Put-Call Parity

Put-call parity is the benchmark in option trading for judging whether the call and put at the same strike price are rich or cheap against each other. It is also used to build the same position as futures out of a call and a put.

Building the same profit or loss as a forward

If you buy a call option (call) and sell a put option (put) with the same underlying asset, strike price and expiry, the profit or loss at expiry is the same as that of a forward contract to buy the underlying asset at the strike price¹. Put-call parity is the relationship among the prices of the call, the put and the underlying asset that follows from this match. How calls and puts work, and what in the money and out of the money mean, are covered in Option Basics: Calls and Puts.

Take a strike price of 100 and look at the payments at expiry, leaving out the option premium. If the underlying price at expiry is 120, the call buyer exercises and receives 20, and the put buyer does not exercise. If the price is 80, the call buyer does not exercise, and the put buyer exercises, so the put seller pays 20. In both cases the payment is "price at expiry − 100", the same as for the buyer of a forward to buy the underlying asset at 100. The short put takes the loss when the price falls, and the long call takes the profit when the price rises¹.

Figure description

This shows the payoff at expiry of buying one call with a strike price of 100 and selling one put with the same strike price and expiry. Premiums are not included. The horizontal axis is the underlying price at expiry, and the vertical axis is the profit or loss per unit. The profit or loss is a single straight line that rises one for one with the price. It is 0 at a price of 100, a profit above 100 and a loss below 100. Because the position includes a short put, the payoff is negative when the underlying price is below the strike price. At a price of 60 the loss is 40, and at 140 the profit is 40. The line has the same shape as the profit or loss of the buyer of a forward to buy the underlying asset at 100.

This combination is called a synthetic forward². When calls and puts are combined to create the same position as buying or selling futures, the combination is called a synthetic futures position³. Conversely, selling a call and buying a put gives the same profit or loss as selling a forward.

The same relationship also works when you rearrange the combination. For example, buying JGB futures and selling a call at the same time gives the same shape of profit or loss as selling a put⁴.

Equal payments mean equal value today

Two combinations whose payments at expiry are the same whatever the price must also have the same value today⁵. This idea gives the relationship between call and put prices. Here, take an underlying asset that pays no dividends or other income before expiry. The first combination is buying a call and selling a put, both with a strike price of $K$. The second is buying the underlying asset today at its current price $S$ and borrowing money on a promise to repay $K$ at expiry. The amount you can borrow today is the present value of $K$. With $S_T$ as the underlying price at expiry, the payments at expiry are as follows.

The total payment at expiry is $S_T - K$ for both the first combination (long call and short put) and the second (underlying asset and loan repayment). The money needed to set up the first combination today is the call price $C$ minus the put price $P$. For the second, it is the cost of the underlying asset minus the amount borrowed. Setting the two equal gives the following formula⁵.

$$C - P = S - K \times DF$$

$C$ and $P$ are the prices of a call and a put with the same underlying asset, strike price $K$ and expiry¹. Both are European options⁵. $S$ is the current price of the underlying asset, $DF$ is the discount factor from today to expiry, and $K \times DF$ is the present value of the strike price. The formula for an underlying asset that pays dividends appears in a later section.

Deep diveThe formula with a continuously compounded rate and a dividend yield

With a continuously compounded annual interest rate $r$, a continuous annual dividend yield $q$ on the underlying asset and $T$ years to expiry, the discount factor is $e^{-rT}$. The value today of the right to receive the underlying asset at expiry is $S e^{-qT}$, because the holder does not receive the dividends paid along the way. The relationship is then as follows⁵.

$$C - P = S e^{-qT} - K e^{-rT}$$

With $q = 0$, this is the same as the formula in the main text.

Worked example: discount factor 0.98

Consider a hypothetical example in which the current underlying price is 100, expiry is one year away and the one-year discount factor is 0.98. For a call and a put with a strike price of 100, the present value of the strike price is 100 × 0.98 = 98. So the call price is 100 − 98 = 2 higher than the put price. If the put price is 6, the call price is 8.

When the strike price changes, the difference changes as follows.

The lower the strike price, the more expensive the call, and the difference is 0 when the strike price is 100 ÷ 0.98 ≈ 102.04. This 102.04 is the same as the price of a forward that delivers the underlying asset at expiry (for an underlying asset with no income such as interest)⁵. In a synthetic forward with a strike price of 100, you pay 2 today and buy the underlying asset at 100 at expiry. The 2 paid today is worth 2 ÷ 0.98 ≈ 2.04 at expiry, so the purchase price in expiry terms is 102.04, the same as buying through a forward.

If the interest rate is 0 and the discount factor is 1, call price − put price is 100 − strike price, which is 0 for a strike price of 100 and 10 for 90.

The simulator below calculates under this zero-rate condition, so the values it shows differ from the worked example above (discount factor 0.98). At the initial strike price of 100 and a volatility of 20%, the call and put premiums are both 7.97, and the net premium is 0.00. At a strike price of 90, the call is 13.59 and the put is 3.59, so you pay a net 10.00. At any strike price, the profit or loss at expiry including the premiums is the straight line "price at expiry − 100", and the break-even point is 100.

Arbitrage when the relationship breaks down

If the difference between the call and put prices moves away from the formula, you can lock in a profit whatever the price at expiry through arbitrage, selling the expensive side and buying the cheap side. Market participants would quickly take any such trade, so the price relationship returns to the formula⁶.

Under the conditions of the previous section (underlying 100, strike price 100, discount factor 0.98), suppose the call trades at 8 and the put at 5. The difference is 3, which is 1 more than the 2 from the formula, so the call is expensive relative to the put. In that case, you make the following four trades at the same time.

You receive 1 today, and the payment at expiry is 0 whether the price is 80 or 120. Together, the short call, the long put and the long underlying asset receive 100 at expiry at any price, which repays the 100 on the loan. If the difference is smaller than the formula gives (the put is expensive), you make the opposite trades. You buy the call, sell the put, borrow and sell the underlying asset, and lend 98 of the proceeds until expiry, which locks in the difference in the same way. This calculation leaves out trading commissions, the bid-ask spread, and the cost of borrowing the underlying asset to sell it.

Estimating the price of an option with no quoted price

Put-call parity is used to find the price of one option when the price of the other, the call or the put with the same strike price, is known. For example, the price of an illiquid in-the-money put that has no quotes on the order book can be estimated from the price of the out-of-the-money call with the same strike price⁷.

For options on futures, the relationship can be written simply as call price − put price = futures price − strike price. Strictly, an interest adjustment such as taking the present value of the strike price is needed⁷. Consider a hypothetical example in which the futures price is 100 and a call with a strike price of 102 is 0.5. Ignoring interest, the put price can be estimated as 0.5 − 100 + 102 = 2.5.

How the relationship changes with dividends and American options

For an underlying asset that pays dividends before expiry, you replace the underlying price in the formula with the underlying price minus the present value of the dividends to expiry¹⁰.

$$C - P = S - \text{PV of dividends} - K \times DF$$

Under the conditions of the worked example, if the present value of the dividends to expiry is 1, the difference between the call and put prices with a strike price of 100 is 100 − 1 − 98 = 1, which is 1 smaller than with no dividends. For currency options, the foreign interest rate plays the role of the dividends, and you use the current exchange rate discounted at the foreign interest rate over the period to expiry¹⁰.

For American options, which can be exercised before expiry, the relationship becomes an inequality rather than an equality. For an underlying asset that pays no dividends, the difference between the American call price $C_A$ and the American put price $P_A$ falls within the following range¹¹.

$$S - K \le C_A - P_A \le S - K \times DF$$

Under the conditions of the worked example, the difference is between 0 and 2. With no dividends and a positive interest rate, exercising an American call before expiry never pays, so the call price is the same as for a European option. An American put, on the other hand, can be worth exercising early for the buyer, so the equality breaks down¹². With dividends, you replace $S$ in the lower bound with $S$ minus the present value of the dividends, and keep the upper bound as it is¹³. The differences between exercise styles are covered in Exercise Styles: European, American and Bermudan.

Notes

1. Japan Exchange Group, "国債先物オプション入門" (Introduction to JGB Futures Options, in Japanese), by Takahiro Hattori, "４．１ 先物の損益をプットとコールに分解" (Breaking down the profit and loss of futures into puts and calls), pages 16–17 (pages 17–18 of the PDF). The document explains this with charts that exclude the option premium. ↩ ↩² ↩³

2. The University of Texas at Austin, "9. Parity and Other Option Relationships", "“Synthetic” Forwards", page 5. ↩

3. Japan Exchange Group, "合成先物（用語集）" (Glossary: synthetic futures, in Japanese). ↩

4. Japan Exchange Group, "国債先物オプション入門" (Introduction to JGB Futures Options, in Japanese), by Takahiro Hattori, "バイライト（国債先物買い＋コール売り）" (Buy-write: buying JGB futures and selling a call), page 33 (page 34 of the PDF). ↩

5. The Ohio State University, "Derivatives and Their Portfolios, Put-Call Parity", "one price principle", "Put-call Parity" and "Versions of Put-Call", page 1, and the forward price formula under "Review". The document's $F^P_{0,T}(S)$ corresponds to $S$ here for an underlying asset with no dividends, and $PV_{0,T}(K)$ corresponds to $K \times DF$. ↩ ↩² ↩³ ↩⁴ ↩⁵

6. The Wharton School, "Handout 19: Numerical Examples of Put-Call Parity and Minimum Value", page 2. ↩

7. Japan Exchange Group, "国債先物オプション入門" (Introduction to JGB Futures Options, in Japanese), by Takahiro Hattori, "４．２ プット・コール・パリティの使用例" (An example of using put-call parity) and note 19, pages 17–18 (pages 18–19 of the PDF). ↩ ↩²

8. Martin Haugh, "A Brief Review of Derivatives Pricing & Hedging" (Columbia University), note 11, page 10. ↩

9. Martin Haugh, "A Brief Review of Derivatives Pricing & Hedging" (Columbia University), page 11. ↩

10. The Ohio State University, "Derivatives and Their Portfolios, Put-Call Parity", "Versions of Put-Call", pages 1–2. Currency options are the formula under "If S is a foreign currency" in the same section. ↩ ↩²

11. David Cai, "American Options" (New York University, lecture 8), "Put-call parity for American options", pages 4–5. The document's $B(0,T)$ corresponds to $DF$ here. ↩

12. David Cai, "American Options" (New York University, lecture 8), item 4 of 1.1 (page 2), the proof of "Put-call parity for American options" (pages 4–5), and item 1 of "Results" in 1.3 "American Puts" (page 10). ↩

13. David Cai, "American Options" (New York University, lecture 8), pages 5–6. The document writes the lower bound as $P.V.[S(T)] - K$, and $P.V.[S(T)]$ with a single dividend as $S(0) - D(t_1)B(0,t_1)$. ↩