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 '16 at 15:00
  • $\begingroup$ That is essentially what I am asking. $\endgroup$ – user1559897 Aug 6 '16 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 '16 at 22:24
  • $\begingroup$ Is it not a portfolio of two forward contracts $For(t, S_t) - For(t+1, S_{t+1})$? $\endgroup$ – user1559897 Aug 7 '16 at 14:37

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!


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