# 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?

The time step size is more like a byproduct of the convergence of Monte Carlo simulation rather than something to be decided a priori. It depends on the accuracy threshold. In general, one should keep refining the grid (number of paths * number of time steps) until her error metric goes under the threshold. To the simplest, this metric could be the difference between two consecutive runs.

As a side note, there are many ways to speed up the convergence, I'll leave it out for now.

UPDATE:

I described the general approach in determining the optimal uniform time step. It may differ problem by problem, depends on the accuracy demand. On the other hand, here's a paper that looks relevant to what OP was looking for. I think a good approach would be to first establish robust method with uniform time step, and then explore ways to improve efficiency (non-uniform time step being one of them).

• 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
• I think you meant the other way. I would expected it to be less sensitive to time step when the time step gets closer to 0. I realize that there are actually approaches with non-uniform time steps. Like this paper: home.thep.lu.se/~anders/papers/jcp14.pdf It definitely make sense to explore the efficiency of these approaches once you have a baseline approach of uniform time steps and convergence profile. – Will Gu Dec 7 '16 at 0:42
• 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