Black-Scholes-Merton (BSM) Model

19Nov 0

Black-Scholes-Merton (BSM) Model

Last Updated on 2024-12-13 by Admin

The Black-Scholes-Merton Model is one of the most famous and widely used models for pricing European-style Options. It was developed by economists Fischer Black and Myron Scholes in 1973, with contributions from Robert Merton. It revolutionized the field of financial markets by providing a way to calculate the theoretical price of options. The model is based on the assumption that financial markets behave in a specific way and that asset prices follow a stochastic (random) process.

Overview

The Black-Scholes model provides a theoretical framework for pricing options based on several key variables. The model assumes that the underlying asset price follows a geometric Brownian motion, which incorporates both a drift (average return) and a random component (volatility). The most widely known formula from this model is used to calculate the price of a European call option (the right to buy an asset at a predetermined price) and the price of a European put option (the right to sell an asset at a predetermined price).

Assumptions

European-style options: These options can only be exercised at expiration, not before.
No dividends: The model assumes that the underlying asset does not pay dividends during the life of the option.
Efficient markets: The market for the underlying asset is efficient, meaning that all information is immediately reflected in the asset’s price.
No transaction costs: There are no costs for buying or selling the asset or for trading the options.
Constant volatility: The volatility of the underlying asset is constant over the life of the option.
Constant risk-free interest rate: The risk-free rate, often represented by the rate on government bonds, remains constant over the life of the option.
Log-normal distribution: The price of the underlying asset follows a log-normal distribution, meaning the asset prices change according to a random walk but can’t fall below zero (they are strictly positive).

Formulas

1. Call Option Price (C)

$C = S_0 N(d_1) – X e^{-rT} N(d_2)$

$$C=S_0Φ(d_1)−Xe^{−rT}Φ(d_2)$$

Where:

C = Price of the call option
S₀ = Current price of the underlying asset
Φ(d) = Cumulative distribution function (CDF) of the standard normal distribution, which gives the probability that a random variable with a normal distribution is less than or equal to 𝑑
K = Strike price of the underlying asset
r = Risk-free rate
T = Time to expiration or maturity (years)
σ = Volatility of the underlying asset (annualized)

2. Put Option Price (P)

$$𝑃=𝐾𝑒^{-rT}Φ(−𝑑_2)−𝑆_0Φ(−𝑑_1)$$

Where:

P = Price of the put option
K = Strike price of the underlying asset
r = Risk-free rate
T = Time to expiration or maturity (years)
S₀ = Current price of the underlying asset
Φ(d) = Cumulative distribution function (CDF) of the standard normal distribution, which gives the probability that a random variable with a normal distribution is less than or equal to 𝑑
σ = Volatility of the underlying asset (annualized)
ln = Natural logarithm

The terms 𝑑₁ and 𝑑₂ are intermediate values that are calculated as follows:

$$d_1=\frac{ln(\frac{S_0}{X})+(r+\frac{σ^2}{2})T}{σ\sqrt{T}}$$

$$𝑑_2=𝑑_1−𝜎\sqrt{T}$$

Where:

C = Price of the call option
P = Price of the put option
S₀ = Current price of the underlying asset
K = Strike price of the underlying asset
r = Risk-free rate
T = Time to expiration or maturity (years)
σ = Volatility of the underlying asset (annualized)
Φ(d) = Cumulative distribution function (CDF) of the standard normal distribution, which gives the probability that a random variable with a normal distribution is less than or equal to 𝑑

Interpretation

S₀: This is the current price of the underlying asset (e.g., stock). It is a critical input in determining the option’s price. The higher the price of the underlying asset relative to the strike price, the higher the call option price will be.
X: The strike price of the option. For a call option, the lower the strike price relative to the asset’s price, the more valuable the option is. For a put option, the higher the strike price relative to the asset’s price, the more valuable the option is.
r: The risk-free rate is typically based on government bond yields and reflects the time value of money.
T: The time to expiration is the time remaining until the option expires. The longer the time to expiration, the more valuable the option, as there is more time for the underlying asset to move in the desired direction.
𝜎: Volatility is the standard deviation of the asset’s price returns. It is a measure of how much the asset’s price fluctuates. A higher volatility increases the value of both call and put options because it increases the likelihood that the option will end up in the money.
Φ(𝑑₁) and Φ(𝑑₂): These are the cumulative standard normal distribution functions for 𝑑1 and 𝑑2, respectively. They represent the probabilities of certain outcomes, helping to model the likelihood of the option being exercised profitably.

The Intuition Behind the Formula

The Black-Scholes Model is based on the principle of arbitrage-free pricing. In an efficient market, there must be no opportunity for riskless profit. The model assumes that the underlying asset follows a log-normal distribution, meaning that the price of the asset over time evolves in a random manner, with a certain expected drift (average return) and volatility.
Delta-Hedging: One of the key insights of the Black-Scholes model is that the option price can be replicated by holding a portfolio of the underlying asset and a risk-free bond. This portfolio must be continuously rebalanced to remain “delta-neutral,” which means that changes in the price of the underlying asset do not affect the portfolio’s value. The delta of an option, which is the rate of change of the option price with respect to the price of the underlying asset, is a critical component of this rebalancing strategy.

The Black-Scholes model is derived using stochastic calculus and assumptions about stock price behavior. The key assumptions of the model are:

Lognormal Distribution of Prices: The model assumes that stock prices follow a lognormal distribution, meaning their logarithms are normally distributed. This means stock prices cannot become negative and typically grow exponentially over time.
No Arbitrage: The model assumes that markets are efficient and free of arbitrage (i.e., there are no opportunities to make riskless profit).
Constant Volatility: Volatility is assumed to remain constant over the life of the option, although in reality, it may change over time (this is often accounted for with models like the Implied Volatility Surface).
European Options: The model is designed for European options, which can only be exercised at expiration (as opposed to American options, which can be exercised anytime before expiration).
No Dividends: The basic Black-Scholes model assumes that the underlying asset does not pay dividends. However, there are variations of the model that account for dividends.
Continuous Trading: The model assumes continuous trading of the underlying asset and the ability to continuously adjust portfolios, including borrowing and lending at the risk-free rate.

Limitations

While the Black-Scholes Model is widely used and important, it has several limitations:

Constant volatility assumption: The model assumes that volatility is constant over the life of the option, which is not always true in real markets. In practice, volatility can change over time.
No dividends: The model assumes that the underlying asset does not pay dividends, but many stocks do pay dividends, and this can affect the option price.
European options only: The model applies only to European-style options, which can only be exercised at expiration. It does not account for American-style options, which can be exercised at any time before expiration.
Market inefficiencies: The model assumes that markets are efficient, meaning that all information is instantly reflected in the asset’s price, but in reality, markets may be subject to inefficiencies, such as delays in information dissemination or irrational behavior by investors.

Explanation

1. Current Stock Price (S₀)

The current stock price 𝑆₀, is the price of the underlying asset today. This is a critical factor in determining the value of the option, as the option’s price is directly related to the current price of the asset. If the stock price is higher than the strike price, the call option becomes more valuable (in-the-money).

2. Strike Price (𝑋)

The strike price 𝑋, is the price at which the option holder can buy the underlying asset. It is the predetermined price set in the option contract. The relationship between the stock price and strike price determines whether the option is “in the money” (profitable) or “out of the money” (not profitable).

3. Risk-Free Interest Rate (𝑟)

The risk-free interest rate 𝑟 is typically based on the yield of government bonds, often considered a “safe” investment with minimal risk. It is used to calculate the time value of money — essentially, the present value of future cash flows.
The term 𝑒^−𝑟𝑇 in the formula represents the discounting factor, which adjusts the strike price for the time value of money over the life of the option.

4. Time to Maturity (𝑇)

The time to maturity 𝑇 is the amount of time left before the option expires. It is crucial because the longer the time to expiration, the more time the option has to become profitable (i.e., the stock price may move in the favorable direction).
Time is expressed in years, so if an option has 6 months until expiration, 𝑇=0.5.

5. Volatility (𝜎)

Volatility 𝜎 represents the annualized standard deviation of the asset’s returns. It is a measure of how much the price of the underlying asset fluctuates over time. Higher volatility increases the likelihood of the asset’s price moving favorably for the option holder (e.g., moving above the strike price for a call option).
In the Black-Scholes model, volatility is assumed to be constant over the life of the option.

6. Cumulative Distribution Function Φ(𝑑)

Φ(𝑑₁) and Φ(𝑑₂) represent the cumulative probabilities under a standard normal distribution. These values give us the likelihood of the option finishing in-the-money, accounting for the randomness of the stock’s price movements.
Φ(𝑑₁) gives the probability that the option will be exercised, and Φ(𝑑₂) helps adjust the strike price for the time value of money. The standard normal CDF Φ(𝑑) gives the probability that a standard normally distributed random variable is less than or equal to 𝑑. This is a crucial concept in the Black-Scholes model because financial markets are assumed to follow a log-normal distribution (i.e., the logarithm of the asset price follows a normal distribution).

7. 𝑑₁ and 𝑑₂

d₁ and 𝑑₂ are intermediate variables that incorporate the relationship between the current stock price, strike price, time to maturity, interest rate, and volatility.
𝑑₁ represents the normalized difference between the current price and the strike price, adjusted for time and volatility. It can be interpreted as a measure of how far the stock price is expected to move, adjusted for the time value and volatility.
𝑑₂ is simply 𝑑₁ minus the volatility term 𝜎√𝑇, adjusting for the time remaining to expiration. 𝑑₂ helps estimate the probability that the option will be exercised at expiration.

Conclusion

The Black-Scholes Model has become a cornerstone of modern financial theory and practice, providing a way to price European options based on certain key factors, such as the current price of the asset, the strike price, time to expiration, volatility, and the risk-free interest rate. While the model has its limitations, it is still widely used for pricing and hedging options in financial markets today, and it laid the foundation for much of the options trading strategies employed by institutions and individuals alike. The Black-Scholes model is widely used for pricing options because it provides a closed-form solution, making it easy to calculate the theoretical price of options in real-time. However, due to its assumptions (such as constant volatility and no dividends), the model may not always capture market realities perfectly, especially during periods of high volatility or when stocks pay dividends.

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