Approximation of SDE

Question

I have been struggling with the following problem: If you want to find a numerical result by simulating the paths of a stochastic differential equation, in particular a geometric brownian motion I have found literature saying you should go for the exact solution (continuous time) and literature stating that you should approximate the GBM (e.g. with the Euler-Maruyama Scheme).

From what I understand, there are SDE's who can be modelled exactly, and SDE's who can't. What's true for a GBM?

Additional info:

From Glasserman, P. (2004). Monte Carlo Methods in Fin. Engineering:

"Chapter 3 gives examples of continuous-time stochastic processes which can be simulated exactly [...]" (p. 339). Chapter 3 then lists the GMB in 3.2 (pp. 93ff.).

From Günther & Jüngel (2010). Finanzderivate mit Matlab. :

given $$dS_t = rS_tdt + \sigma S_tdW_t$$

transform this with the Euler-Maruyama Method to

$$\Delta S_t = rS_t\Delta t + \sigma S_t \Delta W_t$$

where $$\Delta W_t = Z \cdot \sqrt{\Delta t}$$ and $$Z\sim\mathcal{N}(0,1)$$ (pp. 102f.)

I hope that helps clarifying my issue. If not I'll happily give more details. Thanks

Answer 1

You can sample from the exact distribution

For a GBM following SDE $$ \frac{\mathrm{d}X_t}{X_t} = \mu \mathrm{d}t + \sigma \mathrm{d}W_t $$ (under some physical measure $\mathbb{P}$) with initial value $X_0$ almost surely, we have the solution $$ X_t = X_0 \exp\left(\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right) $$ where for simplicity we have assumed $W_t$ to be a standard Brownian motion (Weiner process) and $\mu$ and $\sigma$ to be constants (extending to the time dependent case is not too hard).

So why would you ever simulate the path?

If all you're interested in is the terminal value, then you would just sample the exact process, as above. This is the case in several applications, such s pricing European options in finance.

However, there are several applications where we are interested not only in what the final value is, but also how it got there. This would be the case if we wanted to ask if the solution was always positive (yes for GBM), did it exceed a certain threshold value, what does/did the running average value look like, etc. This frequently crops up when pricing more exotic options in finance with path dependent payoffs, such as Asian options or barrier options. In such a scenario you may need to generate the entire path.