The Black–Scholes market is the simplest model rich enough to price options and
simple enough to solve in closed form. It contains exactly two tradeable assets: a
riskless bond (the money-market account) and a risky stock.
The bond 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. Idealised, yes — but the
idealisation is exactly what makes the option price computable.
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:
-
Bond: dB_t = r B_t\,dt, hence
B_t = e^{rt} — riskless growth at rate
r.
-
Stock:
dS_t = \mu S_t\,dt + \sigma S_t\,dW_t — geometric Brownian
motion with drift \mu, volatility
\sigma.
-
Discounted stock:
d\tilde S_t = (\mu - r)\,\tilde S_t\,dt + \sigma\,\tilde S_t\,dW_t,
where \tilde S_t = e^{-rt}S_t — discounting trades
\mu for the excess drift \mu - r.
-
Assumptions: constant r and
\sigma, no dividends, no transaction costs, continuous trading,
no arbitrage.
Every assumption is a knowingly false simplification; the model's enduring value is that it is
wrong in understood ways. The most consequential cracks:
-
Constant volatility. Real \sigma moves, and
options of different strikes imply different volatilities — the celebrated
volatility smile. Markets price deep out-of-the-money options as if extreme
moves were more likely than the lognormal allows.
-
Continuous paths. GBM never jumps, but prices gap on news and crashes. Jump
and Lévy models add the discontinuities Black–Scholes smooths over.
-
Frictionless, continuous trading. Real rebalancing costs money and happens
in discrete time, so a perfect delta-hedge is unattainable; transaction-cost models trade a
little hedging error for far less trading.
Practitioners keep the framework and patch it — feeding in an implied-volatility surface rather
than a single \sigma — because its structure (replication,
no-arbitrage, risk-neutral expectation) survives every patch.
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.