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The book by J.C. Hull gives a proof that if interest rate is constant, futures price is same as forward price (no-arbitrage price).

In the proof outlined in the book, scenario of daily settlement is taken. The buyer enteres into a long futures position on end of day 0 and the future expires on day N. At the end of day 1 the settlement is F_1 - F_0 (where F_i is future price at the end of day i). Similarly for other days F_i - F_{i-1}.

My question is to make the value of futures contract zero at the end of day 1, shouldn't (F_1 - F_0)*e^{-r(N-1)} ie. the present value (r is continuously compounded constant interest rate with time in days) be the settlement at the end of day 1, as that is what the value of a forward contract would be on the end of day 1 (having same maturity date and negotiated at end of day 0).

  • "My question is to make the value of futures contract zero at the end of day 1" This doesn't seem to make much sense. Can you elaborate / clarify? – ThatDataGuy Jan 6 at 16:25
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The time value of money element is already part of the settlement price of the futures contract - the market has to include that aspect of valuation when deciding how much to buy and sell the contract for. It may indeed be the value you provided, if nothing else happens in the market, and no new information comes to light. Of course this never happens, so the price is the old classic "the price is what the market will pay". If people want more gold, gold futures go up, etc...

In other words, your valuation estimate makes sense from one day to the next, if literally nothing else happens.

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