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.

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 [

Analytical valuation formulae for multiasset options are available only in the standard Black-Scholes [

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

discontinuous variations in the equity prices,

stochastic evolution of the interest rate,

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.

This article is organized as follows: Section

In the sequel,

The short rate is driven by the following two-factor, time-dependent, mean-reverting stochastic differential equation:

This is an extended Vasicek [

Let

Thus, the model used for equity is an extension of Merton [

As a consequence of the theory of nonarbitrage valuation of contingent claims [

Using the previously defined notations, let

See Appendix

Hence, the instantaneous rates of change of

In order to price options on

Solutions (

Let

See Appendix

Lemma

At any given time

See Appendix

Thus, a combination of Lemmas

We can now turn to the computation of (

Denoting by

See Appendix

As a consequence of Lemma

Given

To compute the expectations of the first kind in (

Let

See Appendix

Applying the five previous lemmas, it only remains to sum over the joint distribution of

Consider first the value of a put option on the minimum of two assets with strike price

To compute conditional expectation (

Let

The numerical implementation of Formula

Values of put options on the minimum of two assets

3-month expiry put option on minimum | 3-month expiry put option on minimum | 1-year expiry put option on minimum | 1-year expiry put option on minimum | |
---|---|---|---|---|

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 |

The “constant interest rate” setting is defined by taking an interest rate equal to 3%.

The “stochastic interest rate” setting is defined by taking

The “low intensity jumps” setting is defined by taking

The “high intensity jumps” setting is defined by taking

The values of put options on the minimum of

For all numerical values reported in Table

Next, the case of a spread option is handled. Lemma

Let

Value of a spread option

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 |

The “low intensity jumps” setting is defined by taking

The “high intensity jumps” setting is defined by taking

Finally, the product option is dealt with. Lemma

Let

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

Since the processes

It is a classical result from the theory of continuous-time processes that

Equation (

From (

Let

The author declares that there are no competing interests regarding the publication of this paper.