# An Analytically Tractable Model for Pricing Multiasset Options with Correlated Jump-Diffusion Equity Processes and a Two-Factor Stochastic Yield Curve

Authors: Tristan Guillaume
Journal: Journal of Applied Mathematics
 (2016)

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## 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.


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## Body

## 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: Section2 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,Wti stands for a standard Brownian motion and the constant correlation coefficient between Wti and Wtj is denoted by ρi.j,i,j∈N2; also, the indicator function is denoted by Ia,bx when it tests the inclusion of a real number x within the range a,b and by IZ when it tests the past realization of an event defined as a subset Z of a given σ-algebra.The short rate is driven by the following two-factor, time-dependent, mean-reverting stochastic differential equation:(1)drt=abt-rtdt+σR1dWt1+σR2dWt2,where a,σR1,σR2∈R+3 and bt is a deterministic function of t 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, σR1 and σR2 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 bt.LetSt1,t≥0 and St2,t≥0 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 S1 and S2, under the real measure of the financial market denoted by P, are given, respectively, by the following stochastic differential equations (SDEs):(2)dSt1St-1=μ1-δ1dt+σS1dWt3+It1dNt1(3)dSt2St-2=μ2-δ2dt+σS2dWt4+It2dNt2,where μ1 and μ2 are the rates of return demanded by the market to invest on S1 and S2, respectively, typically including a risk premium; δ1 and δ2 are constant payout rates associated with assets S1 and S2, respectively, for example, continuous-time equivalent dividend rates inferred from actual dividends paid at discrete dates; σS1 is the constant diffusive volatility of S1 and σS2 is the constant diffusive volatility of S2; that is, σS1 and σS2 are the standard deviations of the returns of S1 and S2, respectively, as measured with respect to the continuous part of the variations of S1 and S2; Nt1 and Nt2 are two Poisson processes with intensities λ1 and λ2 which admit the following decomposition cases: (4)Nt1=Zt1+Zt12,Nt2=Zt2+Zt12,where Zt1, Zt2, and Zt12 are independent Poisson processes with intensities λ1′, λ2′, and λ12′, respectively; thus, Nt1 and Nt2 have positive correlation given by (5)ρ1.2N=λ12′λ1′+λ12′λ2′+λ12′,Iti=∑nUniIτn-1i,τnit,∀n∈N, where τni=inf⁡t≥0,Nti=n and Uni is a sequence of independent, identically distributed random variables taking values in -1,+∞; in other words, Iti is a right-continuous process providing the magnitude values of the jumps of asset Sti. Set Jni=ln⁡1+Uni and assume that Jni is normally distributed with mean ξi and variance εi2; then, (6)EUni≜κi=exp⁡ξi+εi22-1.It is assumed that all Brownian and compound Poisson processes implied by the model are defined on an adequate probability product space S in which the smallest σ-algebra generated by the random variables Wsi, Nsi,∀s≤t, and UniIn≤Nti, ∀n≥1, is denoted by Ft.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 t0, of European call or put options written on S1 and S2 with expiry T, can be expressed as linear combinations of conditional expectations of two kinds:(7)EQexp⁡-∫0TrtdtfST1,ST2IZ∣F0(8)EQexp⁡-∫0TrtdtIZ∣F0.In (7) and (8), Z represents an event involving ST1 and/or ST2 that has to occur for the option to have strictly positive value at expiry T, while f is a function implied by the specific payoff under consideration, and Q is an equivalent martingale measure. Let us first give examples of Z and f, assuming Q 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 K and the minimum between two equity asset prices at expiry T. 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:(9)KEQexp⁡-∫0TrtdtI1∣F0(10)-EQexp⁡-∫0Trtdtα1ST1I1∣F0(11)+KEQexp⁡-∫0TrtdtI2∣F0(12)-EQexp⁡-∫0Trtdtα2ST2I2∣F0,where α1∈0,1 and α2∈0,1 are predefined weights attached to S1 and S2, respectively, and I1 and I2 are indicator functions defined by(13)I1=Iα1ST1<K,α1ST1<α2ST2=Iln⁡ST1/S01<ln⁡K/α1S01,ln⁡ST1S02/ST2S01<ln⁡α2S02/α1S01(14)I2=Iα2ST2<K,α2ST2<α1ST1=Iln⁡ST2/S02<ln⁡K/α2S02,ln⁡ST2S01/ST1S02<ln⁡α1S01/α2S02.Our 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 T. Its Black-Scholes price was first obtained by Margrabe [10]. In our model, the no-arbitrage value of this contract can be written as(15)EQexp⁡-∫0Trtdtα1ST1I3∣F0(16)-EQexp⁡-∫0Trtdtα2ST2I3∣F0(17)+EQexp⁡-∫0Trtdtα2ST2I4∣F0(18)-EQexp⁡-∫0Trtdtα1ST1I4∣F0,where I3 and I4 are indicator functions defined by(19)I3=Iα1ST1>α2ST2=Iln⁡ST1S02/ST2S01>ln⁡α2S02/α1S01I4=Iα2ST2>α1ST1=Iln⁡ST2S01/ST1S02>ln⁡α1S01/α2S02.Our 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, S1 and S2, and a prespecified strike price K at expiry T. Usually, in a product option, S1 is a foreign exchange rate, while S2 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 as(20)EQexp⁡-∫0TrtdtST1ST2I5∣F0(21)-KEQexp⁡-∫0TrtdtI5∣F0,where I5 is an indicator function defined by(22)I5=IST1ST2>K=Iln⁡ST1ST2/S01S02>ln⁡K/S01S02.Let us now move on to the question of the pricing measure Q. As a result of the theory of no-arbitrage valuation, Q must be the equivalent martingale measure under which the numeraire is the so-called money market account βt, defined by βt=exp⁡∫0trsds,t≥0. 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 P is said to be martingale equivalent for a given numeraire N if the price of any asset divided by N is a martingale under P [12]. The following lemma is introduced in order to determine the dynamics of S1 and S2 under Q.Lemma 1.
Using the previously defined notations, letSt,t≥0 be a process driven by dSt/St-=μtdt+σdWt+ItdNt, where μt is a continuous Ft-adapted process verifying the usual assumptions for the latter stochastic differential equation to be well defined. Then, denoting by λ the intensity of the Poisson process Nt,t≥0, a necessary condition for the discounted asset price S~T≜exp⁡-∫0TrtdtST to be a martingale is to have μt=rt-λκ.Proof of Lemma1.
See AppendixA.Hence, the instantaneous rates of change ofS1 and S2 under Q are driven, respectively, by the following equations:(23)dSt1St-1=rt-δ1-λ1κ1dt+σS1dWt3+It1dNt1(24)dSt2St-2=rt-δ2-λ2κ2dt+σS2dWt4+It2dNt2.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 λ12′: the lower this parameter is, the more the jump risk becomes idiosyncratic. The magnitude of the parameters λ1′ and λ2′ 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 λ12′.


## 3. Analytical Method of Valuation

In order to price options onST1 and ST2, SDEs (23) and (24) must be solved, conditional on NT1 and NT2. An application of Ito’s lemma to ln⁡St1 and to ln⁡St2 yields the following solutions to (23) and (24) at time T: (25)ST1∣NT1=S01exp⁡∫0Trtds-δ1+λ1κ1+σS122T+σS1WT3+∑i=1NT1Ji1ST2∣NT2=S02exp⁡∫0Trtdt-δ2+λ2κ2+σS222T+σS2WT4+∑i=1NT2Ji2,where ∑i=1NT1Ji1∣NT1 and ∑i=1NT2Ji2∣NT2 are normally distributed with E∑i=1NTkJik∣NTk=NTkξk and var⁡∑i=1NTkJik∣NTk=NTkεk2,  ∀k∈1,2.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 S1 and S2 on the Brownian motions W1and W2. Consequently, neither do they give the correlation structure between the continuous parts of the returns on S1 and S2, conditional on the correlation between fixed income and equity, that is, the correlation between W3 and W4 conditional on W1 and W2. First, to determine explicitly the correlation between fixed income and equity, let us focus on the time integral of rt on 0,T, that is, on ∫0Trtdt. By taking the stochastic differential of exp⁡atrt and integrating it on 0,t, (1) yields (26)rt=r0e-at+a∫0te-at-sbsds+σR1∫0te-at-sdWs1+σR2∫0te-at-sdWs2.The integral ∫0Trtdt is normally distributed since rt is a Gaussian process. Fubini’s theorem yields E∫0Trtdt=μ-r, as given in the Appendix. The variance of ∫0Trtdt can be written as var⁡∫0Trtdt=∫0T∫0tcovrt,rsdsdt. To tackle this integral, the following lemma is introduced.Lemma 2.
LetWs1,s≥0 and Ws2,s≥0 be two standard Brownian motions with constant correlation coefficient ρ. The natural filtration generated by Ws1,Ws2 is denoted by Fs. Let Ys1,s≥0 and Ys2,s≥0 be two right-continuous, Fs-adapted processes such that E∫0TYs12ds<∞ and E∫0TYs22ds<∞. Then, ∀0≤t≤T, one has (27)cov⁡∫0tYs1dWs1,∫0TYs2dWs2=ρ∫0tEYs1Ys2ds.Proof of Lemma2.
See AppendixA.Lemma2 yields the covariance function of rt,t≥0: (28)cov⁡rt,rs=e-as+te2as-12aσR12+σR22+2ρ1.2σR1σR2,so that the variance of the time integral of rt,t≥0 is equal to(29)var⁡∫0Trtdt=1a2σR12+σR22+2ρ1.2σR1σR2T+2e-aT-1a+1-e-2aT2a.Thus, for fixed T≥0, the following equality holds in law:(30)∫0Trtdt=μ-r+σ-rWT1σR1+σR2ρ1.2+σR2σ2∣1W-T2,where (31)σ-r=1aTT-2a1-e-aT+12a1-e-2aT.The next step is to determine explicitly the correlation structure between the continuous parts of the returns on S1 and S2, conditional on the correlation between fixed income and equity. This can be achieved by using the following lemma.Lemma 3.
At any given timet≥0, Wt2, Wt3, and Wt4 can be written as follows:(32)Wt2=ρ1.2Wt1+σ2∣1W-t2t(33)Wt3=ρ1.3Wt1+ρ2.31W-t2+σ3∣1.2W-t3(34)Wt4=ρ1.4Wt1+ρ2.4∣1W-t2+ρ3.4∣1.2W-t3+σ4∣1.2.3W-t4,where W-t2,t≥0, W-t3,t≥0, and W-t4,t≥0 are standard Brownian motions independent both of one another and of Wt1,t≥0 and where the following definitions hold:(35)σ2∣1=1-ρ1.22ρ2.3∣1=ρ2.3-ρ1.2ρ1.3σ21σ3∣1.2=1-ρ1.32-ρ2.312ρ2.4∣1=ρ2.4-ρ1.2ρ1.4σ2∣1ρ3.4∣1.2=ρ3.4-ρ1.3ρ1.4-ρ2.3∣1ρ2.4∣1σ3∣1.2σ4∣1.2.3=1-ρ1.42-ρ2.4∣12-ρ3.4∣1.22.Proof of Lemma3.
See AppendixA.Thus, a combination of Lemmas1, 2, and 3 yields, under the pricing measure Q, the following strong solutions to SDEs (23) and (24), conditional on NT1 and NT2, in which all the correlations are explicit:(36)ln⁡ST1S01=μ-r-δ1+λ1κ1+σS122T+WT1σ-rσR1+σR2ρ1.2+σS1ρ1.3+W-T2σ-rσR2σ2∣1+σS1ρ2.3∣1+W-T3σS1σ3∣1.2+∑i=1NT1Ji1ln⁡ST2S02=μ-r-δ2+λ2κ2+σS222T+WT1σ-rσR1+σR2ρ1.2+σS2ρ1.4+W-T2σ-rσR2σ2∣1+σS2ρ2.4∣1+W-T3σS2ρ3.4∣1.2+W-T4σS2σ4∣1.2.3+∑i=1NT2Ji2.Since all the random variables summed in (36) are normally distributed and pairwise independent, the conditional distributions of ln⁡ST1/S01 and ln⁡ST2/S02 under Q are Gaussian, and so are those of ln⁡ST1S02/ST2S01, ln⁡ST2S01/ST1S02, and ln⁡ST1ST2/S01S02. 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 byZ a subset of the σ-algebra generated by the random variables defined on the space S, one has(37)EQexp⁡-∫0TrtdtIZ∣F0=B0,TEPBTIZ∣F0,where(38)Bt,T=exp⁡-At,Trt+Ct,T,0≤t≤TAt,T=1a1-e-aT-tCt,T=-a∫tT∫tue-au-sbsdsdu+1a2σR12+σR22+2ρ1.2σR1σR2T-t+2e-aT-t-1a+1-e-2aT-t2aand PBT is a measure equivalent to Q.Proof of Lemma4.
See AppendixA.As a consequence of Lemma4, we obtain (39)WT1=ZT1-σR1+σR2ρ1.2∫0TAs,Tds=ZT1+σR1+σR2ρ1.21-e-aT-Ta2W-T2=Z-T2-σR2σ2∣1∫0TAs,Tds=Z-T2+σR2σ2∣11-e-aT-Ta2,where Z1 and Z-2 are independent standard Brownian motions under PBT.GivenNT1 and NT2, one can then easily find strong solutions for ST1 and ST2 under PBT 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.
LetX~NμX,σX2,Y~NμY,σY2, and Z~NμZ,σZ2 be three normal random variables with constant correlation coefficients denoted by ρX.Y,ρX.Z,  and  ρY.Z. Let a and b be two real numbers. Then, (40)Eexp⁡XIY≤a,Z≤b=exp⁡μX+σX22N2a-μY-ρX.YσXσYσY,b-μZ-ρX.ZσXσZσZ;ρY.ZEexp⁡XIY≤a=exp⁡μX+σX22Na-μY-ρX.YσXσYσY,where Na is the univariate standard normal cumulative distribution function with upper bound a∈R and N2a,b;ρ is the bivariate standard normal cumulative distribution function with upper bounds a∈R and b∈R, and correlation coefficient ρ∈-1,1.Proof of Lemma5.
See AppendixA.Applying the five previous lemmas, it only remains to sum over the joint distribution ofNT1,NT2, which is the product of the distributions of ZT1,ZT2, and ZT12, 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 priceK, as described in Section 2. The pairs of random variables lnST1S02/ST2S01,lnST1/S01 and lnST2S01/ST1S02,lnST2/S02 are both bivariate normal and the implied pairwise conditional covariances, given by Φ3 and Φ6 in Appendix B, are easily derived using (36). To compute conditional expectations (9) and (11), Lemma 4 is applied to obtain EPBTI1∣F0 and EPBTI2∣F0, respectively.To compute conditional expectation (10), Lemma 5 is applied with(41)X=-∫0Trtdt+ln⁡ST1S01,Y=ln⁡ST1S01,Z=ln⁡ST1S02ST2S01.To compute conditional expectation (12), Lemma 5 is applied with(42)X=-∫0Trtdt+ln⁡ST2S02,Y=ln⁡ST2S02,Z=ln⁡ST2S01ST1S02.Generalizing 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.
LetVα1,α2;S01,S02;K;T denote the no-arbitrage value at time t0=0 of a put or a call on the minimum or on the maximum at expiry T of two weighted assets α1S1 and α2S2 with strike price K. Within the modeling framework defined in Section 2, we have(43)Vα1,α2;S01,S02;K;T=∑n1=0∞∑n2=0∞∑n12=0∞e-λ1′T-λ2′T-λ12′Tλ1′Tn1λ2′Tn2λ12′Tn12n1!n2!n12!Υα1S01exp⁡ν1+γ122N2Υln⁡α1S01/K+ν2+Φ1γ2,θϑ1,2+ν3+Φ2γ3;ΥθΦ3γ2γ3+Υα2S02exp⁡ν4+γ422N2Υln⁡α2S02/K+ν5+Φ4γ5,θϑ2,1-ν3+Φ5γ3;ΥθΦ6γ5γ3-ΥB0,TKN2Υln⁡α1S01/K+ν6γ2,θϑ1,2+ν7γ3;ΥθΦ3γ2γ3+N2Υln⁡α2S02/K+ν8γ5,θϑ2,1-ν7γ3;ΥθΦ6γ5γ3,where(44)Υ=1calloption-1putoptionθ=1optiononmaximum-1optiononminimum.All 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 Formula1 is easy using the algorithm of Drezner and Wesolowsky [13], or its slightly improved version by Genz [14], to evaluate the N2·,·;· 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 σS1 of asset S1 is assumed to be higher than the diffusive volatility σS2 of asset S2. The intensity of the jump process N2 is greater than the intensity of the jump process N1. The variance of the jumps of S2, that is, the parameter ε2, is greater than the variance ε1 of the jumps of S1. Thus, the variability of S1 is higher than that of S2 in the continuous world, but the discontinuous variations of asset S2 are more frequent and more pronounced than those of asset S1. The continuous part of the correlation between S1 and S2, that is, the coefficient ρ34, 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 W1 and W2, are assumed to have a positive correlation coefficient ρ1.2. The correlation between fixed income and equity is negative, whether it is the correlation with the short end of the yield curve (coefficients ρ1.3 and ρ1.4) or the correlation with the long end (coefficients ρ2.3 and ρ2.4).Table 1
Values of put options on the minimum of two assetsa.
3-month expiry put option on minimumK=100 3-month expiry put option on minimumK=90 1-year expiry put option on minimumK=100 1-year expiry put option on minimumK=90 Model 1: Black-Scholes 9.304886727 3.630237975 17.0518359 10.66664224 Model 2: no jumps, stochastic interest rate 9.62905761 2.87924378 19.1872084 11.5275656 Model 3: low intensity jumps, constant interest rate 11.2923396 4.64155701 21.0275563 13.5964336 Model 4: low intensity jumps, stochastic interest rate 10.4286542 3.61484021 20.6860149 12.9592418 Model 5: high intensity jumps, constant interest rate 11.8892729 5.15613322 22.3691619 14.8593002 Model 6: high intensity jumps, stochastic interest rate 11.0618998 4.16206372 22.0902324 14.307958 aAll values other than Black-Scholes were computed by means of Formula 1 with the following inputs: α1=α2=1  S01=S02=100, σS1=35%, σS2=28%, δ1=δ2=0, ρ1.2=0.3, ρ1.3=-0.25, ρ1.4=-0.28, ρ2.3=-0.22, ρ2.4=-0.26, ρ3.4=-0.36, ξ1=ξ2=0, ε1=0.2, and ε2=0.3. The “constant interest rate” setting is defined by taking an interest rate equal to 3%. The “stochastic interest rate” setting is defined by takingr0=0.03, bt=0.03, a=0.6, σR1=0.08, and σR2=0.12. The “low intensity jumps” setting is defined by takingλ1′=0.2, λ2′=0.3, and λ12′=0. The “high intensity jumps” setting is defined by takingλ1′=0.4, λ2′=0.6, and λ12′=0.The values of put options on the minimum ofS1 and S2 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 S1 and S2 according to (23) and (24), with two different levels of intensity and with or without stochastic interest rate.For all numerical values reported in Table1, a convergence level of 10-8 was obtained by truncating the infinite series at n1=n2=n12=10, 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 10-5 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. Lemma5 is applied with the following inputs:(45)X=-∫0Trtdt+ln⁡ST1S01,Y=ln⁡ST1S02ST2S01to  compute15X=-∫0Trtdt+ln⁡ST2S02,Y=ln⁡ST1S02ST2S01to  compute16X=-∫0Trtdt+ln⁡ST2S02,Y=ln⁡ST2S01ST1S02to  compute17X=-∫0Trtdt+ln⁡ST1S01,Y=ln⁡ST2S01ST1S02to  compute18.Deriving the required covariances (given in Appendix B), the following formula can then be obtained.Formula 2.
LetVα1,α2;S01,S02;T denote the no-arbitrage value at time t0=0 of a zero-strike spread option on two weighted assets α1S1 and α2S2 with expiry T. Within the modeling framework defined in Section 2, we have(46)Vα1,α2;S01,S02;T=∑n1=0∞∑n2=0∞∑n12=0∞e-λ1′T-λ2′T-λ12′Tλ1′Tn1λ2′Tn2λ12′Tn12n1!n2!n12!α1S01expν1+γ122Nϑ1,2+ν3+Φ2γ3-α2S02exp⁡ν4+γ422Nϑ1,2+ν3+Φ7γ3+α2S02exp⁡ν4+γ422Nϑ2,1-ν3+Φ5γ3-α1S01exp⁡ν1+γ122Nϑ2,1-ν3+Φ8γ3.The 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.Table 2
Value of a spread optiona.
3-month expiry spread option 6-month expiry spread option Model 1: Black-Scholes 20.6206953 28.9473327 Model 2: low intensity jumps 21.7144031 30.6466976 Model 3: high intensity jumps 22.7485628 32.2521365 aAll values other than Black-Scholes were computed by means of Formula 2 with the following inputs: α1=α2=1  S01=S02=100, σS1=35%, σS2=28%, δ1=δ2=0, ρ1.2=0.3, ρ1.3=-0.25, ρ1.4=-0.28, ρ2.3=-0.22, ρ2.4=-0.26, ρ3.4=-0.36, ξ1=ξ2=0, ε1=0.2, and ε2=0.3. The “low intensity jumps” setting is defined by takingλ1′=0.2, λ2′=0.3, and λ12′=0. The “high intensity jumps” setting is defined by takingλ1′=0.4, λ2′=0.6, and λ12′=0.Finally, the product option is dealt with. Lemma4 is applied to compute (21) while Lemma 5 is applied to compute (20) with X=-∫0Trtdt+ln⁡ST1ST2/S01S02 and Y=ln⁡ST1ST2/S01S02. Deriving the required covariances (given in Appendix B), the following formula can then be obtained.Formula 3.
LetVS01,S02;K;T denote the no-arbitrage value at time t0=0 of a product option on two assets S1 and S2 with strike price K and expiry T. Within the modeling framework defined in Section 2, we have(47)VS01,S02;K;T=∑n1=0∞∑n2=0∞∑n12=0∞e-λ1′T-λ2′T-λ12′Tλ1′Tn1λ2′Tn2λ12′Tn12n1!n2!n12!ΥS01S02exp⁡ν9+γ622NΥln⁡S01S02/K+ν10+Φ9γ7-ΥKB0,TNΥln⁡S01S02/K+ν11γ7Υ=1calloption-1putoption.


## 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 Lemma3 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.



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