Short Maturity Implied Volatility

Specialeforsvar ved Mikael K. Kristoffersen

Titel: Short Maturity Implied Volatility

Resume: By letting implied volatility be modelled as a stochastic process, we derive an option pricing PDE directly in implied volatility. This implied volatility PDE is derived for a broad class of two dimensional stochastic volatility models. In the short maturity limit, when using implied normal volatility, the implied volatility PDE collapses into a polynomial structure. This particular structure enables us to derive a handful of theoretical results that relate the short maturity at-the-money implied volatility with the underlying stochastic volatility model. E.g. we find how to derive a short maturity, model independent, minimum variance delta. We also show how to use the short maturity at-the-money implied volatility to specify the underlying stochastic volatility process. Applying the latter result, we derive a reconstruction formula that instructs how to estimate a joint smile. The reconstruction formula is tested empirically on foreign exchange option data, e.g., we reconstruct the EURUSD smile using the at-the-money implied volatility quote along with the EURGBP and the GBPUSD smile. The empirical
investigation shows that the reconstruction formula only holds sporadically.

Vejleder: Rolf Poulsen
Censor: David Skovmand