Forwards and Futures

Among all derivatives, forwards and futures are the simplest.
They are linear contracts whose payoffs vary one-for-one with the underlying price.
Understanding their pricing through the lens of no-arbitrage provides the mathematical foundation for all derivative valuation.

1. What Are Forwards and Futures?

Forward Contract

A forward is a private OTC agreement to exchange an asset at a fixed price $K$ on a future date $T$.

• Long forward: commits to buy at $K$.
• Short forward: commits to sell at $K$.

At maturity, the long receives:

\[\Pi_T = S_T - K.\]

Futures Contract

A futures contract is the exchange-traded equivalent:

• standardized contract size and expiry,
• daily mark-to-market settlement through a clearinghouse,
• margin requirements that limit credit exposure.

2. The No-Arbitrage Pricing Principle

Pricing of forwards and futures rests on the idea that identical future payoffs must cost the same today.

Assume:

• Current spot price of the asset: $S_0$
• Continuous risk-free rate: $r$
• Continuous yield on the asset (dividend, foreign interest, or convenience yield): $q$
• Time to maturity: $T$

We construct replicating portfolios to enforce price consistency.

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2.1 Cash-and-Carry Arbitrage

Imagine the forward is over-priced ($F_0 > S_0 e^{(r - q)T}$).
We can lock in risk-free profit via a cash-and-carry strategy:

1. Borrow $S_0$ cash at rate $r$.
2. Buy one unit of the underlying asset at $S_0$.
3. Short one forward at price $F_0$ (agree to sell later).

At maturity $T$:

• Deliver the asset into the forward and receive $F_0$.
• Repay the loan $S_0 e^{rT}$.
• Receive any income (dividends or yield) of $S_0(e^{qT}-1)$.

The total terminal payoff:

\[\text{Profit} = F_0 - S_0 e^{(r - q)T}.\]

If $F_0$ is indeed higher than $S_0 e^{(r - q)T}$, this profit is positive and arbitrage exists.
Competition pushes $F_0$ down until equality holds.

2.2 Reverse Cash-and-Carry Arbitrage

Conversely, if the forward is under-priced ($F_0 < S_0 e^{(r - q)T}$):

1. Short sell one unit of the underlying at $S_0$.
2. Invest the proceeds at rate $r$.
3. Go long one forward at $F_0$ (agree to buy later).

At $T$:

• Buy back the asset via the forward for $F_0$.
• Collect the matured investment $S_0 e^{rT}$.
• Pay any carry cost or yield $S_0(e^{qT}-1)$.

The payoff is:

\[\text{Profit} = S_0 e^{(r - q)T} - F_0.\]

If positive, arbitrageurs buy forwards and short spot, driving prices back to parity.

2.3 Equilibrium Relation

In equilibrium (no free lunch), both strategies yield zero profit:

\[F_0 = S_0 e^{(r - q)T}.\]

This is the cost-of-carry model — the cornerstone of forward and futures pricing.

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2.4 Example: Equity Forward on a Dividend-Paying Stock

Suppose:

• Current stock price $S_0 = 100$
• Annual risk-free rate $r = 5\%$
• Continuous dividend yield $q = 2\%$
• Maturity $T = 0.5$ years

Then

\[F_0 = 100\, e^{(r - q)T} = 100\, e^{(0.05 - 0.02)\times0.5} = 101.51.\]

Arbitrage Intuition

If a forward were quoted at $F_0 = 103$:

• You would short the forward, buy the stock, and borrow $100.
• In 6 months:
  receive $103$, repay $100 e^{0.05\times0.5}=102.53$, receive dividends ≈ $1.
• Net profit ≈ $1.47 risk-free → arbitrage.

2.5 Visualization

The diagram illustrates how arbitrage connects spot, carry, and forward value:
if the forward is too high → cash-and-carry (sell forward, buy spot);
too low → reverse cash-and-carry (buy forward, sell spot).

3. Futures Pricing and Convexity

For deterministic interest rates, futures and forwards have the same fair value.
However, with stochastic rates:

\[E_t[F_T] \ne F_t,\]

because futures are marked to market daily.
If $\text{corr}(r_t, S_t) > 0$, the long futures tends to outperform the forward — a positive convexity bias.

4. Market Usage

Sector	Purpose	Typical Instrument
Corporates	Hedge commodity or FX exposure	Oil futures, FX forwards
Investors	Adjust equity exposure	Index futures
Banks	Construct forward rate curves	FRA, Eurodollar futures
Hedge funds	Relative-value arbitrage	Calendar spreads

These instruments underpin price discovery across asset classes.

5. Summary

• No-arbitrage ensures $F_0 = S_0 e^{(r - q)T}$.
• Cash-and-carry and reverse carry strategies enforce this relation.
• Forwards and futures bridge present and future market prices.
• Small deviations signal funding or collateral effects rather than arbitrage.

Next up: Options: Basics and Put–Call Parity