Futures Convergence, Carry, and Arbitrage-Based Pricing

# Does the future's price decay to the spot?


# Does the future's price decay to the spot?












I am confused with how the the spot process, $S_t$ and futures process $F_t$ evolve through time. From what I understand:

- The discounted-spot process, $\frac{S_t}{B_t}$, is a $\mathbb{Q}$-martingale

- The futures' is represented as the expected payoff, $F(t;T,X) = \mathbb{E}^{\mathbb{Q}}(X|\mathcal{F_t})$

- At terminal time, $T$, we have $F(T;T) = S_T$.

But I am trying to get a better intuition around futures with a heuristic viewpoint. (Ignoring cost-of-carry and dividends), if you wanted to buy a future on a stock, $S_t=\\\$100$ with a 1 year expiry where interest-rates are $r=0.10$, you would happily buy the futures until it hit $F=\\\$110$, because if $F=\\\$105$, you could buy the the future and then invest the spot money into $B_t$ bonds. If the stock has a 50/50 chance of going up or down, then on average your PnL would be the difference between the bond rate and forward-rate of the future (which is > 0 in this case).

Then it seems the conclusion from this point that when $t \to T$, $F_t\searrow S_t$, that is the future decays to the spot. But point 1 suggests that $S_t\nearrow F_t$, since the non-discounted spot, $S_t$ has upwards-drift at the rate of the bond.

Although, I somewhat understand that point 1 is specifically that the discounted-process is a martingale under the risk-neutral measure, rather than the real-world measure, which is what a derivative's price wants to be compensated for selling a derivative (the premium).

Question 1: Is it the spot price drifting towards the future, or the future's price decaying to the spot?

Question 2: The extra layer of confusion on the future is that both the short-seller and buyer are on margin (for cash-settled futures?) and are marked-to-market. (Is why we don't need to discount the future to be martingale since both the underwriter and buyer can put the value of the spot in a bond and adjust their margin daily, some clarity on this also would be appreciated). So then wouldn't they both be happy to trade under the $\\\$110$ price tag? Since they can both put value of the spot into a bond? Or is the futures risk-neutral pricing under the assumption that the seller already holds the asset and so there is no cash to buy a bond?

## Answer by dm63 (score 2, accepted)

https://quant.stackexchange.com/a/80938

Question 1: for derivatives pricing it doesn’t matter whether you assume spot->futures or the other way round. All that matters is the relative pricing. For real world you can never specify which one is ‘correct’, because the actual outcome is random. Historical data presumably shows spot-> futures more often , since stocks generally have returns at least equal to the risk free rate.

Question 2: the futures price is determined by arbitrage , so both seller and buyer should agree on the 110. The point is that there is no initial cash outlay to transact the future. Consider the arbitrage starting from a flat position : if futures >110, borrow cash , buy stock, sell futures. If futures <110, sell stock, invest cash , buy futures.