History
The groundbreaking work of economists Fischer Black and Myron Scholes in 1968 laid the foundation for a significant breakthrough in financial economics. They introduced the concept of a dynamic portfolio revision, highlighting that eliminating the expected return of security would mitigate risk, thus formulating what became known as the risk-neutral argument. Their pioneering ideas were built upon previous research by notable market researchers and practitioners such as Louis Bachelier, Sheen Kassouf, and Edward O. Thorp.
However, when Black and Scholes attempted to apply their formula to the markets, their lack of adequate risk management led to financial losses. Subsequently, in 1970, they decided to return to the academic sphere to further refine their model.
After three years of rigorous efforts, the formula, which was named in their honor for its public introduction, was finally published in 1973 in an article titled “The Pricing of Options and Corporate Liabilities” in the Journal of Political Economy. This formula revolutionized financial markets, offering a groundbreaking method for pricing options and corporate liabilities.
Robert C. Merton significantly contributed by furthering the mathematical understanding of the options pricing model. He expanded upon the initial work of Black and Scholes, ultimately coining the term “Black–Scholes options pricing model”.
The introduction of the Black-Scholes-Merton (BSM) model marked a turning point in financial markets, sparking a notable surge in options trading and fundamentally altering the industry landscape. Its arrival brought crucial mathematical legitimacy to the workings of pivotal institutions like the Chicago Board Options Exchange and numerous other options markets worldwide.
The BSM model’s impact was far-reaching, profoundly reshaping the financial markets by providing a robust and standardized framework for pricing options. It revolutionized how financial instruments were perceived, valued, and traded. By offering a systematic method for valuing options, the model brought clarity to the intricate relationship between option prices and key market factors.
This model unlocked a deeper understanding of how underlying asset prices, time until expiration, volatility, and the risk-free interest rate influence option values. It essentially democratized options pricing, providing both professionals and investors with a common language and a structured approach to evaluating and trading these instruments.
The BSM model’s legacy lies in its ability to establish a cohesive and universally accepted methodology for pricing options. Its influence reverberates throughout the financial industry, playing a pivotal role in shaping strategies, risk management practices, and the broader dynamics of financial markets.
In 1997, the Nobel Memorial Prize in Economic Sciences was conferred upon Myron Scholes and Robert C. Merton for their pioneering contributions in developing “a new methodology for assessing the worth of derivatives.” Although Fischer Black had passed away two years prior, the Nobel Prizes, by their rules, are not awarded posthumously. However, the Nobel committee acknowledged Black’s integral role in formulating the Black-Scholes model.
Introduction
The Black–Scholes–Merton model stands as a mathematical framework designed for understanding the dynamics within financial markets, particularly involving derivative investment instruments. This model operates under several underlying assumptions. At its core lies the Black–Scholes equation, a parabolic partial differential equation, from which emerges the Black–Scholes formula. This formula offers a theoretical estimation of the value of European-style options, illuminating that an option possesses a unique price considering the security’s risk and expected return.
The fundamental principle governing this model revolves around option hedging achieved through a meticulously orchestrated process of buying and selling the underlying asset. This strategic maneuver, termed “continuously revised delta hedging,” serves as the bedrock for more intricate hedging strategies adopted by entities such as investment banks and hedge funds.
Central to the Black-Scholes framework is the assumption that financial instruments, including stock shares or futures contracts, conform to a lognormal price distribution. This occurs through a random walk characterized by a consistent drift and volatility. By employing this presumption alongside other crucial variables, the equation derives the price of a European-style call option.
Five key variables are essential in the Black-Scholes equation. These inputs encompass volatility, the underlying asset’s price, the option’s strike price, the duration until the option’s expiration, and the risk-free interest rate. Armed with these variables, options sellers can theoretically establish rational prices for the options they offer.
Moreover, the model forecasts that heavily traded assets adhere to a geometric Brownian motion marked by unchanging drift and volatility. When applied specifically to a stock option, the model seamlessly integrates the stock’s constant price fluctuation, the time value of money, the option’s strike price, and the duration until the option’s expiry.
Key Assumptions
- Log-normal Asset Price Dynamics: The model presumes that the movement of the underlying asset’s price, like stock price, adheres to a continuous diffusion process, exhibiting lognormal behavior. This assumption implies that the logarithm of the asset’s price follows a normal distribution over time.
- Constant Volatility: Throughout the lifespan of the option, the asset price volatility remains fixed and unchanging. This assumption assumes a stable level of market fluctuation without alterations in the asset’s price volatility.
- Continuous Trading, Absence of Transaction Costs, and No Arbitrage Opportunities: The model assumes a scenario where investors can seamlessly trade the asset and options continuously without incurring any fees or transaction costs. This condition allows for smooth and frictionless trading activities.
- Option Type: The options are European i.e., they can only be exercised at expiration.
- Constant Risk-Free Interest Rate: The model operates on the premise that there exists a singular risk-free interest rate for borrowing and lending funds. This assumption implies that this interest rate remains constant and unaffected by market fluctuations throughout the option’s lifespan.
Black-Scholes Equation
The Black-Scholes equation is a differential equation for the value of an option as a function of the underlying asset and time.
The basic equation is
∂V/∂t + rS ∂V/∂S + ½ σ² S² ∂²V/∂S² -rV= 0
where V(S, T) is the option value as a function of asset price S and T.
Characteristics of the Black-Scholes Equation
- Derivation from Assumptions and Hedging Principle: The equation arises from specific assumptions and a mathematical, financial rationale involving the concept of hedging.
- Linearity and Homogeneity: The equation possesses linearity and homogeneity properties, allowing for the valuation of a portfolio comprising multiple derivative contracts by aggregating the values of individual contracts. This property simplifies the assessment of the combined value of diverse derivatives.
- Partial Differential Equation (PDE): Being a PDE, the equation involves more than one independent variable—here, S (the underlying asset’s price) and t (time). The presence of multiple variables necessitates a differential equation approach for the solution.
- Parabolic Nature: The equation demonstrates a parabolic type, reflecting specific characteristics in its structure. It signifies that one variable, typically time (t), exhibits a first derivative term, while the other variable, such as the asset price (S), comprises a second derivative term. This parabolic structure defines the equation’s behavior and informs its solution methodology.
- Backward Time Evolution: Characterized as a backward-type equation, the process begins by defining a final condition, which represents the option’s payoff at the expiry. The equation is then solved backward in time, allowing for the determination of the option’s current value based on its future potential payoffs. This approach is distinctive and essential in estimating the option’s present value using future conditions.
The equation contains four terms
- ∂V/∂t = time decay, how much the option value changes if the stock price doesn’t change.
- ½ σ² S² ∂²V/∂S² = convexity term, how much a hedged position makes on average from stock moves.
- rS ∂V/∂S = drift term allowing for the growth in the stock at the risk-free rate.
- rV = the discounting term, since the payoff is received at expiration but you are valuing the option now.
Black Scholes Formula
Notation
C (S,t) is the price of a European call option.
P (S,t) is the price of a European put option.
T is the time of option expiration.
T-t is the time until maturity.
K is the strike price of the option.
N(x) denotes the standard normal cumulative distribution function
N’(x) denotes the standard normal probability density function
The Black–Scholes formula serves as a tool for computing the prices of European put and call options. The prices calculated through this formula align with the solutions of the Black–Scholes equation. This connection is established as the formula can be derived by solving the equation while incorporating the relevant terminal and boundary conditions.
The value of a call option for a non-dividend-paying underlying stock in terms of the Black–Scholes parameters is:
The price of a corresponding put option based on put-call parity with discount factor is:
Benefits and Limitations
Benefits
1. Standardized Pricing: It provided a standardized and widely accepted method for pricing options, leading to increased transparency and efficiency in financial markets.
2. Foundation for Derivatives: It laid the groundwork for the development of various derivative pricing models and option trading strategies, forming the basis for quantitative finance and risk management.
3. Risk Management: Enabled the development of risk management strategies by providing insights into hedging and managing risks associated with options and portfolios.
4. Influence on Financial Theory: The BSM model significantly influenced financial theory and academic research, fostering advancements in financial economics and quantitative finance.
Limitations
1. Assumptions: The model is built upon several assumptions, including constant volatility, no dividends, efficient markets, no transaction cost, and a constant risk-free rate, which might not always hold in real-world scenarios. Deviations from these assumptions can affect the accuracy of the model’s predictions.
2. European Options Only: BSM is specifically designed for European options, which can be exercised only at expiration. It’s less applicable to American options that can be exercised at any time before expiration.
3. Market Conditions: During periods of extreme volatility, market crises, or when the underlying asset exhibits non-standard behavior, the BSM model might not perform accurately.
4. Sensitivity to Inputs: BSM outputs are highly sensitive to the inputs used, such as volatility estimates and interest rates. Small changes in these parameters can significantly affect the calculated option prices.
Example 1
Below is an example of a 3D plot illustrating the relationship between option prices, stock prices, and time to expiry using the Black-Scholes-Merton model:
Interpretation
- Call Option Prices Plot: The first 3D plot represents the relationship between call option prices, stock prices (on the x-axis), and time to expiry (on the y-axis). As the stock price and time to expiry change, the value of the call option fluctuates. Higher stock prices or longer time to expiry generally lead to higher call option prices.
- Put Option Prices Plot: The second 3D plot illustrates the relationship between put option prices, stock prices (on the x-axis), and time to expiry (on the y-axis). Similar to the call option plot, changes in stock price and time to expiry impact the value of the put option. However, put option prices tend to rise when stock prices decrease or as the time to expiry extends. Both plots provide a visual understanding of how call and put option prices change concerning variations in stock prices and time to expiry. Observing these plots allows for an intuitive grasp of how different factors influence option pricing according to the Black-Scholes-Merton model. Analysts and investors can utilize these visualizations to anticipate option price movements, discern optimal entry or exit points, and devise strategies based on varying market conditions, stock price fluctuations, and time to expiry in the options market.
Example 2
By utilizing the inputs within the Black-Scholes-Merton (BSM) formula, we can effectively determine the prices of call-and-put options, accounting for variations in dividend yield and also without considering dividend yield.
Inputs
- Stock Price = 100
- Strike Price = 110
- Time to expiry = 1
- Risk free rate = 0.05
- Volatility = 0.2
- Dividend yield = 0.05
Calculating Option Value Without Dividend Yield
Call Option Price: 6.04
Put Option Price: 10
Calculating Option Value Considering Dividend Yield
Call Option Price with Dividend Yield Adjustment: 4.08
Put Option Price with Dividend Yield Adjustment: 13.59
Interpretation
- Call Option with Dividend Yield: When a dividend is expected on the underlying stock, it can affect the pricing of call options. Typically, a dividend payment reduces the stock price by the amount of the dividend before the ex-dividend date. This reduction in the stock price impacts the call option price, particularly reducing it because the stock price decreases due to the dividend payout.
- Put Option with Dividend Yield: Dividend yield can also influence put option prices. Generally, if a stock pays dividends, the put option holder might anticipate lower dividends as the stock price drops after the ex-dividend date. This anticipation could impact the put option price, potentially increasing it due to the expected decline in stock price.
“The Python code for the above examples is accessible in the Python code category.“
