Managing volatility risk Innovation of financial derivatives,stochastic models and their analytical implementation.
详细信息
文摘
This dissertation investigates two timely topics in mathematical finance. In particular, we study the valuation, hedging and implementation of actively traded volatility derivatives including the recently introduced timer option and the CBOE (the Chicago Board Options Exchange) option on VIX (the Chicago Board Options Exchange volatility index). In the first part of this dissertation, we investigate the pricing, hedging and implementation of timer options under Heston's (1993) stochastic volatility model. The valuation problem is formulated as a first-passage-time problem through a no-arbitrage argument. By employing stochastic analysis and various analytical tools, such as partial differential equation, Laplace and Fourier transforms, we derive a Black-Scholes-Merton type formula for pricing timer options. This work motivates some theoretical study of Bessel processes and Feller diffusions as well as their numerical implementation. In the second part, we analyze the valuation of options on VIX under Gatheral's double mean-reverting stochastic volatility model, which is able to consistently price options on S&P 500 (the Standard and Poor's 500 index), VIX and realized variance (also well known as historical variance calculated by the variance of the asset's daily return). We employ scaling, pathwise Taylor expansion and conditional Gaussian moments techniques to derive an explicit asymptotic expansion formula for pricing options on VIX. Our method is generally applicable for multidimensional diffusion models. The convergence of our expansion is justified via the theory of Malliavin-Watanabe-Yoshida. In numerical examples, we illustrate that the formula efficiently achieves desirable accuracy for relatively short maturity cases.