Option Pricing via Simulation under General Model
Under general SDE for the underlying price process, simulate the entire price path
For some specific SDE, e.g. the GBM, we can derive nice expression of the as
where under the GBM.
Then, construct
depending on whether the risk-free interest rate is constant or not.
Finally, report the average and standard error or .
Accuracy v.s. Sample Size
The central limit theorem implies
is a confidence interval for .
This implies that the accuracy (i.e. length of the CI) of the estimate of scales down like and in order to improve the accuracy by 2 fold, we need to quadruple the sample size or somehow reduce the variance () by 4 fold.
The width of the 95% confidence interval for the expected value is . For example, for an accuracy of 0.01, we need
Without other explicit variance reduction techniques, we might need tremendous amount of data.
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