Risk-Neutral Drift, Carry Costs, and Futures Calibration
Summary
This note clarifies when the risk-neutral drift equals the risk-free rate and why market forwards or futures may be used to set drifts in pricing and risk simulations. For an asset with no dividends, carry expenses, or repo costs, the risk-neutral drift is the risk-free rate. When those features are present, the drift must account for them, so the simple equality no longer applies. Forward or futures prices provide a market-based way to incorporate these effects when the relevant instruments are available.
The response distinguishes the pricing measure from the real-world drift question raised in the discussion. It explains that forwards correspond to expected asset values under the forward measure, while futures correspond to expectations under the risk-neutral measure. The note does not explain how to estimate real-world drifts; its focus is the adjustment of risk-neutral drift and the use of market-implied prices. The statements are general and do not cover details such as futures convexity adjustments or specific asset classes.
Key ideas
- For an asset without dividends or carrying costs, the risk-neutral drift equals the risk-free rate.
- Dividends, repo costs, and other carry effects require adjustments to that drift.
- Forwards and futures can provide market-based inputs for risk-neutral modeling.
- Forward and futures expectations are associated with different pricing measures.
Tags
Full text
# Risk neutral drift vs real world # Risk neutral drift vs real world I was of the understanding that risk neutral drift was always the risk free rate. A section from Gregory's book on Credit Value Adjustment seems to say risk neutral drifts are typically estimated from futures. Have I just misunderstood the text (I'm hoping so, because it sounds completely wrong to me)? If risk neutral drifts can be different than the risk free rate and risk neutral drift can be estimated from futures, how then do we observe real world drifts? Here is an excerpt of the text: > One area where risk-neutral parameters tend to be used even for risk management simulations is the determination of the drifts of underlying risk factors, which are typically calibrated from forward rates. ... Despite the above problems with drifts, most PFE (potential future exposure) and CVA calculations will calibrate to forward rates in the market. From the CVA point of view, this is justified by hedging. For PFE purposes, this is often done more for convenience’s sake, since it means that simple instruments are by construction priced properly and circumvents the need to attempt to estimate the “real-world” drift of risk factors. ## Answer by Antoine Conze (score 7, accepted) https://quant.stackexchange.com/a/20833 The risk neutral drift is the risk free rate for an asset with no dividends, no cost of carry, no repo cost, etc. Otherwise the drift has to be adjusted to take these into account, and the easiest way to do it (when available) is to use forwards (equal to the expected asset value under the forward measure) or futures (equal to the expected asset value under the risk neutral measure).
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.