This model has become the gold standard for calculating what an option should be worth based on current stock prices and volatility, as well as other key variables. Read on to gain an easy overview explaining The Black Scholes Model, its pricing options and its formula.
What is the Black Scholes model?
The Black-Scholes model, developed by economists Fischer Black, Myron Scholes and Robert Merton in 1973, is one of the most influential theories in modern finance. Though the model itself involves complex mathematical equations, its implications for trading, banking and corporate management are far-reaching.
Black-Scholes is a method to estimate the value of options contracts that allow the right to buy or sell an asset in the future. Even with its flaws and limitations, Black-Scholes is the most widely adopted approach globally for options pricing.
How Does Options Pricing Help?
Options contracts allow traders to mitigate risk in stock portfolios. Similarly, company executives use real options theory to evaluate expansion decisions and new projects with flexibility. The problem is options do not have an obvious intrinsic value like bonds or stocks do. Their payoff depends on the price path of the underlying till expiry.
So, what constitutes a fair value becomes crucial but highly subjective. This is where the black scholes option pricing model comes in very useful with its structured framework.
Black Scholes Formula
The pricing equation is, without a doubt, complex-looking. Fortunately, options traders do not require derivations and need the end formula:
C = S * N(d1) - K * e^ (-r * t) * N(d2)
Where,
d1 = (In(S/K) (r s^2/2) * t) / (s * vt)
And,
d2 = d1 - s * vt
And where:
C=Call option price
S= Current stock (or other underlying) price
K: Strike price
r: Risk-free interest rate
t: Time to maturity
N=A normal distribution
What Assumptions Is the Model Based On?
The Black-Scholes options pricing model approach makes some important assumptions regarding markets, interest rates and asset prices. Relaxing these can reduce its accuracy.
Firstly, it assumes stocks follow a random walk model with lognormal distribution in prices. Next, there are no arbitrage opportunities, transaction costs or taxes in trading options or stocks. For simplicity, options are priced in European style, exercisable only on the expiry date.
Besides, key inputs like volatility and risk-free interest rates remain constant over the option's lifetime, which may not be true in practice. Maintaining these strict assumptions lets you derive an elegant closed-form pricing equation.
Where is the Black and Scholes Model Applied?
1. Options Trading Desks
Traders at exchanges use Black-Scholes to evaluate if options are over or underpriced relative to the computed fair value. This allows them to execute buy, sell or hold strategies accordingly to benefit from mispricing’s.
2. Over-the-Counter Markets
Banks employ quantitative analysts who build Black-Scholes software models to price exotic or customized options products before offering them to clients.
3. Corporate Treasury Functions
From employee stock options to supply chain contracts, the treasury department relies on Black-Scholes math to reduce business risks through appropriate hedging products.
4. Capital Investment Decisions
Strategic project evaluations, venture capital funding, patent filing, etc., leverage real options theory enabled by the Black-Scholes approach to gain a competitive advantage.
Thus, the Black and Scholes model enables decision-making under uncertainty across the financial services industry and in corporate planning functions through a consistent options pricing framework.
Criticisms and Shortcomings of the Theory
Given its fame, the Black-Scholes model has inevitably drawn criticism over the decades. Academics and traders claim it makes overly simplistic assumptions that don't match real market dynamics.
Its inability to account for fat-tailed asset returns, changing volatility and early exercise of options are flagged as flaws. The model yields prices consistently different from actual market quotes.
Besides, market crashes, lack of liquidity during crises and behavioural factors affecting investor decisions cannot be modelled easily. Scores of alternative pricing approaches attempt to address these, but Black-Scholes remains popular.
Through various tweaks and adjustments, practitioners mitigate its limitations reasonably well for practical implementation across financial institutions.
Conclusion
The Black-Scholes model combines advanced mathematics and economics to transform markets. It began as a theory in an academic paper and is now crucial for modern options trading and corporate strategy planning worldwide. Despite criticisms, it enables fairer price discovery and transparency. It has been in use for almost 50 years and is unlikely to be replaced soon, given its pervasive adoption and strong foundation.
