The Pricing of Financial Derivatives

We review the construction of a self-financing portfolio in a complete market and appeal to the Fundamental Theorems of Asset Pricing, along with the no-arbitrage principle, to arrive at the martingale characterisation of the value of a financial derivative trading in that market. We then show how the Black-Scholes-Merton PDE arises from the Feynman-Kac Theorem and discuss the boundary conditions of this PDE for European options. Computationally we implement the Black-Scholes pricing formulae for such options and their hedge parameters, and by simulation of individual asset prices show how the writer of a European option may dynamically hedge against the risk associated with the sale of the option. Finally we examine how to approximate the market value of the volatility parameter.