Constructing Arbitrage Trades From Parity Violations

From a Parity Mismatch to an Arbitrage

Put-call parity gives a no-arbitrage relationship between the prices of European calls and puts with the same underlying asset, strike, and expiration. If observed prices do not satisfy that relationship, the mismatch identifies two portfolios with the same future cash flows but different current costs. The trade is to buy the cheaper portfolio and sell the more expensive one.

The central task is not merely to notice that a quoted call or put looks high. A candidate must compare the complete parity portfolios, including the underlying asset, the amount needed at expiration, and any known income from the asset. The price difference is the amount received at the start of the trade; the offsetting future cash flows show why that amount is locked in under the stated assumptions.

Definition: An arbitrage from a parity violation is a set of trades that requires no net initial investment (or produces a positive initial receipt), has no future net loss, and produces a strictly positive gain in at least one state. When one parity portfolio is overpriced, buy the other portfolio and sell the overpriced one.

The Two Portfolios and Their Values

Consider European call and put options on the same asset, each with strike $K$ and expiration at time $T$. Let $C$ and $P$ be their current prices, and let $S_0$ be the current asset price. Assume the options have identical contract terms, the asset can be bought or sold at the quoted price, borrowing and lending use the same risk-free rate, and there are no transaction costs, taxes, or short-sale constraints.

First suppose the asset pays no income before expiration and the annual effective risk-free rate is $i$. The discount factor for $T$ years is $v^T=(1+i)^{-T}$. Compare these portfolios:

Portfolio A: one call and a risk-free investment that pays $K$ at time $T$. Its current value is $C+Kv^T$.
Portfolio B: one put and one unit of the underlying asset. Its current value is $P+S_0$.

At expiration, Portfolio A pays $\max(S_T-K,0)+K$, while Portfolio B pays $\max(K-S_T,0)+S_T$. If $S_T\geq K$, both pay $S_T$. If $S_T<K$, both pay $K$. Thus their expiration cash flows agree for every possible $S_T$.

$$ C+Kv^T=P+S_0 $$

If the asset pays known cash dividends before expiration, let $I$ be the present value of those dividends, discounted at the risk-free rate to time 0. The corresponding parity portfolios are a call, a risk-free payment of $K$ at expiration, and claims to the known dividends, versus a put and the underlying asset. Their values satisfy:

$$ C+Kv^T+I=P+S_0 $$

Here the dividend claims pay the specified dividends on their payment dates. They match the cash received by holding the underlying asset. This version assumes that the dividend amounts and dates are known and that the discounting rate is appropriate for the stated risk-free cash flows. Do not add $I$ if the asset pays no income, and do not treat uncertain income as a known dividend claim.

Why the Trade Locks In the Difference

The parity relation follows from equal future cash flows. If the two portfolios had different current values, an investor could buy the less expensive portfolio and sell the more expensive one. The purchase and sale create a positive initial receipt equal to the price difference. Since the portfolios' future cash flows match, the long and short positions cancel at every relevant date.

For the no-income case, if Portfolio A is more expensive, sell A and buy B. The initial receipt is $C+Kv^T-(P+S_0)$. At expiration, the short A position pays $\max(S_T-K,0)+K$, while the long B position receives $\max(K-S_T,0)+S_T$. The net expiration cash flow is zero in either region $S_T\geq K$ or $S_T<K$. If Portfolio B is more expensive, reverse the trades. With known dividends, the dividend claim and the dividends received on the long or short underlying position must also be included in the cash-flow check.

Set the contract terms.
Confirm that the call and put have the same underlying, strike, expiration, and settlement conventions. Identify the rate basis and any known income dates.

Value each complete portfolio.
Calculate $C+Kv^T+I$ and $P+S_0$, using $I=0$ when there is no income. Keep the portfolios intact rather than comparing $C$ and $P$ alone.

Buy the cheaper side and sell the dearer side.
The difference between the expensive and inexpensive portfolio values is the initial receipt before any costs or constraints.

Verify all cash-flow dates.
Check the interim income payments and expiration payoff in the regions above and below $K$. A positive initial receipt with zero net future cash flows is the arbitrage in the frictionless model.

Worked Arbitrage Trades

Worked Example: No-Income Asset and an Overpriced Put-Stock Portfolio

A non-dividend-paying asset is priced at $S_0=\$100$. A European call and put both have strike $K=\$105$ and expire in two years. The annual effective risk-free rate is $i=5\%$. The call price is $C=\$12.20$, and the put price is $P=\$9.00$. Identify the overpriced portfolio and specify the trade.

Method. Compare $C+Kv^T$ with $P+S_0$. There is no dividend income, so $I=0$.

The two-year accumulation factor is:

$$ (1.05)^2=1.1025 $$

Therefore the present value of the strike payment is:

Kv^T=\frac{105}{1.1025}=\$95.2381

The portfolio values are:

C+Kv^T=12.20+95.2381=\$107.4381 $$ $$ P+S_0=9.00+100=\$109.00

Portfolio B, the put plus the asset, is overpriced by $\$109.00-\$107.4381=\$1.5619$. Buy Portfolio A: buy one call and invest $\$95.2381$ in a risk-free asset that pays $\$105$ in two years. Sell Portfolio B: sell one put and short one unit of the underlying asset. The initial receipt is $\$109.00-\$107.4381=\$1.5619$.

Expiration check. If $S_T\geq105$, the long call and bond pay $S_T-105+105=S_T$; the short put has no payment, and the short asset costs $S_T$ to close. Net expiration cash flow is zero. If $S_T<105$, the call expires without value and the bond pays $\$105$; the short put costs $105-S_T$, and closing the short asset costs $S_T$. The net is $105-(105-S_T)-S_T=0$. The initial receipt is therefore retained in either case.

Check. The observed parity difference is $C+Kv^T-(P+S_0)=107.4381-109.00=-\$1.5619$, confirming that Portfolio B is dearer. The common mistake is to see a negative difference and assume the call should be sold; the correct action is to sell the entire overpriced portfolio, not an isolated option.

Worked Example: Known Dividend and an Overpriced Put-Stock Portfolio

An asset is priced at $S_0=\$80$. It will pay a known cash dividend of $\$3$ exactly one year from now. European options with strike $K=\$82$ expire in two years. The annual effective risk-free rate is $i=4\%$, the call price is $C=\$8.40$, and the put price is $P=\$9.50$. Find the arbitrage trade and check the dividend-date cash flows.

Method. Use the parity relation with the present value of the known dividend. The dividend claim and the underlying position must both appear in the trade.

The discount factors are $1.04^{-1}$ for the dividend and $1.04^{-2}$ for the strike. Thus:

I=\frac{3}{1.04}=\$2.8846,\qquad Kv^T=\frac{82}{(1.04)^2}=\frac{82}{1.0816}=\$75.8136

The complete portfolio values are:

C+Kv^T+I=8.40+75.8136+2.8846=\$87.0982 $$ $$ P+S_0=9.50+80=\$89.50

Portfolio B is overpriced by $\$89.50-\$87.0982=\$2.4018$. Buy Portfolio A: buy the call, buy the risk-free claim paying $\$82$ at year 2 for $\$75.8136$, and buy the dividend claim for $\$2.8846$. Sell Portfolio B: sell the put and short the asset. The initial receipt is $\$2.4018$.

Interim cash flow. At year 1, the dividend claim pays $\$3$, while the short asset position requires payment of the $\$3$ dividend to the asset lender. These amounts offset exactly.

Expiration check. At year 2, the dividend claim has already paid, and the comparison is the call plus the $\$82$ bond payment against the put plus the asset value. If $S_T\geq82$, the long call and bond pay $S_T-82+82=S_T$, exactly covering the cost to close the short asset; the put expires worthless. If $S_T<82$, the bond pays $\$82$, the short put costs $82-S_T$, and closing the short asset costs $S_T$, for a net of $82-(82-S_T)-S_T=0$. Including the year-1 dividend, the net future cash flow is zero.

Check. The dividend was discounted from its actual payment date, while the strike was discounted from year 2. Discounting both amounts for two years would misstate the portfolio value and the arbitrage amount.

Worked Example: An Overpriced Call-Bond Portfolio

A non-dividend-paying asset costs $S_0=\$120$. A European call and put have strike $K=\$115$ and expire in one year. The annual effective risk-free rate is $i=5\%$. The observed prices are $C=\$14.00$ and $P=\$2.00$. Determine the trade, its initial receipt, and why its expiration payoff cancels.

Method. Compare both sides of parity; do not infer the trade direction from the call price alone. The strike's present value is:

Kv^T=\frac{115}{1.05}=\$109.5238

The two portfolio values are:

C+Kv^T=14.00+109.5238=\$123.5238 $$ $$ P+S_0=2.00+120=\$122.00

Portfolio A is overpriced by $\$123.5238-\$122.00=\$1.5238$. Sell A by shorting the call and borrowing the present value of the strike payment, which requires repayment of $\$115$ at expiration. Buy B by purchasing the put and one unit of the asset. The initial receipt is $\$1.5238$.

If $S_T\geq115$, the short call costs $S_T-115$ to settle, repayment of the borrowing costs $\$115$, and the long asset is worth $S_T$; the put expires worthless. The net expiration cash flow is $-(S_T-115)-115+S_T=0$. If $S_T<115$, the short call expires worthless, repayment costs $\$115$, the put pays $115-S_T$, and the asset is worth $S_T$. The net is $-115+(115-S_T)+S_T=0$.

Check. The price difference is positive in the direction $C+Kv^T-(P+S_0)$, so Portfolio A must be sold. A frequent error is to buy the call because its quoted price seems high; the arbitrage trade concerns the whole call-bond portfolio, which is more expensive than the put-stock portfolio.

Common Mistakes and Conditions

Comparing only option prices. Parity compares complete portfolios. A call price cannot be judged against a put price without accounting for the underlying and the discounted strike.
Reversing the trade. If $C+Kv^T+I>P+S_0$, sell the left-hand portfolio and buy the right-hand one. If the inequality is reversed, buy the left and sell the right.
Using the wrong discount period. Discount each known cash flow from its own payment date. The strike is due at expiration; a dividend paid earlier has a different discount period.
Checking only expiration. A known dividend creates an interim cash flow. Include a matching dividend claim or otherwise the proposed trade may have an unbalanced cash flow before expiration.
Ignoring model assumptions. Transaction costs, bid-ask spreads, taxes, short-sale restrictions, mismatched option terms, or uncertain dividends can prevent a theoretical parity difference from being an executable arbitrage.

Parity-based arbitrage checks are useful because they turn a pricing equation into a precise trading decision. A reliable solution reports the portfolio values, the direction and size of the initial receipt, and the cash-flow cancellation at every payment date and expiration state. This is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the more expensive portfolio, state which side to buy and which to sell, and calculate the initial receipt.

A non-dividend-paying asset has $S_0=\$64$. European options with $K=\$60$ expire in one year. The annual effective rate is $i=5\%$, $C=\$9.00$, and $P=\$3.00$. State the trade and initial receipt. (Answer: $C+Kv^T=\$66.1429$, $P+S_0=\$67.00$; buy the call-bond portfolio, sell the put-stock portfolio, and receive $\$0.8571$ initially.)
An asset with $S_0=\$50$ pays a known $\$4$ dividend in one year. European options with $K=\$52$ expire in two years; $i=5\%$, $C=\$6.50$, and $P=\$8.20$. Find the arbitrage receipt and direction. (Answer: $I=\$3.8095$, $Kv^T=\$47.1655$; Portfolio A is $\$57.4751$, Portfolio B is $\$58.20$; buy A, sell B, and receive $\$0.7249$ initially.)
A non-dividend-paying asset has $S_0=\$90$, $K=\$95$, and one-year European options. The annual effective rate is $i=4\%$, $C=\$7.50$, and $P=\$10.00$. Which portfolio is overpriced, and what is the initial receipt? (Answer: $Kv^T=\$91.3462$, Portfolio A is $\$98.8462$, and Portfolio B is $\$100.00$; buy A, sell B, and receive $\$1.1538$ initially.)
A non-dividend-paying asset costs $\$75$. A call and put with $K=\$70$ expire in two years; $i=5\%$, $C=\$13.00$, and $P=\$5.00$. State the arbitrage direction and verify that the expiration cash flow is zero in both regions. (Answer: $Kv^T=70/(1.05)^2=\$63.4921$; Portfolio A is $\$76.4921$, Portfolio B is $\$80.00$; buy A and sell B, receiving $\$3.5079$ initially. For $S_T\geq70$, both sides pay $S_T$; for $S_T<70$, both sides pay $\$70$.)

What You'll Learn