# Euler discretisation error for stochastic volatility model

Given the following model$$dS_t=S_t(\mu dt+\sigma(t,S_t)dW_t)$$

Using Monte Carlo Pricing method, I want to determine the price of the option. However I have been encountered the following problems:

1. How would I choose the coordinate? Log return or normal $S_t$, which is more appropriate here?
2. How can I tame the discretization error? I am using Euler-Scheme for the discretisation of the stochastic differential equation. The discretization is bounded strongly by $E[|X_t-\bar X_t|]<K\delta^{1/2}$ and weakly by $K\delta$. I want choose the stepsize $\delta$ such that the scheme converges. But I dont know how to verify this. As the exact $X_t$ is not know, I cannot simulate the error $E[|X_t-\bar X_t|]$. I can only compute $E[\bar X_t^{\delta}]$ for different step size. I have tried to compute $E[\bar X_t^{\delta}]-E[\bar X_t^{1/2\delta}]$, but the result doesn't show the weak convergence of order 1. The result jumps from one to other value. I assume this is caused by statistical error(standard error of monte carlo simulation) and rounding off error. Therefore what is the appropriate way to find the accuracy of my scheme.

Edit

I plotted the $E[\bar X_t^{\delta}]-E[\bar X_t^{1/2\delta}]$ on the graph. The horizontal axis shows the number of steps and the vertical axis shows $E[\bar X_t^{\delta}]-E[\bar X_t^{1/2\delta}]$. For example, for the point 100 on the horizontal axis, I calculated $|E[\bar X_t^{T/50}]-E[\bar X_t^{T/100}]|$. How can I explain the peak in the beginning and the graph doesn't really show a convergence as one would expect from weak convergence of Euler Scheme

• Can the book "Monte Carlo simulation in financial engineering" by Glasserman help you? – Gordon Mar 22 '17 at 15:27
• @Gordon The book from glasserman is more from a more theoretical aspect. I need a practical advice how to recognise the promised order 1 convergence in simulation – quallenjäger Mar 22 '17 at 15:30
• @Gordon As my error in the simulation is like a mixture of all kind of error sources, i.e. Statistical error, discretisation error, rounding off error and error of pseudo-random number. I want to investigate the behaviour of discretisation error from this big cocktail of errors. I need to be sure that the scheme I used is stable. – quallenjäger Mar 22 '17 at 15:40
• this really local vol rather than stochastic vol. I would use log coordinates here. – Mark Joshi Mar 23 '17 at 0:14
• @MarkJoshi Thank you! Could you emphasize the difference between log coordiantes and normal? When do I need use log coordinates? – quallenjäger Mar 23 '17 at 15:43

I figured it out. The problem is just, that I am not taking the same driving brownian motion. That would mean, if I am calculating $E[\bar X^{\delta}]$ and $E[\bar X^{1/2\delta}]$. The sample of path is complete different and thus not comparable. For weak convergence, one need to be sure, that the sample path are almost surely the same. As a result, the variance of the sample of data will dominate the discretisation error. For that purpose, one might need to use the same driving brownian motion. I generate 2 random variables for $1/2\delta$ and add them in pairs in order to compute the brownian motion for the $\delta$. As the path are close to each other, the variance is also reduced.