Why Futures Prices Are Risk-Neutral Martingales but Have Real-World Drift
Summary
The document addresses why a futures price is modeled as a martingale under the risk-neutral measure even though its expected path under the real-world probability measure may rise. Daily settlement and the absence of a futures contract’s funding cost distinguish futures from holding the underlying asset. Under risk-neutral pricing, the futures price’s expected changes are consistent with no arbitrage and the relevant discounting framework.
The responses explain that this pricing property does not imply zero expected return under the true measure. A futures contract can share the underlying’s risk exposure, so investors may require compensation for bearing risk; consequently, the real-world expected futures price can move upward. A second explanation uses portfolio risk: investors may accept an expected loss on a futures position when it reduces portfolio variance, while risk-increasing positions require expected compensation. These are conceptual illustrations, not a full derivation, and the portfolio argument depends on assumptions about correlation and investor behavior.
Key ideas
- A futures price can be a martingale under the risk-neutral measure because of its pricing and settlement properties.
- Risk-neutral expectations and real-world expectations answer different questions.
- A futures position can carry a risk premium even when it has zero initial value.
- Portfolio hedging demand can affect the expected return investors require on futures positions.
- The portfolio-risk explanation depends on assumptions about correlations and investor preferences.
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Full text
# Why is future price process defined to be a martingale under the risk neutral measure?
# Why is future price process defined to be a martingale under the risk neutral measure?
In Shreve's book, future process is defined to be a stochastic process that satisfies the following two conditions:
(1) $Fut_s(T,T) = S(T)$ where $Fut_s(T,T)$ is the future price at expiration and S(T) is the price of the underlying.
(2) At any time $t_k$, the value of the (daily settlement) payment to be received at time $t_{k+1}$ and indeed all future times is zero.
As a consequence of the second condition, the book claims that the future price process has to follow the following equation:
I understand that the expectation of the right hand side should be zero by condition (2). Can anyone explain why is it not zero under the true probability measure instead of the risk neutral measure?
Note: This is question is from page 243 of Shreve's Stochastic Calculus for Finance
## Answer by Mats Lind (score 1)
https://quant.stackexchange.com/a/29529
Your equation describes how the future price is risk adjusted expected to stay constant as it has no interest rate related funding cost (in contrast to the stock). It bears however the same risk as the stock and should hence be compensated by the same true probability updrift. Say S is at 100 now and 2-yr F at 102 because r =1%. With risk premium of 5% we expect stock at approx 106 and 112 in 1 and 2 years. Hence we expect under true prbobability F to rise to 107 in one year and by (1) to 112 in 2 years. Thats the stocks 5 per year in risk premium!
## Answer by Rodolfo Oviedo (score 0)
https://quant.stackexchange.com/a/73342
Suppose the futures price F is positively correlated with the portfolio of investors (whose composition equals that of the market portfolio of risky assets under CAPM).
Because of the positive correlation, adding a long futures contract to the investor's portfolio increases their risk. (The futures contract is a zero value investment, therefore no other asset is taken out of the portfolio to make room for the futures contract, so there is no related risk reduction. Assume that the initial margin is posted using the assets already in the portfolio.)
- Because of the increase in the risk of the portfolio produced by the inclusion of the long futures contract, a portfolio manager will buy a futures contract only if the expected spot price at expiration is higher than the current futures price thus generating an expected profit. Result: F<E[S(T)] where E[S(T)] is the expected (E) spot price (S) at expiration (T).
- Remember we assumed positive correlation between F and the portfolio of investors. Another portfolio manager should be willing to sell futures at a price F<E[S(T)] thus generating an expected loss because the futures contracts acts as a partial hedge by reducing the variance of the portfolio.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.