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Financial Math Seminar

Mathematics behind volatility index and trading on volatility

Abstract: Volatility of the stochastic dynamics of an asset price process is the standard deviation of the logarithm of asset price returns. The volatility of financial assets is a hidden stochastic process derived from observable asset prices. Fund managers may explore investment opportunities on trading volatility instead of directional moves of asset prices. The Chicago Board of Options Exchange launched the volatility index of the S&P 500 stock index (ticker symbol VIX), which is formally defined as the square root of the risk neutral expectation of the integrated variance of the S&P 500 over the next 30 calendar dates. The goal of the VIX is to compute model free volatility measure implied by the traded stock option prices. The dynamics of VIX is highly correlated with market anxiety and sentiment, so VIX bears the nickname “fear gauge”. In this talk, we discuss the mathematics behind the construction of VIX, and how VIX can be expressed in terms of weighted average of traded option prices. Under certain model assumption on stochastic volatility, VIX is related to the instantaneous variance. We also discuss pricing models of variance derivative products that facilitate trading on volatility, like the corridor variance swaps and timer options. The payoff structures of these variance derivatives are dependent on realized variance of the underlying asset price process. We explore analytic pricing of variance derivatives under various choices of stochastic volatility models.