No-Arbitrage and Replication

People in pricing and risk management use the ideas of no-arbitrage and replication when they check the assumptions behind the prices a valuation model produces.

Same cash flows, same price

A trade that starts with no capital, cannot lose money, and can make a profit is called an arbitrage opportunity. The absence of such opportunities in the market is called no-arbitrage, and the theory of derivative prices starts from this condition¹². Under no-arbitrage, two things whose future cash flows are the same in every outcome also have the same price today. This is called the law of one price. If their prices differed, you could buy the cheaper one and sell the more expensive one to make a profit, which would be an arbitrage opportunity³.

Consider a fictional example in which the 1-year risk-free rate (a rate with no credit risk or similar risks) is 2% a year. Suppose you can borrow and deposit money at that same rate. You can create a certain receipt of 1,020 yen in one year by depositing 1,000 yen today at 2%. So a zero-coupon bond with no credit risk that repays 1,020 yen in one year is priced at 1,000 yen. If this bond were sold for 990 yen, you could borrow 990 yen and buy the bond. In one year you would receive 1,020 yen and repay 990 × 1.02 = 1,009.80 yen on the loan, leaving 10.20 yen with no capital. No-arbitrage theory prices instruments on the assumption that such opportunities do not exist. Finding the value of fixed cash flows by discounting them at an interest rate is covered in Present Value and Discount Factors.

The replication cost is the price

Replication means building, by trading instruments with known prices, a combination whose future cash flows match those of an instrument of unknown value in every outcome. The combination you build is called a replicating portfolio⁴. By the law of one price, in a market with no arbitrage the price of the instrument equals the price of the replicating portfolio (the total price of the instruments in it)². The zero-coupon bond in the previous section could be replicated with a 1,000-yen deposit, so its price was 1,000 yen. A bond with cash flows on several dates can also be replicated by collecting zero-coupon bonds for each date⁵.

Finding a price by replication takes three steps.

1. Write out the future cash flows of the instrument you want to value, outcome by outcome.
2. Combine instruments with known prices (the underlying, deposits, borrowing and so on) into a combination whose cash flows are the same in every outcome.
3. Add up the cost of building that combination today.

A zero-coupon bond, whose cash flows depend only on interest rates, could be replicated with a deposit. For a derivative, whose future cash flows change with the price of the underlying asset, the combination includes the underlying itself. The calculation below uses a forward contract as an example.

Worked example: the forward price of a stock by replication

Company A's stock pays no dividends. You can find the price of a forward contract on it for delivery in one year (the forward price) by replicating the forward with a purchase of the stock and a loan. Consider a fictional example in which Company A's stock price today is 1,000 yen and the 1-year risk-free rate is 2% a year. The rate is simple interest for exactly one year, so you discount by dividing by 1.02. Assume there are no transaction costs, and that you can borrow the stock and sell it (short selling)⁶.

At maturity, a long forward pays "stock price at maturity − forward price." To build the same cash flow, borrow 1,000 yen today and buy one share. Today, the 1,000 yen borrowed and the 1,000 yen paid for the stock net to 0 yen. In one year, you hold the stock and repay 1,000 × 1.02 = 1,020 yen on the loan.

At every stock price, the total of the combination is "stock price at maturity − 1,020 yen." This is the same cash flow as a long forward with a forward price of 1,020 yen. No money changes hands when a forward is entered into, and the combination also costs 0 yen to build, so their values today match as well. The forward price is therefore 1,020 yen⁶. Figure 1 plots the total of this combination at each stock price at maturity.

Figure description

The horizontal axis is the stock price at maturity, and the vertical axis is the total of the combination of buying one share and borrowing 1,000 yen. The figure covers stock prices from 800 yen to 1,200 yen. The line is "stock price at maturity − 1,020 yen" and rises from left to right: it is −120 yen at a stock price of 900 yen and 80 yen at 1,100 yen. The total is 0 yen when the stock price at maturity is 1,020 yen, and the figure labels this point "Breakeven 1,020". The actual combination is a stock bought for 1,000 yen plus a loan, and it gives the same line as a long forward with a forward price of 1,020 yen, so the cost of building the combination and the value of the forward match.

$$\text{Forward price} = \text{Stock price today} \times (1 + \text{interest rate}) = 1{,}000 \times 1.02 = 1{,}020$$

The calculation uses no forecast of the stock price at maturity and no probability of the stock rising. This is because the combination's cash flow matches the forward at every stock price.

Arbitrage steps when the forward price is off

If the forward price moves away from 1,020 yen, you can lock in a profit that does not depend on the stock price at maturity through arbitrage: selling the expensive side and buying the cheap side⁶. Under the same conditions as the previous section, a forward price of 1,050 yen makes the forward expensive relative to the combination of the stock and a loan. In that case, sell the forward at 1,050 yen, borrow 1,000 yen and buy one share. Today's cash flows net to 0 yen.

At maturity, you deliver the stock you hold to the buyer of the forward and receive 1,050 yen, then repay 1,020 yen on the loan. Whether the stock price is 900 yen or 1,100 yen, 30 yen is left. If the forward price is 1,000 yen and the forward is cheap, you do the reverse. Borrow the stock and sell it for 1,000 yen, deposit the proceeds at 2%, and buy the forward at 1,000 yen. At maturity the deposit has grown to 1,020 yen. You pay 1,000 yen under the forward, receive the stock and return it to the lender, so 20 yen is left whatever the stock price⁶.

This calculation leaves out trading commissions and the cost of borrowing the stock.

Calculations that use replication

Beyond forwards, replication underlies calculations that derive price relationships, and it is the basis for option valuation. The relationship between the prices of a call option and a put option can also be derived from replication. For European options (options that can be exercised only at maturity) with the same underlying, strike price and maturity, two combinations have the same cash flow at maturity. This relationship between call and put prices is put-call parity. An FX forward rate can also be found by replicating it with deposits in two currencies and a spot FX trade.

An option can also be replicated by combining the underlying with lending or borrowing money (deposits or loans). In a model where the underlying price at the end of the period can take only two values (the binomial model), you can choose the amount of the underlying and the amount to deposit (or borrow) so that the cash flows at the end of the period match the option. The price found this way does not depend on the probability that the underlying price rises⁷. This calculation is covered in Binomial Trees.

The binomial model calculation involves probabilities determined by the two possible underlying prices and the interest rate (risk-neutral probabilities). Taking the expected value of the cash flows under these probabilities and discounting it at the interest rate is called risk-neutral valuation⁷. In the Black-Scholes model, the underlying price moves continuously, and the option is replicated by constantly rebalancing positions in the underlying and in cash⁸.

When no-arbitrage does not fix the price

In real markets, no-arbitrage does not hold exactly. This is because of "frictions" such as transaction costs and institutional factors like accounting and tax rules⁹. Pricing theory that relies on no-arbitrage is built on the assumption that the effect of these frictions is small⁹. In the forward price example, if commissions and other costs add up to more than 30 yen, arbitrage leaves no profit even when the forward price is 1,050 yen.

When short selling is not possible, the range that no-arbitrage pins down also changes. You can still carry out the arbitrage when the forward price is too high (buy the stock and sell the forward), but the arbitrage when it is too low requires selling the stock short. So for an underlying that cannot be sold short, no-arbitrage only gives an upper bound: "the forward price is 1,020 yen or less"¹⁰. This applies when the underlying cannot be borrowed, as with a commodity in short supply or an asset whose owners are reluctant to lend it¹⁰.

When the instruments needed for replication are not available in the market, replication alone does not fix the price. Even for a bond with fixed cash flows, if zero-coupon bonds for an intermediate date are not traded, the cash flow on that date cannot be replicated. In that case, the cash flows are discounted with an added charge for risk (a risk premium) estimated with a model¹¹.

Notes

1. 重川一郎, "数理ファイナンス入門" (Introduction to mathematical finance), Kyoto University (intensive lectures at the Faculty of Science, Ehime University, 2009), "無裁定条件" (No-arbitrage condition), page 4 (in Japanese). The source calls a trade that starts from zero capital and earns a positive gain an arbitrage trade (arbitrage opportunity), and calls the absence of such trades the no-arbitrage condition. ↩

2. 小田信之 (Nobuyuki Oda), "信用リスクを反映した金融商品のプライシング" (Pricing financial instruments that reflect credit risk), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (March 1999), "（１）プライシングの基本" (Basics of pricing) and note 4, page 68 (in Japanese). The source explains the no-arbitrage condition as the proposition that no one can earn a profit without taking risk, and states that under this condition the market price of a portfolio with the same cash flows as a transaction must equal the market price of that transaction. ↩ ↩²

3. 重川一郎, "数理ファイナンス入門," the paragraph just before "命題 2.7" (Proposition 2.7), page 20 (in Japanese). ↩

4. 重川一郎, "数理ファイナンス入門," "ポートフォリオ" (Portfolio), page 5 (in Japanese). The source calls building a portfolio whose value at maturity equals the option's payoff replication (the source uses the English word "duplication"). ↩

5. 小田信之 (Nobuyuki Oda), "信用リスクを反映した金融商品のプライシング," "（１）プライシングの基本" (Basics of pricing), the paragraph giving a concrete example of the replication method, page 70 (in Japanese). The source states that a portfolio of zero-coupon bonds with different maturities, each held in the amount of the cash flow on its date, has the same cash flows as a newly issued bond and is in a replicating relationship with it. ↩

6. Martin Haugh, "Forwards, Swaps, Futures and Options" (Columbia University), the paragraph following Definition 1 in Section 1 (page 1; the forward price is set so that the contract's value is 0 when it is entered into), and equation (1) in Section 1.1, its proof, and note 3 (page 2). The discount factor d(0,T) in the source's formula F = S/d(0,T) corresponds to 1 ÷ 1.02 in the text. ↩ ↩² ↩³ ↩⁴

7. Martin Haugh, "Forwards, Swaps, Futures and Options" (Columbia University), the replicating portfolio in the 1-period binomial model in Section 4, and "the price of the derivative does not depend on p!" (page 13). ↩ ↩²

8. Martin Haugh, "A Brief Review of Derivatives Pricing & Hedging" (Columbia University), Section 5 "Delta Hedging," the first paragraph and note 15 (page 18). ↩

9. 小田信之 (Nobuyuki Oda), "信用リスクを反映した金融商品のプライシング," "（１）プライシングの基本" (Basics of pricing), the paragraph that follows the concrete example of the replication method and begins "ところで実際の市場では" (In actual markets, however), page 70 (in Japanese). ↩ ↩² ↩³

10. Martin Haugh, "Forwards, Swaps, Futures and Options" (Columbia University), equation (4) in Section 1.3 "Tight Markets" (page 3). ↩ ↩²

11. 小田信之 (Nobuyuki Oda), "信用リスクを反映した金融商品のプライシング," the paragraph on the case where the market is not complete (when some of the zero-coupon bonds are not traded) (page 70), and the second paragraph of "（１）プライシングの基本" (Basics of pricing) (page 68; the source states that because the market is not complete, cash flows must be discounted using a risk premium estimated with a model). ↩ ↩²