SDE simulation: P or Q?

Question

Let's take a GBM under $P$:

$dS=\mu dt+\sigma dW_{t}^{P}$

and then under $Q$

$dS=r dt+\sigma dW_{t}^{Q}$, where $dW_{t}^{Q} = dW_{t}^{P} + (\mu - r)/\sigma dt $

Now, let's say that I have calibrated my model on the mkt option prices (using B&S) getting the parameters that i need. Question:

Do I have to simulate the path subtracting from $W^{Q}$ the market price of risk? Or what i only need is a brownian motion (knowing that $r$ in the drift part is already the result of the change of measure)?

Thanks.

Answer 1

It depends on the purpose of your simulation.

If you want to model the asset price path for pricing some derivative then you need the risk-neutral measure (thus you take the risk-less rate as drift). Why? Because the risk-neutral measure makes your pricing compatible with the pricing of other contracts in the market. It makes the prices consistent.

If you want to do some kind of risk management or portfolio optimization then you need the real world probability measure. Why? Because if you want to model the asset statistically then you need the real drift (or some ad-hoc drift, or no drift at all).

Check the equation again: your SDE is not GBM - it should say $dS/S$ on the left-hand side.