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: enter image description here

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

  • $\begingroup$ The use of the Risk Neutral Expectation is warranted when the P&L can be hedged. What is the hedge here? $\endgroup$
    – Alex C
    Aug 6, 2016 at 15:00
  • $\begingroup$ That is essentially what I am asking. $\endgroup$ Aug 6, 2016 at 16:28
  • $\begingroup$ What happens if you sell the future and buy the commodity in the spot market at time 0, then hold this position until time T? $\endgroup$
    – Alex C
    Aug 6, 2016 at 22:24
  • $\begingroup$ Is it not a portfolio of two forward contracts $For(t, S_t) - For(t+1, S_{t+1})$? $\endgroup$ Aug 7, 2016 at 14:37
  • $\begingroup$ @user1559897 Hi, would you give me feedback on the answer I have just posted? $\endgroup$ Sep 28, 2022 at 18:17

2 Answers 2


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!


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.)

  1. 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).

  2. 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.


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