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Essays on volatility derivatives and portfolio optimization

Ashish Jain, 2007
Faculty Advisor: Mark Broadie
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This thesis is a collection of four papers: (1) Discrete and continuously sampled volatility and variance swaps, (2) Pricing and hedging of volatility derivatives, (3) VIX index and VIX futures, and (4) Asset allocation and generalized buy and hold trading strategies. The first three papers answer various questions relating to the volatility derivatives. Volatility derivatives are securities whose payoff depends on the realized variance of an underlying asset or an index. These include variance swaps, volatility swaps and variance options. All of these derivatives are trading in over-the-counter market. With the popularity of these products and increasing demand of these OTC products, the Chicago Board of Options Exchange (CBOE) changed the definition of VIX index and launched VIX futures on VIX index. The new definition of VIX index approximates the one month variance swap rate. In second chapter we investigate the effect of discrete sampling and asset price jumps on fair variance swap strikes. We calculate the fair discrete volatility strike and the fair discrete variance strike in different models of the underlying evolution of the asset price: the Black-Scholes model, the Heston stochastic volatility model, the Merton jump-diffusion model and the Bates and Scott stochastic volatility model with jumps. We determine fair discrete and continuous variance strikes analytically and fair discrete and continuous volatility strikes using simulation and variance reduction techniques and numerical integration techniques in all models. Numerical results are provided to show that the well known convexity correction formula doesn't work well to approximate volatility strikes in the jump-diffusion models. We find that, for realistic contract specifications and realistic risk-neutral asset price processes, the effect of discrete sampling in minimal while the effect of jumps can be significant.

In the third chapter we present pricing and hedging of variance swaps and other volatility derivatives, e.g., volatility swaps and variance options, in the Heston stochastic volatility model using partial differential equation techniques. We formulate an optimization problem to determine the number of options required to best hedge a variance swap. We propose a method to dynamically hedge volatility derivatives using variance swaps and a finite number of European call and put options.

In the fourth chapter we study the pricing of VIX futures in the Heston stochastic volatility (SV) model and the Bates and Scott stochastic volatility with jumps (SVJ) model. We provide formulas to price VIX futures under the SV and SVJ models. We discuss the properties of these models in fitting VIX futures prices using market VIX futures data and SPX options data. We empirically investigate profit and loss of strategies which invest in variance swaps and VIX futures empirically using historical data of the SPX index level, VIX index level and VIX futures data. We compare the empirical results with theoretical predictions from the SV and SVJ model.

In fifth chapter we present the generalized buy-and-hold (GBH) portfolio strategies which are defined to be the class of strategies where the terminal wealth is a function of only the terminal security prices. We solve for the optimal GBH strategy when security prices follow a multi-dimensional diffusion process and when markets are incomplete. Using recently developed duality techniques, we compare the optimal GBH portfolio to the optimal dynamic trading strategy. While the optimal dynamic strategy often significantly outperforms the GBH strategy, this is not true in general. In particular, when no-borrowing or no-short sales constraints are imposed on dynamic trading strategies, it is possible for the optimal GBH strategy to significantly outperform the optimal dynamic trading strategy. For the class of security price dynamics under consideration, we also obtain a closed-form solution for the terminal wealth and expected utility of the classic constant proportion trading strategy and conclude that this strategy is inferior to the optimal GBH strategy.

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