How to Price a Futures Contract

[lsusr]

All investments are risky. The techniques explained in this series are exceptionally risky. I am not a registered investor, attorney, advisor, broker, banker, lawyer, dealer or anything of the sort. All opinions expressed here are for my personal exploration. I present it to you for entertainment. This post may help you understand policies related to financial regulation for the purposes of more informed voting. This post is not investment advice. This post may contain errors.

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A security is a tradable financial asset. A future is the obligation to buy or sell a security at a predetermined strike price $k$.

Suppose a security has price $S_0$ right now and the risk-free bond interest rate is $r$. What is the market equilibrium strike price $k$ of a security at temporal displacement $T$ into the future? You may assume the existence of an idealized free market.

$$k=S_0{e}^{rT}$$

The price is forced to satisfy the above equation due to arbitrage.

What is the market equilibrium price $S_0$ right now of a future with strike price $k$?

$$S_0=k{e}^{-rT}$$

You can simulate a future by short-selling the underlying security and buying a bond with the revenue. You can simulate short-selling the same future by borrowing money (selling a bond) and using the money to buy the underlying security.

If the strike price of a future is anything other than $k=S_0{e}^{rT}$ then you can earn money from it.

How do you extract risk-free profit from an future with strike price $k^{-}<k$?

The strike price $k$ is too low compared to $S_0$. We can earn money by longing the future. In the case of a future, “longing” means agreeing to buy the underlying asset at time $T$.

It is not enough to merely earn money on average. We care about risk-adjusted returns. The fact that the future price deviates from $S_0=k{e}^{-rT}$ means we can extract money risk free. We can hedge our long of the future by shorting the underlying asset. Then we can use the bond market to bridge the temporal distance between shorting the underlying asset now and when the future contract expires.

Perform the following three actions simultaneously.

• Long the future. You agree to buy the underlying security at time $T$ for price $k^{-}$. This contract costs you nothing upfront.

• Short-sell the underlying security for $k{e}^{-rT}$ in cash.

• Lend the cash $k$ as a bond with interest rate $r$.

When the future expires, perform the following three actions simultaneously.

• Collect the principal and interest from your bond for revenue $k{e}^{rT}$.

• Fulfill your end of the future contract by purchasing the underlying security at price $k^{-}$.

• Immediately hand off that underlying security to fulfill your end of the short-selling contract.

You pocket a profit of $(k-k^{-})e^{rT}$.

How do you extract risk-free profit from an overpriced future with strike price $k^{+}>k$?

Same as the previous question except you short the future, borrow the cash and buy the underlying security. Your net profit is $(k^{+}-k)e^{rT}$.

Of course, there are limitations on this in real life. Real markets have transaction costs. You cannot borrow money at the risk-free rate. The equation ignores margin calls and other limits on your own short-selling.

Despite these limitations, this analysis shows some interesting principles. In particular, the trading strategies outlined here require no underlying equity. They are constructed from pure leverage.

This is great for increasing market efficiency. It also sets the stage for liquidity crises.

[Richard_Kennaway]

Can you expand on what this step means, in the same way you said what “Long the future” means? Who does what, and when?

Short-sell the underlying security for $k{e}^{-rT}$ in cash.

[lsusr]

Short-selling a security means borrowing it and then immediately selling it with the intention to repurchase it later at a lower time-discounted price right before you return the principal (in the form a a security instead of cash) to your creditor. In this case, you borrow exactly enough of the underlying security such that you will acquire $k{e}^{-rT}$ in cash by immediately selling it.

[boulevardier]

Thanks, great post and clearly explained. One question – doesn’t longing imply that we are agreeing to buy the security for the strike price $k^{-}$ at time $T$?

Long the future. You agree to sell the underlying security at time $T$ for price $k^{-}$. This contract costs you nothing upfront.

[lsusr]

Fixed. Thanks.

[papetoast]

How do you extract risk-free profit from an underpriced future with strike price $k^{+}>k$?

This should read overpriced, not underpriced, right?

[lsusr]

Fixed. Thanks.