Forward Arbitrage Bounds With Market Frictions

Why Forward Prices Have Bounds, Not Always One Exact Value

In a frictionless market, the forward price of an asset with no income is tied to its spot price and the cost of financing that price until delivery. Actual trading can involve a spot bid-ask spread, different borrowing and lending rates, transaction costs, and restrictions on short selling. Those frictions can prevent one or both arbitrage strategies from being carried out. The result is generally a range of forward prices consistent with the trades an investor can actually make.

The bounds below apply to a single asset with no income or carrying costs before delivery. Assume the asset and forward contract have the same delivery date, the forward is settled by delivery of one asset for the agreed price, and there is no default risk. Rates and costs are known in advance. The examples also assume that the investor can trade the stated quantities and meet any collateral requirements, except where a restriction is explicitly introduced.

Definition: Let $S_b$ and $S_a$ be the spot bid and ask prices: the price received when selling the asset and the price paid when buying it. Let $F_b$ and $F_a$ be the forward bid and ask prices, with the same convention. For delivery in $T$ years, let $A_L$ be the accumulation factor available on lending or investing cash, and $A_B$ the factor on borrowing. If the rates are constant effective annual rates $i_L$ and $i_B$, then $A_L=(1+i_L)^T$ and $A_B=(1+i_B)^T$. Usually $i_B \geq i_L$.

Two Strategies Create the Bounds

The upper bound comes from cash-and-carry: buy the asset now, finance the purchase, and agree to sell the asset forward. Because the asset must be bought at the spot ask and the borrowing must be repaid at maturity, the comparison is with $S_aA_B$, not with a mid-market spot price accumulated at a single assumed rate.

The lower bound comes from reverse cash-and-carry: sell the asset short now, invest the proceeds, and agree to buy the asset forward. The investor sells at the spot bid and earns the available lending return, so the relevant amount at maturity is $S_bA_L$. This strategy requires that short selling be permitted and that the short-sale proceeds can be invested as assumed.

F_b \leq S_a A_B \qquad\text{and}\qquad F_a \geq S_b A_L

The first inequality prevents a profitable cash-and-carry trade: the forward bid must not exceed the maturity cost of buying and financing the asset. The second prevents a profitable reverse cash-and-carry trade: the forward ask must not be below the maturity value of the short-sale proceeds. If the forward has one executable price $F$, the corresponding no-arbitrage interval is $S_bA_L \leq F \leq S_aA_B$, provided both strategies are feasible. With separate bid and ask prices, the two inequalities must be checked against the appropriate side of the forward quote.

Important condition: The lower bound is conditional on being able to short the asset and use the proceeds as specified. If short selling is prohibited, or the proceeds are locked up and cannot earn the assumed return, this reverse strategy does not establish the stated lower bound.

Why the Bounds Hold

The argument is a comparison of maturity cash flows. If the cash flow from one side of an arbitrage can be produced for less than the amount received on the other side, the difference is a risk-free profit under the assumptions of the model.

Test the upper bound.
Borrow $S_a$ at time 0 and use it to buy the asset at the spot ask. At time $T$, deliver the asset under a short forward and receive $F_b$. The loan repayment is $S_aA_B$, so the net maturity cash flow is $F_b-S_aA_B$. If $F_b>S_aA_B$, this is positive.

Test the lower bound.
At time 0, short the asset at the spot bid and invest the $S_b$ proceeds. Also enter a long forward. At time $T$, the investment is worth $S_bA_L$; use the forward to acquire the asset for $F_a$ and return it to the lender. The net maturity cash flow is $S_bA_L-F_a$. If $F_a<S_bA_L$, this is positive, provided short selling and the investment are feasible.

This cash-flow comparison also explains why a single frictionless formula should not be applied mechanically when market terms differ. The upper strategy uses the borrowing rate and spot ask; the lower strategy uses the lending rate and spot bid. Replacing both rates with one rate or both spot prices with a midpoint can erase the very frictions that determine whether an arbitrage is possible.

Worked Example: Testing an Overpriced Forward

A hypothetical asset pays no income and has spot bid $ \$98 $ and spot ask $ \$100 $. An investor can borrow at an effective annual rate of 10%. A forward expiring in two years has a dealer bid of $ \$123 $. Assume no transaction fees, no storage costs, and no short-sale requirement for the cash-and-carry strategy. Is there an arbitrage using the quoted forward bid?

Method. Compare the executable forward bid with the borrowing-funded cost of purchasing the asset. This avoids the mistake of using the spot bid or the lending rate in an upper-bound test.

At time 0, borrow $ \$100 $ and buy one asset at the spot ask of $ \$100 $. The two-year accumulation factor on the borrowing is

$$ A_B=(1.10)^2=1.21. $$

The loan repayment at time 2 is

\$100(1.21)=\$121.

Enter a short forward at the dealer bid of $ \$123 $. At maturity, deliver the asset and receive $ \$123 $. After repaying the loan, the net cash flow is

\$123-\$121=\$2.

The cash-and-carry trade earns a certain $ \$2 $ per asset at maturity under the stated assumptions. The upper-bound test confirms it: $F_b=\$123>S_aA_B=\$121$. Check: the asset purchased for $ \$100 $ is delivered under the forward, and the only remaining maturity cash flows are receipt of $ \$123 $ and repayment of $ \$121 $.

Worked Example: Testing an Underpriced Forward With a Short-Sale Fee

A hypothetical asset has spot bid $ \$96 $ and spot ask $ \$99 $. An investor can invest at an effective annual rate of 5% for two years. A dealer offers a forward ask of $ \$104 $. Short selling is allowed, the short-sale proceeds are available to invest, and a $ \$1 $ fee per asset is payable at maturity. Determine whether the reverse strategy produces an arbitrage.

Method. Short the asset at its bid, invest the proceeds at the lending rate, and buy the asset forward. Include the stated fee as a maturity cost. The common mistake is to use the spot ask as the short-sale proceeds or to omit the fee.

At time 0, short one asset for $ \$96 $ and invest the proceeds. The investment grows to

\$96(1.05)^2 =\$96(1.1025) =\$105.84.

Enter a long forward at the ask of $ \$104 $. At maturity, pay $ \$104 $ to receive the asset, return that asset to close the short position, and pay the $ \$1 $ fee. The net maturity cash flow is

\$105.84-\$104-\$1=\$0.84.

The reverse cash-and-carry strategy earns $ \$0.84 $ per asset at maturity. Equivalently, the lower threshold after the maturity fee is $S_bA_L-\$1=\$104.84$, and the forward ask of $ \$104 $ is below that threshold. Check: the proceeds of the investment cover the forward payment and fee, with $ \$0.84 $ remaining.

Adjusting Bounds for Market Frictions

Transaction costs can be included by comparing the forward proceeds or payment with the full maturity-equivalent cost of each strategy. If a specified cash-and-carry fee has maturity value $C_{cc}$, then its relevant cost is $S_aA_B+C_{cc}$. An executable forward bid above that amount can still support the cash-and-carry arbitrage. If a reverse-trade fee has maturity value $C_{rc}$, the reverse strategy's available amount is $S_bA_L-C_{rc}$; an executable forward ask below this amount can support the reverse arbitrage. A fee paid earlier must be accumulated to maturity at the appropriate funding rate before it is compared with maturity cash flows.

These adjustments are specific to the costs and financing available to the trader. They are not a universal allowance that can be added to any quoted forward price. For example, a short-sale fee affects the reverse strategy, while a purchase commission affects the cash-and-carry strategy. A restriction on borrowing may also prevent the upper-bound strategy even if the arithmetic suggests a positive payoff. In each case, first check whether the trades can actually be completed on the stated terms.

Worked Example: A Short-Sale Restriction Removes the Lower Bound

A hypothetical asset has spot bid $ \$50 $ and spot ask $ \$52 $. The borrowing rate is an effective annual 8%, and the lending rate is an effective annual 3%. Delivery is in one year. The forward quote is bid $ \$48 $, ask $ \$49 $. Short selling is prohibited. What bound can be supported by the stated trades, and does the quote breach it?

Method. Test the cash-and-carry upper bound using the borrowing rate and spot ask, then separately assess whether the reverse strategy is available. This avoids treating a theoretical lower bound as enforceable when its required short sale is forbidden.

The borrowing accumulation factor is $1.08$. The maturity cost of buying the asset at the spot ask and financing it is

S_aA_B=\$52(1.08)=\$56.16.

The forward bid is $ \$48 $, which does not exceed $ \$56.16 $. Therefore, the cash-and-carry upper-bound test does not identify an arbitrage. If short selling were allowed and its proceeds could be invested, the candidate lower threshold would be

S_bA_L=\$50(1.03)=\$51.50.

The forward ask of $ \$49 $ is below $ \$51.50 $, but the reverse strategy requires shorting the asset. That trade is prohibited, so this calculation does not establish an arbitrage in the stated market. The valid conclusion from these strategies is an upper bound of $ \$56.16 $, with no lower bound established by reverse cash-and-carry. Check: the quote does not breach the available upper-bound test, while the apparent lower-bound violation depends on an unavailable trade.

Common Exam Traps

Using the wrong side of a quote. Buying the spot asset costs $S_a$, while short selling it raises only $S_b$. A trader selling forward receives $F_b$; a trader buying forward pays $F_a$.
Using one interest rate for both strategies. Borrowing finances the cash-and-carry purchase, so use $A_B$. The reverse strategy invests the short-sale proceeds, so use $A_L$.
Ignoring timing. Compare amounts at the same date. A time-0 fee must be accumulated before subtracting it from a time-$T$ amount.
Assuming short-sale proceeds are freely investable. If proceeds are restricted or earn a different return, replace $S_bA_L$ with the amount actually available at maturity.
Calling a theoretical bound an arbitrage when a required trade is unavailable. A price comparison is not enough; the investor must be able to execute every leg under the stated market rules.

Key takeaway: The practical upper bound compares the forward bid with the cost of buying at the spot ask and borrowing to delivery. The practical lower bound compares the forward ask with the short-sale proceeds invested at the lending rate, but it applies only when that short-sale strategy is feasible.

Check Your Understanding

For each question, assume a no-income asset, no default risk, and the stated trades are available unless a restriction is specified.

The spot ask is $ \$80 $, the borrowing rate is an effective annual 6%, and delivery is in three years. What is the cash-and-carry upper threshold? (Answer: $ \$95.28 $ per asset, rounded to cents.)
The spot bid is $ \$120 $, the lending rate is an effective annual 4%, and delivery is in two years. What is the reverse cash-and-carry lower threshold, assuming no fees and unrestricted short selling? (Answer: $ \$129.79 $ per asset, rounded to cents.)
The spot ask is $ \$90 $, the borrowing rate is 8% for one year, and a dealer's forward bid is $ \$103 $. What is the cash-and-carry arbitrage profit per asset at maturity, ignoring all other costs? (Answer: $ \$5.80 $.)
The spot bid is $ \$70 $, the lending rate is 5% for two years, and a short-sale fee of $ \$2 $ is payable at maturity. A dealer's forward ask is $ \$74 $. If short selling is permitted and the proceeds are investable, what is the reverse-strategy profit per asset at maturity? (Answer: $ \$1.18 $, rounded to cents.)
The spot bid is $ \$190 $, the spot ask is $ \$200 $, the borrowing rate is 7% for one year, and short selling is prohibited. What bound is supported by the stated strategies? (Answer: the cash-and-carry upper bound is $ \$214.00 $; the reverse strategy does not establish a lower bound.)

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