Mathematical Finance and Probability Seminars (Since covid these events are taking place online.)

Define a variance swap on an underlying share price S to pay at time T the quadratic variation of log(S). According to the standard theory (which underpins, for example, the widely-quoted VIX volatility index), a variance swap has the same value as a contract paying -2 log(S_T/S_0), assuming that S has continuous paths. Empirically, however, stock prices jump.

We generalize the valuation theory to variance swaps on arbitrary exponential Levy dynamics stochastically time-changed by an arbitrary continuous clock having arbitrary correlation with the driving Levy process (subject to integrability conditions). Moreover, extending to discrete sampling under an independence condition, we prove that discretization increases variance swap values. Joint work with Peter Carr.

Speaker: Roger Lee, University of Chicago