Recursive Algorithms For Pricing Discrete Variance Options And Volatility Swaps Under Time-Changed Lévy Processes

We propose robust numerical algorithms for pricing variance options and volatility swaps on discrete realized variance under general time-changed Lévy processes. Since analytic pricing formulas of these derivatives are not available, some of the earlier pricing methods use the quadratic variation approximation for the discrete realized variance. While this approximation works quite well for long-maturity options on discrete realized variance, numerical accuracy deteriorates for options with low frequency of monitoring or short maturity. To circumvent these shortcomings, we construct numerical algorithms that rely on the computation of the Laplace transform of the discrete realized variance under time-changed Lévy processes. We adopt the randomization of the Laplace transform of the discrete log return with a standard normal random variable and develop a recursive quadrature algorithm to compute the Laplace transform of the discrete realized variance. Our pricing approach is rather computationally efficient when compared with the Monte Carlo simulation and works particularly well for discrete realized variance and volatility derivatives with low frequency of monitoring or short maturity. The pricing behaviors of variance options and volatility swaps under various time-changed Lévy processes are also investigated.