Arbitrage, Replication, and the Cost of Carry

Arbitrage refers to buying an asset in a cheaper market and simultaneously selling it in a more expensive market to make a risk-free profit. Traders endeavor to exploit arbitrage opportunities when there are short-lived market differences between assets in the same or different markets. An arbitrageur will buy assets in a market with low prices and sell in another market at a higher price to make a profit. Arbitrage opportunities disappear quickly. When multiple traders buy low-priced assets and sell high-priced assets simultaneously, it increases the demand and price for the former while decreasing the demand and price for the latter. The markets will continue to work in that fashion until prices converge, thereby eliminating arbitrage.

Example: Arbitrage Opportunity

Company ABC’s stock trades on the New York Stock Exchange for $10.15 and the equivalent of $10.25 on the London Stock Exchange. How does this set up a perfect, risk-free arbitrage opportunity?

Solution

The ‘arbitrageur’ can buy ABC’s stock on the New York Stock Exchange for $10.15 and simultaneously sell the stock on the London Stock Exchange for $10.25, making a ‘riskless’ profit of $0.10 per share. This action by other market participants would force the two prices to converge to one price.

Arbitrage and the Law of One Price

The law of one price postulates that assets that produce identical results have only one true market price. In layman’s language, it means identical things should have the same prices. Intuitively, arbitrage opportunities exist if the law of one price does not hold.

Arbitrage Opportunities in Derivative Contracts

Remember that the value of derivative contracts is derived from future cash flows linked to the underlying assets. As such, arbitrage opportunity results in the following ways:

• Case 1: Two assets with identical future cashflow trade at different prices, or
• Case 2: An asset with a definite future price does not trade at the present value of its future price, calculated at an appropriate discount rate.

Example: Arbitrage Opportunity (Case 1)

Bonds X and Y have the same maturity dates, payment at par, and default risk. Bond X has a price of $80 at the initiation. Bond Y has a price of $80.30 at the initiation. If both bonds have an expected price of $100, show how arbitrage opportunity is created.

Solution

At initiation, sell bond Y at $80.30 and buy bond X at $80 to receive a cash inflow of $0.30. At maturity, receive $100 from bond X and buy bond Y at $100 to cover the short position. Cashflows offset each other, earning an investor a riskless profit of $0.30.

Example: Arbitrage Opportunity (Case 2)

Mkate Bakeries wishes to enter a one-year forward contract to buy 100 bags of wheat at an agreed price of $40 per bag. Today’s spot price for wheat is $35 per bag, and the applicable risk-free interest rate is 5%. Assume that Mkate Bakeries can borrow at the risk-free rate of interest, and the wheat is stored at no cost. Show how Mkate Bakeries can make a riskless profit.

Solution

Note that the future price of a bag of wheat does not equate its present value. Using discrete compounding: $$\begin{align*}\text{PV}&=\text{FV}(1+\text{r})^{-\text{N}}\\&=40(1.05)^{-1}\\&=\$38.10\neq\$35\end{align*}$$ At time \(t=0\), Mkate Bakeries borrows \(\$3,500 (=35×100)\) and buys 100 bags of wheat at today’s spot price. Then, Mkate Bakeries enters a forward contract to sell the wheat at $40 per bag in one year. At maturity (\(t=T\)), Mkate Bakeries delivers 100 bags of wheat and receives \(\$4000 (= 40×100)\). Moreover, Mkate Bakeries repays the loan of \(\$3,675 (=3,500(1.05)^1)\). Riskless profit is equal to the forward sale proceeds minus the repayment of the loan: $$\text{Riskless profit}=\$4,000-\$3,675=\$325$$ The riskless profit is equivalent to $3.25 per bag of wheat. Therefore, for Mkate Bakeries to earn a riskless profit,  it must enter a forward contract to sell the wheat due to the discrepancy between spot and future wheat prices. In conclusion, the no-arbitrage conditions for pricing derivatives with the underlying with no additional cash flows include:

• Identical assets (assets with identical cashflows) traded at the same time must have the same price.
• Assets with known future prices must have a spot price equal to the present value of the future price discounted at risk-free of interest.

Replication

Replication refers to a strategy in which a derivative’s cash flow stream may be recreated using a combination of long or short positions in an underlying asset and borrowing or lending cash. Replication mirrors or offsets a derivative position, given that the law of one price holds and arbitrage does not exist. It implies that a trader can take opposing positions in a derivative and the underlying, creating a default risk-free hedge portfolio and replicating the payoff of a risk-free asset. For example, the following combinations produce the equivalent single asset: $$\text{Long asset}+\text{Short derivatives}=\text{Long risk-free asset}$$ We can rearrange the same formula as: $$\text{Long asset}+\text{Short risk-free asset}=\text{Long derivatives}$$ $$\text{Short derivative}+\text{Short risk-free asset}=\text{Short asset}$$ If assets are priced correctly to prohibit arbitrage, replication would seem to be a pointless exercise. However, if we relax the no-arbitrage assumption, we may identify opportunities where replication may be more profitable or have lower transaction costs.

Example: Replicating Long Forward Commitment

Consider a long forward contract with a forward price of $1,600. The underlying spot price is $1,560, and the risk-free interest rate is r% (r > 0). Show how the cash flow stream of the forward contract can be replicated using borrowing funds at a risk-free interest rate.

Solution

At time \(t=0\):

• Borrow $1,560 at a risk-free rate of interest and buy the underlying at \(S_0=\$1,560\).

$$\text{Cashflow}=\$1,560-\$1,560=\$0$$

• Enter a forward contract to buy the underlying at \(F_0(T)=\$1,600\)

$$\text{Cashflow}=\$0$$

At Maturity t=T

• Sell the underlying at the \(S_T\) and repay the loan of \(S_0(1+r)^{T})=\$1,600\).

$$\text{Cashflow}=\text{S}_{\text{T}}-\$1,600$$

• Settle the forward contract at \(S_T-F_0(T)=S_T-\$1,600\), and thus,

$$\text{Cashflow}=\text{S}_{\text{T}}-\$1,600$$ As such, we have replicated a long derivative with a long asset plus a short risk-free asset: $$\begin{array}{l|c|l} \textbf{Cash Market}&&\textbf{Long Forward Contract}\\ \hline\textbf{Time t=0:}&&\textbf{Time t = 0:}\\ \ \ \ \ \ \ \ \text{Borrow \$1,560 and buy at the}&&\ \ \ \ \ \ \ \ \text{Agree to buy the underlying at}\\ \ \ \ \ \ \ \ \text{underlying at} S_0 =\$1,560&&\ \ \ \ \ \ \ F_0(T)=\$1,600\\\ \ \ \ \ \ \ \textbf{Cashflow}=\$1,560-\$1,560=\$0&\Huge =&\ \ \ \ \ \ \ \ \textbf{Cashflow}=\$0\\\textbf{Time t = T:}&&\textbf{Time t =T:}\\\ \ \ \ \ \ \ \  \text{Repay} S_0(1+r)^T=\$1,560(1.026)^1&&\ \ \ \ \ \ \ \text{Settle the contract and sell at spot,}\ S_T\\ \ \ \ \ \ \ \ =\$1,600\ \text{and sell at spot}, S_T &&\ \ \ \ \ \ \ S_T-F_0(T)=S_T-\$1,600\\ \ \ \ \ \ \ \ \textbf{Cashflow}=S_T-\$1,600&& \ \ \ \ \ \ \ \textbf{Cashflow}=S_T-\$1,600\\ \hline \ \ \ \\ \ \ \ \ \ \ \ \ \ \ \ \ \big \uparrow &&\ \ \ \ \ \ \ \ \ \ \ \ \ \big \uparrow\\ \textit{Long Asset + Short Risk-Free Asset}&&\textit {Long Derivative}\\\end{array}$$

Question

Which statement best describes arbitrage? A. Arbitrage is the opportunity to make consistent abnormal returns due to market inefficiency. B. Arbitrage refers to the ability to profit from price mismatches that last a very short time. C. Arbitrage allows market participants to recreate using a combination of long or short positions in an underlying asset and borrowing or lending cash.

Solution

The correct answer is B. Arbitrage refers to buying an asset in the cheaper market and simultaneously selling that asset in the more expensive market to make a risk-free profit. A is incorrect. Arbitrage opportunities allow investors to make risk-free returns without capital commitment. However, such opportunities do not persist for any length of time and cannot be consistently captured. C is incorrect. It’s a description of replication.