Chapter 4 introduces the elegant and general theory of arbitrage pricing by risk-neutral discounting. Chapter 5 discusses the rather special setting of Black-Scholes-Merton and presents the famous analytical valuation formula for European options. In the first case, the generality of the approach is what is appealing. In the second case, the highly specific but very useful valuation formula is the advantage.
The question is whether there is an approach to derive formulas as useful as the BSM one in more general settings, thereby bridging the gap between generality of risk-neutral pricing and the specificity of the BSM formula. Fortunately, there is an approach: Fourier-based option pricing. This approach allows the use of semi-analytic valuation formulas for European options whenever the characteristic function of the stochastic process representing the underlying is known.
The Fourier approach, presented in this chapter, has three main advantages:
- generality: as pointed out, the approach is applicable whenever the characteristic function of the process driving uncertainty is known; and this is the case for the majority of processes/models applied in practice
- accuracy: the semi-analytic formulas can be evaluated numerically in such a way that a high degree of accuracy is reached at little computational cost (e.g. compared to simulation techniques)
- speed: the formulas can in general be evaluated very fast such that 10s, ...