Different Borrowing And Lending Rates In Parity Trades

Why One Financing Rate Is Not Enough

With one risk-free rate, the standard European put-call parity relation gives a single benchmark. When a trader can borrow only at a higher rate than the rate earned on lending, the two directions of a parity trade have different financing costs. The benchmark therefore becomes a band: a quote outside it may permit an executable arbitrage, while a quote inside it does not.

The key is to identify the financing in each trade rather than insert one rate indiscriminately. A strategy that invests cash uses the lending rate. A strategy that raises cash by borrowing uses the borrowing rate. The derivation below makes the connection between each rate and its bound explicit.

Definition: Let $S_0$ be the current price of a non-dividend-paying asset, and let $C$ and $P$ be the prices of European call and put options on that asset with the same strike $K$ and expiration $T$. Write $i_l$ for the effective annual lending rate and $i_b$ for the effective annual borrowing rate, with $i_b \ge i_l$. Under annual effective compounding, the amounts at time 0 corresponding to a payment of $K$ at time $T$ are $K/(1+i_l)^T$ when lending and $K/(1+i_b)^T$ when borrowing.

Assume the options are European, have matching strikes and expiration dates, and are on the same underlying asset. Assume there are no dividends before expiration, no transaction costs or taxes, and no restrictions on short selling or borrowing. Rates are available over the full term $T$, and the same borrowing and lending rates apply to the traders’ relevant cash flows. These assumptions matter: changing dividends, option exercise rights, or trading frictions can change the analysis.

Deriving the Two Bounds

At expiration, a long call and a short put have a combined payoff of $S_T-K$. Subtracting one share of the asset leaves a fixed payoff of $-K$, regardless of whether $S_T$ is above or below $K$. Thus the portfolio consisting of a long call, a short put, and a short share has expiration payoff $-K$, and its initial cost is $C-P-S_0$.

First consider the trade that lends. Combine that options-and-share portfolio with an investment that pays $K$ at time $T$. The investment costs $K/(1+i_l)^T$. The combined expiration payoff is zero. Its initial cost is $C-P-S_0+K/(1+i_l)^T$. If this cost were negative, the trader would receive money initially and owe nothing at expiration. To rule out that arbitrage, the cost must be nonnegative.

Next consider the trade that borrows. Reverse the options-and-share portfolio: long a share, long a put, and short a call. It pays $K$ at expiration. Borrow $K/(1+i_b)^T$ at time 0; the loan repayment is $K$ at expiration. Together, the positions have zero expiration payoff. Their initial net cash flow is $C-P-S_0+K/(1+i_b)^T$. If this were positive, the trader would receive money initially and have no net payment at expiration. To rule out that arbitrage, this cash flow must be nonpositive.

C + \frac{K}{(1+i_b)^T} \;\le\; P+S_0 \;\le\; C + \frac{K}{(1+i_l)^T}

This is the borrowing-and-lending no-arbitrage band. Since $i_b \ge i_l$, the amount $K/(1+i_b)^T$ is no greater than $K/(1+i_l)^T$, so the lower endpoint cannot exceed the upper endpoint. If borrowing and lending rates are equal, the two endpoints coincide and the usual single parity relation is recovered.

Calculate the lower endpoint.
Use the borrowing rate: $C+K/(1+i_b)^T$. A quote below this endpoint can create the arbitrage that requires borrowing.

Calculate the upper endpoint.
Use the lending rate: $C+K/(1+i_l)^T$. A quote above this endpoint can create the arbitrage that requires investing cash.

Compare the market combination with both endpoints.
Use $P+S_0$ for the quoted put-and-share side. A value inside the band does not give either of these parity arbitrages under the stated assumptions.

Verify the initial and expiration cash flows.
For a claimed arbitrage, identify the actual borrowing or lending and confirm that the expiration payoffs cancel in every stock-price outcome.

Worked Arbitrage Examples

Worked Example: A Quote Below the Lower Endpoint

A non-dividend-paying share is priced at $S_0=\$80$. A European call and put have strike $K=\$84$, expiration in $T=0.5$ years, and prices $C=\$6$ and $P=\$5$. The effective annual borrowing rate is $i_b=6\%$, and the effective annual lending rate is $i_l=2\%$. Determine whether the quote is outside the parity band and, if so, construct the trade.

Method: calculate the band and then verify the portfolio cash flows. The quoted put-and-share value is $P+S_0=\$5+\$80=\$85$.

The borrowing-rate discount factor for half a year is $(1.06)^{0.5}\approx1.029563$, so $K/(1+i_b)^T=\$84/1.029563\approx\$81.588$. The lower endpoint is therefore $C+K/(1+i_b)^T=\$6+\$81.588=\$87.588$.

For comparison, $(1.02)^{0.5}\approx1.009950$, so $K/(1+i_l)^T=\$84/1.009950\approx\$83.173$. The upper endpoint is $C+K/(1+i_l)^T=\$6+\$83.173=\$89.173$.

Since $\$85<\$87.588$, the quote is below the lower endpoint. Use the borrowing-rate trade: buy one share and one put, sell one call, and borrow $\$81.588$. The initial net cash flow is $C-P-S_0+K/(1+i_b)^T=\$6-\$5-\$80+\$81.588=\$2.588$, a receipt of approximately $\$2.59$.

At expiration, if $S_T\ge \$84$, the share and put pay $S_T$, while the short call costs $S_T-\$84$; the option-share package pays $\$84$. If $S_T<\$84$, the share is worth $S_T$, the put pays $\$84-S_T$, and the call expires worthless; the package again pays $\$84$. The loan repayment is $\$84$, so the final net payoff is zero in either case. The initial receipt is an arbitrage under the assumptions.

Common mistake avoided: This trade needs borrowed funds, so its test uses $i_b$, not $i_l$. The lending rate still determines the upper endpoint, but it does not determine this trade’s borrowing cost.

Worked Example: A Quote Above the Upper Endpoint

A share costs $S_0=\$96$. A European call and put with strike $K=\$104$ and one year to expiration cost $C=\$6$ and $P=\$12$. The effective annual borrowing rate is $8\%$, and the effective annual lending rate is $4\%$. Test the quote and show the executable trade if one exists.

Method: test the upper endpoint because the quote may be unusually high. The put-and-share value is $P+S_0=\$12+\$96=\$108$.

The lower endpoint uses the borrowing rate: $C+K/(1+i_b)^T=\$6+\$104/1.08=\$6+\$96.296296\ldots=\$102.296296\ldots$. The upper endpoint uses the lending rate: $C+K/(1+i_l)^T=\$6+\$104/1.04=\$6+\$100=\$106$.

Because $\$108>\$106$, the quote is above the upper endpoint. Buy one call, sell one put, short one share, and lend $\$100$, which grows to $\$104$. The options-and-share portfolio initially costs $C-P-S_0=\$6-\$12-\$96=-\$102$; that is, it generates $\$102$. After paying $\$100$ to lend, the initial net receipt is $\$102-\$100=\$2.00$.

At expiration, the call-minus-put-minus-share package pays $-\$104$, as shown by the payoff identity. The investment pays $+\$104$, leaving zero. Thus the $\$2.00$ initial receipt is not offset by a future liability.

Check: The upper endpoint is $\$106$, while the quoted put-and-share value is $\$108$; the gap is $\$108-\$106=\$2.00$. The cash-flow construction gives the same surplus.

Common mistake avoided: Using the borrowing rate at the upper endpoint would give the wrong test. This trade invests cash, so the relevant endpoint discounts the strike at the lending rate.

Worked Example: A Quote Inside the Band

A share is priced at $S_0=\$100$. European options with strike $K=\$100$ and one year to expiration are quoted at $C=\$9$ and $P=\$5$. The effective annual borrowing rate is $8\%$, and the effective annual lending rate is $3\%$. Is there a parity arbitrage using these quotes?

Method: compare the put-and-share value with both endpoints. First, $P+S_0=\$5+\$100=\$105$.

The lower endpoint is $C+K/(1+i_b)^T=\$9+\$100/1.08=\$9+\$92.592593\ldots=\$101.592593\ldots$. The upper endpoint is $C+K/(1+i_l)^T=\$9+\$100/1.03=\$9+\$97.087379\ldots=\$106.087379\ldots$.

The market value $\$105$ lies between approximately $\$101.59$ and $\$106.09$. Neither inequality is violated, so neither of the two parity portfolios produces a positive initial receipt with a zero expiration payoff under the stated assumptions.

Common mistake avoided: Do not compare the quote with only one discounted-strike value. The lending and borrowing rates create two distinct boundaries; an apparent gap against a single-rate parity value may still be inside the executable band.

Interpreting the Band and Avoiding Traps

The band is a no-arbitrage condition for the specified parity portfolios, not a promise that every price inside it is a fair value or that no other trading opportunity exists. It says that these two zero-expiration-payoff constructions do not produce a positive initial receipt when the quoted $P+S_0$ lies between the endpoints. A quote outside the band is a candidate arbitrage; the trade’s cash flows must still be checked against the actual assumptions.

A reliable way to remember the rate placement is to focus on the cash strategy that completes each portfolio. For the lower endpoint, the trade borrows now and repays $K$, so use the borrowing rate. For the upper endpoint, the trade invests now and receives $K$, so use the lending rate. Since borrowing is at least as expensive as lending, its present value of a fixed repayment is smaller; that is why the borrowing-rate endpoint is the lower one.

Key takeaway: With European options on a non-dividend-paying asset, the parity comparison is $C+K/(1+i_b)^T \le P+S_0 \le C+K/(1+i_l)^T$, under the stated assumptions. A lower-side violation is tested with borrowing; an upper-side violation is tested with lending. Always confirm both the initial surplus and the expiration payoff.

Check Your Understanding

For each item, use the stated annual effective rates and the matching-option assumptions above. Decide whether the quote is outside the band and identify the relevant financing direction where requested.

A share is $\$96$; $C=\$6$, $P=\$12$, $K=\$104$, and $T=1$. The lending rate is $4\%$ and the borrowing rate is $8\%$. Is there an upper-limit arbitrage, and what is its initial surplus? (Answer: Yes; upper-limit arbitrage, with an initial surplus of $\$2.00$.)
A share is $\$105$; $C=\$7$, $P=\$7$, $K=\$108$, and $T=0.5$. The lending rate is $4\%$ and the borrowing rate is $12\%$. Is $P+S_0=\$112$ outside the band? (Answer: No; the band is approximately $\$109.05$ to $\$112.90$, so $\$112$ lies inside it.)
A share is $\$45$; $C=\$4$, $P=\$0.20$, $K=\$50$, and $T=1$. The lending rate is $4\%$ and the borrowing rate is $10\%$. Is the quote below the lower endpoint? If so, give the initial surplus to the nearest cent. (Answer: Yes; the lower endpoint is approximately $\$49.45$, versus $P+S_0=\$45.20$, giving a borrowing-side surplus of approximately $\$4.25$.)
A share is $\$120$; $C=\$15$, $P=\$8$, $K=\$110$, and $T=1$. The lending rate is $5\%$ and the borrowing rate is $10\%$. Which boundary is violated, and what is the initial surplus? (Answer: The upper boundary is violated; the surplus is approximately $\$128-\$119.76=\$8.24$.)
A share is $\$70$; $C=\$9$, $P=\$4$, $K=\$64$, and $T=2$. The lending rate is $2\%$ and the borrowing rate is $6\%$. Is the quote inside the band? (Answer: No; $P+S_0=\$74$, above the upper endpoint $9+64/(1.02)^2\approx\$70.51$.)

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