DER 4 Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives

A forward commitment fixes today the price at which an underlying asset will change hands on a future date. That price is not free to be anything. It is anchored to the current spot price by a single discipline: in a market without transaction costs, prices should never hand out a riskless profit. Pricing that respects this discipline is called arbitrage-free pricing, and it underpins everything that follows.

The starting rule is the law of one price. Two assets, or two bundles of positions, that deliver identical future cash flows must sell for the same amount at the same instant. If they do not, a trader buys the cheaper one, sells the dearer one, and books an immediate gain that carries no risk. Acting on that gain, arbitrageurs bid the low price up and push the high price down until the two meet and the opportunity closes.

For instruments whose value comes from an underlying asset, arbitrage can surface in two forms:

1. Two assets with identical future cash flows trade at different prices.
2. An asset with a known future price is quoted away from the present value of that price, found by discounting at an appropriate rate.

Identical cash flows, different prices

Take two zero-coupon bonds from the same issuer, maturing on the same day, each repaying par of EUR100 and sharing the same default risk. Bond A is quoted at EUR99 and Bond B at EUR99.15, yet both are expected to be worth EUR100 at maturity. The prices disagree while the future cash flows do not, so an arbitrage exists. Selling Bond B short brings in EUR99.15, buying Bond A costs EUR99, and the trader keeps the EUR0.15 difference at once. At maturity each bond pays EUR100, so the proceeds from Bond A exactly fund the purchase that closes the short in Bond B, leaving only the EUR0.15 already earned. As others copy the trade, Bond B is sold down and Bond A is bid up until the two prices align and the profit vanishes.

A known future price and its present value

The second form rests on the time value of money. The future value of a single amount under discrete compounding over N periods, and under continuous compounding over time T, are:

Here r is the stated rate per period and N the number of compounding periods; the continuous form uses the natural exponential as the period length shrinks toward zero. These lessons apply discrete compounding to a single underlying asset, and continuous compounding when the underlying is a portfolio such as an equity, bond, commodity, or credit index or when two currencies and their interest rates are involved. For an asset with no extra costs or benefits of ownership, the correct discount rate is simply the risk-free rate. Example 1 shows what happens when the spot price sits below that discounted future price.

Two arbitrage conditions for an underlying with no extra cash flows follow from these examples:

1. Identical assets, or assets with identical cash flows, trading at the same time must carry the same price, so that S0A equals S0B.
2. An asset with a known future price must have a spot price equal to that price after discounting at the risk-free rate, so that S0 equals ST(1 + r) raised to minus T.

For a positive risk-free rate and an asset with no income or costs, these conditions place the forward price above the spot price: the longer the horizon or the higher the rate, the wider the gap. In practice the relevant risk-free rate for most participants is the repo rate, at which borrowing is secured by highly liquid collateral. A forward commitment also has a symmetric payoff: at settlement the seller receives the difference F0(T) minus ST, gaining when the market price ends below the agreed price and losing when it ends above.

Replication rebuilds the cash-flow stream of a derivative from a combination of a long or short position in the underlying and borrowing or lending cash at the risk-free rate. The arbitrage examples above exploited a mispricing; replication instead assumes the law of one price holds, and it serves to mirror or offset a derivative and to pin down the fair forward price itself.

Long forward versus borrow-and-buy

Compare two ways to end up owning gold in three months. A long forward to buy at F0(T) settles for ST minus F0(T). Alternatively, borrowing S0 today, buying the asset at spot, then later selling at ST and repaying the loan, yields ST minus S0(1 + r) raised to T. The two routes deliver the same payoff for every possible ST only when the forward price equals the future value of the spot price:

This is the no-arbitrage forward price for an asset with no extra costs or benefits. Example 2 confirms that the two routes track each other whatever gold does.

Holding the asset and selling it forward

A related construction holds the asset outright and sells it forward at the same time. The long cash position gains when ST rises above S0; the short forward gains when ST falls below F0(T). Combined, the two offset each other, and the net result is a fixed amount that does not depend on the final price of the underlying.

The risk-free rate is the opportunity cost of tying up money in an asset, and it applies whether or not the buyer actually borrows. With no other cash flows, the spot and forward prices are linked by that rate alone, under discrete or continuous compounding:

Most assets, though, bring extra costs or benefits to whoever holds them. The net of these, together with the opportunity cost, is the cost of carry, and it must be built into the forward price so that no arbitrage opens between the spot and derivative markets. Writing benefits (income) as I and costs as C when they are known present-value amounts at t = 0, and as rates i and c when expressed as returns over the life of the contract, the relationships become:

The signs tell the story. A cost of ownership raises the forward price, because a seller who has carried the asset expects to recover that outlay, giving a higher forward sometimes written F0+(T). A benefit of ownership lowers the forward price, because the forward buyer gives up income the spot holder would have collected, giving a lower forward written F0-(T). The opportunity cost of a positive risk-free rate always pushes in the cost direction.

Equities and equity indexes

Individual shares may pay a dividend, a cash benefit expressed as a known amount, while an index is usually treated with a dividend yield, a benefit expressed as a rate. The two examples below apply Equation 5 and Equation 6 in turn.

Interest rates and credit carry a term structure of different rates by maturity, covered in a later lesson.

Two asset classes deserve a closer look, because their carry works differently: foreign exchange, where both sides of the trade earn interest, and physical commodities, where storage costs and a non-cash benefit called the convenience yield come into play.

Foreign exchange forwards

An exchange rate is quoted as the number of units of a price currency (foreign, f) per single unit of a base currency (domestic, d). A USD/EUR rate of 1.20 means USD1.20 buys EUR1, with the dollar as the price currency and the euro as the base. An FX forward locks in a rate F0,f/d(T) at which the two currencies will be swapped later; a long position buys the base currency and sells the price currency. Each currency carries its own opportunity cost, the foreign rate rf and the domestic rate rd, so replicating the position means borrowing one currency and lending the other. The no-arbitrage forward rate follows:

What matters is the interest-rate differential, not the level of rates. When the foreign rate exceeds the domestic rate, the forward rate rises above spot and the foreign currency trades at a forward discount; when it is lower, the forward falls below spot and the foreign currency trades at a premium.

Commodities and the convenience yield

Physical commodities usually carry real costs: storage, insurance, transport, and, for soft commodities, spoilage. A holder who will deliver later expects the forward price to compensate for those costs, so they push the forward above spot. Working the other way, holding the physical good rather than a derivative can bring a non-cash benefit called the convenience yield. It appears when inventories run low and users value having the commodity on hand: if crude oil stocks are scarce, refiners may bid the spot price up so far that forward prices no longer fully reflect storage costs and interest.