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Unlike the valuation of risky assets (which discounts expected future cash flows at a risk-adjusted rate), derivative valuation rests on a no-arbitrage condition. Arbitrage is a transaction in which an investor buys one asset (or portfolio) and simultaneously sells another asset (or portfolio) with identical future payoffs at a higher price, locking in a risk-free gain.

Arbitrage opportunities are rare and quickly exploited. We therefore use the no-arbitrage condition to determine the current value of a derivative from a known portfolio of assets that has the same future payoffs as the derivative under all future states. Where transaction costs exceed the potential arbitrage gain, small price differences can persist.

Consider a 1-year forward contract on an Acme share that pays no dividends and currently trades at $S_0 = \$30$. Let $F_0(1)$ be the forward price. Two strategies that both deliver one Acme share at $t=1$:

• Portfolio 1: Buy a pure-discount bond yielding 5% that pays $F_0(1)$ at $t=1$ (cost today $= F_0(1)/1.05$), and enter a long forward at $F_0(1)$ (zero cost). Total cost today $= F_0(1)/1.05$. At $t=1$ the bond pays $F_0(1)$, which is used to buy one share under the forward; payoff $= S_1$.
• Portfolio 2: Buy one Acme share at $S_0 = 30$ and hold it. Cost today $= 30$; payoff at $t=1$ is $S_1$.

By the law of one price, two portfolios with identical payoffs must have identical costs today: $$\frac{F_0(1)}{1.05} = 30 \;\Rightarrow\; F_0(1) = 30(1.05) = 31.50$$

To see why $F_0(1) = 31.50$ is the only no-arbitrage price, examine deviations:

• If $F_0(1) = 32$ (too high): sell the forward and buy the share. Borrow $30$ at 5%, buy one share. At $t=1$, deliver the share under the forward (receive $32$), and repay the loan ($31.50$). Risk-free profit $= 32 - 31.50 = 0.50$ with zero initial cost.
• If $F_0(1) = 31$ (too low): buy the forward and short the share. Invest the $30$ short-sale proceeds at 5% (grows to $31.50$ at $t=1$). At $t=1$, take delivery under the forward at $31$ and close the short. Risk-free profit $= 31.50 - 31 = 0.50$.

When the forward price is too high → sell forward, buy underlying. When too low → buy forward, short underlying. Arbitrageurs' actions push the forward price back to $S_0(1+R_f)^T$.

Replication means constructing a cash-market portfolio whose payoffs match those of a derivative under all future states of the underlying.

• Replicate a long forward: borrow $30$ at 5% and buy one Acme share. At settlement, payoff $= S_1 - 30(1.05) = S_1 - 31.50$, identical to a long forward struck at $31.50$.
• Replicate a short forward: short one share and invest the $30$ proceeds at 5%. At settlement, payoff $= 31.50 - S_1$, identical to a short forward struck at $31.50$.

The replication argument gives the no-arbitrage forward price for an asset with no carrying costs or benefits: $$F_0(T) = S_0(1+R_f)^T$$

The result $F_0(T) = S_0(1+R_f)^T$ assumes the only cost of holding the asset is the opportunity cost of funds (the risk-free rate). In practice, an asset may have:

• Costs of holding — storage, insurance, spoilage (mainly for commodities; usually negligible for financial assets).
• Monetary benefits — dividends on equities, coupon interest on bonds.
• Non-monetary benefits (convenience yield) — value from owning a hard-to-short commodity, or from holding inventory when shortages may make near-term sale advantageous.

Let $\text{PV}_0(\text{cost})$ and $\text{PV}_0(\text{benefit})$ be the present values (at $R_f$) of all carrying costs and benefits between $t=0$ and $t=T$. Adjusting the no-arbitrage argument:

• Costs only: $F_0(T) = [S_0 + \text{PV}_0(\text{cost})](1+R_f)^T$ — costs raise the forward price.
• Benefits only: $F_0(T) = [S_0 - \text{PV}_0(\text{benefit})](1+R_f)^T$ — benefits lower the forward price.
• Both: $F_0(T) = [S_0 + \text{PV}_0(\text{cost}) - \text{PV}_0(\text{benefit})](1+R_f)^T$.

The cost of carry = benefits of holding − costs of holding.

The same relationships expressed with continuously compounded rates (recall $FV = Se^{rT}$, $PV = Se^{-rT}$):

• No costs / benefits: $F_0(T) = S_0\,e^{rT}$, where $r$ is the continuously compounded risk-free rate.
• Continuously compounded storage cost rate $c$: $F_0(T) = S_0\,e^{(r+c)T}$.
• Continuously compounded benefit rate $b$ (e.g., a dividend yield): $F_0(T) = S_0\,e^{(r+c-b)T}$.

Forward Contracts on Currencies

From Economics, the no-arbitrage currency forward satisfies: $$F_{p/b} = S_{p/b}\times \frac{1+R_{\text{price}}}{1+R_{\text{base}}}$$ where $p/b$ is the price-currency-per-base-currency quote.

Suppose the euro risk-free rate is 3%, the USD risk-free rate is 2%, and the spot USD/EUR is $1.10$. A US-based arbitrageur borrows $\text{\$}100$ at 2%, exchanges to euros at $1.10$, invests at 3% for one year, and converts back. After one year:

• Euros held: $\dfrac{100}{1.10}\times 1.03 = 93.64\;\text{€}$.
• USD owed: $100 \times 1.02 = 102$.

For zero arbitrage, the forward must convert $93.64\;\text{€}$ into exactly $\text{\$}102$, i.e., the forward USD/EUR rate is $$F_{\text{USD/EUR}}=\frac{102}{93.64}=1.0893$$ Equivalently, $F = 1.10\times \dfrac{1.02}{1.03} = 1.0893$. The euro's depreciation in the forward exactly offsets the higher euro interest rate.

If the forward rate $> 1.0893$, an arbitrage profit exists (borrow USD, lend EUR, convert forward); if $< 1.0893$, the opposite trades are profitable.

Converting effective annual rates to continuously compounded equivalents: $r_{\text{USD}}=\ln 1.02 = 1.98\%$, $r_{\text{EUR}}=\ln 1.03 = 2.96\%$. Then $$F_{\text{USD/EUR}} = 1.10 \times e^{(0.0198 - 0.0296)} = 1.0893$$