# Time Step Size for Heston Model for Different Option Maturity

Suppose we are to price with Monte Carlo method two options differing only in the maturity time, with the same, say, call option payoff, or Asian option payoff with a fixed averaging window, with the underlying stock price following the Heston model. We know the distribution of variance in the Heston model approaches a stationary one as time approaches infinity. To achieve the same accuracy,

1) can we use longer time step size for the option with longer maturity than the one with shorter maturity?

2) What about using variable time step size with step size growing towards the time of maturity?

• But my question can be simply rephrased into an equivalent form along the line of your preferred approach: as you refine the time axis partition, while holding number of paths constant, would you expect the result (option price) to be more sensitive with respect to the decreasing of the time step closer to time $0$ than that farther away from $0$? – Hans Dec 6 '16 at 4:33
• Oh I see. You meant closer to time 0 on the timeline. That probably makes sense. – Will Gu Dec 7 '16 at 0:43
• Yes, that is exactly what I meant. I should have said "... decreasing of the time step length as the time gets closer to time $0$ than that as the time moves farther away from $0$". Thank you for the paper. It looks pertinent. If you see more in similar vein, please send them my way. – Hans Dec 7 '16 at 9:20