How to Calculate Put Option Price From Call Option Price

When trading options, understanding the relationship between put and call options is crucial. The put-call parity formula allows you to calculate the price of a put option from a call option, given the underlying asset's price and the risk-free interest rate.

Introduction

Options trading involves two primary types of options: calls and puts. A call option gives the holder the right to buy an asset at a specified price (strike price), while a put option gives the right to sell the asset at that price. The put-call parity relationship shows that the price of a put option can be derived from the price of a call option, the underlying asset's price, and the risk-free interest rate.

This relationship is particularly useful for arbitrage strategies and for understanding the theoretical relationship between call and put options.

Put-Call Parity Formula

The put-call parity formula is expressed as:

Put Option Price = Call Option Price + (Underlying Asset Price × e^{-(r × T)}) - Strike Price × e^{-(r × T)}

Where:

Call Option Price - Current market price of the call option
Underlying Asset Price - Current market price of the underlying asset
Strike Price - Strike price of the option
r - Risk-free interest rate (annualized)
T - Time to expiration (in years)

This formula accounts for the time value of money and the risk-free interest rate, which are key factors in options pricing.

Calculation Steps

Determine the current market price of the call option.
Identify the current market price of the underlying asset.
Note the strike price of the option.
Estimate the risk-free interest rate (you can use the current yield on 10-year Treasury bonds as a proxy).
Calculate the time to expiration in years.
Plug these values into the put-call parity formula to calculate the put option price.

Note: The put-call parity formula assumes no dividends are paid on the underlying asset. If dividends are expected, the formula becomes more complex.

Worked Example

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

Call Option Price: $5.00
Underlying Asset Price: $100.00
Strike Price: $105.00
Risk-free Interest Rate: 2% (0.02)
Time to Expiration: 6 months (0.5 years)

Using the put-call parity formula:

Put Option Price = $5.00 + ($100.00 × e^{-(0.02 × 0.5)}) - $105.00 × e^{-(0.02 × 0.5)}

First, calculate e^{-(0.02 × 0.5)}:

e^-0.01 ≈ 0.99005

Now plug this back into the formula:

Put Option Price = $5.00 + ($100.00 × 0.99005) - $105.00 × 0.99005

Put Option Price = $5.00 + $99.005 - $103.955

Put Option Price ≈ $0.05

The calculated put option price is approximately $0.05. This means the put option is slightly out of the money, which aligns with the given values.

Key Assumptions

The put-call parity formula makes several key assumptions:

No dividends: The formula assumes the underlying asset does not pay dividends. If dividends are expected, the formula must be adjusted.
No transaction costs: The formula does not account for brokerage fees or other transaction costs.
No arbitrage opportunities: The formula assumes there are no arbitrage opportunities in the market.
Continuous compounding: The formula uses continuous compounding for the risk-free interest rate.

These assumptions are important to understand when applying the put-call parity formula in real-world scenarios.

FAQ

What is the difference between a call and a put option?

A call option gives the holder the right to buy an asset at a specified price, while a put option gives the right to sell the asset at that price. The put-call parity formula shows how these two options are related.

Can I use put-call parity to make money?

Put-call parity is primarily used for arbitrage strategies. If you find that the calculated put price differs significantly from the market price, you can potentially profit by buying the cheaper option and selling the more expensive one.

What happens if the underlying asset pays dividends?

If the underlying asset pays dividends, the put-call parity formula must be adjusted to account for the expected dividends. The formula becomes more complex and requires knowledge of the dividend schedule.