Implied non-recombining trees and calibration for the volatility smile

In this paper we capture the implied distribution from option market data using a non-recombining (binary) tree, allowing the local volatility to be a function of the underlying asset and of time. The problem under consideration is a non-convex optimization problem with linear constraints. We elaborate on the initial guess for the volatility term structure and use nonlinear constrained optimization to minimize the least squares error function on market prices. The proposed model can accommodate European options with single maturities and, with minor modifications, options with multiple maturities. It can provide a market-consistent tree for option replication with transaction costs (often this requires a non-recombining tree) and can help pricing of exotic and Over The Counter (OTC) options. We test our model using options data for the FTSE 100 index obtained from LIFFE. The results strongly support our modelling approach.