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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.

Editenter image description here

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

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  • $\begingroup$ Can the book "Monte Carlo simulation in financial engineering" by Glasserman help you? $\endgroup$ – Gordon Mar 22 '17 at 15:27
  • $\begingroup$ @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 $\endgroup$ – quallenjäger Mar 22 '17 at 15:30
  • $\begingroup$ @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. $\endgroup$ – quallenjäger Mar 22 '17 at 15:40
  • $\begingroup$ this really local vol rather than stochastic vol. I would use log coordinates here. $\endgroup$ – Mark Joshi Mar 23 '17 at 0:14
  • $\begingroup$ @MarkJoshi Thank you! Could you emphasize the difference between log coordiantes and normal? When do I need use log coordinates? $\endgroup$ – quallenjäger Mar 23 '17 at 15:43
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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.

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