Arbitrage Portfolios For Both Directions Of A Parity Violation

Why Compare the Two Portfolios?

European put-call parity gives two ways to combine a European call, a European put, the underlying stock, and a risk-free bond. The call-plus-bond portfolio and the put-plus-stock portfolio have the same value at expiration in every possible stock-price state. Under the standard no-arbitrage assumptions, they must therefore have the same market value today.

This tutorial turns that equality into a trading rule. If one portfolio is priced higher than the other, the arbitrage direction is to buy the cheaper portfolio and short the more expensive portfolio. The option position alone does not identify the trade: the stock and the bond positions must also be reversed along with the relevant portfolio.

Assumptions: The options are European, have the same underlying stock, strike $K$, and expiration time $T$, and the stock pays no dividends before expiration. The risk-free rate is a fixed effective annual rate $i$, and a default-free bond paying $K$ at time $T$ can be bought or sold at its no-arbitrage value. Assume frictionless trading, including the ability to short the relevant assets, with no transaction costs or taxes. If these assumptions fail, the equality or the arbitrage trades described here may not apply as stated.

The Portfolios and Their Values

Let $S_0$ be the stock price today, $C$ the market price of the call, and $P$ the market price of the put. The value today of a risk-free bond that pays $K$ at time $T$ is $Kv^T$, where $v=1/(1+i)$. Thus, the two portfolio values are

A=C+Kv^T,\qquad B=P+S_0.

Portfolio $A$ consists of one long call and a long bond paying $K$ at expiration. Portfolio $B$ consists of one long put and one share of stock. Under the assumptions above, put-call parity states that $A=B$. In an arbitrage comparison, calculate both market values using the quoted option premiums and the correctly discounted strike, then compare them.

Definition: For this tutorial, $A=C+Kv^T$ is the market value of the call-plus-bond portfolio, and $B=P+S_0$ is the market value of the put-plus-stock portfolio. If $A>B$, buy $B$ and short $A$. If $B>A$, buy $A$ and short $B$. The positive difference is the cash surplus received at initiation for one option contract unit and one share, before any contract multiplier is applied.

Why the Expiration Payoffs Match

Write $S_T$ for the stock price at expiration. A European call pays $\max(S_T-K,0)$, a European put pays $\max(K-S_T,0)$, and the long bond pays $K$. The total payoff of $A$ is therefore

\max(S_T-K,0)+K=\max(S_T,K).

The put-plus-stock portfolio pays $\max(K-S_T,0)+S_T$. If $S_T\leq K$, this is $K-S_T+S_T=K$; if $S_T>K$, it is $0+S_T=S_T$. In either case, it equals $\max(S_T,K)$. Thus, the two portfolios have identical payoffs for every possible value of $S_T$.

The equality of current values follows from the no-arbitrage principle. If the same future payoff could be purchased for two different prices, a trader could buy the cheaper version and sell the more expensive version. The future payoffs would cancel state by state, while the initial proceeds would be positive. That is the reason for comparing complete portfolios rather than looking at a call or put premium in isolation.

When the Call-Plus-Bond Portfolio Is More Expensive

If $A>B$, short portfolio $A$ and buy portfolio $B$. In component terms, short the call, short the bond, buy the put, and buy the stock. Shorting the bond means receiving its price today and owing its face amount $K$ at expiration. The trades generate $A-B$ at initiation. At expiration, the long put-plus-stock payoff is exactly offset by the short call-plus-bond payoff.

Worked Example: Call Plus Bond Is Richer

A hypothetical non-dividend-paying stock is priced at $S_0=\$100$. A one-year European call with strike $K=\$105$ is quoted at $C=\$12$, and the matching put is quoted at $P=\$10$. The effective annual risk-free rate is $i=5\%$. Determine the trade and verify its expiration result. Method: compare the complete portfolio values, then buy the cheaper side and short the more expensive side. The calculation avoids treating the call premium alone as the parity comparison.

Step 1: Discount the strike. Here $T=1$, so $v=1/1.05$ and

Kv^T=\frac{105}{1.05}=\$100.

Step 2: Compare market values.

A=C+Kv^T=12+100=\$112,\qquad B=P+S_0=10+100=\$110.

Since $A-B=112-110=\$2$, portfolio $A$ is more expensive by $\$2$. Buy $B$: spend $\$10$ on the put and $\$100$ on the stock. Short $A$: receive $\$12$ from shorting the call and $\$100$ from shorting the bond. Net initiation cash is $12+100-10-100=\$2$.

Step 3: Check expiration. The long side is put plus stock; the short side is call plus bond. If $S_T=\$80$, the long side pays $(105-80)+80=\$105$, while the call and bond together pay $0+105=\$105$. If $S_T=\$105$, each side pays $\$105$. If $S_T=\$130$, the long side pays $\$130$, while the call and bond pay $(130-105)+105=\$130$. The net expiration payoff is zero in each case. The $\$2$ initiation surplus is therefore not offset by a later net payment from these positions.

When the Put-Plus-Stock Portfolio Is More Expensive

If $B>A$, reverse every position: buy portfolio $A$ and short portfolio $B$. This means buy the call and the bond, then short the put and the stock. The short put and short stock together have the opposite payoff of the long put-plus-stock portfolio. The initial surplus is $B-A$, and the expiration payoffs again cancel in every stock-price state.

Worked Example: Put Plus Stock Is Richer

A hypothetical stock with no dividends before expiration has $S_0=\$90$. A two-year European call and put with strike $K=\$100$ are quoted at $C=\$8$ and $P=\$3$, respectively. The effective annual risk-free rate is $i=25\%$. Find the arbitrage direction and check three expiration prices. Method: discount the bond payment over the full two years, compare portfolio values, and take the long position in the cheaper portfolio. This avoids using the undiscounted strike as the bond's current value.

Step 1: Find the bond value. With $T=2$, the two-year accumulation factor is $(1.25)^2=1.5625$, so

Kv^T=\frac{100}{(1.25)^2}=\frac{100}{1.5625}=\$64.

Step 2: Compare the portfolios.

A=C+Kv^T=8+64=\$72,\qquad B=P+S_0=3+90=\$93.

Portfolio $B$ is more expensive by $93-72=\$21$. Buy $A$: pay $\$8$ for the call and $\$64$ for the bond. Short $B$: receive $\$3$ for the put and $\$90$ for the stock. The net initiation receipt is $3+90-8-64=\$21$.

Step 3: Verify expiration. If $S_T=\$60$, portfolio $A$ pays $0+100=\$100$, and portfolio $B$ pays $40+60=\$100$. If $S_T=\$100$, both pay $\$100$. If $S_T=\$130$, portfolio $A$ pays $30+100=\$130$, and portfolio $B$ pays $0+130=\$130$. The long $A$ and short $B$ position therefore has zero net expiration payoff at all three prices, as required.

Applying the Rule with a Different Term

The comparison does not require a one-year option or a round-number bond value. Discount the strike for the actual term $T$, using the stated effective annual rate. In particular, for a fractional or multi-year term, use $Kv^T=K/(1+i)^T$, not $K/(1+i)$ unless $T=1$.

Worked Example: A Two-Year Parity Check

A hypothetical non-dividend-paying stock has $S_0=\$130$. Matching two-year European options have strike $K=\$146.41$, call price $C=\$25$, and put price $P=\$18$. The effective annual rate is $i=21\%$. Determine the positions and check the expiration payoff when $S_T=\$120$. Method: calculate the two-year present value of the strike before comparing values. This avoids applying a one-year discount factor to a two-year bond.

Step 1: Discount the strike over two years. Since $(1.21)^2=1.4641$,

Kv^T=\frac{146.41}{(1.21)^2}=\frac{146.41}{1.4641}=\$100.

Step 2: Calculate and compare.

A=25+100=\$125,\qquad B=18+130=\$148.

The put-plus-stock portfolio is more expensive by $148-125=\$23$. Buy $A$ by buying the call for $\$25$ and the bond for $\$100$. Short $B$ by shorting the put for a $\$18$ receipt and shorting the stock for a $\$130$ receipt. The net initiation receipt is $18+130-25-100=\$23$.

Step 3: Check the specified expiration price. At $S_T=\$120$, the long call pays $0$ and the bond pays $\$146.41$, so $A$ pays $\$146.41$. The put and stock pay $(146.41-120)+120=26.41+120=\$146.41$, so $B$ pays the same amount. Buying $A$ and shorting $B$ gives a net expiration payoff of zero at this price, consistent with the payoff identity.

Common Mistake: Reversing the Trade

A positive difference $A-B$ means that $A$ is expensive relative to $B$; it does not mean that $A$ should be bought. For example, in the first worked example, $A-B=112-110=\$2$. The incorrect trade would be to buy $A$ for $\$112$ and short $B$ for $\$110$, producing an initial cash flow of

110-112=-\$2.

That is a $\$2$ outflow, not an arbitrage receipt. Its expiration payoffs still cancel, so it leaves a certain loss at initiation. The correction is to short the expensive $A$ and buy the cheaper $B$, giving $112-110=\$2$ at initiation. When $B>A$, reverse the positions: buy $A$, short $B$, and receive $B-A$.

Key takeaway: Compute $A=C+Kv^T$ and $B=P+S_0$. If $A>B$, short the call and bond and buy the put and stock. If $B>A$, buy the call and bond and short the put and stock. The paired expiration payoffs are equal, so the correct direction leaves a positive initiation surplus without a net expiration obligation from the combined positions.

Check Your Understanding

For each problem, assume European options on a non-dividend-paying stock, frictionless trading, and the stated effective annual risk-free rate. Work out the discounted strike, compare $A$ and $B$, and identify the trade.

A one-year call and put have $K=\$80$, $C=\$6$, and $P=\$11$. The stock is $S_0=\$72$, and $i=5\%$. Which portfolio is more expensive, and what is the initiation surplus? (Answer: $A=6+80/1.05=\$82.19$, $B=\$83.00$; buy $A$, short $B$, and receive approximately $\$0.81$.)
A one-year call and put have $K=\$100$, $C=\$14$, and $P=\$5$. The stock is $S_0=\$105$, and $i=4\%$. Give the trade direction and initiation surplus. (Answer: $A=14+100/1.04\approx\$110.15$, $B=5+105=\$110.00$; short $A$, buy $B$, and receive approximately $\$0.15$.)
A two-year call and put have $K=\$121$, $C=\$7$, and $P=\$4$. The stock is $S_0=\$98$, and $i=10\%$. Identify the expensive portfolio and the component positions to take. (Answer: $Kv^T=121/(1.10)^2=\$100$, so $A=\$107$ and $B=\$102$; short the call and bond, buy the put and stock, and receive $\$5$.)
A two-year call and put have $K=\$100$, $C=\$8$, and $P=\$3$. The stock is $S_0=\$90$, and $i=25\%$. If $S_T=\$130$, what are the expiration payoffs of $A$ and $B$, and which side should be bought? (Answer: $A=72$ and $B=93$ today, so buy $A$, short $B$, and receive $\$21$; at $S_T=\$130$, both portfolios pay $\$130$.)
A two-year call and put have $K=\$146.41$, $C=\$25$, and $P=\$18$. The stock is $S_0=\$130$, and $i=21\%$. State the initiation cash surplus and verify the net expiration payoff if $S_T=\$120$. (Answer: $Kv^T=\$100$, $A=\$125$, $B=\$148$; buy $A$, short $B$, and receive $\$23$. At expiration, each portfolio pays $\$146.41$, so the net payoff is $\$0$.)

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