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Black-Scholes (European Put Option Price)

Calculates the theoretical price of a European put option using the Black-Scholes model.

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P = X e^{- r T} N (- d_{2}) - S_{0} N (- d_{1}) where d_{1} = \frac{ln ( S _{0} / X ) + ( r + σ ^{2} /2 ) T}{σ T} and d_{2} = d_{1} - σ T

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Core idea

Overview

The Black-Scholes model is a fundamental tool in financial mathematics for pricing European-style options. For a put option, it estimates its fair value by considering five key inputs: the current stock price, the option's strike price, the time until expiration, the risk-free interest rate, and the volatility of the underlying asset. This model assumes a log-normal distribution for asset prices and continuous trading, providing a theoretical benchmark for option valuation.

When to use: Use this formula when you need to determine the theoretical fair price of a European put option, given the current market conditions and the option's characteristics. It is applicable for options that can only be exercised at expiration and when the underlying asset does not pay dividends.

Why it matters: The Black-Scholes model revolutionized financial markets by providing a consistent framework for option pricing, enabling more efficient trading and risk management. It is crucial for traders, portfolio managers, and risk analysts to value options, hedge portfolios, and understand market implied volatility.

Symbols

Variables

S_0 = Current Stock Price, X = Strike Price, T = Time to Expiration, r = Risk-Free Interest Rate, \sigma = Volatility

S_{0}

Current Stock Price

X

Strike Price

T

Time to Expiration

y e a r s

r

Risk-Free Interest Rate

d ec ima l

σ

Volatility

d ec ima l

N (d)

Cumulative Standard Normal Distribution Function

u ni tl ess

P

Put Option Price

Walkthrough

Derivation

Formula: Black-Scholes (European Put Option Price)

The Black-Scholes formula for a European put option is derived using principles of risk-neutral pricing and stochastic calculus.

The stock price follows a geometric Brownian motion with constant drift and volatility.
The risk-free interest rate is constant and known.
There are no dividends paid during the option's life.
The option is European-style (exercisable only at expiration).
No transaction costs or taxes.
Short selling is permitted with full use of proceeds.

Start with the Stochastic Differential Equation:

The stock price $S_t$ is assumed to follow a geometric Brownian motion, where $\mu$ is the drift, $\sigma$ is the volatility, and $dW_t$ is a Wiener process.

d S_{t} = μ S_{t} d t + σ S_{t} d W_{t}

Apply Ito's Lemma:

Ito's Lemma is used to find the stochastic process for the option price $V(S,t)$, which is a function of the stock price and time.

d V = (\frac{\partial V}{\partial t} + r S \frac{\partial V}{\partial S} + \frac{1}{2} σ^{2} S^{2} \frac{\partial ^{2} V}{\partial S ^{2}}) d t + σ S \frac{\partial V}{\partial S} d W_{t}

Construct a Risk-Free Portfolio:

A portfolio $\Pi$ is constructed by holding one option and shorting $\Delta = \frac{\partial V}{\partial S}$ shares of the underlying stock. This portfolio is instantaneously risk-free.

Π = V - Δ S

Derive the Black-Scholes PDE:

Since the portfolio is risk-free, its return must equal the risk-free rate. This leads to the Black-Scholes Partial Differential Equation (PDE).

\frac{\partial V}{\partial t} + r S \frac{\partial V}{\partial S} + \frac{1}{2} σ^{2} S^{2} \frac{\partial ^{2} V}{\partial S ^{2}} - r V = 0

Solve the PDE with Boundary Conditions:

The PDE is solved subject to the terminal condition for a European put option, which is its payoff at expiration $T$. This involves complex mathematical techniques, often using a change of variables to transform it into a heat equation.

P (S, T) = max (X - S_{T}, 0)

Final Put Option Formula:

The solution to the Black-Scholes PDE with the put option boundary condition yields the formula, where $N(d)$ is the cumulative standard normal distribution function, and $d_1$ and $d_2$ are specific terms involving the input variables.

P = X e^{- r T} N (- d_{2}) - S_{0} N (- d_{1})

Result

P = X e^{- r T} N (- d_{2}) - S_{0} N (- d_{1})

Source: Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637-654.

Visual intuition

Graph

Graph unavailable for this formula.

The graph of the Black-Scholes European put option price as a function of the underlying asset price S_0 is a convex, monotonically decreasing curve that approaches zero as S_0 increases. It exhibits an asymptotic decay toward the horizontal axis for high asset prices, while approaching the discounted strike price (K * e^(-rT)) as S_0 nears zero. This shape reflects the financial reality that the value of a put option diminishes as the underlying stock price rises, eventually becoming worthless when the stock is deep out-of-the-money.

Graph type: exponential

Why it behaves this way

Intuition

The Black-Scholes model for a put option can be visualized as a balance between the discounted strike price and the discounted expected stock price, where each is weighted by the probability of the option finishing

P

Theoretical fair price of a European put option

This is the value an investor would pay today for the right to sell the underlying asset at a fixed price in the future.

X

Strike price (or exercise price) of the option

The predetermined price at which the option holder can sell the underlying asset upon expiration. A higher strike price generally increases a put option's value.

S_{0}

Current market price of the underlying asset

The starting price of the asset. A lower current price generally increases a put option's value.

r

Risk-free interest rate (continuously compounded)

Represents the return on an investment with no risk, used to discount future cash flows. A higher 'r' generally decreases a put option's value because the present value of the strike price is lower.

T

Time to expiration of the option, in years

The remaining duration until the option can be exercised. A longer time 'T' generally increases a put option's value due to more time for the underlying asset's price to fall.

σ

Volatility of the underlying asset's returns

A measure of the expected fluctuation of the asset's price. Higher volatility 'sigma' generally increases a put option's value because there's a greater chance of extreme price movements (downwards for a put).

N (x)

Cumulative standard normal distribution function

Represents the probability that a standard normal random variable will be less than or equal to 'x'. In Black-Scholes, it converts the 'd' values into probabilities.

e^{- r T}

Continuous compounding discount factor

Used to bring the future strike price (X) back to its present value, accounting for the time value of money.

Signs and relationships

-rT: The negative sign in the exponent `e^{-rT}` indicates discounting. It reduces the future value of the strike price (X) to its present value, reflecting the time value of money.
- S_0 N(-d_1): The overall subtraction of `S_0 N(-d_1)` reflects the put option's payoff structure. A put option's value is the present value of receiving the strike price (first term)
-d_1, -d_2: For a put option, we are interested in the probability that the stock price falls below the strike price. The negative arguments `(-d_1)` and `(-d_2)` in the cumulative normal distribution function `N(x)` correctly yield
+ (r + \sigma^2/2)T: This positive term in the numerator of `d_1` represents the expected drift of the underlying asset's price under the risk-neutral measure. It combines the risk-free rate (`r`)
\sigma\sqrt{T}: This term, appearing in the denominator of `d_1` and `d_2` and in their difference, represents the standard deviation of the logarithm of the asset price over the time period `T`.

Free study cues

Insight

Canonical usage

All monetary values (option price, strike price, stock price) must be in the same currency unit. The risk-free rate, time to expiration, and volatility must use consistent time units, typically annual.

Common confusion

A common mistake is using inconsistent time units for 'r', 'T', and 'sigma' (e.g., 'r' as an annual rate but 'T' in months or days). Another frequent error is using percentages directly for 'r' and 'sigma' instead of

Dimension note

The terms d_1 and d_2 are dimensionless. The cumulative standard normal distribution function N() takes a dimensionless input and returns a dimensionless probability. The exponential term e^(-rT) is also dimensionless.

Unit systems

$P$ Currency (e.g., USD) · The calculated theoretical price of the European put option.

$X$ Currency (e.g., USD) · The strike price of the option.

$S_{0}$ Currency (e.g., USD) · The current price of the underlying asset.

$r$ year^-1 · The annualized risk-free interest rate, expressed as a decimal (e.g., 0.05 for 5%). Must be consistent with the time unit of T.

$T$ year · The time to expiration of the option, expressed in years or a fraction of a year (e.g., 0.5 for six months). Must be consistent with the time unit of r and sigma.

$s i g ma$ year^-1/2 · The annualized volatility of the underlying asset's returns, expressed as a decimal (e.g., 0.20 for 20%). Must be consistent with the time unit of T.

Ballpark figures

Quantity:
Quantity:
Quantity:

One free problem

Practice Problem

A European put option has a strike price of $100 an d e x p i r es in 0.5 y e a r s . T h ec u r r e n t s t oc k p r i ce i s$ 105, the risk-free interest rate is 3% (annualized), and the stock's volatility is 25% (annualized). Calculate the theoretical price of this put option.

Current Stock Price105 $

Strike Price100 $

Time to Expiration0.5 years

Risk-Free Interest Rate0.03 decimal

Volatility0.25 decimal

Solve for: P

Hint: Remember to calculate d1 and d2 first, then find N(-d1) and N(-d2) using a standard normal distribution table or calculator.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

A hedge fund manager uses the Black-Scholes model to price a European put option on a stock to decide whether to buy or sell it, or to hedge an existing position.

Study smarter

Tips

Ensure all inputs (T, r, sigma) are annualized.
Volatility (sigma) is often the most challenging input to estimate accurately.
The model assumes no dividends, constant risk-free rate and volatility, and efficient markets.
N(d) represents the cumulative standard normal distribution function, which requires a statistical table or calculator.

Avoid these traps

Common Mistakes

Using incorrect units for time (e.g., days instead of years).
Misinterpreting volatility (annualized standard deviation of returns).
Confusing put option formula with call option formula.
Incorrectly calculating N(d) values.

Common questions

Frequently Asked Questions

The Black-Scholes formula for a European put option is derived using principles of risk-neutral pricing and stochastic calculus.

Use this formula when you need to determine the theoretical fair price of a European put option, given the current market conditions and the option's characteristics. It is applicable for options that can only be exercised at expiration and when the underlying asset does not pay dividends.

The Black-Scholes model revolutionized financial markets by providing a consistent framework for option pricing, enabling more efficient trading and risk management. It is crucial for traders, portfolio managers, and risk analysts to value options, hedge portfolios, and understand market implied volatility.

Using incorrect units for time (e.g., days instead of years). Misinterpreting volatility (annualized standard deviation of returns). Confusing put option formula with call option formula. Incorrectly calculating N(d) values.