Black-Scholes-Merton (BSM) Model Assumptions and Formula The Black-Scholes-Merton (BSM) model is a widely used option pricing model that relies on several as...
The Black-Scholes-Merton (BSM) model is a widely used option pricing model that relies on several assumptions to calculate the fair value of an option. These assumptions play a crucial role in determining the option's price and are essential for accurate option pricing.
Key Assumptions:
Geometric Brownian motion: The underlying asset's price follows a geometric Brownian motion process, where the standard deviation and mean of the process are constant.
Constant volatility: The volatility of the underlying asset is constant over time.
No trading costs: There are no costs associated with entering or exiting the option, such as brokerage fees or transaction costs.
Perfect market: The underlying asset is perfectly liquid and can be traded at any price.
No dividend payments: The underlying asset does not pay dividends.
Formula:
The BSM model uses the following formula to calculate the fair value of an option:
Fair Value = S * (e^(rt) - e^(rt)) + K * e^(-rt)
where:
S is the current stock price
K is the strike price of the option
r is the risk-free rate
t is the time to expiration of the option
S is the underlying asset price
Assumptions and Limitations:
The BSM model is primarily used for European options, where the underlying asset has a single exchange.
It assumes that the underlying asset follows a normal distribution.
The model is not suitable for options with very short or very long expiration dates.
It requires a relatively high level of market data for accurate pricing.
Implications:
The fair value of an option depends on the underlying asset's price, strike price, time to expiration, and risk-free rate.
An increase in the stock price will cause the fair value of an option to increase.
An increase in the strike price will also cause the fair value of an option to increase.
An increase in the time to expiration will cause the fair value of an option to decrease.
An increase in the risk-free rate will also cause the fair value of an option to decrease