Model Answer
0 min readIntroduction
Options contracts, granting the right but not the obligation to buy or sell an asset at a predetermined price, are fundamental instruments in modern finance. The need for a robust and mathematically sound method to determine the fair price of these options led to the development of the Black-Scholes Option Pricing Model (BSM) in 1973 by Fischer Black, Myron Scholes, and Robert Merton. This model revolutionized financial markets, providing a framework for risk management and derivative pricing. While subsequent models have built upon it, the BSM remains a foundational concept for understanding option valuation and its applications extend beyond simple options trading.
Understanding Options and the Need for a Pricing Model
An option contract gives the buyer the right, but not the obligation, to buy (call option) or sell (put option) an underlying asset at a specified price (strike price) on or before a specified date (expiration date). The price of an option is determined by several factors, including the price of the underlying asset, the strike price, the time to expiration, the volatility of the underlying asset, and the risk-free interest rate. Before the BSM, option pricing was largely subjective and lacked a standardized approach.
The Black-Scholes Option Pricing Model: Formula and Variables
The BSM formula for pricing a European-style call option is:
C = S * N(d1) - X * e-rT * N(d2)
Where:
- C = Call option price
- S = Current stock price
- X = Strike price
- T = Time to expiration (in years)
- r = Risk-free interest rate
- e = Base of the natural logarithm (approximately 2.71828)
- N(x) = Cumulative standard normal distribution function
- d1 = [ln(S/X) + (r + σ2/2)T] / (σ√T)
- d2 = d1 - σ√T
- σ = Volatility of the stock’s returns
The formula for a European-style put option is:
P = X * e-rT * N(-d2) - S * N(-d1)
Where P is the put option price and all other variables are as defined above.
Assumptions of the Black-Scholes Model
The BSM relies on several key assumptions:
- The underlying asset price follows a log-normal distribution.
- The volatility of the underlying asset is constant over the option’s life.
- The risk-free interest rate is constant and known.
- There are no dividends paid on the underlying asset during the option’s life.
- The market is efficient (no arbitrage opportunities).
- European-style options are considered (can only be exercised at expiration).
Illustrative Example
Let's consider a stock currently trading at $100 (S = $100). A call option with a strike price of $105 (X = $105) expires in 6 months (T = 0.5 years). The risk-free interest rate is 5% (r = 0.05), and the volatility is 20% (σ = 0.20). Calculating d1 and d2, and then using the cumulative normal distribution function, we can arrive at a call option price (C). (Detailed calculation omitted for brevity, but can be easily performed using financial calculators or software). The resulting call option price might be approximately $6.22.
Applications of the Black-Scholes Model
- Option Pricing: The primary application is determining the theoretical fair value of options.
- Hedging: The model provides the ‘delta’ – the sensitivity of the option price to changes in the underlying asset price – which is crucial for creating hedging strategies to mitigate risk.
- Risk Management: Financial institutions use the BSM to assess and manage the risks associated with options portfolios.
- Implied Volatility: By inputting the market price of an option into the BSM, one can calculate the implied volatility, which reflects the market’s expectation of future price fluctuations.
- Corporate Finance: Valuation of employee stock options and real options (e.g., the option to expand a project) can utilize BSM principles.
Limitations of the Black-Scholes Model
Despite its widespread use, the BSM has limitations:
- Assumption Violations: Real-world markets often deviate from the model’s assumptions (e.g., volatility is rarely constant).
- Fat Tails: The log-normal distribution underestimates the probability of extreme price movements (fat tails).
- American Options: The model is designed for European options and requires adjustments for American options (which can be exercised at any time).
- Dividend Paying Stocks: The basic model doesn’t account for dividends, requiring modifications for dividend-paying stocks.
Conclusion
The Black-Scholes Option Pricing Model remains a landmark achievement in financial economics, providing a foundational framework for understanding and pricing options. While its assumptions are often violated in practice, it serves as a benchmark and a starting point for more sophisticated models. Its applications extend far beyond simple option trading, influencing risk management, corporate finance, and the broader understanding of financial markets. Continuous refinement and adaptation of the model are essential to address its limitations and maintain its relevance in an evolving financial landscape.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.