How to Calculate Put Option Price From Call Option

When trading options, you may need to calculate a put option price from a call option price. The put-call parity relationship provides a mathematical relationship between call and put options that can be used for this calculation. This guide explains the formula, provides a calculator, and includes practical examples.

Introduction

Options trading involves buying and selling contracts that give the holder the right, but not the obligation, to buy (call options) or sell (put options) an underlying asset at a specified price (strike price) before a certain date (expiration date).

The put-call parity relationship is a fundamental concept in options pricing that establishes a mathematical relationship between the prices of call and put options with the same strike price and expiration date. This relationship can be used to calculate the price of a put option when you know the price of a call option.

Put-Call Parity Formula

The put-call parity formula is expressed as:

Put Option Price = Call Option Price + Strike Price × e^(-rT) - Underlying Asset Price

Where:

Call Option Price - The price of the call option
Strike Price - The strike price of the option
r - The risk-free interest rate
T - The time to expiration in years
Underlying Asset Price - The current price of the underlying asset

The formula accounts for the time value of money by discounting the strike price using the risk-free interest rate and the time to expiration. This ensures that the put option price is correctly valued relative to the call option price.

It's important to note that put-call parity assumes no arbitrage opportunities exist in the market. If the calculated put option price does not match the market price, it indicates an arbitrage opportunity.

Worked Example

Let's calculate the put option price using the following values:

Call Option Price: $5.00
Strike Price: $50.00
Risk-Free Interest Rate (r): 2% or 0.02
Time to Expiration (T): 0.5 years
Underlying Asset Price: $55.00

Using the put-call parity formula:

Put Option Price = $5.00 + ($50.00 × e^(-0.02 × 0.5)) - $55.00

First, calculate the discount factor:

e^(-0.02 × 0.5) ≈ 0.99005

Then, multiply by the strike price:

$50.00 × 0.99005 ≈ $49.50

Now, plug the values back into the formula:

$5.00 + $49.50 - $55.00 = $0.50

The calculated put option price is $0.50. This means that according to put-call parity, the put option should be priced at $0.50 to eliminate any arbitrage opportunities.

FAQ

What is put-call parity?

Put-call parity is a relationship between the prices of call and put options with the same strike price and expiration date. It establishes that the price of a call option plus the present value of the strike price should equal the price of the put option plus the price of the underlying asset.

When is put-call parity useful?

Put-call parity is useful for calculating the theoretical price of a put option when you know the price of a call option. It's also used to identify arbitrage opportunities when the market prices of options do not satisfy the put-call parity relationship.

What factors affect the put option price calculated using put-call parity?

The put option price calculated using put-call parity is affected by the call option price, strike price, risk-free interest rate, time to expiration, and the underlying asset price. Changes in any of these factors will affect the calculated put option price.

Can put-call parity be used for all types of options?

Put-call parity is typically used for European-style options, which can only be exercised at expiration. It may not be directly applicable to American-style options, which can be exercised at any time before expiration.

What happens if the calculated put option price does not match the market price?

If the calculated put option price does not match the market price, it indicates an arbitrage opportunity. Traders can exploit this discrepancy by buying the cheaper option and selling the more expensive one to lock in a risk-free profit.