The Black–Scholes Market Model

The Black–Scholes market is a small world with sharp rules. It contains exactly two tradeable assets — one risky stock following geometric Brownian motion, one riskless bond — and trading in them is frictionless, continuous and unlimited. Nothing else exists: no second stock, no dividends, no fees, no gaps, no defaults. It is unmistakably a toy. But inside this toy world something remarkable happens: every option has exactly one fair price — not a range, not an estimate, but a single number forced on all traders by the rules of the world itself, whatever they each believe about the stock's future. That toy, published in 1973, turned out to run a multi-trillion-dollar industry: the formulas priced on the world's derivatives desks today are all descendants of this two-asset model.

Here are the rules. The bond (the money-market account) grows deterministically at the risk-free rate r,

dB_t = r B_t\,dt, \qquad B_0 = 1 \;\Rightarrow\; B_t = e^{rt},

while the stock follows geometric Brownian motion,

dS_t = \mu S_t\,dt + \sigma S_t\,dW_t, \qquad S_0 > 0,

with constant drift \mu and constant volatility \sigma. Six standing assumptions pin the model down: constant r and \sigma, no dividends, no transaction costs, continuous trading, and no arbitrage. Every one of them is knowingly false — and each buys something specific. The rest of this page walks the world's two assets, stress-tests the assumptions one by one, and computes the single quantity (the discounted stock) on which the whole pricing theory turns.

What the stock actually does: the GBM solution

The stock SDE has an explicit solution — one of the few SDEs that does. Applying Itô's lemma to \ln S_t (the second-derivative term -\tfrac{1}{2S^2}\,(dS)^2 = -\tfrac{\sigma^2}{2}\,dt is where the correction enters) gives

S_T = S_0\,\exp\!\Big(\big(\mu - \tfrac{\sigma^2}{2}\big)T + \sigma W_T\Big).

Since W_T \sim N(0, T), the log-price is normal and S_T itself is lognormal — always positive, skewed to the right. Two summary numbers matter constantly, and they are not the same:

\mathbb{E}[S_T] = S_0\,e^{\mu T}, \qquad \operatorname{median}(S_T) = S_0\,e^{(\mu - \sigma^2/2)T}.

Why the -\sigma^2/2? The exponential is convex, so upside wiggles help more than downside wiggles hurt: a few lucky paths that compound enormously drag the mean up to S_0 e^{\mu T}, while the typical path — the median — grows at only \mu - \sigma^2/2. This gap is volatility drag. Concretely, take \mu = 10\% and \sigma = 50\%: the mean grows at 10\% a year, but \mu - \sigma^2/2 = 0.10 - 0.125 = -0.025 — the typical path loses about 2.5\% a year even while the average gains. A GBM can have a rising mean and a falling median at the same time; more than half of all paths lose money while the average portfolio "does great". Forgetting the drag is one of the most expensive arithmetic slips in finance.

The two assets, side by side

The two engines of the model on one set of axes: the deterministic bond B_t = e^{rt} climbing smoothly, and a representative stock path S_t = S_0\,e^{(\mu - \frac12\sigma^2)t + \sigma W_t} jittering around it. The Brownian path underneath is frozen, so the sliders reshape the same randomness rather than rolling new dice — which makes the parameters' separate jobs visible. Push \sigma up and the stock's wiggles amplify (and, thanks to the -\sigma^2/2 drag, the path's overall level sags even though \mu hasn't moved — watch for it at high \sigma). Push \mu up and the whole stock path tilts upward without getting any noisier. Push r up and the bond curve steepens — the hurdle the stock must beat. Tune all three and ask: which asset wins, and how sure are you?

The assumptions, stress-tested one by one

A model earns its keep by what its assumptions buy. Here is each Black–Scholes assumption, what it purchases for the mathematics, and where reality breaks it:

The pattern to take away: each assumption removes a source of risk or friction so that exactly one source of randomness remains — the Brownian motion W_t — and one risky instrument to trade against it. One risk, one hedge: that is the balance which makes the price unique.

The discounted stock, line by line

The single most important quantity in the model is the discounted stock

\tilde{S}_t = e^{-rt} S_t = \frac{S_t}{B_t},

the stock priced in units of the bond — what a share is worth after stripping out the risk-free growth that any idle cash would have earned anyway. Its dynamics set the entire stage for risk-neutral pricing, so we compute d\tilde S_t carefully with the Itô product rule.

Step 1 — name the two factors. Write \tilde S_t = f(t)\,S_t with the deterministic discount factor f(t) = e^{-rt}, whose ordinary derivative is

df(t) = -r\,e^{-rt}\,dt.

Step 2 — apply the product rule. For a deterministic factor times an Itô process there is no cross-variation term (df carries no dW, so df\,dS = 0), and the Itô product rule reduces to the ordinary one:

d\tilde S_t = df(t)\,S_t + f(t)\,dS_t = -r\,e^{-rt}S_t\,dt + e^{-rt}\,dS_t.

Step 3 — substitute the GBM dynamics dS_t = \mu S_t\,dt + \sigma S_t\,dW_t:

d\tilde S_t = -r\,e^{-rt}S_t\,dt + e^{-rt}\big(\mu S_t\,dt + \sigma S_t\,dW_t\big).

Step 4 — factor out e^{-rt}S_t. Every term carries the common factor e^{-rt}S_t = \tilde S_t:

d\tilde S_t = e^{-rt}S_t\big(-r\,dt + \mu\,dt + \sigma\,dW_t\big) = \tilde S_t\big((\mu - r)\,dt + \sigma\,dW_t\big).

Step 5 — read off the discounted dynamics. Distributing \tilde S_t gives the SDE the whole theory turns on:

d\tilde S_t = (\mu - r)\,\tilde S_t\,dt + \sigma\,\tilde S_t\,dW_t.

Discounting has done one beautifully simple thing: it replaced the drift \mu by the excess drift \mu - r, the stock's growth over and above the risk-free rate, while leaving the volatility \sigma untouched. The discounted stock is itself a geometric Brownian motion — with drift \mu - r.

Here is the punchline that the next lessons cash in. If only the drift were zero, the discounted stock would be a martingale — a fair game — and prices would be plain expectations. The excess drift \mu - r is the only thing in the way, and it is exactly what a change of measure (Girsanov) will rotate away, turning \mu - r into 0 and \tilde S_t into a martingale under the risk-neutral measure. Step 5 is the doorway to risk-neutral pricing.

The model consists of two assets and a set of standing assumptions:

A startling feature of the model: the option price will not depend on the stock's drift \mu at all. Step 5 foreshadows why. Pricing is done by replication, which only ever touches the discounted stock, and the change of measure that makes \tilde S_t a martingale erases the excess drift \mu - r entirely — the stock is forced to drift at the risk-free rate under the pricing measure. What remains is the volatility \sigma and the rate r. Two investors who violently disagree about \mu (will the stock soar or stagnate?) must nonetheless agree on every option price — a fact that still surprises newcomers, and the single deepest idea in derivative pricing.

The road from model to formula

This page sets the stage; the derivation itself unfolds across the next lessons. Here is the whole route in one view, so each step lands in context when you get there:

  1. Set up the hedge portfolio. Hold the option and trade against it a self-financing portfolio of \Delta_t shares and some bond — a portfolio that never needs fresh cash after time zero.
  2. Apply Itô. Write the option value as a smooth function V(t, S_t) and expand dV with Itô's lemma — the \tfrac12 \sigma^2 S^2 V_{SS}\,dt term appears here, and it is the term that separates option pricing from ordinary calculus.
  3. Cancel the risk, invoke no-arbitrage. Choosing \Delta_t = \partial V/\partial S kills the dW term: the hedged position is momentarily riskless, so no-arbitrage forces it to earn exactly r. That equality is the Black–Scholes PDE — and, true to the vignette above, \mu has already cancelled out of it.
  4. Attach boundary conditions. The PDE alone fits every derivative on this stock; the contract enters only through its terminal payoff — for a call, V(T, S) = \max(S - K, 0) — plus the behaviour at S = 0 and S \to \infty.
  5. Solve. A change of variables turns the PDE into the heat equation, and out comes the closed-form Black–Scholes formula, C = S_0\,\Phi(d_1) - K e^{-rT}\,\Phi(d_2). Its sensitivities — the Greeks — are what a trading desk actually watches.

There is a second, parallel route — price as a discounted risk-neutral expectation and evaluate the lognormal integral — and the Feynman–Kac theorem is the bridge proving the PDE route and the expectation route give the same answer. The model on this page is the ground both routes stand on.

What the model's prices look like

To feel what the machinery outputs, run the finished formula on a concrete contract — a one-year at-the-money call, S_0 = K = 100, r = 5\% — under two volatilities:

Read what just happened: nothing observable about today changed. The entire \$7.57 difference is a statement about the width of the future — more volatility means more chance of finishing deep in the money, while the loss on the downside is capped at the premium. An option is convex, so it loves volatility, and in this model \sigma is the only free parameter carrying that information: \mu appears nowhere in either number. This is why options traders describe their job as trading volatility — the direction of the stock is hedged away; the width of its distribution is the product. It is also why the market runs the formula backwards: given an option's traded price, solve for the \sigma that reproduces it. That number — the implied volatility — is how option prices are quoted the world over.

Black–Scholes does not say that markets follow geometric Brownian motion. It says: if a stock followed GBM with constant \sigma in a frictionless market, then each option on it would have this one arbitrage-free price. It is a conditional, an implication — and attacking the model for the falsity of its hypothesis misses how it is actually used. Three things to hold straight:

Fischer Black and Myron Scholes could barely get the paper published. The Journal of Political Economy rejected it; so did the Review of Economics and Statistics — too specialised, not enough economics. Only after Merton Miller and Eugene Fama leaned on the JPE's editors did "The Pricing of Options and Corporate Liabilities" appear, in 1973. The timing was uncanny: the Chicago Board Options Exchange opened the same year — the first marketplace for standardised options — and suddenly thousands of traders needed exactly the number the rejected paper computed. Within a few years Texas Instruments was shipping a handheld calculator module with the Black–Scholes formula built in; Black joked that he received no royalties. Robert Merton, working in parallel, recast the argument in continuous time — the elegant replication-and-Itô derivation this course follows is really the Black–Scholes–Merton model. In 1997 Scholes and Merton received the Nobel Prize in economics for it; Fischer Black had died in 1995, and the Nobel is never awarded posthumously — the committee took the unusual step of naming him in the citation. From twice rejected to a Nobel, an exchange, and a pocket calculator: not bad for a toy world with two assets.

See it explained