Analytic techniques for option pricing under a hyperexponential Lévy model

We develop series expansions in powers of q−1 and q−1∕2 of solutions of the equation ψ(z)=q, where ψ(z) is the Laplace exponent of a hyperexponential Lévy process. As a direct consequence we derive analytic expressions for the prices of European call and put options and their Greeks (Theta, Delta, and Gamma) and a full asymptotic expansion of the short-time Black-Scholes at-the-money implied volatility. Further we demonstrate how the speed of numerical algorithms for pricing exotic options, which are based on the Laplace transform, may be increased.