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Drift of futures prices, Risk Neutral word
Posted: February 1st, 2024, 2:24 am
by cdsharm75
Reference: Hull 8th edition, page 369
Hello - would appreciate some insight into this topic. I've taken it for granted all this time and finally started paying a little more attention and getting confused
Hull states - "if we enter into a long futures contract at time 0...it's value is zero". Then this statement/fact is used to show that under RN expectations, the drift is zero. I'm not sure what is special about this argument is special to the RN world; if I enter into a long futures contract at time 0 in the real world, it's value at the moment of initiation is zero as well. Also, there is no initial investment to enter the futures contract in the real world either. So why should this be a reason to conclude 0 drift in the RN world? Would appreciate some other readings that discuss in more detail...Hull can sometimes be very concise.
Screenshot attached for easy reference.
Re: Drift of futures prices, Risk Neutral word
Posted: February 1st, 2024, 3:18 pm
by bearish
The rabbit goes into the hat in the expression you underlined, where the discounting is done outside the expectation. This is only justified for deterministic interest rates, and it leads to the conclusion that futures prices are equal to the equivalent forward prices, as first shown by Fischer Black in (I think) 1976. If you were to try the same approach under the P measure you’d need to calculate the expectation of the product of the futures price change and the stochastic discount factor, generally given as the reciprocal of the value of the growth optimal portfolio. In a model with stochastic interest rates under the standard risk neutral measure you also need to discount with the money market account inside the expectation, which gives rise to a covariance term between the short rate and the change in the futures price. For long dated futures contracts on rates, e.g. SOFR, this covariance term can be worth several basis points and is generally (if a bit sloppily) referred to as a convexity effect.
Re: Drift of futures prices, Risk Neutral word
Posted: February 1st, 2024, 3:52 pm
by katastrofa
Concise but precise

I think it's the RN *pricing* (taking into account only the value of money - vide "very short term interest rate" - no risk premium) that leads to 0 drift. Not the assumption you highlighted. But better wait for the finance gurus like bearish to give you the right answer.
Re: Drift of futures prices, Risk Neutral word
Posted: February 2nd, 2024, 1:22 am
by cdsharm75
Thank you both! @katastofa - hehe; yes precise for sure.....but students like me could always use a little more help:). Hull is still my starting reference point- solid resource to have.
@bearish - I'll check the paper by Black...could you recommend any others that discuss this at a more accessible level; especially the convexity effect. I can live with the level of Hull...but Duffie will put me in a daze. Thanks sir!
Re: Drift of futures prices, Risk Neutral word
Posted: March 20th, 2024, 4:41 pm
by skafetaur
@bearish -- Simple question on your comment
".... In a model with stochastic interest rates under the standard risk neutral measure you also need to discount with the money market account inside the expectation, which gives rise to a covariance term between the short rate and the change in the futures price. ....", did you mean that the covariance term will arise from using Ito's Calculus if interest rates were stochastic (which they are)? I recently completed the course Pricing Options with Mathematical Models - Caltech - Course Info | Coursera and hence my question to join the dots.
Re: Drift of futures prices, Risk Neutral word
Posted: March 21st, 2024, 9:39 pm
by bearish
I don’t think your statement is literally right, in the sense that the covariance term will still be there in models that do not rely on Ito calculus per se, e.g. some kind of discrete lattice model. But in a continuous time/state setting, yes it will have the flavor of an Ito co-quadratic variation term. Which may or may not be helpful, but I’d like to think it is correct.