Abstract
This paper shows how to value multiasset options analytically in a modeling framework that combines both continuous and discontinuous variations in the underlying equity or foreign exchange processes and a stochastic, two-factor yield curve. All correlations are taken into account, between the factors driving the yield curve, between fixed income and equity as asset classes, and between the individual equity assets themselves. The valuation method is applied to three of the most popular two-asset options.
1. Introduction
Multiasset options, that is, options whose payoff depends on more than one underlying risky asset, are actively traded in the financial markets. They are valuable to investors as tools for diversification and because they allow trading not only volatility but also correlation. They are useful to companies that need to hedge complex positions across various asset classes. For more background on these topics, the reader may refer, for example, to Bouzoubaa and Osseiran [1].
Analytical valuation formulae for multiasset options are available only in the standard Black-Scholes [2] modeling framework. The latter is notoriously based on a number of assumptions that are not empirically verified, as explained, for example, by Epps [3]. In this paper, it is shown how to price two-asset options analytically without having to rely on two of the most spurious assumptions of the Black-Scholes model, namely, the continuity of asset price paths and the nonvariability of interest rates. The former of these two assumptions is clearly invalidated by observed data in times of financial stress, especially during the so-called market crashes, in which stock prices do exhibit jumps. It results in dangerously underestimated option prices, especially for short-lived, out-of-the-money contracts. As for the assumption that interest rates are constant, that is, the yield curve is flat and perfectly predictable, it becomes all the more misleading as the option expiry increases, or as the option payoff is a function of the correlation between equity and fixed income.
To overcome these two major flaws in the Black-Scholes theory, this article puts forward a model that combines a bivariate jump-diffusion equity component with a two-factor, time-dependent, stochastic yield curve. The originality of this model is to allow the simultaneous introduction of(i)discontinuous variations in the equity prices,(ii)stochastic evolution of the interest rate,(iii)cross-correlation of all random factors, that is, between jumps, between the factors driving the yield curve, between fixed income and equity, and between the individual stocks themselves.These features are introduced without having to resort to approximating numerical schemes. Complete analytical tractability is preserved, resulting in closed form formulae for option prices in terms of functions that can be numerically evaluated with the accuracy and the efficiency required for all practical purposes. To achieve analytical tractability in spite of the relatively high number of correlated random factors, the following mathematical tools are introduced: a general result on the covariance of two Ito integrals, an explicit rule for the orthogonal decomposition of a multivariate Gaussian random vector, and a computational technique allowing bypassing the use of a change of measure under certain conditions. The key to the derivation of the formulae is the ability to find strong solutions to the stochastic differential equations driving the variations of the underlying equity or foreign exchange assets, thus allowing obtaining the covariances between the random variables involved at the option expiry in a straightforward manner.
This article is organized as follows: Section 2 describes the model and formulates the valuation problem; Section 3 explains the method followed to solve the problem; Section 4 provides analytical formulae for three of the most widely traded multiasset options, as well as numerical results; Appendix A includes the proofs of all the lemmas introduced in Sections 2 and 3; Appendix B lists the definitions of terms used in the formulae of Section 4.
2. Description of the Model and Formulation of the Problem
In the sequel, stands for a standard Brownian motion and the constant correlation coefficient between and is denoted by ,; also, the indicator function is denoted by when it tests the inclusion of a real number within the range and by when it tests the past realization of an event defined as a subset of a given -algebra.
The short rate is driven by the following two-factor, time-dependent, mean-reverting stochastic differential equation: where and is a deterministic function of satisfying a linear growth condition.
This is an extended Vasicek [4] model, similar to the G2++ model of Brigo and Mercurio [5]. Compared to the original Vasicek model, a second Brownian motion has been added. This is because statistical studies of the yield curve have consistently pointed out the need to introduce at least two factors in order to reproduce the observed variability of market rates (see, e.g., [5]). In this respect, and are the sensitivities of the interest rate to the first and the second random factors affecting the yield curve. The mean-reversion feature has been preserved, as it is validated by empirical data, but it has been made time-dependent, so that calibration to the current market prices can be achieved by fitting an appropriate function .
Let and be the price processes of two risky assets of equity or foreign exchange type. It is assumed that they evolve in time as semimartingales with both continuous and noncontinuous sources of randomness. More precisely, the instantaneous rates of change of and , under the real measure of the financial market denoted by , are given, respectively, by the following stochastic differential equations (SDEs):where and are the rates of return demanded by the market to invest on and , respectively, typically including a risk premium; and are constant payout rates associated with assets and , respectively, for example, continuous-time equivalent dividend rates inferred from actual dividends paid at discrete dates; is the constant diffusive volatility of and is the constant diffusive volatility of ; that is, and are the standard deviations of the returns of and , respectively, as measured with respect to the continuous part of the variations of and ; and are two Poisson processes with intensities and which admit the following decomposition cases: where , , and are independent Poisson processes with intensities , , and , respectively; thus, and have positive correlation given by ,, where and is a sequence of independent, identically distributed random variables taking values in ; in other words, is a right-continuous process providing the magnitude values of the jumps of asset . Set and assume that is normally distributed with mean and variance ; then, It is assumed that all Brownian and compound Poisson processes implied by the model are defined on an adequate probability product space in which the smallest -algebra generated by the random variables , ,, and , , is denoted by .
Thus, the model used for equity is an extension of Merton [6], consisting in the introduction of a stochastic interest rate as well as of a bivariate framework allowing for multicorrelation.
As a consequence of the theory of nonarbitrage valuation of contingent claims [7, 8], the prices, at the current time denoted by , of European call or put options written on and with expiry , can be expressed as linear combinations of conditional expectations of two kinds:In (7) and (8), represents an event involving and/or that has to occur for the option to have strictly positive value at expiry , while is a function implied by the specific payoff under consideration, and is an equivalent martingale measure. Let us first give examples of and , assuming is known. To begin with, consider a put option on the minimum of two assets. This popular contract pays out the difference, if positive, between a prespecified strike price and the minimum between two equity asset prices at expiry . Its Black-Scholes price is given by Stulz [9]. In our model, the no-arbitrage value of this contingent claim can be decomposed as follows:where and are predefined weights attached to and , respectively, and and are indicator functions defined byOur second example of actively traded instrument is the (zero-strike) spread option, which yields the difference between the higher and the lower of two asset prices at expiry . Its Black-Scholes price was first obtained by Margrabe [10]. In our model, the no-arbitrage value of this contract can be written aswhere and are indicator functions defined byOur third example is a call option on a product, which pays out the difference, if it is positive, between the product of two asset prices, and , and a prespecified strike price at expiry . Usually, in a product option, is a foreign exchange rate, while is a foreign currency denominated equity price, and the contract is often called a quanto option. The Black-Scholes value of the latter can be found in Zhang [11]. In our model, the no-arbitrage value of this contract can be written aswhere is an indicator function defined byLet us now move on to the question of the pricing measure . As a result of the theory of no-arbitrage valuation, must be the equivalent martingale measure under which the numeraire is the so-called money market account , defined by . Let us recall that a numeraire is a financial asset whose price can be modeled by a strictly positive stochastic process, while a probability measure is said to be martingale equivalent for a given numeraire if the price of any asset divided by is a martingale under [12]. The following lemma is introduced in order to determine the dynamics of and under .
Lemma 1. Using the previously defined notations, let be a process driven by , where is a continuous -adapted process verifying the usual assumptions for the latter stochastic differential equation to be well defined. Then, denoting by the intensity of the Poisson process , a necessary condition for the discounted asset price to be a martingale is to have .
Proof of Lemma 1. See Appendix A.
Hence, the instantaneous rates of change of and under are driven, respectively, by the following equations:It is emphasized that the condition given in Lemma 1 is necessary but not sufficient. Actually, the introduction of jumps makes the market incomplete; that is, perfect hedging is not possible. Merton [6] argues that jump risk is diversifiable and therefore not rewardable with excess return. However, that argument is debatable because of the existence of industrywide and countrywide shocks. In this respect, two kinds of jumps can be considered in our model. The “idiosyncratic” ones refer to shocks that specifically affect a single company instead of the market or a whole economic sector. This is the kind of jumps that can be diversified. On the other hand, “systemic” jumps affect all stocks, as can be observed during market crashes. The nature of risk can be defined by adjusting the parameter : the lower this parameter is, the more the jump risk becomes idiosyncratic. The magnitude of the parameters and reflects the degree of diversification: in a perfectly diversified portfolio, these parameters should tend to zero, so that the compound Poisson processes driving the two stocks have the same market-driven intensity .
3. Analytical Method of Valuation
In order to price options on and , SDEs (23) and (24) must be solved, conditional on and . An application of Ito’s lemma to and to yields the following solutions to (23) and (24) at time : where and are normally distributed with and , .
Solutions (25) are not explicit enough to compute option values. They are only representations of the solutions to (23) and (24) in terms of the integrated interest rate. They do not give the correlation structure between fixed income and equity, that is, the dependence of assets and on the Brownian motions and . Consequently, neither do they give the correlation structure between the continuous parts of the returns on and , conditional on the correlation between fixed income and equity, that is, the correlation between and conditional on and . First, to determine explicitly the correlation between fixed income and equity, let us focus on the time integral of on , that is, on . By taking the stochastic differential of and integrating it on , (1) yields The integral is normally distributed since is a Gaussian process. Fubini’s theorem yields , as given in the Appendix. The variance of can be written as . To tackle this integral, the following lemma is introduced.
Lemma 2. Let and be two standard Brownian motions with constant correlation coefficient . The natural filtration generated by is denoted by . Let and be two right-continuous, -adapted processes such that and . Then, , one has
Proof of Lemma 2. See Appendix A.
Lemma 2 yields the covariance function of : so that the variance of the time integral of is equal toThus, for fixed , the following equality holds in law:where The next step is to determine explicitly the correlation structure between the continuous parts of the returns on and , conditional on the correlation between fixed income and equity. This can be achieved by using the following lemma.
Lemma 3. At any given time , , , and can be written as follows:where , , and are standard Brownian motions independent both of one another and of and where the following definitions hold:
Proof of Lemma 3. See Appendix A.
Thus, a combination of Lemmas 1, 2, and 3 yields, under the pricing measure , the following strong solutions to SDEs (23) and (24), conditional on and , in which all the correlations are explicit:Since all the random variables summed in (36) are normally distributed and pairwise independent, the conditional distributions of and under are Gaussian, and so are those of , , and . The parameters of these distributions follow easily from stability under addition and are listed in Appendix B.
We can now turn to the computation of (7) and (8). A classical way of turning an expectation involving an indicator function into a probability, that is, of getting rid of any random function multiplying the indicator function inside the expectation operator, is to switch to a new measure. In this respect, the following lemma is instrumental in computing the expectations of the second kind in (8).
Lemma 4. Denoting by a subset of the -algebra generated by the random variables defined on the space , one haswhereand is a measure equivalent to
Proof of Lemma 4. See Appendix A.
As a consequence of Lemma 4, we obtain where and are independent standard Brownian motions under .
Given and , one can then easily find strong solutions for and under by substituting (39) into (36).
To compute the expectations of the first kind in (7), one can draw on the joint normality of the involved random variables and use the following lemma.
Lemma 5. Let , and be three normal random variables with constant correlation coefficients denoted by , and . Let and be two real numbers. Then, where is the univariate standard normal cumulative distribution function with upper bound and is the bivariate standard normal cumulative distribution function with upper bounds and , and correlation coefficient .
Proof of Lemma 5. See Appendix A.
Applying the five previous lemmas, it only remains to sum over the joint distribution of , which is the product of the distributions of , and , to obtain closed form option pricing formulae, as the next section shows.
4. Formulae and Numerical Results
Consider first the value of a put option on the minimum of two assets with strike price , as described in Section 2. The pairs of random variables and are both bivariate normal and the implied pairwise conditional covariances, given by and in Appendix B, are easily derived using (36). To compute conditional expectations (9) and (11), Lemma 4 is applied to obtain and , respectively.
To compute conditional expectation (10), Lemma 5 is applied withTo compute conditional expectation (12), Lemma 5 is applied withGeneralizing to a put or a call option on the minimum or on the maximum of two assets, the following formula can be obtained.
Formula 1. Let denote the no-arbitrage value at time of a put or a call on the minimum or on the maximum at expiry of two weighted assets and with strike price . Within the modeling framework defined in Section 2, we havewhereAll the symbols and notations in Formula 1 that have not been previously defined are given in Appendix B. The same remark applies to Formulas 2 and 3.
The numerical implementation of Formula 1 is easy using the algorithm of Drezner and Wesolowsky [13], or its slightly improved version by Genz [14], to evaluate the function with the accuracy and the efficiency required for all practical purposes. The infinite series can be truncated in a simple manner by setting a convergence threshold such that no further terms are added once the difference between two successive finite sums becomes smaller than that prespecified level. In Table 1, a few numerical results are reported. Formula 1 is applied and a comparison is made with option prices obtained in a standard Black-Scholes framework. The diffusive volatility of asset is assumed to be higher than the diffusive volatility of asset . The intensity of the jump process is greater than the intensity of the jump process . The variance of the jumps of , that is, the parameter , is greater than the variance of the jumps of . Thus, the variability of is higher than that of in the continuous world, but the discontinuous variations of asset are more frequent and more pronounced than those of asset . The continuous part of the correlation between and , that is, the coefficient , is negative, according to the way two-asset options are usually structured. The random factors driving the short end and the long end of the yield curve, that is, the Brownian motions and , are assumed to have a positive correlation coefficient . The correlation between fixed income and equity is negative, whether it is the correlation with the short end of the yield curve (coefficients and ) or the correlation with the long end (coefficients and ).
The values of put options on the minimum of and are computed, both at-the-money and out-of-the-money and with two different maturities. The first modeling environment considered (Model 1) is the standard Black-Scholes one; that is, interest rates are constant and there are no discontinuous variations in equity asset prices. The second modeling environment considered (Model 2) is one in which the yield curve is driven by (1), but the variations of equity asset prices remain purely continuous. The other modeling environments considered (Model 3, Model 4, Model 5, and Model 6) include jumps in and according to (23) and (24), with two different levels of intensity and with or without stochastic interest rate.
For all numerical values reported in Table 1, a convergence level of was obtained by truncating the infinite series at , which resulted in a computational time of approximately 0.2 seconds on an ordinary personal PC. A Monte Carlo simulation algorithm was also implemented as an alternative approximate valuation method to Formula 1. It took more than two hours to achieve a convergence level with a single value obtained by applying Formula 1. This shows how developing exact analytical formulae can result in dramatic accuracy and efficiency gains when using models implying several correlated continuous and noncontinuous sources of randomness. Monte Carlo simulation is particularly slow and inaccurate in this case, as SDEs (1), (23), and (24) need to be discretized, and the valuation algorithm has to generate a lot of interarrival times between jumps consistent with the option’s specifications.
Next, the case of a spread option is handled. Lemma 5 is applied with the following inputs:Deriving the required covariances (given in Appendix B), the following formula can then be obtained.
Formula 2. Let denote the no-arbitrage value at time of a zero-strike spread option on two weighted assets and with expiry . Within the modeling framework defined in Section 2, we haveThe numerical implementation of Formula 2 is even easier than that of Formula 1, as only univariate standard normal cumulative distribution functions are involved instead of bivariate ones. A few numerical results are reported in Table 2. There is no model specification with a stochastic interest rate as the spread option is not a function of interest rate.
Finally, the product option is dealt with. Lemma 4 is applied to compute (21) while Lemma 5 is applied to compute (20) with and . Deriving the required covariances (given in Appendix B), the following formula can then be obtained.
Formula 3. Let denote the no-arbitrage value at time of a product option on two assets and with strike price and expiry . Within the modeling framework defined in Section 2, we have
5. Conclusion
In this article, three of the most widely traded two-asset options are analytically valued in a modeling framework allowing discontinuous variations in the equity prices, stochastic two-factor evolution of the yield curve, and cross-correlation between all the random factors. The method used can be applied just as easily to other payoffs involving two assets, as long as they are not path-dependent (e.g., barrier or lookback option) or of American type (i.e., the option can only be exercised at expiry). For instance, one could easily handle an option based on a weighted average of the returns of two stocks at expiry; such a payoff belongs to a category known as “basket options.” The same method can also be applied to options written on more than two assets, by extending Lemma 3 to a vector space of dimension greater than four. One important point should be noticed, though, when extending the method to more assets: some options, such as the product option, will still admit valuation formulae in terms of linear combinations of univariate or bivariate cumulative distribution functions, whose numerical evaluation is easy, but others, such as the option on the maximum or on the minimum, will admit valuation formulae in terms of multivariate normal cumulative distribution functions of increasing dimension, which rapidly become uneasy to evaluate with accuracy and efficiency. Also, the number of summation operators will increase, thus slowing down computational time. Hence, the extension to options written on a higher number of assets is feasible but more or less advisable, depending on the computed payoff.
Appendix
A. Proofs of Lemmas
Proof of Lemma 1.
Proof of Lemma 2. Since the processes and are square integrable and predictable with respect to , the integrals and are Ito integrals, and they are martingales with respect to . Therefore, we have By conditioning with respect to and using elementary properties of the Ito integral, we obtainSince and are right-continuous, square integrable processes, they can be approximated, respectively, by the following simple processes: where is the number of points in a sequence of partitions of ; and are constants if is a pair of deterministic processes; and are square integrable, -adapted, random variables if is a pair of random processes; and .
It is a classical result from the theory of continuous-time processes that and can be approximated byrespectively, where the approximating sums in (A.5) converge in the mean square sense [15]. Hence, as ,By conditioning on , the sum can be rewritten as follows:Denoting by a standard Brownian motion independent of defined on the same filtered probability space as , one can apply the well-known orthogonal decomposition of with respect to in the following manner: If and are two natural integers such that , then ; similarly, if , then . Thus, Hence, . Turning to the sum , we obtainwhere the last sum converges in mean square to .
Proof of Lemma 3. Equation (32) is a well-known result, used already in the proof of Lemma 2. Since any finite-dimensional Gaussian space is a vector space with respect to a basis of pairwise independent standard normal Gaussian random variables, the random variable admits the following orthogonal decomposition:where and are real-valued scalars; must be positive since, by definition of the multivariate normal random vector , has to be the standard deviation of conditional on and . From the definition of linear correlation and the bilinearity of covariance, we obtainFrom the standard deviation of and from the stability under addition of the normal distribution, we getThe orthogonal decomposition of in (34) is derived in a similar manner.
Proof of Lemma 4. From (30) and from the formula for the moment generating function of a normal random variable, we get as given by Lemma 4; is the price, at time , of a zero-coupon bond maturing at . Applying Ito’s lemma to under and then integrating on , one can obtainwith Hence, we havewhere is the equivalent martingale measure under which the numeraire is the zero-coupon bond, whose Radon-Nikodym derivative is given by
Proof of Lemma 5. Let denote the joint density of , and . Then,Apply the following chain of changes of variables: and then use the identity to obtain Lemma 5.
B. Definitions of Terms Involved in the Formulae
(1) The -Terms. This collection of terms refers to expectations involved in Formulas 1, 2, and 3:
(2) The -Terms. This collection of terms refers to standard deviations involved in Formulas 1, 2, and 3:
(3) The -Terms. This collection of terms refers to covariances involved in Formulas 1, 2, and 3:
(4) Miscellaneous. One has
Competing Interests
The author declares that there are no competing interests regarding the publication of this paper.