Why Frictionless Price Tests Can Mislead
In a frictionless model, a price outside a no-arbitrage relation may appear to offer a straightforward trade: buy the underpriced position, sell the overpriced one, and lock in a gain. In practice, an investor may pay the ask price to buy and receive only the bid price to sell. Trades may also incur fees, borrowing may cost more than lending earns, and a short sale may be prohibited. A price gap is therefore not enough. The proposed trade must be feasible at the quoted prices, after all costs, and under the stated constraints.
This tutorial focuses on testing apparent forward and option arbitrage under those conditions. The central habit is to write the trade first, price every initial cash flow at its executable quote, and then verify its payoff in every relevant outcome.
Forward Bounds With Bid–Ask Prices and Funding Rates
Consider a one-year forward contract on an asset that pays no income. Let \(S_{\text{bid}}\) and \(S_{\text{ask}}\) be the current spot bid and ask prices, and let \(F_{\text{bid}}\) and \(F_{\text{ask}}\) be the forward bid and ask prices. Suppose the annual effective lending rate is \(i_L\) and the annual effective borrowing rate is \(i_B\), with \(i_L \leq i_B\). Assume the asset can be short-sold, there are no transaction fees, and the forward is settled by delivery at the end of the year. These assumptions will be changed explicitly when an example includes fees or restrictions.
To create a long forward synthetically, short the asset at its bid price and invest the proceeds at the lending rate. At maturity, the investment is \(S_{\text{bid}}(1+i_L)\). It is enough to cover the forward purchase if the forward price is no greater than that amount. To create a short forward synthetically, buy the asset at its ask price using borrowed funds. Repaying the borrowing costs \(S_{\text{ask}}(1+i_B)\). A higher forward sale price can cover that cost.
These are quote-level tests: the forward ask is the price paid to buy a forward, and the forward bid is the price received to sell one. If the ask is below the lower bound, a trader may buy the forward and short the spot asset. If the bid is above the upper bound, a trader may sell the forward and buy the asset. Transaction fees change the inequalities, and a short-sale restriction can remove the first strategy altogether.
Worked Example: A Forward Above the Borrowing-Cost Bound
A non-dividend-paying asset has a spot ask price of $100. The one-year borrowing rate is 7% effective annually. A forward dealer’s bid is $108.50. The trader may borrow and short sales are not needed. Buying the asset and selling the forward each incurs a $0.25 fee, paid immediately. Determine whether the apparent forward premium produces an arbitrage. Assume fees are financed at the borrowing rate and the forward has no additional settlement fee.
Method. Compare the forward bid with the cost of buying the asset and financing both the purchase and the initial fees. Since the proposed strategy buys spot and sells the forward, use the spot ask and the forward bid.
The initial borrowing needed is the asset price plus both fees:
The amount due after one year is:
At maturity, deliver the asset into the forward and receive $108.50. After repaying the loan, the net profit is:
The initial fees have already been included in the borrowed amount, and there is no initial net investment because the borrowing funds the purchases and fees. The terminal profit is positive and certain, so this is an arbitrage under the stated assumptions. The $0.965 result is exact to the displayed precision; reported to the nearest cent, the profit is $0.97.
Common mistake. Ignoring the fees gives \(108.50-100(1.07)=1.50\), which overstates the profit. Comparing the forward bid with the asset’s spot bid would also use a price unavailable when buying the asset.
Option Parity: Test the Executable Portfolio
For European call and put options on the same non-dividend-paying asset, with the same strike \(K\) and expiry, the expiry identity is \(C_T-P_T=S_T-K\). This identity motivates put–call parity: a call plus a risk-free payment of \(K\) at expiry has the same expiry value as a put plus one asset. With a single effective annual rate \(i\), the frictionless relation is \(C+Kv=S+P\), where \(v=1/(1+i)\). In a market with quotes and constraints, compare the costs and payoffs of the two portfolios using executable prices rather than midpoints.
For example, a candidate trade can buy the call, invest enough at the lending rate to produce \(K\) at expiry, sell the put, and short the asset. If all these actions are permitted, the option and asset payoffs cancel at expiry. Its initial cost must be calculated using the call ask, put bid, and stock bid, with the bond deposit valued at the applicable lending rate. If the short sale is forbidden, that candidate trade is not admissible, even if the frictionless parity equation shows a gap.
Worked Example: A Parity Gap Blocked by a Short-Sale Restriction
One-year European options on a non-dividend-paying asset have strike $100. The asset spot bid and ask are both $100. The call bid and ask are $10.50 and $11, and the put bid and ask are $6.50 and $7. The effective annual lending and borrowing rates are both 5%. Short sales of the asset are prohibited, but option trades are allowed. Check the apparent parity signal and explain whether the parity trade can be executed.
Method. First compare executable prices for the portfolio that would exploit a call-minus-put gap below the frictionless parity value. Then check whether every leg is permitted.
The present value of $100 payable in one year, using the 5% rate, is:
The proposed portfolio buys the call and the payment of $100 at expiry, while shorting the put and the asset. Its initial cost before any fees is:
The negative cost appears to be an initial credit of about $0.26. If the trades were permitted, the expiry payoff would be \(C_T-P_T+100-S_T=0\) in every state, by the option payoff identity. The numerical parity signal therefore looks like a credit for a zero-payoff portfolio.
However, the portfolio requires shorting the asset, and the stated rules prohibit that trade. It is not an admissible arbitrage. The calculation identifies a theoretical price gap and a blocked candidate strategy; it does not establish that an arbitrage is available under the actual constraints. No rounding is needed to reach this conclusion.
Common mistake. Treating the apparent $0.26 credit as certain arbitrage overlooks the required short sale. A theoretical portfolio is not an executable portfolio when one of its legs is prohibited.
Transaction Costs Can Remove a Genuine-Looking Gap
Even when every leg is allowed, the parity gap must be large enough to survive all execution costs. Quotes may make the apparent arbitrage smaller than the cost of buying and selling the required positions. A reliable test uses the price paid for each purchase, the proceeds received for each sale, the correct funding rate for deposits or borrowing, and every stated fee. It then checks the payoff across the full range of possible asset prices.
The next example illustrates a parity trade that remains profitable after fees. As in the previous example, assume European options on a non-dividend-paying asset, identical strike and expiry, and physical settlement consistent with the payoff identity. Assume all short sales and option positions are permitted, there is no margin cost, and the deposit earns the stated lending rate.
Worked Example: An Option Parity Arbitrage After Fees
A one-year European call and put have strike $100. The asset spot bid and ask are both $100. The call ask is $8, and the put bid is $5. The effective annual lending rate is 4%. Shorting the asset and the put is permitted. Each of the three traded securities—the call, put, and asset—has a $0.10 transaction fee. Depositing cash has no fee. Determine whether the parity trade remains an arbitrage.
Method. Buy the call, deposit the present value of $100 at expiry, short the put, and short the asset. The option positions use the call ask and put bid; the short asset uses the spot bid. Include the three fees in the initial cash flow.
The deposit needed to produce $100 after one year is:
The cash paid for the call, deposit, and fees, less the proceeds from shorting the put and asset, is:
Thus the strategy produces an initial credit of about $0.5462. At expiry, the net value of the option and asset positions together with the deposit is:
The initial credit can be retained or invested. If invested at 4%, it grows to \(0.546153\ldots(1.04)=0.568\) after one year. The portfolio’s derivative and asset payoff is zero in every state, so the credit, or its accumulated value, is the certain profit. Rounded to the nearest cent, the terminal profit is $0.57. The trade is an arbitrage under the stated assumptions.
Common mistake. Using option midpoints or ignoring the $0.30 in fees can make a marginal parity signal appear larger than it is. Here the correct quote-side calculation still leaves a positive credit; if costs had exceeded that credit, the parity gap would not have produced this arbitrage.
A Practical Arbitrage Checklist
A disciplined test keeps the pricing relation separate from the feasibility test. The relation suggests a candidate trade; the feasibility test decides whether it qualifies. For an option portfolio, explicitly write its initial cost and expiry payoff. For a forward, compare executable forward quotes with the financing cost of acquiring or shorting the underlying asset. In either case, apply the constraints before drawing a conclusion.
Identify the expiry, settlement, asset income, rate period, and any assumptions about exercise, short sales, or collateral.
For each purchase use the ask; for each sale or short sale use the bid. Do not assume a midpoint is executable.
Use the lending rate for invested proceeds and the borrowing rate for financed purchases. Add every fee at the time it is paid.
Confirm every leg is allowed, then show that the net payoff is nonnegative in every state and strictly positive somewhere, or that an initial credit remains after costs.
The borrowing and lending rates need not be equal. A strategy that invests cash must use the rate actually available to the investor; a strategy that needs borrowed money must use the borrowing rate. Replacing both with a single quoted market rate can turn a valid bound into an invalid one. Similarly, a frictionless equality is a useful benchmark, not proof that either side of the trade can be executed at that equality.
Common Exam Traps
- Using the wrong side of a quote. Buying uses the ask; selling uses the bid. A trade that relies on buying at the bid or selling at the ask is not available at the quoted prices.
- Ignoring funding direction. Use the lending rate for invested cash and the borrowing rate for financed positions. A spread between these rates widens the no-arbitrage region.
- Forgetting transaction fees. Fees reduce proceeds or increase costs and may eliminate an apparent surplus. Include them at the time they are charged, and finance them if the problem says they are borrowed.
- Calling a blocked trade an arbitrage. A price relation may identify a theoretical opportunity, but a prohibited short sale or unavailable financing makes that particular strategy inadmissible.
- Checking only one outcome. A true arbitrage must satisfy the payoff condition in every state, not merely have a favorable expected or likely outcome.
Check Your Understanding
For each problem, use the stated quotes and restrictions rather than frictionless prices.
- A non-income-paying asset has spot bid $49 and ask $50. The one-year lending and borrowing rates are 3% and 6%. Forward bid and ask quotes are $50.20 and $51.00. Shorting is allowed and there are no fees. Do these quotes establish an arbitrage? (Answer: No. The lower bound is \(49(1.03)=\$50.47\), and the upper bound is \(50(1.06)=\$53.00\); the forward ask is not below the lower bound and the forward bid is not above the upper bound.)
- A one-year forward bid is $55 on an asset with spot ask $50. The borrowing rate is 8%. Buying the asset and selling the forward each costs a $0.20 fee, paid immediately and financed at the borrowing rate. Find the certain terminal profit from buying the asset and selling the forward. (Answer: Initial borrowing is \(50+0.20+0.20=\$50.40\); repayment is \(50.40(1.08)=\$54.432\); profit is \(55-54.432=\$0.568\), or \(\$0.57\) to the nearest cent.)
- One-year European options with strike $80 have call ask $7 and put bid $4; the asset bid and ask are both $80. The lending rate is 5%, all short sales are allowed, and each call, put, and asset trade incurs a $0.10 fee. Is the trade long call, deposit the present value of $80, short put, and short asset profitable? (Answer: Yes. Its initial cost is \(7+80/1.05+0.30-4-80=-\$0.509523\ldots\), so it gives an initial credit of about $0.51 and has zero combined payoff at expiry.)
- Using the same prices and rates as in Question 3, suppose shorting the asset is prohibited. Does the calculated credit alone establish an admissible arbitrage? (Answer: No. The proposed parity trade requires a short sale of the asset, so it is blocked; the credit identifies a theoretical gap, not an admissible arbitrage under that restriction.)
- A non-dividend-paying asset has spot bid $198 and ask $200. The one-year lending and borrowing rates are 4% and 9%, and a forward dealer quotes bid $217 and ask $218. Shorting is allowed and there are no fees. Is there an arbitrage based on the forward quote? (Answer: No. The lower bound is \(198(1.04)=\$205.92\), and the upper bound is \(200(1.09)=\$218\); the forward ask is not below the lower bound and the forward bid is not above the upper bound.)