Calculate Value of Put From Call in Bsm

When analyzing options strategies, it's often useful to calculate the value of a put option based on the value of a call option, especially when using the Black-Scholes-Merton (BSM) model. This guide explains how to perform this calculation, the underlying formulas, and practical applications.

Introduction

The Black-Scholes-Merton model is a mathematical framework used to determine the theoretical value of European-style options. While the model provides formulas for both call and put options, sometimes you may need to derive the put value from a call value, particularly when analyzing options strategies or comparing different instruments.

This guide will walk you through the process of calculating a put option's value from a call option's value using the BSM framework. We'll cover the key formula, assumptions, and provide a practical example to illustrate the calculation.

Black-Scholes-Merton Formula

The BSM model provides two main formulas: one for call options and one for put options. The general form of the BSM formula for a call option is:

C = S₀N(d₁) - Xe^(-rT)N(d₂)

Where:

C = Price of the call option
S₀ = Current stock price
X = Strike price
r = Risk-free interest rate
T = Time to expiration (in years)
N(d) = Cumulative distribution function of the standard normal distribution
d₁ = [ln(S₀/X) + (r + σ²/2)T] / (σ√T)
d₂ = d₁ - σ√T
σ = Volatility of the underlying asset

The corresponding formula for a put option is:

P = Xe^(-rT)N(-d₂) - S₀N(-d₁)

Notice that the put formula is derived from the call formula using the put-call parity relationship. This relationship shows that the value of a put option can be calculated from the value of a call option and the underlying asset's price.

Calculating Put from Call

To calculate the value of a put option from a call option using the BSM model, you can use the put-call parity relationship. The relationship is expressed as:

C + Xe^(-rT) = P + S₀

Rearranging this equation to solve for P (put option price) gives:

P = C + Xe^(-rT) - S₀

This formula shows that the put option's value is equal to the call option's value plus the present value of the strike price minus the current stock price. This relationship is fundamental in options pricing and allows traders to derive the value of one option type from another.

Note: The put-call parity relationship assumes that the options are European-style (exercise only at expiration) and that there are no dividends or transaction costs. These assumptions may not hold in all real-world scenarios.

Worked Example

Let's walk through a practical example to illustrate how to calculate the value of a put option from a call option using the BSM model.

Example Scenario

Suppose we have the following parameters:

Current stock price (S₀) = $50
Strike price (X) = $55
Risk-free interest rate (r) = 5% or 0.05
Time to expiration (T) = 0.5 years
Price of the call option (C) = $4.20

Step 1: Apply the Put-Call Parity Formula

Using the formula P = C + Xe^(-rT) - S₀, we can calculate the put option's value:

P = $4.20 + ($55 × e^(-0.05 × 0.5)) - $50

P = $4.20 + ($55 × 0.9753) - $50

P = $4.20 + $53.14 - $50

P = $7.34

Result

Based on the given parameters, the calculated value of the put option is $7.34. This means that, according to the put-call parity relationship, the put option should be priced at $7.34 given the current call option price and other market conditions.

In practice, market prices may differ slightly due to factors like transaction costs, bid-ask spreads, and market imperfections. The BSM model provides a theoretical value, while actual market prices may reflect additional real-world considerations.

Frequently Asked Questions

What is the difference between a call and 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 relationship shows how the values of these options are related.
Can I use the put-call parity formula for American options?: The put-call parity formula is specifically derived for European options, which can only be exercised at expiration. For American options, which can be exercised early, the relationship is more complex and may not hold.
What factors can cause the put-call parity relationship to break down?: Several factors can cause deviations from the put-call parity relationship, including transaction costs, dividends, early exercise premium, and market imperfections. These factors are not accounted for in the basic BSM model.
How does volatility affect the put-call parity relationship?: Volatility is a key input to the BSM model, and it affects both the call and put option prices. Higher volatility generally increases the value of both options, but the relationship between them remains consistent as long as other parameters are held constant.
Can I use the put-call parity formula for options on futures contracts?: Yes, the put-call parity relationship can be applied to options on futures contracts, as long as the assumptions of the model are met. The formula remains the same, but the underlying asset is a futures contract rather than a stock.