Fundamentals of Derivative Pricing

In this chapter, we discuss the basics for the pricing of European options and futures in a generalized setting. We begin with some technical preliminaries to provide a framework which is based on an underlying price process with subordinated stochastic volatility process. This framework can also be generalized to a multifactor model without changing the solution methods, as done in Chap. 6. Black and Scholes (1973) and Merton (1973) showed in their seminal papers that a derivative security can be priced by creating a replicating portfolio, i.e. a portfolio of primitive securities which matches the payoff of the derivative at maturity. Since both the replication portfolio and the derivative offer the same payoff at maturity, they have to have the same price at any preceding time. Deviations from this equality lead to arbitrage possibilities. Hence, the pricing by duplication procedure inhibits arbitrage by construction. Harrison and Kreps (1979) (in a discrete time setting) and Harrison and Pliska (1981) (in a continuous time setting) demonstrate that the replication-based price is equivalent to the calculation of the discounted expected value of the derivative’s payoff under the equivalent martingale measure $$\mathbb{Q}$$ . Delbaen and Schachermayer (1994, 1998) extend Harrison and Pliska to more sophisticated unbounded stochastic processes.