Choosing Real-World or Risk-Neutral Drift for SDE Simulation

Summary

The discussion explains that the probability measure for a simulated asset path should match the simulation’s purpose. For derivative valuation, it recommends the risk-neutral measure, using the risk-free rate as the drift so simulated prices are consistent with market pricing. For statistical risk analysis or portfolio optimization, it calls for the real-world measure, using an estimated real drift or another deliberately chosen assumption.

The answer also corrects the question’s equation: a geometric Brownian motion model needs the asset price in the denominator of the return increment. The post gives the conceptual distinction but does not provide a full derivation, calibration procedure, or numerical example. Its guidance therefore identifies which drift framework to use, while leaving the details of estimating real-world dynamics and implementing a simulation to the reader.

Key ideas

• Use risk-neutral dynamics with the risk-free rate as drift when simulating paths to price derivatives.
• Use real-world dynamics for statistical risk analysis or portfolio optimization.
• The appropriate probability measure depends on the purpose of the simulation.
• A geometric Brownian motion equation should express proportional price changes, not absolute price changes.

Tags

# SDE simulation: P or Q?


# SDE simulation: P or Q?












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 by Richi Wa (score 10)

https://quant.stackexchange.com/a/9176

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.

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