July 8th, 2015, 12:44 pm
I'm not sure how general that is. It's relatively easy to prove for a simple diffusion. Take [$]dX_t = a(X_t) dW_t[$]. Changing your notation somewhat, the value function [$]V(t,x) = E_{t,x}[\Phi(X_T)][$] satisfies [$]-V_t = \frac{1}{2} a^2(x) V_{xx}[$].Let [$]H(t,x) \equiv V_{xx}[$]. Differentiate the value function pde twice to get a pde for H. This second pde has a probabilistic representation for itssolution and the convexity follows readily from that. See Appendix 2.3 of 'Option Valuation under Stochastic Volatility' for details anda generalization to stochastic volatility processes.
Last edited by
Alan on July 7th, 2015, 10:00 pm, edited 1 time in total.