# When to use the real world drift and when the risk neutral one for a Monte-Carlo simulation?

Under what conditions should the drift be real world and when risk neutral when simulating

1. Delta Hedging
2. option pricing
4. any other?

For 2. it should be risk neutral. For 1., it could be either as it should result in correct option pricing. For 3. it should be real world. Do you agree?

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In theoretical terms, if you use the Growth Optimal Portfolio as numéraire, the risk neutral and the real world probability measure coincide. – Marco Breitig May 26 '14 at 9:40
Thanks, do you have an example or reference for this? With your interest in econometrics, this time series question you might want to reply. – user12348 May 26 '14 at 20:55
I'll give you some references tomorrow in a short answer. Now I do not have any time. – Marco Breitig May 27 '14 at 16:29
I still do not have enough time, I'll hope to add some references later on. – Marco Breitig May 28 '14 at 16:05

In general these are the two basic approaches to QuantFinance:

Sell side (market maker, risk neutral): You use risk-neutral probabilities ("$\mathbb{Q}$") e.g. in option pricing (to e.g. calculate your greeks and hedge your portfolio), so that you live on the spread.

Buy side (market/risk taker): You use real-world probabilites ("$\mathbb{P}$") for e.g. trading strategies.

'P' Versus 'Q': Differences and Commonalities between the Two Areas of Quantitative Finance by Attilio Meucci.

From the abstract:

There exist two separate branches of finance that require advanced quantitative techniques: the "Q" area of derivatives pricing, whose task is to "extrapolate the present"; and the "P" area of quantitative risk and portfolio management, whose task is to "model the future."

We briefly trace the history of these two branches of quantitative finance, highlighting their different goals and challenges. Then we provide an overview of their areas of intersection: the notion of risk premium; the stochastic processes used, often under different names and assumptions in the Q and in the P world; the numerical methods utilized to simulate those processes; hedging; and statistical arbitrage.

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such a great reference !! (once again only one upvote possible) thank you very much. I was not aware that the buy side uses the real-wordl measure. To my knowledge they just use their own "risk-neutral" model and try to make profit on the pricing errors of others. – Probilitator May 23 '14 at 14:53
@Probilitator: Thank you. I wouldn't rule that out, the markets are huge and there is a place for many diverse strategies out there! – vonjd May 23 '14 at 15:07
by the way you might want to take a look at meta – Probilitator May 23 '14 at 15:50
Thanks, Perfect. That agrees with my thoughts. Hedging can use P but the results will be Q also means you can use Risk free rate for drift because there are many possible P drifts possible depending on the views. So, starting with Q drift should also do. Any thoughts?(notify @Probilitator ) – user12348 May 23 '14 at 17:42

To my knowledge the real world drift plays a crucial role in risk management. The reason being that one is not interested in risk adjusted paths but in real-world scenarios that might actually occure.

Still you should be aware that "the real world drift" is a somewhat controversial topic in quant circles. Nobody knows exactly how to get it. Mostly you end up doing what you consider to be more or less theoretically and empirically justified.

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Where in Risk Management do you use it? – user12348 May 23 '14 at 10:28
Value at Risk, Testing your hedges. Mostly in models that are used to calculate the Risk Based Capital (RBC) - this is often needed for regulatory reasons (e.g. Basel III) – Probilitator May 23 '14 at 10:31
Agree on VaR and RBC. do you see hedges can be either or always RW drift, and why? – user12348 May 23 '14 at 10:35
I have to admit that I am no hedging expert. Among the plethora of hedging approaches there are some that use the real-world measure and other that use the risk-neutral one. I suggest you post it as a stand-alone questions. I think hedging (or at least testing of the hedges) under ther risk neutral measure can be considered somewhat conservative. – Probilitator May 23 '14 at 10:41
Rethinking, for VaR you actually use the actual returns in Historical simulation, not RW drift, for MC VaR simulation, empirical or parametric distribution, so I dont see where the underlying is simulated with RW drift? – user12348 May 23 '14 at 10:44