Application of power series approximation techniques to valuation of European style options

Abstract

Fourier transforms provide versatile techniques for pricing financial derivative securities. In applying such techniques, a typical derivative valuation expression is often written as an inner product of the Fourier transform of payoff and the characteristic function of the underlying asset dynamics. Some modelling specifications imply that it might be challenging to find a closed-form expression for the characteristic function. In such situations, numerical approximations have to be employed. This paper utilises the power series approximation technique in finding explicit expressions of the characteristic function for the underlying stochastic variables. We analyse the convergence and accuracy of this technique in the context of valuing European style options written on underlying securities whose dynamics evolve under the influence of multiple Heston-type stochastic volatilities [Heston, S., A closed-form solution for options with Stochastic volatility with applications to Bond and currency options. Rev. Financ. Stud., 1993, 6, 327–343] and Cox–Ingersoll–Ross stochastic interest rates [Cox, J., Ingerson, J. and Ross, S., A theory of the term structure of interest rates. Econometrica, 1985, 53, 385–407]. This paper contributes to the existing literature four-fold by: (i) adapting the valuation technique to long-dated instruments; (ii) providing an adjustment to a series of points around which the power series expansion is performed; (iii) analysing the performance of different strategies for hedging European call options; (iv) applying the power series approach to the valuation of guaranteed minimum accumulation benefit riders embedded in variable annuity contracts. Our results demonstrate a high computational efficiency of the power series approximation method for evaluation of derivative prices and hedge ratios.