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Joint Applied Math and Probability Seminar
Robust Replication of Volatility Derivatives
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Let S be a positive asset price process. A contract whose payoff depends only on S_T can be replicated by a static portfolio consisting of the underlying asset and vanilla options at the appropriate strikes. Such a portfolio cannot replicate a contract which pays realized variance, defined to be the quadratic variation of log(S_t) on [0,T]. However, introduction of dynamic trading in the underlying asset does allow replication of this payoff. Known since the 1990s (Carr-Madan, Derman, and others), this model-independent result assumes, in essence, only that S is an Ito process. Does this extend to other functions of realized variance? In particular, it is of practical interest to replicate contracts which pay realized volatility, the square root of realized variance. Previous replication efforts sacrificed robustness, by imposing specific models on the S-dynamics, and requiring estimation of the models' parameters. We show that introduction of dynamic trading in options allows robust replication of contracts which pay general functions of realized volatility. We do make a correlation assumption, but we also propose how to neutralize, to leading order, deviations from the correlation condition. This work is joint with Peter Carr. |