The Black-Scholes Model, sometimes referred to as the Black-Scholes-Merton Model, is a mathematical model using six variables to calculate the theoretical value of a European-style option contract. Developed in 1973, the Black-Scholes model for option pricing remains relevant today for hedging options portfolios to lower the risk profile by buying and selling the underlying asset. It is also known as continuously revised delta hedging.
The Black-Scholes Model Explained
According to the Black-Scholes model, asset prices have a lognormal distribution, experience ongoing drift, and volatility, and follow the Random Walk Theory, suggesting technical analysts cannot predict future price movements. It remains a widely discussed topic, but the Black-Scholes model start with a parabolic partial differential equation referred to as the Black-Scholes equation, from which users can deduce the Black-Scholes formula.
The Black-Scholes model requires the following parameters:
- The current stock price
- Expected dividends
- The option strike price
- Expected interest rates
- Expiration time
- Expected volatility
The Black-Scholes model simple explanation is that it provides a theoretical estimate of European-style options prices. It aims to illustrate that an option contract has a unique price irrelevant to the risk of the asset and its expected return. The Black-Scholes model replaces the anticipated return of the underlying asset with a risk-neutral rate. Some describe the Black-Scholes model as a second-order partial differential equation.
A Historical Overview of the Black-Scholes Model
American economist Fischer Black and Canadian-American financial economist Myron Scholes developed the Black-Scholes model in 1973. They introduced it in their 1973 paper “The Pricing of Options and Corporate Liabilities,” published in the Journal of Political Economy.
American economist Robert Merton helped edit the paper and published his article “Theory of Rational Option Pricing,” in The Bell Journal of Economics and Management Science. He coined the term Black–Scholes Theory of Options Pricing and has expanded the mathematical understanding and applications of the model, which remains in use today, primarily for hedging and risk mitigation strategies, with variation and rules relaxations.
Scholes and Merton received the Nobel Memorial Prize in Economic Sciences in 1997, as Black passed away in 1995, and the Nobel Foundation does not award prizes posthumously. It did recognize the contributions Black made to developing the Black-Scholes model.
Applications of the Black-Scholes Model
The Black-Scholes model is used in one of its numerous modifications and model expansion for hedging and risk mitigation by investment banks, hedge funds, professional traders, brokers, and advanced retail traders.
Some of the applications of the Black-Scholes model include:
- Put and call options
- Futures contracts
- OTC derivatives
- Asset with continuous yielding dividends (modified model), for example, index options
- Asset with discrete proportional dividends (modified model), for example, single stock options
- American options (modified model)
- Perpetual put options
- Binary options and their relationship with vanilla options
- Cash-or-nothing call/put options
- Asset-or-nothing call/put options
- Forex options
The Black-Scholes Formula
Looking at any Black-Scholes model example is daunting, as the Black-Scholes model formula is extremely complex and intimidating. Most will struggle to make sense of it, but plenty of calculators and trading platform tools exist that handle the math and give traders the results.
The Black-Scholes parabolic partial differential equation formula is:
Here is the Black-Scholes formula for a call option (C) of a non-dividend-paying asset (S):
Formulas for inputs d₁ and d₂ are:
Where:
- S = Price of the underlying asset
- T = Expiration date
- t = Current date
- X = Strike price
- r = Risk-free interest rate
- σ = Volatility (standard deviation of the underlying asset)
- N(*) = Cumulative distribution function for a normal (Gaussian) distribution
Black-Scholes Model Assumptions Explained
Before using the Black-Scholes model, traders and risk managers must understand the assumptions of the Black-Scholes model, which correlates to the limitations of the Black-Scholes model.
The core Black-Scholes model assumptions are:
- Lognormal distribution (following the Brownian motion)
- No dividends (many modifications and model expansions exist to account for dividends)
- Random Walk Theory (a highly disputed theory suggesting technical analysis cannot predict future price movements)
- Frictionless market (an unrealistic assumption that no trading costs exist)
- Risk-free interest rate (a false assumption that interest rates are a constant)
- Normal distribution (a false assumption that asset volatility is constant)
- No arbitrage (a false assumption that arbitrage trading does not exist)
The Black-Scholes Model Limitations
The most notable limitations of the Black-Scholes model are its false or unrealistic assumptions about a frictionless market, the risk-free interest rate, normal distribution, and lack of arbitrage mentioned above. Assuming variables as constants and ignoring actual market conditions will lead to inaccurate results deviating from live price action. Models adjusted for dividends exist. Otherwise, the Black-Scholes model would deliver unreliable options pricing.
Black-Scholes Model Conclusion
The original Black-Scholes model consists of four false or unrealistic assumptions, one highly disputed one and one debatable one. Therefore, many applications of the Black-Scholes model used today include modifications and model expansions of the core formula, which some consider the most efficient options pricing model available.
FAQs
Do traders use the Black-Scholes model?
Traders and risk managers use the Black-Scholes model for hedging and risk mitigation despite several false assumptions in the original formula.
How accurate is the Black-Scholes model?
The accuracy depends on modifications to the original formula to correct false or unrealistic assumptions. Many consider a modified Black-Scholes model as the most efficient options pricing tool.