Connecting Derivative Prices Written on the Same Underlying

Often, different financial derivatives are subject to the same sources of risk.  In such cases, the prices of these derivatives ought to be related.  In this talk, we will explore the relationship between a variance swap, which is a financial derivative whose payoff depends on the entire path of a stock $S$ over a fixed time interval $[0,T]$, and a European contract, which is a financial derivative whose payoff depends only on the value of $S$ at the terminal time $T$.  We will prove that, when a stock is modeled as a time-changed Markov process, the variance swap has the same value as a European contract whose payoff function is the solution of an integro-differential equation (which we will solve).  The significance of this result is that the path-dependent variance swap contract can be priced relative to liquidly traded (and efficiently priced) path-independent European options in a semi-nonparametric fashion.