Interpreting Risk-Neutral Covariance in Forward–Futures Spreads
Summary
The discussion asks how to interpret the covariance between an underlying asset and the discount factor under the risk-neutral measure, in connection with the forward–futures spread. One answer recommends interpreting risk-neutral quantities from the viewpoint of a hypothetical risk-neutral agent: that agent uses risk-neutral probabilities, which need not match anyone’s real-world beliefs, while remaining consistent with market prices. Under that perspective, the covariance is related to the difference between forward and futures prices.
A second response questions the framing and points to covariance’s familiar role in portfolio diversification, but it does not directly resolve the pricing-measure question. The exchange offers an intuition rather than a derivation: it gives no formula, assumptions, or worked example, and the competing reply may blur risk-neutral pricing with portfolio-risk analysis. Readers should treat the first answer as a conceptual guide and consult a pricing derivation for precise conditions.
Key ideas
- Risk-neutral probabilities are a pricing device and need not represent real-world beliefs.
- Interpret risk-neutral moments from the perspective of a hypothetical agent using those probabilities.
- Covariance between the underlying and discount factor is connected to the forward–futures spread.
- Portfolio covariance for diversification is a different context from covariance under a pricing measure.
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Full text
# What is a good way to interpret covariance under risk neutral measure? # What is a good way to interpret covariance under risk neutral measure? Shreve mentioned that the forward and futures spread depends on the covariance of the underlying and discount factor under the risk neutral measure. Can anyone explain how to interpret this covariance under risk neutral measure? What does that mean intuitively? ## Answer by Igor Pozdeev (score 1) https://quant.stackexchange.com/a/36720 Interpreting risk-neutral moments is tricky. Personally, I think the only good way to avoid heuristics and confusion in doing so is to always use the phrase "from the perspective of a risk-neutral agent". A risk-neutral agent is just a weirdo who thinks that the probability of a hurricane breaking out tomorrow is 10%. You would disagree with him on that, of course, but - interestingly - agree on the price of the hurricane insurance! For this weirdo, the covariance between the underlying of a futures and the discount factor is equal to the forward-futures spread; for you, it is not. ## Answer by user31219 (score 0) https://quant.stackexchange.com/a/37550 I don’t understand the complication of “risk neutrality” either temporally (risk, in the sense of uncertainty, is obviously dynamic) or in terms of agency (two agents may agree or disagree on a risk assessment; if they agree to disagree we call them counter parties). Covariance is most important as a step in building a diversified portfolio, you explore the covariance of potential assets to reduce overall portfolio risk while maintaining the expected value of overall portfolio return.
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