Black-Scholes-Merton (BSM) Model

Last Updated on October 19, 2025 by Admin

Notes

• Used to determine the fair market value of a European Call or Put option

The original model was published by;

• Fischer S. Black, Sloan School of Management, Massachusetts Institute of Technology (MIT)
• Myron S. Scholes, Booth School of Business, University of Chicago
• Paper: “The Pricing of Options and Corporate Liabilities”
• Journal: Journal of Political Economy
• Publisher: University of Chicago Press
• Publication Date: May 1973
• Significance: This is the original paper introducing the Black-Scholes model, providing a formula for pricing European options based on no-arbitrage principles and continuous-time finance.

The model was extended and published by;

• Robert C. Merton, Sloan School of Management, Massachusetts Institute of Technology (MIT)
• Paper: “Theory of Rational Option Pricing”
• Journal: The Bell Journal of Economics and Management Science
• Publisher: The RAND Corporation
• Publication Date: Spring 1973
• Significance: This paper extended and generalised the Black-Scholes framework, provided a more rigorous mathematical derivation, and showed how it could be applied to a wide range of contingent claims giving rise to the Black-Scholes-Merton model.

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Assumptions

• Risk-free interest rate
• Constant volatility

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Formula

$$Call\;Option\;\left( C \right)=SN(d_{1}) – Ke^{-rt}N(d_{2})$$

• C = Call Option Price
• N = Cumulative Distribution Function (CDF)
• S = Spot Price
• K = Strike Price
• r = Risk-Free Interest Rate
• T = Time to Maturity (Expiration)
• v = Volatility

$$Put\;Option\;(P) = Ke^{-rT}N\left(-d_{2} \right)- SN\left( -d_{1} \right)$$

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Black-Scholes-Merton (BSM) Calculator

Disclaimer:
This Black-Scholes-Merton (BSM) calculator is currently in a testing phase and is intended for educational purposes only. The results it produces may not be accurate and should not be relied upon for financial, investment, or trading decisions. By using this calculator, you agree to independently verify all outputs and use this tool at your own discretion and risk.