Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives

This reading explains how derivatives are priced by arbitrage and replication, and how the costs and benefits of holding the underlying set a forward price. A candidate must be able to compute a no-arbitrage forward price with discrete or continuous compounding, including carrying costs, benefits and currency interest rates, and describe the trades that exploit a mispriced forward.

LOS 73.a — Arbitrage and replication in derivative pricing

Risky assets are usually valued as the present value of expected cash flows at a risk-adjusted rate. Derivatives are instead priced with a no-arbitrage condition.

• Arbitrage means simultaneously buying an asset (or portfolio) and selling, at a higher price, another with identical future payoffs in every state. It locks in a risk-free gain with no net investment.
• Arbitrage opportunities may be rare, but when they appear they are exploited rapidly. This forces assets with identical payoffs to the same price (the law of one price). Small mispricings can persist when the gain is smaller than the transaction costs of capturing it.
• Replication means building a portfolio of cash market positions (the underlying plus risk-free borrowing or lending) that has the same payoff as the derivative for every possible value of the underlying. The derivative must cost the same as its replicating portfolio. A position that combines the derivative with the opposite replicating position has a certain payoff, so the relevant discount rate is the risk-free rate, and no assumption about investors' risk preferences is needed.

Replicating forwards (underlying with no costs or benefits of holding)

Matching these payoffs with the forward's payoff $S_T-F_0(T)$ gives the no-arbitrage forward price, which is the future value of the spot price at the risk-free rate:

Owning the asset and selling it forward locks in $F_0(T)$: the combined position has a certain payoff, so it must earn the risk-free rate.

Exploiting a mispriced forward

Arbitrage trading pushes the forward price back toward its no-arbitrage level. The overpriced case can also be read as a return comparison: buying the asset at $S_0$ and delivering it at $F_0(T)$ earns more than the risk-free rate, so borrowing at $R_f$ to fund the purchase leaves a riskless surplus.

LOS 73.b — Spot versus expected future price, and the cost of carry

The no-arbitrage forward price is not a forecast of the expected future spot price. It is the cost of buying the underlying today and carrying it to $T$: the financing cost at $R_f$, plus other costs of holding, minus the benefits of holding.

• Costs of holding arise mostly for commodities: storage, insurance and spoilage. Their present value is $P{V}_0(\text{cost})$. For financial assets these costs are negligible.
• Benefits of holding may be monetary (dividends, coupon interest) or non-monetary. The non-monetary benefit is the convenience yield, the advantage of physically holding a commodity, such as being able to sell into a temporary shortage, especially when the asset is hard to short. Their present value is $P{V}_0(\text{benefit})$.

Equivalently, the forward price is $S_0(1+R_f{)}^T$ plus the future value of the holding costs minus the future value of the benefits. A $2 dividend paid at settlement, for instance, lowers the forward price by exactly $2.

• With net costs (costs greater than benefits), the forward price is above $S_0(1+R_f{)}^T$; with net benefits, it is below.
• Carrying costs and benefits affect the forward price at initiation and the forward's value during its life. At expiration the value is simply $S_T-F_0(T)$.

A stated annual rate $r$ with continuous compounding corresponds to an effective annual rate of $e^r-1$. Over $T$ years, $FV=S{e}^{rT}$ and $PV=S{e}^{-rT}$. With continuously compounded annual rates (risk-free $r$, cost $c$, and benefit $b$ such as a dividend yield):

The first applies with no costs or benefits, the second with costs only, and the third with both.

Compared with the spot price itself, what matters is the total carry, financing included: $F_0(T)=S_0$ when $r+c=b$, $F_0(T)>S_0$ when $r+c>b$, and $F_0(T)<S_0$ only when the benefits exceed financing plus storage costs.

Examples.

• Discrete: spot 200, $P{V}_0(\text{cost})=3$, $P{V}_0(\text{benefit})=1$, $R_f=5\%$, $T=0.5$. Then $F_0=(200+3-1)(1.05{)}^{0.5}=202\times 1.024695=206.99$.
• Continuous: an index at 5,000, dividend yield 2%, risk-free rate 4%, $T=0.5$. Then $F_0=5,000e^{(0.04-0.02)(0.5)}=5,000e^{0.01}=5,050.25$.

Currency forwards

For an exchange rate quoted as price currency per unit of base currency, the no-arbitrage forward rate reflects the interest rate differential:

Here $R_{\text{price}}$ and $R_{\text{base}}$ are annual effective risk-free rates and $T$ is in years. For a one-year forward ($T=1$), the discrete formula reduces to $S_0(1+R_{\text{price}})/(1+R_{\text{base}})$. If the rates are instead stated as returns over the forward's own horizon, they are used without the exponent.

The currency with the higher interest rate trades at a forward discount, which offsets its higher interest income. Otherwise, borrowing in one currency and investing in the other would earn an arbitrage profit. If the quoted forward rate (price/base) is above the no-arbitrage rate, borrow the price currency, convert it to the base currency at spot, invest at the base-currency rate and sell the base currency forward. If the quoted rate is below it, reverse every leg. Rates converted to continuous compounding ($r=\ln (1+R)$) give the same forward rate.