

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data and certain key descriptive statistics on our data. Section IV provides several measures of co-

movement from these multivariate distributions and compares their performance to similar measures

obtained from historical data and univariate distributions. We look at three different criteria: conditional

distributions, VaR measures, and probability measures of co-movement. Section V summarizes and

concludes.

II.- Methodology

Several methods have been proposed in the literature two estimate risk-neutral density functions

from option prices. These methods can be divided into two large groups. The first group uses some

estimation technique to fit a smooth and differentiable line to observed option prices and then computes the

second derivative of this function with respect to the strike price to obtain the risk-neutral density function

[Shimko (1993), Campa and Chang (1996, 1998)]. The second group starts by assuming a flexible

functional form for the underlying risk-neutral distribution and then estimating the parameters of this

distribution so as to minimize some criterion function which depends on the difference between the

observed option prices and those predicted by the parameterized distribution function. A number of

different distributions have been used previously in the literature including a mixture of lognormals [Mellik

and Thomas (1996)], a binomial [Jackstein and Rubenstein (1996)], a nonparametric distribution [Ait-

Sahalia and Lo (1999)]. Below we will describe the extensions to the multivariate case for three of the

more widely used univariate methods: the implied binomial tree, the mixture of two normal distributions,

and the “smile” method of fitting a continuous function to the volatility smile of the options.

II.A. Three Alternative Approaches to the PDF

(1) "Implied Binomial Tree" Approach of Rubinstein (1994). Under the implied binomial tree approach,

one solves directly for the risk-neutral probability pi associated with each of the n+1 terminal nodes of an

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n-step binomial tree. Inputs to this approach are the m observed call options, where normally one chooses

n >> m. For our five option observations, for example, we choose a 100-step tree. Under this method,

probabilities are chosen to minimize the sum of squared difference between a prior probability p'i and the

new estimated probabilities pi, subject to these constraints: each call must equal its expected value based

on these probabilities, the probabilities must sum to one, the expected value of the future spot rate must

equal the forward rate (denoted as ForwardT), and pi ≥ 0 for all i.

Mathematically, one minimizes a "loss function," in this case quadratic, solving the following

constrained optimization:

subject to the constraints

(1) ) p - p( p

2ijij

n

j=0

n

=0ii

′∑∑Min

(2)

,0)-(

,0)-(

,0)-(

KSSp i+1

1=Call

KSp i+1

1=Call

KSp i+1

1=Call

si jij

n

=0i

n

j=TT ,K

sTj,ij

n

=0i

n

Kj=TT ,K

sTi,ij

n

j=0

n

K=iTT ,K

j

j

j

21

0

2

1

/Max

Max

Max

∑∑

∑∑

∑∑

(3) njn.,0,1,2,3,..=i all for 0p ij ,...3,2,1,0=≥

7

Typically, the prior probabilities p'ij are based on the lognormal distribution, so that provided the constraints

are satisfied, the distribution appears as close as possible to lognormal. Jackwerth and Rubinstein (1996)

empirically compare, for the S&P 500, a number of loss functions other than quadratic (e.g. goodness-of-

fit, absolute difference, maximum entropy, smoothness), and find similar estimated pdf's. When the loss

function is altered, the effects on the resultant pdf are pronounced only at the tails, where, in any event,

very little absolute probability is found. In general, the further from the observed strike prices, the more

the derived probabilities depend on the chosen parameterization than on the actual data. The main

problem of implementing this problem in the case of a multivarirate distribution is that the number of

probability points that exits in the problem grows exponentially with the number of joint distributions being

estimated. For instance, for the case of a bivariate distribution, with a grid of n points results in (n+1)2

probability points that need to be calculated.

Empirically, one difficulty implementing the binomial approach is irregularities in the tails: outliers

often have a probability orders of magnitude larger than under the lognormal benchmark. Probability

sometimes rises inexplicably deep into the tails at nodes far from observed strike prices. Since the

probability assigned to these outliers is low in absolute terms, one or more orders of magnitude in fact have

(4) 1=pij

n

j=0

n

=0i∑∑

(5)

ForwardSSp

ForwardSp

ForwardSp

TTj,Ti,ij

n

j=0

n

=0i

TTj,ij

n

j=0

n

=0i

TTi,ij

n

j=0

n

=0i

=

=

=

321

22

11

/∑∑

∑∑

∑∑

8

little effect in constraints (2)-(4), where outliers enter only linearly. For higher moments such as kurtosis,

these irregularities take on far greater importance: for example, Campa, Chang and Reider (1998) report

that the estimtated pdfs calculated by this method using options from 48 weekly one-month dollar-mark

options leads to pdf's with an average (excess) kurtosis of 19. One can reduce the excess kurtosis by

"trimming" the distribution, or truncating the distribution by eliminating the ends in the tails (see

Rosenberger and Gasko (1983) for details. This approach, however, is very ad hoc.

(2) "Mixture of Lognormals" Approach of Melick and Thomas (1997). Still another approach is to

fit the distribution as a "weighted average" or mixture of lognormal distributions. This approach is widely

used in the univariate case [Leahy and Thomas (1996), Melick and Thomas (1997), and McCauley and

Melick (1996a)]. This method has a natural economic interpretation---that of multiple alternative regimes.

In Melick and Thomas (1997), for example, three different lognormal distributions for the price of oil

correspond to various outcomes of the 1991 Gulf War. There is an obvious way that this method could be

extended to the multivariate case: using a mixture of multivariate normal distributions.

Following this approach, for instance, we solve for a bivariate pdf h(ST) as a mixture of two

bivariate lognormals, expressed as follows:

where h(ST) represents the unknown composite pdf, and g(µ,σ²) represents the pdf of two variables

whose logarithm is normal with a mean vector µ and a variance-covariance matrix σ². The unknown

parameters in this case are 11: π1, plus µ1, and µ2 with dimension (2x1 each), plus σ1, and σ2 which they

both are symmetric 2x2 matrix. The algorithm will solve for this parameters in order to satisfy as well as

possible the constraints on the m observed call options and the three forward rates. In solving this system,

one typically defines a loss function expressed as the sum of squared percentage deviations from these

constraints.

Statistically, one advantage of the mixture of lognormals approach is the smooth behavior of the

(6) ),)g(-(1+),g(=)Sh( 22

2112

11T σµπσµπ

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tails. Probability in the tails declines monotonically and always decays quickly enough to prevent

unreasonable kurtosis. On the other hand, this approach may impose too rigid a structure on the resultant

pdf, unless one believes a priori that the underlying economics are well captured by a weighted sum of

discrete outcomes. However, two provide economic interpretations to the value of the two lognormal

distributions identified in our derived pdf often make little economic sense. The parameters and probability

associated with each distribution fluctuate widely from week to week for no apparent reason.

As an alternative to this mixture of lognormals approach, Rosenweig (1998) proposes an

alternative flexible functional form based on the estimation of a parameterized bivariate lognormal

distribution in which the terms of the variance-covariance matrix are parameterized as polynomial

functions of the points in the distribution. The distribution function to estimate would be g(µ,σ²), where the

diagonal elements of the matrix σ² would be σ²i = a + b Si +c Si2.. This parameterization has the intuitive

approach to the univariate case that the implicit volatility of the estimated distribution may be a quadratic

function of the position of the distribution which resembles the widely observe phenomena that a volatility

smile might exist.

(3) Methods Based on the Volatility-Smoothing Approach of Shimko (1993).

Shimko (1993) introduces a method of fitting the volatility smile with a quadratic function to obtain

a continuum of call prices as a function of strike price within the range covered by the data. The pdf is

obtained by twice differentiating this result then discounting by the riskless interest rate. For strike prices

outside the observed range, one can extrapolate by extending the quadratic function outwards. This

approach is very widely used and has been generalized to many other functional forms such as spline

functions, higher order polinomyals, and others. However, this approach has no generalization to the

multivariate case since its derivation never relies on the correlation that might exist among the different

univariate distributions.

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III.- Data

Our primary data consist of market quotes of over-the-counter options on three actively traded currency

pairs: the dollar-mark, dollar-pound, mark-pound. In particular, we use closing prices on these options, as

well as on exchange rates and Eurocurrency interest rates, recorded by currency option traders at a major

money-center bank. Observations are weekly (every Wednesday) from 3 April 1996 to 5 March 1997, for

a total of 48 weeks. Unfortunately, due to disclosure restrictions imposed on us by the Bank that provided

us with the data most of the results will be illustrated only for a single date, the first day of the sample,

April 3, 1996.

While exact quotes may differ slightly from bank to bank, the over-the-counter options market is

relatively liquid and competitive, especially for the currency pairs we study.3 Relative to exchange-traded

options, our over-the-counter options are far more liquid. For reference, Table 1 provides information on

daily turnover and outstanding notional amounts of over-the-counter and exchange-traded options on these

currency pairs, as reported in the UPDATED 1996 Bank for International Settlements triennial survey of

foreign exchange activity. Relative to exchange-traded options, over-the-counter options typically have an

order of magnitude more turnover and notional amount outstanding. For example, depending on the

currency pair, daily turnover on exchange-traded options is 12% or less of its OTC counterpart, and the

notional amount outstanding is 7% or less.

In the over-the-counter options market, price quotes are expressed as implied volatilities which

traders by agreement substitute into a Garman-Kohlhagen formula (Black-Scholes adjusted for the foreign

interest rate) to determine the option premium.4 Since the volatility is the only unobservable parameter in

3. In earlier research based on over-the-counter option quotes against the U.S. dollar (Campa and Chang(1995)), we found data from different banks to differ insignificantly. Also, the presence of currency option brokers,who use quotes from different banks to provide the best price to the market, contributes to some degree of priceconvergence.

4. The Garman-Kohlhagen formula is: CallK,T= S0*exp(-rT)N(d1)-K*exp(-rTT)N(d2), where d1=[ln(S0/K)+(rT-r+½σ2)T]/σ√T, d2=d1-σ√T, σ is the implied volatility of the option, r and r* are the continuously compounded domesticand foreign interest rates, and N is the cumulative normal distribution function.

11

the Black-Scholes formula, these volatilities--representing traders' subjective assessment of future

movements in the underlying asset--uniquely determine the options' price. Traders do not necessarily

believe that Black-Scholes holds. This method simply represents a one-to-one mapping from implied

volatility quotes to the option's price, which is also referred to as the "option premium." Under this

convention, option quotes do not necessarily require updating even as the spot rate evolves

minute-by-minute. If prices were expressed as an option premium for a fixed strike, most short-run

changes in the option premium would result from innovations in the spot rate, requiring far greater

coordination between spot markets and option markets.

Our observations for a given maturity (1 and 3 months) consist of implied volatility quotes for

at-the-money-forward options, i.e. whose strike price equals the forward rate of the same maturity, as

well as four additional strike prices, two below and two above the forward rate. The two strike prices

below the forward rate correspond to out-of-the-money put options with "delta" (partial derivative of the

option price with respect to the spot exchange rate) equal to -0.10 and -0.25.5 Similarly, the two strike

prices above the forward rate correspond to out-of-the-money call options with delta equal to 0.10 and

0.25. The higher the level of implied volatility, the wider the spread in strike price between the -0.10 delta

put, the lowest strike price, and the 0.10 delta call, the highest strike price.

Although Black-Scholes assumes a constant volatility, the implied volatility quoted by option

traders will typically vary as a function of an option's strike price. This reflects a departure from the

Black-Scholes assumptions, with the result that the probability distribution for the future exchange rate is

not lognormal.

When volatility is stochastic but uncorrelated with changes in the spot rate, it can be shown that

the Black-Scholes implied volatility is lowest at-the-money forward (strike price=forward rate), increasing

5. The "delta" of a call equals exp(-rT)N(d1), and the "delta" of a put equals exp(-rt)[N(d1)-1] where d1 =[ln(S0/K)+(rT-r+½σ2)T]/σ√T, σ is the implied volatility of the option, r and r* are the continuously compoundeddomestic and foreign interest rates, and N is the cumulative normal density function. Given the implied volatilities,interest rates, time-to-expiration, and spot rate, a given delta for a put or call uniquely defines the strike price K.

12

for both in-the-money (call's strike < spot) and out-of-the-money (call's strike > spot) options. This pattern

is referred to as the "volatility smile," so named for the appearance of a graph with implied volatility on the

vertical axis and the option strike price on the horizontal axis. Different shapes of the volatility smile are

consistent with different distributions of the underlying exchange rate. For instance, a symmetric volatility

smile is consistent with leptokurtosis or "fat tails" in the distribution, i.e. higher probabilities than under the

lognormal distribution of larger positive or negatives changes, as would result from returns with stochastic

volatility. By put-call parity, the implied volatility at any strike is the same for call options and put options.

At times, the probability of future exchange rate realizations is not symmetrically distributed

around the at-the-money strike price. For example, for the sterling-mark exchange rate, the market may

perceive a greater probability of a large depreciation of the pound than a similar depreciation of the mark.

When this asymmetry is present, the smile can be transformed into a "smirk," with the options' implied

volatility rising more sharply for strike prices on one side of the forward rate than the other. Figure 1

depicts the volatility smile or smirk, for our three currency-pairs at the three-month horizon on a single day

in our data sample, 3 April 1996.

As the graphs in Figures 1a-1c, showing the volatility smiles for 3-month options, indicate, these

smiles are far from symmetric. Options are in fact capturing skewness of expectations---i.e. asymmetric

market expectations in terms of the direction of potential large exchange rate changes. In this case, all

five exhibit a one-sided rise in implied volatility in the direction of a major dollar depreciation vs. the mark

or yen, or major mark appreciation vs. the three other European currencies, reflecting greater probability

of this realization than a similar movement in the opposite direction.

IV.-Description and Discussion of measures of co-movement: An Illustration

The implied bivariate distribution for the DEM/USD and USD/GBP exchange rates on April 3,

13

1996 estimated from 3-month options using the method proposed by Rosenweig (1998) described above is

drawn in Figure 2a. Figure 2b describes the estimated distribution using only the information from the

ATM cross-rate option to compute the correlation between both distributions. The difference between

these two distributions indicates a higher probability mass in the regions of a joint appreciation of the dollar

relative to both the Mark and the Pound estimated from the full bivariate estimation. This higher joint

appreciation comes from the observed skewness in the volatility smiles of the DEM/GBP and USD/GBP

exchange rates towards a large pound depreciation. This skewness causes the estimated joint distribution

of dollar returns to suggest a larger probability of a dollar apreciation than that indicated by the options on

the bilateral dollar rates.

The implied marginal distributions for the DEM/USD and USD/GBP exchange rates (figures 3a-

3c) also show significantly different behavior from the implied distributions obtained from the univariate

estimation. Again, the implied marginal distribution for the DEM/USD exchange rate shows a hgiher-

skewness towards large DEM depreciations than that estimtated from the univariate distribution, resulting

from the skewness in the cross-rate distribution (Figure 4a). At the same time, the implied marginal

distribution for the cross-rate shows a much smaller variance (although somewhat larger kurtosis) than

that obtained from the univariate distribution. This result reflects that in relative terms the cross-rate

distribution is expected to be very stable. The implied volatility for the ATM option on the cross-rate is

about 60% of the implied volatility in any of the two ATM options in the dollar rates. The joint estimation

of the bivariate distribution takes this into account and results in an expected distribution of the cross-rate

that is relatively more stable.

A comparison of the estimated marginal distributions relative to alternative methods of estimation

shows that there is a significant difference between the estimated univariate distributions under alternative

methods. There are 2 different types of alternative methods. One is differences in the kind of the data

used for the estimation: only the ATM option, the full range of options on a single underlying and, the full

14

range of options for the triad of currencies. The second are differences in the estimation methods for a

given amount of data used (such as those described in section 2 above). The vast majority of the

differences in the estimated distributions comes from the kind of data being used rather than from the

estimation method. As an illustration we plot the implied DEM/USD distributions obtained in four different

ways: (i) from the bivariate distribution above, (ii) from the univariate distribution using a mixture of two

lognormal distributions, and (iii) fitting a cubic spline to the volatility smile, and, finally, (iv) from the ATM

implied volatility assuming a constant volatility (Black-Scholes) model (Figure 5b). The result show that

there are big differences between the estimated implied distributions under Black-Scholes, using univariate

and using a bivariate distributions. By contrast, the differences among univariate distributions are minor

We are interested in the behavior of the distribution over certain regions of the probability space

rather than for the full support of the distribution. To illustrate these differences we report three

alternative measures of comovement: the conditional distributions of the DEM/USD and USD/GBP

exchange rates for a 10% movement of the other rate relative to its current value; the 5% value-at-risk of

a portfolio invested in these two currencies; and the pricing of an option on the better of two currency

returns.

Table 2 reports the Value-at-Risk of three alternative portfolios invested in Mark and Pounds.

We calculate the 5% VaR using three different methodologies: using the historical distribution of DEM and

GBP returns from 1985 to 1995, and using the bivariate implied distribution from option prices and the

univariate distrubution from the ATM implied volatilities. The results show that the implied distributions

significantly suggest the presence of higher tails than the historical distribution. The probability of a large

dollar depreciation is higher. Also the difference between the VaRs from the multivariate distribution and

the at-the-money lognormal is not monotonous. This depends on the relative prices of the different options

in these currencies. For instance, the multivariate distribution show the relative lower probability of a joint

apreciation of the dollar with respect to both the DEM and the GBP.

15

We computed the conditional distributions of the DEM/USD exchange rate for a value of the

USD/GBP of 1.37, which implies a 10% pound depreciation relative to its spot value (1.525) on April 3,

1996. The conditional distribution of the DEM/USD exchange rate from the multivariate distribution has a

much lower standard deviation, with a mode value of 1.583, relative to a mode value on the ATM

distribution of 1.647 (figure 6a). Therefore, the multivariate distribution suggests that the expected

depreciation of the mark conditional on a 10% pound depreciation is somewhat lower. On the other hand,

the conditional distribution of the USD/GBP exchange rate for a 10% mark depreciation to the dollar,

implies a higher GBP depreciation when estimated from the multivariate implied distribution vs. the ATM

distribution (figure 6b).

V. Conclusions

The purpose of this paper has been to present a methodology for the estimation of multivariate

distributions based on options data and to evaluate the information that can be extracted from these

distributions to inform investment decision making. We extended three of the most widely used methods

to calculate univariate risk-neutral distributions to the multivariate case and evaluated the advantages and

problems in estimation faced by each of these methods. We also proposed several measures derived from

this multivariate distributions that might help asset managers to conform a better picture of the underlying

expectations for risk management.

The estimation and use of the full implied multivariate distribution rather than the first and second

moments of the distribution can be particularly useful in certain contents in which we are really interested

primarily on the shape of certain domain of the distribution, not in its full domain. The detailed knowledge

of certain parts of the multivariate distribution is useful in contexts as diverse as risk management, asset

allocation and pricing of exotic derivatives.

We illustrated the methodology with the estimation of an example using option data on the mark-

dollar, mark-sterling and sterling-dollar exchange rates to compute the bivariate distribution on the dollar-

16

mark and dollar-sterling returns. In this context, we compared the measures of correlation, Value-at-Risk

(VaRs), and probability measures of certain comovements and looked at the conditional distribution of

returns on each asset given a certain return on the other asset. Our results provided strong evidence that

the implied multivariate distribution predicts much larger probabilities of large joint depreciations of the

mark and sterling relative to the dollar, than those computed from historical returns, or from the

information obtained using the information from only at-the-money options. This evidence suggests, for

instance, that markets put a stronger expectation on the likelihood of “contagion” occurring than that

previously estimated in the literature using only historical information.

17

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Figures 1a-1c: 3-Month Volatility Smile, triad USD-DEM-GBP, Aril 3, 1996

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Figures 2a-2c: Implied Distributions DEM/USD and USD/GBP, alternative methods

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Figures 3a-3c: Implied marginal distributions DEM/USD, USD/GBP and DEM/GBP, April 3, 1996

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Figures 4a-4c: Comparisons of Implied Distributions: Bivariate vs. Univariate estimation, April 3, 1996

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Figures 5a-5b: Comparisons of Implied Distributions: USD/DEM, April 3, 1996Bivariate vs. Different Univariate Estimation Methods

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Figures 6a-6b: Implied Conditional Distributions: April 3, 1996

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Table 1: Actual and Predicted Values of Forward Rates and Option Premia

Currency/Instrument Actual Predicted Predicted(Bivariate Dist.) (Univariate Distr.)

DEM/USD

Forward 1.47343 1.47949 1.4732410-delta 0.00363 0.00359 0.0036125-delta 0.01083 0.01092 0.01086At-the-money 0.02827 0.02862 0.0296775-delta 0.05710 0.05793 0.0582390-delta 0.09594 0.09894 0.09640

USD/GBP


DEM/GBP


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Table 2: 5% Value-at-Risk of a Mark and Pound Portfolio, April 3, 1996

Portfolio Weigths: Historical Multivariate ATM Log-Normal

DEM GBP Distribution Distribution Distribution

1 0 -0.070 -0.074 -0.078

0 1 -0.051 -0.059 -0.054

0.5 0.5 -0.062 -0.084 -0.089

implied multivariate value-at-risk

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