Why a Parity Violation Matters
A call, a put, the underlying stock, and a risk-free investment can be combined into portfolios with identical expiration payoffs. If those portfolios have different prices today, an investor can sell the more expensive portfolio and buy the cheaper one. When the trades are executable under the model’s assumptions, the initial price difference is a risk-free arbitrage profit: the portfolios’ future obligations cancel.
The key exam task is not merely to notice unequal prices. You must determine which portfolio is overpriced, choose the correct long and short positions, and verify that the expiration cash flows offset. This tutorial considers European options on a non-dividend-paying stock.
The model assumes the call and put are European, have the same underlying stock, strike, and expiration, and are valued at the same time. It also assumes no dividends, taxes, or transaction costs; that shares can be bought and sold and shorted; and that borrowing and lending are available at the same risk-free rate. These assumptions matter: if they do not hold, a displayed price difference may not be a realizable arbitrage.
The Parity Test and the Direction of the Trade
Under these assumptions, the call-side and put-side portfolios must have the same price:
To test quotes, calculate each side using the same units and compare the results. If \(C+Kv^T\) is greater, the call-side portfolio is overpriced and the put-side portfolio is underpriced. If \(P+S_0\) is greater, the put-side portfolio is overpriced. In either case, sell the overpriced portfolio and buy the underpriced one.
| Price comparison | Overpriced portfolio | Initial positions for the arbitrage |
|---|---|---|
| \(C+Kv^T>P+S_0\) | Call plus risk-free investment | Short call; borrow \(Kv^T\); buy put and stock |
| \(P+S_0>C+Kv^T\) | Put plus stock | Short put and stock; buy call; invest \(Kv^T\) |
“Borrow \(Kv^T\)” means receive that amount today and repay \(K\) at expiration at the risk-free rate. “Invest \(Kv^T\)” means pay that amount today and receive \(K\) at expiration. A short stock position means borrowing and selling the shares, with the obligation to return the shares later.
Why the Two Portfolios Match at Expiration
Let \(S_T\) be the stock price at expiration. A European call pays \(\max(S_T-K,0)\), and a European put pays \(\max(K-S_T,0)\). The call-side portfolio also receives the bond payment \(K\); the put-side portfolio includes the stock, worth \(S_T\). Their expiration values are therefore:
The right sides are equal for every possible \(S_T\), not just for one forecast or scenario. If the portfolios have identical future values but different prices today, the less expensive one can be bought and the more expensive one sold. The future cash flows then cancel, leaving the initial price difference.
Compute \(C+Kv^T\) and \(P+S_0\), discounting the strike payment using the stated rate and time to expiration.
The larger value is the overpriced portfolio. The difference per share is the gross initial arbitrage profit per share under the model.
Short the overpriced portfolio and buy the underpriced portfolio. Translate each portfolio into its stock, option, and risk-free borrowing or lending positions.
Show that the long and short portfolio values are equal at expiration, so their net expiration obligation is zero.
Worked Applications
Worked Example: The Call-Side Portfolio Is Overpriced
A non-dividend-paying stock sells for \(S_0=\$100\). A one-year European call and put have strike \(K=\$105\), with prices \(C=\$12\) and \(P=\$10\). The effective annual risk-free rate is \(i=5\%\), with annual compounding. Determine whether parity holds and, if not, give the trades and the profit per share.
Method. Compare the call plus the present value of the strike with the put plus the stock. Since \(T=1\), the strike’s present value is:
The call-side portfolio costs \(C+Kv^T=\$12+\$100=\$112\). The put-side portfolio costs \(P+S_0=\$10+\$100=\$110\). The call-side portfolio is overpriced by \(\$112-\$110=\$2\) per share.
Trades. Short the call and borrow \(\$100\), which requires repayment of \(\$105\) in one year. Use the proceeds from those positions to buy the put for \(\$10\) and the stock for \(\$100\). The initial receipts are \(\$12+\$100=\$112\), and the initial purchases cost \(\$10+\$100=\$110\), leaving a net receipt of \(\$2\) per share.
Expiration check. If \(S_T\geq \$105\), the stock and put are worth \(S_T\), while the short call and loan repayment together cost \((S_T-\$105)+\$105=S_T\). If \(S_T<\$105\), the stock and put are worth \(S_T+(\$105-S_T)=\$105\); the short call costs zero and repaying the loan costs \(\$105\). In either case, the expiration values offset exactly.
The locked-in gross profit is \(\$2\) per share before any costs. The common mistake this method avoids is to short the cheaper portfolio: the arbitrage requires selling the more expensive call-side portfolio and buying the less expensive put-side portfolio.
Worked Example: The Put-Side Portfolio Is Overpriced
A hypothetical stock has \(S_0=\$80\). One-year European options with strike \(K=\$84\) trade at \(C=\$7\) for the call and \(P=\$11\) for the put. The effective annual risk-free rate is \(i=5\%\). Each option contract represents 100 shares. Find the overpriced portfolio and the gross profit from one parity trade using one call contract and one put contract.
Method. Discount the strike payment for one year:
Per share, the call-side portfolio costs \(\$7+\$80=\$87\). The put-side portfolio costs \(\$11+\$80=\$91\). Thus the put-side portfolio is overpriced by \(\$91-\$87=\$4\) per share, or \(\$4(100)=\$400\) for the 100-share contract unit.
Trades. Short one put contract and short 100 shares of stock. These positions bring in \(\$11(100)=\$1{,}100\) and \(\$80(100)=\$8{,}000\), respectively. Buy one call contract for \(\$7(100)=\$700\), and invest \(\$80(100)=\$8{,}000\) at the risk-free rate. Initial receipts total \(\$9{,}100\); initial outlays total \(\$8{,}700\). The net initial receipt is \(\$400\).
Expiration check. The call-plus-investment portfolio pays \(100[\max(S_T-\$84,0)+\$84]\). The short-put-and-short-stock portfolio has value \(-100[\max(\$84-S_T,0)+S_T]\). The bracketed amounts are equal for every \(S_T\), so the combined expiration value is zero.
The gross profit is \(\$400\), before costs. The trap is to treat a contract quote as if it were already the total contract cost: the quoted option prices are per share, so they must be multiplied by the 100-share contract multiplier.
Worked Example: A Small Quote Difference Is Not Enough After Costs
A stock with no dividends sells for \(S_0=\$50\). One-year European options with strike \(K=\$52.50\) have prices \(C=\$6.40\) and \(P=\$3.80\). The effective annual risk-free rate is \(i=5\%\). Assume the total entry costs for executing the relevant parity package—option and stock trading costs together—are \(\$2.75\) per share-equivalent. Decide whether the quote difference leaves a positive arbitrage profit after costs.
Method. First calculate the parity values before costs. The present value of the strike is \(\$52.50/1.05=\$50\). The call-side portfolio costs \(\$6.40+\$50=\$56.40\), and the put-side portfolio costs \(\$3.80+\$50=\$53.80\). The call-side portfolio is overpriced by \(\$56.40-\$53.80=\$2.60\) per share.
The frictionless direction would be to short the call and borrow \(\$50\), while buying the put and stock. But executing that package costs \(\$2.75\) per share-equivalent. The net amount is \(\$2.60-\$2.75=-\$0.15\) per share. The apparent price difference does not cover the stated costs, so these quotes do not provide a positive arbitrage profit after costs.
Expiration check. Before costs, both portfolios still pay \(\max(S_T,\$52.50)\) per share at expiration. That equality does not remove the \(\$2.75\) entry cost. The method avoids treating a gross parity discrepancy as automatically tradable: an arbitrage must remain profitable after the costs required to put on the offsetting positions.
Common Mistakes and Practical Interpretation
A parity calculation is a comparison of two complete portfolios, not a comparison of the call price with the put price alone. The stock and the discounted strike are essential. Use consistent units: if option prices are quoted per share but the trade uses a contract covering 100 shares, scale the options, stock, and bond position to the same number of shares.
- Reversing the trade. Sell the portfolio with the higher calculated price and buy the one with the lower price. Doing the reverse creates an initial outflow rather than the arbitrage receipt.
- Using the undiscounted strike today. The bond position costs \(Kv^T\) now and pays \(K\) at expiration. It does not cost \(K\) today unless the interest rate or time to expiration makes those amounts equal.
- Ignoring financing direction. Selling the call-side portfolio requires borrowing \(Kv^T\) today and repaying \(K\) later. Buying that portfolio requires investing \(Kv^T\) today.
- Checking only one expiration outcome. Verify the payoff in both regions, \(S_T\geq K\) and \(S_T<K\), or use the maximum-value identities. The positions must offset across all possible expiration prices.
- Calling every discrepancy an executable arbitrage. The theoretical identity relies on its assumptions. Bid–ask spreads, fees, short-sale restrictions, and different borrowing and lending rates can eliminate a small apparent profit.
The discipline is to separate three questions: does the quoted pair satisfy parity under the model; which portfolio is more expensive; and does the correctly directed trade still produce a positive profit after relevant costs? This is aligned with the SOA VEE Accounting and Finance syllabus.
Check Your Understanding
For each problem, compare the complete parity portfolios before choosing the trades. Assume the stated prices are executable and there are no costs unless the problem says otherwise.
- A non-dividend-paying stock has \(S_0=\$72\). One-year European options with \(K=\$74.88\) cost \(C=\$8\) and \(P=\$7\). The effective annual rate is \(4\%\). Which portfolio is overpriced, what are the per-share trades, and what is the gross initial profit? (Answer: \(Kv^T=\$72\); the call-side portfolio costs \(\$80\) and the put-side costs \(\$79\). Short the call, borrow \(\$72\), buy the put and stock; gross profit is \(\$1\) per share.)
- A stock sells for \(\$120\). A one-year call and put with strike \(\$126\) are quoted at \(\$13\) and \(\$8\), respectively. The effective annual risk-free rate is \(5\%\). One option contract covers 100 shares. Find the overpriced portfolio and gross profit for one parity trade. (Answer: \(Kv^T=\$120\); the call-side portfolio costs \(\$133\) per share and is overpriced by \(\$5\). Short the call and borrow \(\$120\) per share; buy the put and 100 shares. Gross profit is \(\$500\).)
- A stock with no dividends has \(S_0=\$90\). For one-year European options with \(K=\$99\), \(C=\$11\) and \(P=\$7\). The effective annual rate is \(10\%\). State the trade and explain why the expiration cash flows cancel. (Answer: \(Kv^T=\$90\); the call-side portfolio is overpriced by \(\$4\) per share. Short the call and borrow \(\$90\); buy the put and stock. Both portfolios pay \(\max(S_T,\$99)\) at expiration.)
- A stock sells for \(\$64\); one-year European options with \(K=\$67.20\) have \(C=\$5\) and \(P=\$4.50\). The effective annual rate is \(5\%\), and the total execution costs are \(\$0.65\) per share-equivalent. Is there a positive arbitrage after costs? (Answer: \(Kv^T=\$64\); the call-side portfolio is overpriced by \(\$0.50\) per share, less than the \(\$0.65\) cost. There is no positive arbitrage after costs.)
- A non-dividend-paying stock sells for \(\$100\). Two-year European options with strike \(\$121\) are quoted at \(C=\$18\) and \(P=\$16\). The effective annual rate is \(10\%\). Calculate the gross profit for a parity trade involving 15 share-equivalents and name the direction of the positions. (Answer: \(Kv^T=\$121/(1.10)^2=\$100\); the call-side portfolio costs \(\$118\), compared with \(\$116\) for the put-side portfolio. Short the call and borrow \(\$100\) per share-equivalent; buy the put and stock. Gross profit is \(\$2(15)=\$30\).)