Futures Pricing, Cost of Carry, and Option Hedging

# Why futures pricing not calculated like options?


# Why futures pricing not calculated like options?












I have read about futures and options ( from online resources ). I only have the basic understanding,not math heavy ( for eg. for Black Scholes I know only the intuitive idea from the khan academy video mentioned ). So I decided to go through J.C. Hull on derivates being a complete beginner, to get more better idea. I currently completed going through the first 3 chapters ( majorily introduction and furures pricing). I thought I'll try to answer this stupid question of mine once I reach the chapter on Black Scholes in the book. But seems it will take time.

In european options we assume some distribution of stock returns. Using it we know the probablity $p$ such that $S_T$ the stock price at expiry will be greater than $X$ the strike price. We also get to know $E[S_T|S_T \gt X]$, the expectation given option ends in money. So we price our option at current time to be $p*(E[S_T|S_T \gt X] -X)*d$, where $d$ is discount factor ( discounting to present value).

Likewise why is a future not priced at $E[S_T]$ assuming log normal distribution of stock returns ( like we did for options ). The current future price $F_0 = E[S_T]$ seems fair to me from the persepectives of both seller and buyer.

- From buyer side if price is above $E[S_T]$ (ie. $F_0 \gt E[S_T]$), the buyer is getting kind of a bad deal, as the buyer at this point is agreeing to buy it at $F_0$ at a future time when he expects that at that time the stock price would be less than $F_0$.

- From seller side if $F_0 < E[S_T]$ , the seller is kind of under paid for his asset.

But J.C. Hull also proves that an arbitrage free pricing of a future ( of asset like wheat) would be $S_0 + C$ ( $S_0$ is spot price, and $C$ is cost of carry ). Which also seems correct to me.

Which one is correct fair pricing of futures ?

PS: At the end of 3rd chapter J.C. Hull mentions about relation between $E[S_T]$ and $F_0$ using concept of systematic risk/ speculators and hedgers, which I did not completely understand. PPS: Please forgive me if I used some wrong terminology. Hope with time I'll be able to learn the basics. Also forgive me if this question is really stupid, I could delete it if required.

## Answer by JoshK (score 2, accepted)

https://quant.stackexchange.com/a/41863

It's really simpler then it might sound at first. The concept at the root of all of this is "arbitrage free pricing". That means that there is a price at which the buyer and the seller of the future each are at break even and no one is making an arbitrage.

For futures it's pretty simple. If I can buy an ounce of gold for \$1000 now and interest rates are 10%, then I should be willing to sell you a one year forward (or future) for \$1100. I can buy the gold, borrow the money, and know that my costs are \$1100. Any price above or below would be an arbitrage for the seller or the buyer. (There are nuances to the pricing including storage, trading frictions, mark-to-market tail hedging, etc, but that's down the road).

For futures and forward you would hedge by selling or buying the entire amount that you are trading. One ounce of gold future is (basically) hedged by one ounce of gold spot. For options you need to hedge the expected jumps in the value of the underlying. Black Scholes basically looks at the volatility and assumes that the spot moves are distributed along the normal curve. So it's solving a different problem. Your expected returns are really the expected costs of holding the hedge that you need and the amounts you would gain or lose from the jumps in value.

For one ounce of gold calls struck at the money you would need maybe 1/2 ounce of actual hedge. As spot moves up the call seller would buy more spot and would sell as spot moves down. The expected value that we are talking about now is really the value of all of the gains or losses that you would have from keeping this hedged.

In extreme, imagine selling a \$1 strike call on one ounce of gold while spot is at \$1000. You would hedge it with about a full ounce of gold in that case and even a five standard dev move would still leave you needing that full ounce hedge.

Does that help?