# Write the Mathematical Formula for Put-Call Parity ⎊ Path

**Published:** 2025-12-04
**Author:** Greeks.live
**Categories:** Path

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## Write the Mathematical Formula for Put-Call Parity.

The put-call parity formula for European options is: C + K e^-rT = P + S. Where C is the call price, P is the put price, S is the underlying asset price, K is the strike price, e^-rT is the present value factor (discounting the strike price by the risk-free rate r over time T). This formula shows that a portfolio of a long call and a short put equals a long position in the underlying asset and a short position in a zero-coupon bond.

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## Glossar

### [Underlying Asset](https://learn.greeks.live/area/underlying-asset/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)

Futures Pricing incorporates the cost of carry, which in crypto markets includes funding rates derived from perpetual swap markets and the time value associated with holding the spot asset.

### [Mathematical Formula](https://learn.greeks.live/area/mathematical-formula/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)

Pricing ⎊ Mathematical formulas are fundamental to the valuation of financial derivatives and the operation of decentralized protocols.

### [Put-Call Parity](https://learn.greeks.live/area/put-call-parity/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)

Relationship ⎊ Put-call parity describes a fundamental relationship between the price of a European call option, a European put option, and the underlying asset price, assuming they share the same strike price and expiration date.

### [Parity](https://learn.greeks.live/area/parity/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)

Equivalence ⎊ ⎊ Parity, within financial markets, fundamentally represents a state of equality between two or more assets, rates, or pricing models, often serving as a benchmark for arbitrage opportunities or risk-neutral valuation.

## Discover More

### [What Is the Fundamental Difference between a Call Option and a Put Option in Crypto Trading?](https://learn.greeks.live/path/what-is-the-fundamental-difference-between-a-call-option-and-a-put-option-in-crypto-trading/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![What Is the Fundamental Difference between a Call Option and a Put Option in Crypto Trading?](https://learn.greeks.live/wp-content/uploads/2025/10/Layered-Obscuration-of-Digital-Asset-Collateral-Structures-Simulating-Synthetic-Option-Payoffs.webp)

Call options give the right to buy (bullish), while Put options give the right to sell (bearish), both at a set strike price.

### [How Does the Constant Product Formula (X Y = K) of a Basic AMM Determine Price and Spread?](https://learn.greeks.live/path/how-does-the-constant-product-formula-x-y-k-of-a-basic-amm-determine-price-and-spread/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![How Does the Constant Product Formula (X Y = K) of a Basic AMM Determine Price and Spread?](https://learn.greeks.live/wp-content/uploads/2025/10/Spherical-Financial-Instrument-Decomposition-Revealing-Internal-Gimbaled-Actuator-Mechanism-for-Decentralized-Derivatives-Hedging.webp)

The price is the ratio of tokens (y/x); the spread is determined by the price impact from the trade and the pool's fee.

### [How Does the 'Constant Sum' Formula Differ from the 'Constant Product' Formula in AMMs?](https://learn.greeks.live/path/how-does-the-constant-sum-formula-differ-from-the-constant-product-formula-in-amms/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![How Does the 'Constant Sum' Formula Differ from the 'Constant Product' Formula in AMMs?](https://learn.greeks.live/wp-content/uploads/2025/10/Sleek-Pale-Ceramic-Instrument-Design-Symbolizing-Decentralized-Finance-Portfolio-Volatility-Hedging-Mechanisms.webp)

Constant Product (x y=k) causes exponential slippage; Constant Sum (x+y=k) causes near-zero slippage and is used for 1:1 asset pairs like stablecoins.

### [How Does a Dividend Payment Affect the Put-Call Parity Relationship?](https://learn.greeks.live/path/how-does-a-dividend-payment-affect-the-put-call-parity-relationship/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![How Does a Dividend Payment Affect the Put-Call Parity Relationship?](https://learn.greeks.live/wp-content/uploads/2025/10/Crystalline-Volatility-Surface-Intersected-by-Bearish-Cross-Vectors-Digital-Asset-Hedging.webp)

The present value of expected dividends must be subtracted from the stock price in the formula to maintain the parity relationship.

### [How Does the Sign of Delta Differ between Call and Put Options?](https://learn.greeks.live/path/how-does-the-sign-of-delta-differ-between-call-and-put-options/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![How Does the Sign of Delta Differ between Call and Put Options?](https://learn.greeks.live/wp-content/uploads/2025/10/Abstract-Representation-of-On-Chain-Liquidity-Pool-Mechanisms-Supporting-Decentralized-Exchange-Settlement-Layers.webp)

Call options have positive Delta (0 to +1); put options have negative Delta (-1 to 0).

### [Define "Put-Call Parity" and Its Role in Options Pricing Theory](https://learn.greeks.live/path/define-put-call-parity-and-its-role-in-options-pricing-theory/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![Define "Put-Call Parity" and Its Role in Options Pricing Theory](https://learn.greeks.live/wp-content/uploads/2025/10/Macro-Analysis-of-Tokenized-Asset-Structure-Fluctuation-Indicating-Liquidity-Pool-Instability.webp)

Put-call parity links the prices of a European call, a European put, the underlying asset, and a bond to ensure arbitrage-free pricing.

### [What Is the Formula for Intrinsic Value of a Put Option?](https://learn.greeks.live/path/what-is-the-formula-for-intrinsic-value-of-a-put-option/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![What Is the Formula for Intrinsic Value of a Put Option?](https://learn.greeks.live/wp-content/uploads/2025/10/Crystalline-Volatility-Surface-Projection-Modulating-Decentralized-Finance-Option-Payoff-Structures.webp)

Intrinsic Value = Max(Strike Price - Underlying Price, 0).

### [How Is a 'Synthetic Long Call' Constructed Using the Underlying Asset and a Put Option?](https://learn.greeks.live/path/how-is-a-synthetic-long-call-constructed-using-the-underlying-asset-and-a-put-option/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![How Is a 'Synthetic Long Call' Constructed Using the Underlying Asset and a Put Option?](https://learn.greeks.live/wp-content/uploads/2025/10/Modular-Digital-Asset-Custody-Casing-Illuminating-Inner-Turquoise-Indicator-For-Crypto-Derivates-Settlement.webp)

A Synthetic Long Call is constructed by buying the underlying asset and buying a put option with the same strike/expiry, based on Put-Call Parity.

### [What Is Put-Call Parity and How Is It Used to Identify Arbitrage Opportunities in Options Trading?](https://learn.greeks.live/path/what-is-put-call-parity-and-how-is-it-used-to-identify-arbitrage-opportunities-in-options-trading/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![What Is Put-Call Parity and How Is It Used to Identify Arbitrage Opportunities in Options Trading?](https://learn.greeks.live/wp-content/uploads/2025/10/Abstract-Visualization-of-Cross-Hedging-Structures-in-Non-Linear-Crypto-Derivatives-Pricing-Surfaces.webp)

Put-call parity is an equation for options; if the market doesn't balance the formula, arbitrageurs do.

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