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“The Black-Scholes model (or Black-Scholes-Merton model) is a fundamental mathematical formula that calculates the theoretical fair price of European-style options, using inputs like the underlying stock price, strike price, time to expiration, risk-free interest rate and volatility.” – Black Scholes

Black-Scholes Model (Black-Scholes-Merton Model)

The Black-Scholes model, also known as the Black-Scholes-Merton model, is a pioneering mathematical framework for pricing European-style options, which can only be exercised at expiration. It derives a theoretical fair value for call and put options by solving a parabolic partial differential equation—the Black-Scholes equation—under risk-neutral valuation, replacing the asset’s expected return with the risk-free rate to eliminate arbitrage opportunities.^1,2,5

Core Formula and Inputs

The model prices a European call option ( C ) as:

C = S_0 N(d_1) - K e^{-rT} N(d_2)

where:

• ( S_0 ): current price of the underlying asset (e.g., stock).^3,7
• ( K ): strike price.^5,7
• ( T ): time to expiration (in years).^5,7
• ( r ): risk-free interest rate (constant).^3,7
• (\sigma ): volatility of the underlying asset’s returns (annualised).^2,7
• ( N(\cdot) ): cumulative distribution function of the standard normal distribution.
• d_1 = \frac{\ln(S_0 / K) + (r + \sigma^2 / 2)T}{\sigma \sqrt{T}}
• d_2 = d_1 - \sigma \sqrt{T}^1,2,5

A symmetric formula exists for put options. The model assumes log-normal distribution of stock prices, meaning continuously compounded returns are normally distributed:

\ln S_T \sim N\left( \ln S_0 + \left( \mu - \frac{\sigma^2}{2} \right)T, \sigma^2 T \right)

where ( \mu ) is the expected return (replaced by ( r ) in risk-neutral pricing).²

Key Assumptions

The model rests on idealised conditions for mathematical tractability:

• Efficient markets with no arbitrage and continuous trading.^1,3
• Log-normal asset returns (prices cannot go negative).^2,3
• Constant risk-free rate ( r ) and volatility ( \sigma ).³
• No dividends (original version; later adjusted by replacing ( S_0 ) with ( S_0 e^{-qT} ) for continuous dividend yield ( q ), or subtracting present value of discrete dividends).^2,3
• No transaction costs, taxes, or short-selling restrictions; frictionless trading with a risky asset (stock) and riskless asset (bond).^1,3
• European exercise only (no early exercise).^1,5

These enable delta hedging: dynamically adjusting a portfolio of the underlying asset and riskless bond to replicate the option’s payoff, making its price unique.¹

Extensions and Limitations

• Dividends: Adjust ( S_0 ) to ( S_0 - PV(\text{dividends}) ) or use yield ( q ).²
• American options: Use Black’s approximation, taking the maximum of European prices with/without dividends.²
• Greeks: Measures sensitivities like delta (\Delta = N(d_1)), vega (volatility sensitivity), etc., for risk management.⁴
  Limitations include real-world violations (e.g., volatility smiles, jumps, stochastic rates), but it remains foundational for derivatives trading, valuation (e.g., 409A for startups), and extensions like binomial models.^3,5,7

Best Related Strategy Theorist: Myron Scholes

Myron Scholes (b. 1941) is the most directly linked theorist, co-creator of the model and Nobel laureate whose work revolutionised options trading and risk management strategies.

Biography

Born in Timmins, Ontario, Canada, Scholes earned a BA (1962), MA (1964), and PhD (1969) in finance from the University of Chicago, studying under Nobel winners like Merton Miller. He taught at MIT (1968–1972, collaborating with Fischer Black and Robert Merton), Stanford (1973–1996), and later Oxford. In 1990, he co-founded Long-Term Capital Management (LTCM), a hedge fund using advanced models (including Black-Scholes variants) for fixed-income arbitrage, which amassed $4.7 billion in assets before collapsing in 1998 due to leverage and Russian debt crisis—prompting a $3.6 billion Federal Reserve bailout. Scholes received the 1997 Nobel Prize in Economics (shared with Merton; Black deceased), cementing his legacy. He now advises at Platinum Grove Asset Management and philanthropically supports education.¹

Relationship to the Term

Scholes co-authored the seminal 1973 paper “The Pricing of Options and Corporate Liabilities” with Fischer Black (1938–1995), an economist at Arthur D. Little and later Goldman Sachs, who conceived the core hedging insight but died before the Nobel. Robert C. Merton (b. 1944, Merton’s 1973 paper extended it to dividends and American options) formalised continuous-time aspects, earning co-credit. Their breakthrough—published amid nascent options markets (CBOE opened 1973)—enabled risk-neutral pricing and dynamic hedging, transforming derivatives from speculative to hedgeable instruments. Scholes’ strategic insight: options prices reflect volatility alone under no-arbitrage, powering strategies like volatility trading, portfolio insurance, and structured products at banks/hedge funds. LTCM exemplified (and exposed limits of) scaling these via leverage.^1,2,5

References

1. https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model

2. https://analystprep.com/study-notes/frm/part-1/valuation-and-risk-management/the-black-scholes-merton-model/

3. https://carta.com/learn/startups/equity-management/black-scholes-model/

4. https://www.columbia.edu/~mh2078/FoundationsFE/BlackScholes.pdf

5. https://www.sofi.com/learn/content/what-is-the-black-scholes-model/

6. https://gregorygundersen.com/blog/2024/09/28/black-scholes/

7. https://corporatefinanceinstitute.com/resources/derivatives/black-scholes-merton-model/

8. https://www.youtube.com/watch?v=EEM2YBzH-2U

9. https://www.khanacademy.org/economics-finance-domain/core-finance/derivative-securities/black-scholes/v/introduction-to-the-black-scholes-formula