Hi,does anyone know of closed-form pricing solutions for one-touch digital barrier options, where volatility is stochastic?Thanks very much,Crisky

What do you mean by one touch? Is it American?If it isn't, it is possible to evalute price through numerical quadrature. It is pretty hairy, but much faster than doing MC.

it is American, the option pays a fixed amount when the barrier is first hit.This paper derives the one-touch price under jump-diffusion: Kou and Wang (2003) "First Passage Times of a Jump Diffusion Process," Advances in Applied Probability.They assume a brownian motion where the Poisson jump sizes are doubly-exponential distributed.I am tempted to use the jump-diffusion model instead of stochastic vol-my guess is that for medium and long-dated options similar types of distributions of the underlying can be achieved with the above J.D. model and with a S.V. model such as the SABR model.Are JD and SV really interchangeable for longer-dated options?

QuoteOriginally posted by: criskyAre JD and SV really interchangeable for longer-dated options?Not really. Jumps cannot generate skews over longer maturities. Stoch Vol processes cannot build up the extreme skews seen in short maturities that quickly. You would need something which combines JD and SV like in Bates.