Hi,is there any paper about properties of the distribution of the maximum of a log-normal process under stochastic volatility?I am looking for closed or semi-closed formulas of the probability to cross the barrier in the Heston model.thanks for any advice,Quentin

Hello, Did you get any answer yet?ThxAgnes

In general, there is no closed-form solution. In exeptional case with zero correlation and zero drift (interest rate minus dividend rate) there is closed-form solution (also there is a closed-form for shifted Heston with zero correlation but skew-consistent).I built a PDE solver for forward equation to tackle this kind of problems. Below is output for Heston model with v(0)=theta=0.04, kappa=4, volvol=0.2, S(0)=1 with barrier at S=0.8, T=1 and zero asset drift for different values of correlation. You will also see that although the barrier hitting probability is less dependent on the correlation assumption, the asset price density conitional on the survival does differ remarkably for different values of correlations (see the attached figure) and this is important to take into account for pricing barrier puts and calls.Barrier, rho=-0.75 Survival Prob= 0.65080480Hitting Prob= 0.34919520Barrier, rho=0 Survival Prob= 0.66477466Hitting Prob= 0.33522534Barrier, rho=0.75 Survival Prob= 0.68264408Hitting Prob= 0.31735592Log-normal diffusion with realized variance = 0.04 Survival Prob= 0.705060309Hitting Prob= 0.294939691

For Lévy processes however there are lots of general solutions...If interested have a look at e.g. Financial Modelling with Jump Processes by Cont & Tankov

QuoteOriginally posted by: JeansFor Lévy processes however there are lots of general solutions...If interested have a look at e.g. Financial Modelling with Jump Processes by Cont & TankovWhat is the utility of these general solutions if they represent multiple integrals in complex plane? The only jump process which can be dealt "semi-analytically" for pricing barrier options is the Poisson process with exponental jumps. However, there ia a couple of jump process specifications which can be implemented in a very robust way (no painful fft) through finite deferences, so that a general and reliable analytics can be developed for all kind of barrier options.

QuoteOriginally posted by: sepparIn general, there is no closed-form solution. In exeptional case with zero correlation and zero drift (interest rate minus dividend rate) there is closed-form solution (also there is a closed-form for shifted Heston with zero correlation but skew-consistent).Can you provide any references?