## Time-discretization of stochastic processes

Option pricing is based on modelling the behavior of underlying assets (and any other parameter we wish to acknowledge as stochastic for that matter, such as volatility) using Stochastic Differential Equations (SDE). With the help of Ito's lemma and the usual no-arbitrage assumption, those SDEs can be used to derive deterministic Partial Differential Equations (PDE) which govern an option's price (like the classic Black-Scholes equation which is solved by the pricer's PDE solver). The PDE is then discretized (e.g. via finite differences) and the discretization error is determined using Taylor expansions.

When using the Monte Carlo (MC) method though, we are solving the underlying SDE directly and we need to discretize it and of course again introduce discretization error. Because the solution of an SDE at any point in time is not a single number but a rather a distribution, there are two ways of judging the convergence of a discretized path to the continuous process it tries to emulate. One is to think similar to the case of a PDE and check that as we increase the time steps/points of our time-discretization, we converge to the exact solution at each of these points. The other is that we think in terms of the distributions at each time point and check that the expectation of the underlying (or a function of it) converges to the exact expectation (of the continuous process). In the first case we talk about

**strong**convergence and in the second about

**weak**convergence.

The undelying SDE model here is that of Geometic Brownian Motion (GBM) for which there is actually an exact solution, which we can use to "jump" between any dates of interest within our sample path with no discretization error at all. This removes one source of error/bias and thus enables more precise estimates for every option priced here with MC, leaving the simulation variance and the Longstaff-Schwartz regression-induced error (for Bermudan options). But since there are many more stochastic models used in quantitative finance (for which there is no exact solution), two generic time-discretization schemes are also included, namely the Euler and the Milstein schemes. The former is the easiest to implement and can be used to discretize almost any underlying model stochastic process, in one or multiple dimensions. Its only problem is that it is not very accurate. It is of weak convergence order 1 and strong convergence order 1/2. The Milstein scheme is still generally simple enough to use in one dimension (but less so in multi-dimensions) and represents a theoretical improvement in that it is of both weak and strong order 1. But does that mean it will yield better results? We can use the pricer to perform some basic tests.

We price a European vanilla and a barrier option. The former's value depends only on the asset's distribution at the option's maturity, while the latter is path-dependent. We use a lot of Sobol-driven paths so that the simulations "flattten-out" and we are left with just the discretization error.

Looking at the vanilla option first, it seems like the Milstein scheme may indeed perform better, though it has to be said that this is not always the case and you may well get a completely different picture than the one above. For some option parameters the Euler scheme comes out on top, but overall my brief testing indicates that while it's not a sure win, Milstein is better more times than not. Feel free to experiment for yourselves. So is this what we should be expecting anyway based on their theoretical convergence order? Maybe. Both schemes are of the same weak order 1 and a vanilla (European) option's value is really the (discounted) expectation of the payoff function. Intermediate values that the underlying asset might go through have no effect on the payoff. So based on that we should probably be expecting similar results from both schemes. Although we need to remember that the theoretical order of a scheme does not directly tell us what the discretization error will be for a given number of time points, but rather tells us how fast that error will fall as we increase the number of time points. Feel free to play around and check the actual convergence order.

The situation changes when it comes to pricing the path-dependent barrier option. Here the Milstein scheme is the clear winner almost every time I've found. Which is again probably what we would expect given that it of strong order 1 to Euler's 1/2 and thus should be expected to be more accurate for each value it goes through on its path, for the same number of time steps. This should be an advantage when pricing a barrier option since we need to check at each time point whether the asset value has hit the barrier.

You are invited to see what happens with Asian options which are also path-dependent but in a different, more "avegare" kind of way! Just click on the button below to go to the main page where you can download the pricer (free beta version for Windows PCs) and perform more tests.

Please refer to [1] for proper definitions of strong and weak convergence order and a general, thorough treatment of this subject.

The situation changes when it comes to pricing the path-dependent barrier option. Here the Milstein scheme is the clear winner almost every time I've found. Which is again probably what we would expect given that it of strong order 1 to Euler's 1/2 and thus should be expected to be more accurate for each value it goes through on its path, for the same number of time steps. This should be an advantage when pricing a barrier option since we need to check at each time point whether the asset value has hit the barrier.

You are invited to see what happens with Asian options which are also path-dependent but in a different, more "avegare" kind of way! Just click on the button below to go to the main page where you can download the pricer (free beta version for Windows PCs) and perform more tests.

Please refer to [1] for proper definitions of strong and weak convergence order and a general, thorough treatment of this subject.

## References

[1] Glasserman, P. (2003). Monte Carlo Methods in Financial Engineering. New York, Springer-Verlag.