THE BLACK SCHOLES BARENBLATT EQUATION FOR OPTIONS WITH UNCERTAIN VOLATILITY AND ITS APPLICATION TO STATIC HEDGING

The Black Scholes Barenblatt (BSB) equation for the envelope of option prices with uncertain volatility and interest rate is derived from the Black Scholes equation with the maximum principle for diffusion equations and shown to be equivalent to a readily solvable standard Black Scholes equation with a nonlinear source term. Analogous arguments yield the envelope for the delta of option prices. We then interpret the concept of static hedging for narrowing the envelope in terms of partial differential equations and define the optimal static hedge and computable approximations to it. We apply the BSB equation to find numerically some optimally hedged portfolios of representative European and American options.