Convexity Preserving Jump Diffusion Models for Option Pricing

A model for a set of stock prices is said to be convexity preserving if the price of any convex European claim is convex as a function of the underlying stock prices at all times prior to maturity. As is well-known, this property is intimately connected to certain monotonicity properties of the option price with respect to volatility and other parameters of the model. Generally speaking, if the option price is convex at all fixed times, then it is also increasing in the volatility. This robustness property motivates the study of convexity preserving models in finance. Although these issues have been studied quite extensively during the last decade, compare [?], [?], [?], [?] and [?] for the case of one-dimensional diffusion models, [?] and [?] for several-dimensional diffusion models, [?] for one-dimensional jump-diffusion models and [?] for exponential semimartingale models, the general picture for more advanced models is not yet fully understood. In [?], a sufficient condition for the preservation of convexity in one-dimensional models with jumps is provided. That condition, however, is not a necessary condition for preservation of convexity. The main contribution of the present paper is to give a necessary condition for convexity to be preserved in jump-diffusion models in arbitrary dimensions. We also use this necessary condition to show that, within a large class of possible models, the only higher-dimensional convexity preserving models are the ones with linear coefficients. To analyze the convexity of an option price we employ the characterization of the price as the unique viscosity solution to a parabolic integro-differential equation (1) ut = Au + Bu with an appropriate terminal condition. In this equation, A is an elliptic differential operator associated with the continuous fluctuations of the stock price processes, whereas B is an integro-differential operator associated with the jumps of the stock price processes. Preservation of convexity of the solution to the equation (??) is dealt with using the notion of locally convexity Date: January 4, 2006. 2000 Mathematics Subject Classification. Primary 91B28; Secondary 35B99, 60J75.