Put Option Pricing Calculator

Use this put option pricing calculator to determine the fair value of a put option contract. Put options give the holder the right, but not the obligation, to sell an underlying asset at a specified price on or before a certain date. This calculator uses the Black-Scholes model to estimate put option prices based on current stock price, strike price, time to expiration, risk-free rate, and volatility.

How to Use This Calculator

To calculate the price of a put option:

Enter the current stock price of the underlying asset
Input the strike price of the put option
Specify the time to expiration in years
Enter the risk-free interest rate (annualized)
Provide the volatility of the underlying asset (annualized)
Click "Calculate" to get the put option price

The calculator will display the estimated put option price along with a chart showing how the price changes with different stock prices.

Put Option Pricing Formula

The put option price is calculated using the Black-Scholes formula:

Black-Scholes Put Option Formula

Put Price = S × N(-d1) - K × e^(-r×T) × N(-d2)

Where:

S = Current stock price
K = Strike price
T = Time to expiration (in years)
r = Risk-free interest rate
σ = Volatility of the underlying asset
N(x) = Cumulative standard normal distribution function
d1 = (ln(S/K) + (r + σ²/2)×T) / (σ×√T)
d2 = d1 - σ×√T

The formula accounts for the time value of money, the risk-free rate, and the volatility of the underlying asset. Higher volatility generally increases the price of put options.

Worked Example

Let's calculate the price of a put option with these parameters:

Current stock price (S): $50
Strike price (K): $55
Time to expiration (T): 0.5 years
Risk-free rate (r): 2% (0.02)
Volatility (σ): 30% (0.30)

Using the Black-Scholes formula, we calculate:

Calculation Steps

1. Calculate d1 = (ln(50/55) + (0.02 + 0.30²/2)×0.5) / (0.30×√0.5) ≈ -0.1036

2. Calculate d2 = d1 - 0.30×√0.5 ≈ -0.2286

3. Find N(-d1) ≈ 0.4636 and N(-d2) ≈ 0.4096

4. Put Price = 50 × 0.4636 - 55 × e^(-0.02×0.5) × 0.4096 ≈ $3.32

The calculated put option price is approximately $3.32. This means the fair value of the put option contract is $3.32 per share.

Interpreting Results

The put option price represents the premium you pay for the right to sell the underlying asset at the strike price. Key factors that affect the price include:

Time to expiration: Put options become more valuable as expiration approaches, especially if the stock price is below the strike price
Volatility: Higher volatility increases the price of put options as they become more likely to finish in-the-money
Strike price: Put options with strike prices below the current stock price are more valuable
Risk-free rate: Higher interest rates make put options more expensive as the time value of money increases

If the calculated put price is higher than the market price, it suggests the option is undervalued. If it's lower, the option may be overvalued.

Frequently Asked Questions

What is a put option?

A put option is a financial contract that gives the holder the right, but not the obligation, to sell an underlying asset at a specified price (the strike price) on or before a certain date (the expiration date).

How does the Black-Scholes model work?

The Black-Scholes model is a mathematical model used to estimate the price of options. It assumes that the underlying asset follows a geometric Brownian motion with constant volatility and that there are no arbitrage opportunities.

What factors affect put option prices?

Put option prices are affected by the current stock price, strike price, time to expiration, risk-free interest rate, and volatility of the underlying asset. Higher volatility generally increases put option prices.

When is a put option in-the-money?

A put option is in-the-money when the current stock price is below the strike price. This means the holder can sell the stock at a higher price than the current market price.

How accurate is this calculator?

This calculator provides an estimate based on the Black-Scholes model. Real-world option prices may differ due to market conditions, transaction costs, and other factors not accounted for in the model.