Pricing basket options with the first three moments of the basket: log-normal models and beyond

Options on baskets (linear combinations) of assets are notoriously challenging to price using even the simplest log-normal continuous-time stochastic models for the individual assets. The paper (Borovkova et al. 2007 ) gives a closed form approximation formula for pricing basket options with potentially negative portfolio weights under log-normal models by moment matching. This approximation formula is conceptually simple, methodologically sound, and turns out to be highly accurate. However, it involves solving a system of nonlinear equations which usually produces multiple solutions and which is sensitive to the selection of initial values in the numerical procedures, making the method computationally challenging. In the current paper, we take the moment-matching methodology in (Borovkova et al. 2007 ) a step further by obtaining a closed-form solution for this nonlinear system of equations, by identifying a unary cubic equation based solely on the basket’s skewness, which parametrizes all model parameters, and we use it to express the approximation formula as an explicit function of the mean, variance, and skewness of the basket. Numerical comparisons with the baskets considered in Borovkova et al. ( 2007 ) show a very high level of agreement, and thus of accuracy relative to the true price of the basket option. In the second half of the paper, we apply the same moment-matching approach to the case of a class of time-changed models, ones where the log-normal assets in the basket are subordinated by a common random time change, and we obtain a closed-form approximation formula for basket option prices under these models also. Our formula in the time-changed case also involves a system of nonlinear equations that needs to be solved. We simplify this system of equations and eventually reduce the basket option pricing problem under these models into the problem of finding a root of a real-valued function on the real line. While this equation, which depends explicitly on the moment-generating function of the time-mixing distribution, need not have explicit solutions, we show that in examples of distributions popular in the literature, the equation can be of polynomial type, or of exponential type, whose numerical resolutions pose no difficulties. Our numerical tests, based on financially realistic parameters, show that such closed-form approximations to basket prices for this class of time-changed models also perform highly accurately.