How to Calculate The Price of A Put Option

What is a Put Option?

A put option is a financial contract that gives the buyer the right, but not the obligation, to sell a specific asset at a predetermined price (the strike price) on or before a specified expiration date. Put options are used to hedge against potential losses or to profit from declining stock prices.

The seller of a put option is obligated to buy the asset if the buyer exercises the option. This creates a balance between the two parties, with the buyer paying a premium for the option and the seller receiving the premium in exchange for the obligation.

The Black-Scholes Model

The Black-Scholes model is the most widely used mathematical model for pricing options. It was developed by Fischer Black, Myron Scholes, and Robert Merton in 1973. The model assumes that the underlying asset follows a geometric Brownian motion with constant drift and volatility, and that there are no arbitrage opportunities.

The model uses several key inputs to calculate the price of an option:

• Spot price (S): The current price of the underlying asset
• Strike price (K): The price at which the option can be exercised
• Time to expiration (T): The remaining time until the option expires
• Risk-free interest rate (r): The current risk-free rate of return
• Volatility (σ): The expected volatility of the underlying asset's price

Put Option Formula

The price of a European put option (which can only be exercised at expiration) is calculated using the following formula:

This formula accounts for the time value of money, the potential for the stock price to decline, and the risk of the option expiring worthless.

How to Calculate a Put Option Price

Step-by-Step Calculation

1. Determine the current spot price (S) of the underlying asset
2. Identify the strike price (K) of the put option
3. Calculate the time to expiration (T) in years
4. Find the current risk-free interest rate (r)
5. Estimate the annualized volatility (σ) of the underlying asset
6. Calculate d1 using the formula above
7. Calculate d2 as d1 - σ√T
8. Find the cumulative standard normal distribution for -d1 and -d2
9. Plug these values into the put option formula to get the price

Example Calculation

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

• Spot price (S) = $50
• Strike price (K) = $55
• Time to expiration (T) = 0.5 years
• Risk-free rate (r) = 5% or 0.05
• Volatility (σ) = 30% or 0.30

Step 1: Calculate d1 and d2

Step 2: Calculate N(-d1) and N(-d2)

Using statistical tables or software:

• N(-0.2254) ≈ 0.4106
• N(-0.4375) ≈ 0.3320

Step 3: Calculate the put price

The negative value indicates that the put option is deep out of the money and has little intrinsic value. In practice, you would need to adjust the parameters to get a positive put price.

Factors Affecting Put Option Price

The price of a put option is influenced by several key factors:

• Time to expiration: Put options tend to increase in value as expiration approaches, especially if the stock price is expected to decline
• Volatility: Higher volatility increases the price of put options because there's a greater chance the stock price will decline
• Interest rates: Higher interest rates increase the cost of borrowing and reduce the present value of future cash flows
• Strike price: A higher strike price makes the put option more valuable because it gives the buyer more time to profit from a decline
• Dividends: If the underlying stock pays dividends, the put option price may be affected, especially if the dividends are expected to be paid before expiration