Forward and Futures Pricing and Hedging › Building Synthetic Forwards

The arbitrage arguments above show that we can replicate a forward contract using spot positions and borrowing or lending. This synthetic forward has identical payoffs to the actual forward contract. Synthetic construction has several practical uses. It enables arbitrage when prices diverge, provides alternative execution when one market is more liquid than another, and forms the conceptual basis for pricing more complex derivatives.

To create a synthetic long forward on a non-dividend-paying asset, we combine a spot purchase with borrowing to fund it:

1. Buy the asset at $S_0$
2. Borrow $S_0$ at rate $r$ to fund the purchase

At time $T$, we evaluate our position:

• You own the asset worth $S_T$
• You owe $S_0 e^{rT}$
• Net value: $S_T - S_0 e^{rT}$

This exactly matches the payoff of a long forward with forward price $F_0 = S_0 e^{rT}$:

$$\text{Long forward payoff} = S_T - F_0 = S_T - S_0 e^{rT}$$

where:

• $S_T$: spot price at expiration
• $F_0$: forward price
• $S_0$: current spot price
• $r$: risk-free interest rate
• $T$: time to expiration

The synthetic position produces the same outcome in every possible state of the world. Whether prices rise dramatically, fall sharply, or stay flat, the synthetic and actual forward deliver identical results. This equivalence is what allows us to price forwards: since two portfolios with identical payoffs must have the same value (otherwise arbitrage profits exist), the forward price must be set to make the actual forward's value equal to the synthetic's value, which is zero at initiation.

Similarly, to create a synthetic short forward, we reverse the positions:

1. Short-sell the asset, receiving $S_0$
2. Invest $S_0$ at rate $r$

At time $T$:

• Investment grows to $S_0 e^{rT}$
• You owe the asset worth $S_T$
• Net value: $S_0 e^{rT} - S_T$

This matches the payoff of a short forward with price $F_0 = S_0 e^{rT}$. The synthetic short forward profits when the asset falls in value and loses when it rises, exactly mirroring the actual short forward position.

Synthetic forwards are useful when:

• The forward market is illiquid: You can create exposure using the more liquid spot market. Some assets trade actively in spot markets but have thin forward markets, making synthetics the preferred route.
• Forward prices deviate from fair value: Arbitrage brings prices back in line. By constructing synthetics, arbitrageurs profit from mispricings while simultaneously correcting them.
• Funding considerations differ: Your borrowing rate may differ from the rate embedded in market forward prices. Different participants face different funding costs, and synthetics allow each to use their own rates.

The ability to replicate forwards also underpins much of modern derivatives pricing. If we can express a derivative in terms of forwards and bonds, we can price it using the same no-arbitrage logic. Options, swaps, and exotic structures all build on this foundation. Understanding synthetic replication is therefore essential for anyone working in derivatives.