Forwards

Forwards are a contract that promises to buy/sell an underlying asset at a certain strike price.

Example. A farmer wants to sell their wheat, and a mill wants to buy some wheat. The current price of wheat is \20$

• The farmer is afraid of the price of wheat going down
• The mill is afraid of the price of wheat going down Therefore, they draft up a contract: one year from now, they will transact the wheat at a price, \22$ determined right now. One year from now, they will transact. This is a forward contract.
• Underlying asset: wheat
• $t$: time of contract
• $T$: time of execution
• Forward price (=Strike price): F_{T}(t)=K=\22$
• Spot price: S(0)=\20$
• Contract size: how many units of wheat?
• ! no money/assets changes hands now. Thus the value of a portfolio with a forwards contract is $zero$ on the day they entered. Now, assume you’re neither a farmer nor a mill, and you just want to bet on the price of wheat.
• Short position: think wheat price will increase. At execution, you will buy wheat from the spot market at price $S(T)$ and give it to the counterparty at price $K$ according to the futures contract
  • Payoff: $K-S(T)$
• Long position: think wheat price will decrease. At execution, you will get price the wheat for price $K$ and then sell at the spot market for $S(T)$
  • Payoff: $S(T)-K$

thm. (The fair price of a forwards) Under assumption of No-Arbitrage, the fair strike price of a futures contract entered at time $t$ and executed at future date $T$ is:

$$K=F_T(t)=S(t)e^{(r-q)(T-t)}$$

• $r$ is the risk-free rate
• $q$ is the dividend rate. In many cases $q=0$. It only applies to underlying stocks. One may prove this by contradiction, by assuming it doesn’t hold, and constructing an arbitrage portfolio:

thm. (Value of ongoing forward contract) For a futures contract entered at $t=0$, executed at $T$, the value of this contract at intermediate time $t$ is:

$${\mathfrak{F}}_T(t)=(F_T(t)-F_T(0))e^{-r(T-t)}$$

Intuition.

• ! Value of a forward contract is not same as the fair price.
• $F_T(t)$ is strike price of a hypothetical contract from $t$ to $T$
• $F_T(0)$ is strike price of a hypothetical contract from $0$ to $T$
• The difference of these two prices, discounted at risk-free rate.

Proof. Let portfolios:

• ${\Pi }_A$:
  • long forward, enter at $t=0$, execute $t=T$, with strike price $F_T(0)$
  • Short forward, enter at $t=t$, execute $t=T$, with strike price $F_T(t)$
• ${\Pi }_B$: Deposit cash $(F_T(t)-F_T(0))e^{-r(T-t)}$ at risk-free $r$ Then at time $T$:
• ${\Pi }_A(T)=\underset{\text{ long forward }}{\underbrace{S(T)-F_T(0)}}+\underset{\text{ short forward }}{\underbrace{F_T(t)-S(T)}}=F_T(t)-F_T(0)$
• ${\Pi }_B(T)=F_T(t)-F_T(0)$ Since ${\Pi }_A(T)={\Pi }_B(T)⟹{\Pi }_A(t)={\Pi }_B(t)$ by Law of One Price. Then:

$${\Pi }_A(t)=\underset{\text{ value of long forward position }}{\underbrace{{\mathfrak{F}}_T(t)}}=F_T(t)-F_T(0)={\Pi }_B(t)$$

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