Generalizing the Black–Scholes formula to multivariate contingent claims

Abstract

This paper provides approximate formulas that generalize the Black-Scholes formula in all dimensions. Pricing and hedging of multivariate contingent claims are achieved by computing lower and upper bounds. These bounds are given in closed form in the same spirit as the classical one-dimensional Black-Scholes formula. Lower bounds perform remarkably well. Like in the onedimensional case, Greeks are also available in closed form. We discuss an extension to basket options with barrier.

Valuation of Performance‐Dependent Options

“…Various multivariate option pricing problems not discussed in this paper allow closed form solutions, see, e.g., Zhang [26] or Carmona and Durrleman [3]. A valuation approach for American-style performance-dependent options using a fairly general Lévy model for the underlying securities is presented in Egloff et al [8].…”

Multi-asset spread option pricing and hedging

“…However, until very recently, not much work has been done on this subject. Carmona and Durrleman (2005) propose approximate formulas for the lower and upper bounds of multi-asset spread options by solving a nonlinear optimization problem. They only consider the geometric Brownian motions case.…”

“…We perform the comparisons for four different dimensions N +1, namely, 3, 20, 50, and 150 using an artificial correlation matrix similar to the one used in Carmona and Durrleman (2005). In addition, in order to test various methods using a more plausible correlation matrix, we also apply the methods to two hypothetical spread we will only compare models in this special case.…”

“…This is because when the dimension (the number of assets) is high, numerical approaches, such as numerical integration method, numerical solutions to partial differential equations and Monte Carlo simulation, become extremely slow and often inapplicable. A noticeable work that approximates the multi-asset spread option price is Carmona and Durrleman (2005). While Carmona and Durrleman's method is quite accurate, it suffers from a somewhat major shortcoming.…”

“…Using matrix algebra, we show that the computational cost of the second-order boundary approximation is very low. Compared with the method in Carmona and Durrleman (2005), both our methods are in closed form and only involve arithmetic calculations, thus they are quite straightforward to implement. We also extend both our methods to price hybrid spread-basket options through a technique commonly used in valuing Asian options.…”