Choosing a Pricing Measure for Monte Carlo Derivative Valuation

# Monte Carlo methods: Choosing the best measure


# Monte Carlo methods: Choosing the best measure












When pricing derivatives using Monte Carlo methods, we take outset in the risk neutral pricing formula which states that we need to calculate the expected value of the discounted cashflows. To do this, we simulate the market variables which - depending on the model and the product of consideration - might require that we actually simulate some state variables and then transform these into the market variables.

We can do all sorts of tricks to improve the accuracy / speed of the simulation such as using methods of variance reduction, quasi-MC methods with "different random variables", running the simulations in parallel, etc.

However, we might also use a change of measure: To get the correct price under a new measure, we will have to change the dynamics of the state / market variables and the numeraire accordingly. One example where such change of measure is benificial is for interest rate derivatives where we can avoid simulating the joint probability of the numeraire and the payoff function, $g$, by changning from the risk-neutral measure, $Q$, to the $T$-forward measure $Q^T$, that is

$$V_0 = B(0) \cdot E^Q_0 \left[\frac{1}{B(T)} \cdot g(r_T)\right] = P(0,T)\cdot E^T_0 \left[g(r_T)\right],$$

where $B$ denotes the bank account, $P$ denotes the zero coupon bonds. In a short rate model (such as Vasicek) this would allow us to easily use an exact simulation scheme without first finding the joint distribution of the $B$ and $g(r_T)$ (which both depend on $r$) or using a discretization scheme.

My question is: How do we decide which measure to price a claim under?

Does it entirely depend on the model and the product? Or does there exist any ways of deciding this a prior whichout performing numerical tests?

## Answer by Rylan (score 1)

https://quant.stackexchange.com/a/76764

I think you touch on these ideas in your question, so I'm sure some (or all) of this is review for you. But just to hopefully tie it all together:

The risk-neutral measure is used (loosely speaking) to price any claims we can hedge. It's a shortcut to finding the value of a self-financing portfolio to hedge with, and if the risk neutral pricing formula you mention comes up with some value $V_0$, that means that if we start with $V_0$ and we trade in a certain way (adapting to market conditions) then we can always get the payoff of the option we seek to price.

In the formula you give you your question, we have that the expectations are equal. You have the "discounted expectation of the payoff, under $Q$", which is always going to be the price but it may be very difficult to actually compute, and you have "something that looks easier to calculate, under $Q^T$".

So in other words, with the standard assumptions, you are always pricing under the risk-neutral measure. However, it might be easier to perform that pricing if you rewrite it to be the expectation under some other measure -- that depends typically on the product and model as you stated.