Black-Scholes Model - Pricing Options in Continuous Time

By EC Assets Research Team, Derivatives Research · Published 20 May 2026 · Updated 28 May 2026

Black-Scholes Model — The Black-Scholes model is a continuous-time framework that prices European options by assuming the underlying follows lognormal returns and that a continuously rebalanced hedge eliminates risk. Its 1973 closed-form solution turned option pricing into engineering and made volatility the common language of derivatives markets.

What the Black-Scholes Model Does

The Black-Scholes model is a continuous-time framework for pricing European options. Published in 1973 by Fischer Black and Myron Scholes, with the hedging argument formalised by Robert Merton, it produces a closed-form price for a call or put from five inputs - the spot price, the strike, the time to expiry, the risk-free rate, and the volatility of the underlying. Of those five, only volatility cannot be read directly off a screen, which is why the entire options market ends up revolving around it.

The model's importance is not really the formula. It is the idea behind the formula: an option can be replicated by a continuously rebalanced position in the underlying and cash. If a portfolio of stock and borrowing can be made to track the option's payoff at every instant, then by no-arbitrage the option must cost exactly what that replicating portfolio costs - and the expected return of the stock drops out of the problem entirely.

The Replicating Hedge - The Real Insight

Imagine selling a call and immediately buying delta shares of the underlying to neutralise the directional risk. As the underlying moves, delta changes (that is gamma), so the hedge must be adjusted continuously. Black, Scholes, and Merton showed that if you rebalance instantaneously, the hedged position is riskless over each tiny interval, and therefore must earn the risk-free rate. Setting the hedged portfolio's return equal to the risk-free rate produces a partial differential equation whose solution is the Black-Scholes price.

The consequence is profound and counter-intuitive: the stock's expected return never enters the option price. Two investors who disagree violently about whether a stock will rise or fall must still agree on the option's value, because both can hedge. Valuation happens in a risk-neutral world where every asset drifts at the risk-free rate.

Reading the Formula

For a European call on a non-dividend-paying asset:

C = S·N(d1) − K·e^(−rT)·N(d2)

where d1 = [ln(S/K) + (r + σ²/2)·T] / (σ·√T) and d2 = d1 − σ·√T, and N(·) is the standard-normal cumulative distribution. Two of these pieces have direct trading meaning:

• N(d1) is the call's delta - the hedge ratio, the number of shares to hold per option.
• N(d2) is, loosely, the risk-neutral probability that the option finishes in the money.

So the call price reads as "the value of the shares you would own if exercised, weighted by the chance of exercise, minus the discounted cost of paying the strike, weighted by the same chance."

Worked Example

Take S = 100, K = 100, T = 0.25 years, r = 4%, and σ = 20%. Then:

d1 = [ln(1) + (0.04 + 0.02)·0.25] / (0.20·√0.25) = 0.015 / 0.10 = 0.15 d2 = 0.15 − 0.10 = 0.05

Reading the normal CDF, N(0.15) ≈ 0.560 and N(0.05) ≈ 0.520. The call is:

C = 100·0.560 − 100·e^(−0.01)·0.520 = 56.0 − 51.5 ≈ 4.49

The delta is N(d1) ≈ 0.56 - the at-the-money drift above 0.50 comes from the (r + σ²/2) term. Put-call parity then fixes the matching put at P = C − S + K·e^(−rT) ≈ 4.49 − 0.995 ≈ 3.49.

Where the Model Breaks - The Volatility Smile

Black-Scholes assumes volatility is constant and returns are lognormal. Real markets honour neither. Crashes are larger and more frequent than a normal distribution allows, so out-of-the-money puts trade richer than the model says they should. When traders invert market prices to find the implied volatility, they get different numbers for different strikes - the volatility smile / skew. A single constant σ cannot reproduce the observed surface.

[!warning] A constant-volatility model confronting a non-constant-volatility world is why no serious desk uses a single Black-Scholes volatility. They use the implied volatility for that specific strike and maturity, read off the surface, and layer on local- or stochastic-volatility corrections for path-dependent products.

Why It Still Matters for Institutional Investors

If the assumptions are wrong, why has the model survived for fifty years? Because it is used backwards. Nobody believes the single-volatility forecast; instead, the market quotes an option's price and inverts Black-Scholes to express that price as an implied volatility. That turns the model into a universal translation layer - a way to compare a one-month option on one name against a one-year option on another in the common currency of vol.

• A common language. Implied volatility, the Greeks, and the surface are all defined relative to Black-Scholes. It is the coordinate system of the options world.
• The foundation, not the ceiling. Local-volatility, stochastic-volatility (Heston, SABR), and jump-diffusion models all start from Black-Scholes and correct its shortcomings. You cannot understand them without it.
• Risk and capital. Greeks computed under Black-Scholes feed hedging, risk limits, and regulatory capital. Even an imperfect model, consistently applied, makes exposures comparable across a book.

[!key] Black-Scholes is best understood as a hedging argument, not a crystal ball. Its lasting contribution is that an option's fair price comes from the cost of replicating it - which is why the underlying's expected return is irrelevant and why volatility is the only thing left to argue about.

References

1. Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637-654.
2. Merton, R. C. (1973). Theory of Rational Option Pricing. Bell Journal of Economics and Management Science, 4(1), 141-183.
3. Hull, J. C. (2022). Options, Futures, and Other Derivatives (11th ed.). Pearson. Chapters 15-17.
4. Derman, E., & Miller, M. B. (2016). The Volatility Smile. Wiley.