Option Pricing

The foundations of option pricing were laid down by Back, Scholes, and Merton. However, this theory is very difficult to extend because the general principles (e.g., the risk-neutral valuation) and one particular model (the log-normal random walk) are deeply intertwined. This chapter presents a general option pricing scheme that can accommodate discrete ARCH processes for the underlying asset (i.e., no Itô calculus is used). It is built on the construction of equivalent martingale measures (using a generalized Esscher transform) and on arbitrage-free pricing. The valuation of a European option is given by the expectation of the terminal payoff in the physical measure, with a weight for the price paths given by the change of measure. This formulation removes the need to express the underlying process in the risk-neutral measure. The minimal variance hedging is introduced, leading to the usual delta replication. To turn this general scheme into a practical tool, a small time step expansion is performed, leading to efficient Monte Carlo simulations. In principle, the option prices depend on the risk preference of the issuers and are not unique. Yet, the small time expansion shows that this dependency is mostly irrelevant in practice. Finally, a cross-product approximation of the implied volatility surface can be done, allowing one to compute very efficiently option prices and the related Greeks. This approximation leads to real European option prices, including the risk and cost of hedging. The implied volatility surfaces are presented for processes of increasing complexity for the underlying and compared to the empirical surface of options on the SP500.