Closed-form Approximations for Spread Options in Lévy Markets

Spread options are exotic derivatives with payoffs that are based on the price difference (i.e., the spread) between two (or more) underlying assets. Spread option contracts are ubiquitous in financial markets where they can serve as a speculative device or risk management tool. Similar payoff patterns also appear in credit risk management.

Essentially, spread option valuation amounts to a two-dimensional integration problem with respect to the joint density function of the two assets involved. Because of the two-dimensionality involved, the pricing is a non-trivial task for which analytical solutions are, in general, not readily available for standard market models. Therefore, one has to resort to numerical methods such as Monte Carlo simulations or approximation formulas. However, because the budget for carrying out simulations might be significant, in many cases, approximation formulas are preferred.

In this regard, new closed-form approximation formulas for pricing spread options are presented in the paper. The formulas are based on conditioning techniques and Gaussian quadrature. First, a pricing formula for the seminal Black-Scholes (BS) model, in which asset log-returns are assumed to be normally distributed, is presented. The numerical illustrations show that the presented method performs slightly better than existing (already quite accurate) pricing formulas. However, although the BS model is a corner stone in the finance literature, its deficiencies are also well documented. Specifically, asset returns display fatter tails than can be modeled by a normal distribution. Moreover, there is also the frequent observation that negative returns have heavier tails than positive ones (skewness). And finally, yet another problem with the BS setting is that it implies asset prices to evolve without jumps. Therefore, also pricing formulas for more general market models, i.e. the Variance Gamma (VG) and Normal Inverse Gaussian (NIG) model, are presented, as these make it possible to deal with the aforementioned stylized features of asset returns. In the context of VG and NIG models, our presented pricing formulas have no known analytical competitors and can be regarded as alternatives to existing techniques based on Fast Fourier transforms and computationally expensive Monte Carlo simulations. Numerical illustrations show that the presented pricing formulas provide (very) tight approximations of the “true” option prices. Lastly, in the same regard, the analytical Margrabe formula for pricing exchange options (spread options with zero strike) in the BS model is extended to the more general VG model under a (practically relevant) homogeneity assumption.