Black–Scholes Model

Origins and the core idea

The first mathematical treatment of option prices was Louis Bachelier's (1900) doctoral thesis, which modeled price changes as Brownian motion. Paul Samuelson (1965) introduced geometric Brownian motion so that prices could not turn negative, but the problem of a discount rate that depends on investors' risk preferences remained. The breakthrough of Fischer Black and Myron Scholes (1973) was to show that mixing an option and the underlying asset in the right proportion creates a portfolio whose price risk cancels over a short instant. Such a portfolio is riskless and must therefore earn the risk-free rate, or arbitrage would be possible. This condition yields a partial differential equation that the option price must satisfy, and remarkably its solution contains neither expected returns nor risk preferences. Robert Merton (1973) used stochastic calculus to formalize and generalize the argument.

The formula, assumptions and implied volatility

The Black–Scholes formula expresses the price of a European call in terms of five variables: the underlying price, the strike price, the time to maturity, the risk-free rate and the volatility of the underlying's returns. Only volatility cannot be observed directly, so implied volatility, obtained by inverting the formula from traded option prices, came into wide use. The model makes strong assumptions: no transaction costs, continuous trading, constant volatility and interest rates, and continuous price paths. After the 1987 stock market crash a pronounced volatility smile emerged, with implied volatility varying by strike, revealing where the assumptions diverge from reality. Sensitivities derived from the formula, such as delta, gamma and vega, became the basic language of risk management.

Extensions and influence

Cox and Ross (1976) supplied the intuition of risk-neutral pricing, and the binomial model of Cox, Ross and Rubinstein (1979) offered a discrete-time method that readily handles American options. Harrison and Kreps (1979) established the equivalence between the absence of arbitrage and the existence of a martingale measure, laying the foundation of modern asset pricing theory. Merton's (1976) jump-diffusion model, Heston's (1993) stochastic volatility model and Rubinstein's (1994) implied binomial trees are attempts to explain the smile. Merton (1974) analyzed credit risk by treating a firm's equity as a call option on its assets, a perspective that also led to real options theory. Scholes and Merton received the 1997 Nobel Prize in economics. MacKenzie and Millo (2003) influenced the sociology of finance with an analysis of performativity, showing that market prices moved closer to the model as traders adopted it. With the efficient market hypothesis and modern portfolio theory, the model forms a core of modern financial economics.

FAQ

What is the Black–Scholes model in simple terms?

It uses the fact that options and stock can be combined into a riskless position to calculate a fair option price with a formula.

What is implied volatility?

It is the volatility obtained by plugging a traded option price into the Black–Scholes formula and solving backwards, used as a gauge of how much the market expects prices to move.

Sources reviewed: 2026-10-26

References

1. Bachelier, L. (1900). Théorie de la spéculation. Annales scientifiques de l'École Normale Supérieure, 17, 21–86.
2. Samuelson, P. A. (1965). Rational theory of warrant pricing. Industrial Management Review, 6(2), 13–39.
3. Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637–654.
4. Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4(1), 141–183.
5. Merton, R. C. (1974). On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance, 29(2), 449–470.
6. Cox, J. C., & Ross, S. A. (1976). The valuation of options for alternative stochastic processes. Journal of Financial Economics, 3(1–2), 145–166.
7. Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3(1–2), 125–144.
8. Cox, J. C., Ross, S. A., & Rubinstein, M. (1979). Option pricing: A simplified approach. Journal of Financial Economics, 7(3), 229–263.
9. Harrison, J. M., & Kreps, D. M. (1979). Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory, 20(3), 381–408.
10. Rubinstein, M. (1994). Implied binomial trees. Journal of Finance, 49(3), 771–818.
11. Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies, 6(2), 327–343.
12. MacKenzie, D., & Millo, Y. (2003). Constructing a market, performing theory: The historical sociology of a financial derivatives exchange. American Journal of Sociology, 109(1), 107–145.
13. Hull, J. C. (2018). Options, Futures, and Other Derivatives (10th ed.). New York: Pearson.