A Workout in Computational Finance

12 Characteristic Function Methods for Option Pricing

The conditional expectation of the value of a contract payoff function under the risk neutral measure can be linked to the solution of a partial (integro-) differential equation (PIDE) (0ksendal, 2007). This PIDE can then be solved using discretization schemes, such as Finite Differences (FD) and Finite Elements (FEM), or by Wavelet-based methods, together with appropriate boundary and terminal conditions. A direct discretization of the underlying stochastic differential equation, on the other hand, leads to (Quasi)Monte Carlo (QMC) methods. Both groups of numerical techniques – discretization of the P(I)DE as well as discretization of the SDE – are discussed elsewhere in the book. A third group of methods, which will be discussed in this chapter, directly applies numerical integration techniques to the risk neutral valuation formula for European options (Cox and Ross, 1976)

where V denotes the option value, Δt is the difference between the maturity T and the valuation date t₀, f(y | x) is the probability density of y = ln(S_T/K) given x = ln(S₀/K), and r is the risk neutral interest rate. Direct integration techniques have often been limited to the valuation of vanilla options, but their efficiency makes them particularly suitable for calibration purposes. A large part of state of the art numerical integration techniques relies ...