Option pricing with additive processes

Rama Cont

Abstract: The insufficiency of diffusion models to explain certain empirical properties of asset returns and option prices has led to the development, in option pricing theory, of a variety of models with jumps, ranging from Lévy processes to general semi-martingale models. While Lévy processes offer a tractable class of models, empirical studies have revealed their insufficiency for explaining the term structure of implied volatility in options data.

We present the class of exponential additive (EA) models in which the risk neutral dynamics of an asset is represented as $S_t=S_0 exp X_t$ where $X$ is an additive process - a stochasticaly continuous process with independent increments. We show that exponential additive models are capable of reproducing empirical features of implied volatility surfaces across strikes and maturities and present an algorithm based on relative entropy minimization (jointly developed with Peter Tankov) for constructing an exponential additive model consistent with observed market prices of options. Applying this algorithm to SP500 and DAX options, we extract the Lévy measures implied by the data and examine their properties. Our empirical results show good performance of these models and clearly underline the need for time-inhomogeneity.

By using the notions of spot and forward Lévy measures, we show that forward start options can also be priced in a simple way in such models. The valuation of barrier options leads to integro-differential equations of the type: $\frac{\partial u}{\partial t}(t,x) = L_tu (t,x)$, $u(0,x) = h(x)$ where $(L_t)$ is the time-dependent infinitesimal generator of $X$, with appropriate boundary conditions. Such equations can have non-smooth solutions. We present finite-difference schemes (developed with Ekaterina Voltchkova) for such equations and discuss their convergence using the concept of viscosity solution.

Finally, we describe the dynamics of implied volatility surfaces in an exponential additive model: we show that implied volatilities evolve in time in a simple manner, consistent with the ``sticky delta'' rule used in foreign exchange markets.