Risk-Neutral Measures for Monte Carlo Option Pricing

# How to transform process to risk-neutral measure for Monte Carlo option pricing?


# How to transform process to risk-neutral measure for Monte Carlo option pricing?












I am trying to price an option using the Monte Carlo method, and I have the price process simulations as an inputs. The underlying is a forward contract, so at all times the mean of the simulations is current forward price (also the starting point of the simulations). There is no drift.

However, these simulations are calibrated on market data and exhibit i.e. mean reversion and other best estimates of market behavior. So even though they have zero drift, I believe they are not risk neutral and cannot be used in Monte Carlo valuation.

Is there any way how to convert arbitrary market simulations (in market measure) to risk neutral measure? Without knowledge of the SDE that generated them, since it may have been SDE with some further arbitrary postprocessing.

The reason I believe I cannot use them directly, is that some simulations (particularly simulations of commodity spreads) exhibit mean reverting properties, just like in real market. This means, that the variance of terminal prices for larger times does not rise linearly, but slows down or even approaches a constant. This lowers the option price in comparison to standard GBM model. But for a process with similar properties, i.e. Ohrenstein-Uhlenbeck, the option price rises with the mean reversion speed (claimed by http://web.mit.edu/wangj/www/pap/LoWang95.pdf), which is just the opposite effect.

It seems that I am missing something obvious, because everywhere the Monte Carlo method is promoted as a best method for valuating the options on very complex price processes, but it seems that these cannot be completely arbitrary (have to be risk-neutral) and I haven't found any source on how to verify and/or ensure this.

## Answer by Brian B (score 5)

https://quant.stackexchange.com/a/4348

The use of risk-neutral measure is based on the ability to arbitrage away the instantaneous risk of contingent claims. Although for forward contracts the hedge quantity is 1.0, in the general contingent claims case we must assume it varies instantaneously with the market state.

The Girsanov Theorem tells us what the difference is, instantaneously, between real-world "market measure" (of which your historical paths are realizations) and risk-neutral measure suitable for pricing contingent claims. The difference comes in the form of a drift adjustment.

Since the adjustment is market state dependent, it is inappropriate to simply multiply all your path values by the same constant in order to achieve any particular terminal distribution. Instead, you need to have a way of adjusting instantaneously.

Here, I believe you have two choices:

- Work with some model, e.g. a local volatility model, with the local vols calibrated to agree with local vols observed on your paths

- Group your paths to represent the universe of possible outcomes as a discrete distribution. You can then discretize the Girsanov theorem drift adjustment by the local variance, and modify their drift accordingly.

## Answer by FKaria (score 1)

https://quant.stackexchange.com/a/4351

You don't have to believe anything. The simulations are risk-neutral if the expectation at any date divided by the spot price is the return of a risk free zero-coupon bond. That's everything (the only exception being the dividend paying stocks where you have to subtract the dividends in the period.)

Another issue is if you want to model other features such as mean reversion, stochastic volatility, stacionality, etc. but that does not has anything to do with the pricing measure, only with the model SDE. As long as the expected return is risk-free at every time $T>0$ you are in the risk neutral world.