Reducing Look-Ahead Bias in Least-Squares Monte Carlo
Summary
The document discusses foresight, or look-ahead, bias in least-squares Monte Carlo valuation of American options. In the Longstaff–Schwartz approach, regression on the same simulated paths used for valuation can overfit continuation values and make estimated option values too high. The question asks whether increasing the number of paths reduces this effect; the author reports simulations in which it did, though those results are limited to the models tested.
An answer describes leave-one-out cross-validation: fit the regression without one path, predict on that excluded path, and repeat. For linear regression, adjusted predictions can be calculated analytically, avoiding repeated full regressions. The answer reports that bias is proportional to the number of regression variables divided by the sample size, so more paths or fewer predictors can reduce it. Independent out-of-sample simulation is offered as an alternative. The document points to a related study but gives no detailed assumptions or broad empirical validation.
Key ideas
- In-sample regression in least-squares Monte Carlo can create look-ahead bias in option valuation.
- Leave-one-out predictions reduce the influence of fitting and valuing on the same path.
- For linear regression, adjusted leave-one-out predictions can be computed analytically.
- The reported bias grows with the number of regression variables relative to the sample size.
- An independent simulation can also provide out-of-sample valuation.
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Full text
# Foresight bias in least square monte carlo # Foresight bias in least square monte carlo Foresight bias means we tend to over estimate the American option value. This we observe in other areas of statistics - e.g. in sample test almost always gives better prediction than out of sample tests in linear regression models. My question is: is it known (theoretically or empirically) such bias is reduced by increasing sample size in least square monte carlo algorithm? I have done some simulation myself. From my simulation, at least for the models I have worked on, the answer seems to be yes. Presumably this is simply because bigger sample tends to mean the sampling distribution of the estimator has smaller variance blah blah blah? Is there any theoretical results or empirical studies on this? Edit: foresight bias is also known as look-ahead bias ## Answer by jaehyukchoi49 (score 2, accepted) https://quant.stackexchange.com/a/42303 Recently, I co-authored a paper (Arxiv | SSRN) on the issue. You can efficiently remove the look-ahead bias (a.k.a., foresight bias) using leave-out-out-cross-validation (LOOCV) method. Basically, the procedure is (i) take out one sample ,(ii) run regression using the rest, (iii) get the prediction on the removed sample, and (iv) repeat the process for each sample. This way, you avoid the in-sample overfitting. Luckily for linear regression, we can get the adjusted prediction analytically without running the regression as many times as the sample size. Transitional trick to remove the look-ahead bias is to run an independent simulation for out-of-sample valuation. With LOOCV trick, this is not necessary. Why look-ahead bias happens in Longstaff-Schwartz algorithm and how LOOCV is applied is pretty much illustrated in one figure in the paper. We also show that the amount of the look-ahead bias (LOOCV also enables us to measure it) is proportional to $M/N$, where $M$ is the number of regression variables and $N$ is the sample size (@Lost1, you're right that foresight bias is reduced by increasing sample size $N$). It intuitively makes sense because the overfitting in linear regression increases with $M$. ## Answer by Yian Pap (score 2) https://quant.stackexchange.com/a/23170 This study seems to be on point: http://christian-fries.de/finmath/foresightbias/Fries_ForesightBias.pdf
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