Put Option Calculation with Risky Asset and Risk Free Asset

Introduction

Put options are one of the most common financial derivatives traded in markets worldwide. They provide investors with a way to hedge against potential losses in the value of an asset or to speculate on a decline in asset prices. The price of a put option is influenced by several key factors, including the current price of the underlying asset, the strike price, the time until expiration, the risk-free interest rate, and the volatility of the underlying asset.

This guide explains how to calculate put option prices using the Black-Scholes model, which is the standard mathematical model for pricing options. We'll cover the formula, calculation process, and interpretation of results, along with a practical example.

Put Option Formula

The Black-Scholes formula for put option pricing is:

Where:

• S = Current price of the underlying asset
• K = Strike price of the option
• r = Risk-free interest rate
• T = Time to expiration (in years)
• σ = Volatility of the underlying asset

The formula calculates the theoretical price of a put option based on these inputs. The risk-free interest rate and volatility are key factors that reflect the market's expectations of future returns and price fluctuations.

Calculation Process

To calculate the put option price using the Black-Scholes model, follow these steps:

1. Determine the current price of the underlying asset (S).
2. Identify the strike price of the option (K).
3. Estimate the risk-free interest rate (r) for the time period until expiration.
4. Calculate the time to expiration (T) in years.
5. Estimate the volatility (σ) of the underlying asset.
6. Calculate d1 and d2 using the formulas provided.
7. Use the cumulative distribution function of the standard normal distribution (N) to find N(-d1) and N(-d2).
8. Plug these values into the put option formula to get the price.

This process can be complex without the right tools, which is why our calculator provides a simplified way to perform these calculations accurately.

Worked Example

Let's walk through a practical example to illustrate how to calculate a put option price.

Example Scenario

Suppose you want to buy a put option on a stock with the following characteristics:

• Current stock price (S) = $50
• Strike price (K) = $55
• Risk-free interest rate (r) = 5% (0.05)
• Time to expiration (T) = 6 months (0.5 years)
• Volatility (σ) = 30% (0.30)

Using these values, let's calculate the put option price step by step.

Step 1: Calculate d1 and d2

First, we calculate d1 and d2 using the formulas:

Calculating these values gives us:

• d1 ≈ -0.22
• d2 ≈ -0.35

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

Using the cumulative distribution function of the standard normal distribution, we find:

• N(-d1) ≈ N(0.22) ≈ 0.5886
• N(-d2) ≈ N(0.35) ≈ 0.6368

Step 3: Calculate the Put Option Price

Now, plug these values into the put option formula:

The calculated put option price is approximately $0.24. This means the put option is currently worth $0.24, giving the holder the right to sell the stock at $55 in 6 months.

Interpreting Results

The put option price calculated using the Black-Scholes model provides several insights:

• Intrinsic Value: The difference between the strike price and the current price of the underlying asset.
• Time Value: The portion of the option price that is derived from the time remaining until expiration.
• Impact of Volatility: Higher volatility generally increases the option price, reflecting greater uncertainty and potential for price movements.
• Impact of Interest Rates: Higher risk-free interest rates can increase the present value of the strike price, potentially lowering the put option price.

Understanding these components helps investors make informed decisions about whether to buy or sell put options based on their market expectations and risk tolerance.