Connecting univariate smiles and basket dynamics: a new multidimensional dynamics for basket options

A new approach to modelling and pricing derivative securities based on many underlying assets is developed, with the ultimate, practical aim to properly price such derivatives when each underlying shows a volatility smile/skew. We show that the proposed multidimensional model can indeed account for the observed implied volatility smiles for a range of single securities, when each single-asset volatility smile is modeled according to a density-mixture dynamical model. Extending in an intuitive way the model from the univariate to the multivariate setting, this theory allows to sample from an entirely new type of dynamics that still enjoys an internal consistency with the observed volatility surfaces for the individual securities. The computational implications have a strong impact on the calculation of prices of European options on baskets of securities. Operative proposals for the use of this model to deduce skews on baskets of many stocks/FX rates will be presented. A natural extension to the case of the Libor Market Model would allow computing in a quasianalytical fashion the swap rates’ smile given the smiles in the individual caplets. Introduction It is known that the Black–Scholes model [2] does not price all European options quoted on a given market in a consistent way. In fact, their model lies on the fundamental assumption that the asset price volatility is a constant. In reality, the implied volatility (i.e. the volatility parameter that, when plugged into the Black–Scholes formula, allows to reproduce the market price of an option) generally shows a dependence on both the option maturity and strike. If there were no dependence on strike one could extend the model in a straightforward fashion by allowing a deterministic dependence of the underlying’s instantaneous volatility on time, so that the dynamics could be represented by the following stochastic differential equation (SDE): (1) dSt = μStdt + σtStdWt, σt being the deterministic instantaneous volatility referred to above. In that case, reconstruction of the time dependence of σt would follow by considering that, if v(Ti) denotes the implied volatility for options maturing at time Ti, then (2) v(Ti) Ti = ∫ Ti 0 σ sds. Implied volatility however does indeed show a strike dependence; in the common jargon, this behaviour is described with the term smile whenever volatility has a minimum at the forward asset price level, or skew Corresponding author: e-mail [email protected] One of us (FR) deeply thanks the participants to the Fifteenth Annual Conference of the Financial Options Research Centre of the Warwick Business School and to the Workshop on Multiasset Options organized by Frontiéres en Finance for useful discussions.