Choosing Real-World and Risk-Neutral Drifts for Simulation
Summary
The document distinguishes the probability assumptions used in derivatives pricing from those used to forecast outcomes and manage risk. Under the risk-neutral measure, simulations support option valuation and the calculation of sensitivities used in hedging. Under the real-world measure, simulations represent possible future outcomes for trading strategies and risk management, where expected returns and risk premia matter. The discussion frames these as the “Q” and “P” areas of quantitative finance, respectively.
For delta hedging, the answer says either drift may be used if the purpose is to obtain correct option pricing, while emphasizing risk-neutral probabilities for pricing and hedge calculations. The document offers conceptual guidance rather than a worked example or empirical comparison. It also cautions that estimating real-world drift is difficult and contentious, so results from real-world simulations depend on assumptions that may be theoretically or empirically motivated but remain uncertain.
Key ideas
- Risk-neutral probabilities are used to price derivatives and calculate their sensitivities.
- Real-world probabilities are used to model potential outcomes for trading and risk management.
- The distinction reflects different goals: pricing current claims versus modeling future investment outcomes.
- Real-world drift is uncertain and must be estimated using assumptions.
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Full text
# When to use the real world drift and when the risk neutral one for a Monte-Carlo simulation?
# 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
- Delta Hedging
- option pricing
- trading strategy
- 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?
## Answer by vonjd (score 8, accepted)
https://quant.stackexchange.com/a/11411
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.
See also this excellent article: '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.
## Answer by Probilitator (score 3)
https://quant.stackexchange.com/a/11407
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.