# Put-Call Parity

> Put call parity links the prices of calls, puts, the underlying and interest rates. Learn the formula, a worked example, arbitrage logic and synthetic positions.

Source: https://learn.tradelabsai.com/options/put-call-parity/  
Track: Options · Level: Intermediate · Updated: 2026-10-03  
Publisher: TradeLabs AI (https://tradelabsai.com). Education, not financial advice.  
Cite as: TradeLabs Learn, "Put-Call Parity", https://learn.tradelabsai.com/options/put-call-parity/

Put call parity is a fundamental relationship in options pricing. It says that, for European options with the same strike and expiration, a call and a put must be priced consistently with the underlying asset and interest rates. If they are not, traders can build offsetting positions to lock in a profit, and that buying and selling pushes prices back into line. Hans Stoll described the relationship formally in 1969. It explains why calls and puts move together, how synthetic positions work and how to spot mispricing.

## The formula

For a non dividend paying stock:

```
C - P = S - K / (1 + r)^T
```

- **C:** call price
- **P:** put price
- **S:** current stock price
- **K:** strike price
- **r:** risk free interest rate
- **T:** time to expiration in years

Rearranged: **C + K / (1 + r)^T = P + S**. A call plus cash equal to the present value of the strike is worth the same as a put plus the stock.

With continuous compounding the discount factor becomes e^(minus rT), and with known dividends you subtract their present value from S.

## Why it must hold

Consider two portfolios:

- **Portfolio A:** one call plus cash that will grow to K at expiration.
- **Portfolio B:** one put plus one share.

| Stock at expiry | Portfolio A | Portfolio B |
|---|---|---|
| Above K | Exercise call with cash: own the share, worth S | Put expires; own the share, worth S |
| Below K | Call expires; hold cash K | Exercise put: sell share for K |

Both portfolios are worth the same in every outcome, max(S, K). So they must cost the same today. If they did not, you could buy the cheap one, sell the expensive one and pocket the difference with no risk.

**Example: Checking parity**
A stock trades at $100 and pays no dividend. A one year $100 put costs $5.57, and the interest rate is 5% a year with continuous compounding. The present value of the strike is 100 × e^(minus 0.05) = $95.12, so parity says the call should cost 5.57 + 100 minus 95.12 = $10.45.

Suppose the call instead trades at $10.70. It is then $0.25 rich relative to the put. In theory, a trader could sell the call at $10.70, buy the put and buy the stock (a conversion), locking in about $0.25 per share before costs. In practice, commissions, spreads, borrow costs and dividends often absorb small differences like this, and professional firms close larger gaps within moments.

## Conversions and reversals

| Trade | Positions | Used when |
|---|---|---|
| Conversion | Long stock, long put, short call | Calls are rich relative to puts |
| Reversal (reverse conversion) | Short stock, short put, long call | Puts are rich relative to calls |

These are arbitrage trades used by market makers and professional firms. See [Arbitrage Strategies](https://learn.tradelabsai.com/strategies/arbitrage-strategies/).

## Synthetic positions

Rearranging parity shows how to build one position from others:

| Synthetic | Built from |
|---|---|
| Synthetic long stock | Long call + short put (same strike and expiry) |
| Synthetic short stock | Short call + long put |
| Synthetic long call | Long stock + long put |
| Synthetic long put | Short stock + long call |

See [Synthetic Positions](https://learn.tradelabsai.com/options/synthetic-positions/).

## What parity tells traders

- **Calls and puts share the same implied volatility** at the same strike and expiry, in theory. If they did not, parity would be violated. See [Implied Volatility (IV)](https://learn.tradelabsai.com/volatility/implied-volatility/).
- **Interest rates matter:** higher rates make calls more valuable relative to puts.
- **Dividends matter:** expected dividends lower call values and raise put values.
- **Hard to borrow stocks:** when shorting is costly, puts can trade richer than parity suggests, because the reversal arbitrage requires shorting the stock.

## American options

For American options, early exercise turns parity into a range rather than an exact equation. The relationship still holds approximately and is widely used, but small deviations can persist. See [American vs European Options](https://learn.tradelabsai.com/options/american-vs-european-options/).

## Frequently asked questions

### What is put call parity?

A relationship stating that a call minus a put with the same strike and expiry equals the stock price minus the present value of the strike, for European options.

### Why is put call parity important?

It keeps call and put prices consistent, explains synthetic positions and identifies arbitrage opportunities when prices get out of line.

### Does put call parity work for American options?

Only approximately. Early exercise means it becomes a range of prices rather than an exact equality.

Next, learn to draw any option strategy's profit and loss in [Option Payoff Diagrams](https://learn.tradelabsai.com/options/option-payoff-diagrams/).

## Sources

- Wikipedia, [Put call parity](https://en.wikipedia.org/wiki/Put%E2%80%93call_parity)

## Continue learning

- Next lesson: [Option Payoff Diagrams](https://learn.tradelabsai.com/options/option-payoff-diagrams/)
- Previous lesson: [Early Exercise](https://learn.tradelabsai.com/options/early-exercise/)
- Related: [Early Exercise](https://learn.tradelabsai.com/options/early-exercise/): Early exercise is using an American option before it expires. Learn why it usually loses money and the dividend and interest cases where it makes sense.
- Related: [Synthetic Positions](https://learn.tradelabsai.com/options/synthetic-positions/): Synthetic positions combine options and the underlying to copy another position's payoff. Learn synthetic stock, calls and puts, and why traders use them.
- Related: [Arbitrage Strategies](https://learn.tradelabsai.com/strategies/arbitrage-strategies/): Arbitrage strategies try to profit from price gaps between the same or linked assets. Learn the main types, worked examples and why arbitrage is rarely riskless.
- Related: [Time Value of Money](https://learn.tradelabsai.com/math/time-value-of-money/): A dollar today is worth more than a dollar tomorrow. Learn present and future value, discounting, annuities and NPV, the maths behind bonds, valuations and options.
- Related: [Black-Scholes Model](https://learn.tradelabsai.com/options/black-scholes-model/): The Black Scholes model prices European options from five inputs. Learn the formula, its assumptions, a step by step example and where the model breaks down.
- Related: [Calls and Puts](https://learn.tradelabsai.com/options/calls-and-puts/): A call gives the right to buy and a put gives the right to sell at a set price. Learn how calls and puts work, how they profit and how buyers and sellers differ.