From a discrete spreadsheet random walk to the Wiener process and geometric Brownian motion — the continuous-time model on which the rest of the book is built.
Decks 02 and 03 priced contracts by assuming we knew the distribution of the underlying. Until we write down such a distribution, we have nothing to plug into the binomial recursion or any continuous-time analogue. This chapter does that: it gives the standard model for the random behaviour of a share price.
"Asset prices are random. To price contracts that depend on them, we need a model for that randomness."
— Wilmott, Ch. 4For a convex function $f$ (such as a call payoff $\max(S-K,0)$),
$$\mathbb{E}[f(S)] \;\ge\; f(\mathbb{E}[S]),$$
with strict inequality whenever $S$ is genuinely random. So the expected payoff is not the payoff of the expectation. Pricing a non-linear payoff requires you to know the entire distribution of $S_T$, not just its mean.
For a forward, the payoff is $S_T - F$ — linear in $S_T$. Pricing only needs the mean of $S_T$, which is fixed by no-arbitrage. No model required.
For a call, $f(S_T) = (S_T - K)^+$ is convex — more upside than downside. The more spread out $S_T$ is, the higher the expected payoff. Volatility itself becomes a price input.
Forwards depend only on $\mathbb{E}[S_T]$. Options depend on the whole distribution. That is why the rest of this book is about modelling the dynamics of $S_t$ — not just predicting its end value.
Don't model $S_t$ directly — model its returns. Over a small interval $[t, t+\delta t]$, the return is
$$R = \frac{S_{t+\delta t} - S_t}{S_t} = \frac{\delta S}{S}.$$
Equivalently, in log form, the log-return is $\log(S_{t+\delta t}/S_t) \approx R - R^2/2$. For small $\delta t$ both are nearly identical and the choice is conventional.
Returns are independent and identically distributed, and Normally distributed:
$$\frac{\delta S}{S} \sim \mathcal{N}(\mu\,\delta t,\; \sigma^2\, \delta t).$$
This single assumption is the whole content of the next several chapters.
If a return over $[0,T]$ is the sum of $N$ small iid returns over $[t, t+T/N]$, then by the central limit theorem the total return is approximately Gaussian as $N \to \infty$. Independence + finite variance forces a bell shape.
How should the parameters of the return distribution scale with $\delta t$? Two answers, both forced by the iid-sum structure.
If returns over disjoint intervals are independent with mean $\mu\, \delta t$, the mean over $T$ is $\mu T$. Drift is additive in time.
$$\mathbb{E}\!\left[\frac{\delta S}{S}\right] = \mu\, \delta t.$$
Variances of independent sums add. So $\mathrm{Var}(R_T) = \sigma^2 T$ and
$$\mathrm{std}\!\left(\frac{\delta S}{S}\right) = \sigma\, \sqrt{\delta t}.$$
Combine the two:
$$\frac{\delta S}{S} = \mu\, \delta t + \sigma\, \phi\, \sqrt{\delta t},\qquad \phi \sim \mathcal{N}(0,1).$$
This is the formula you put in a spreadsheet (slide 07) and the discrete prototype of the continuous SDE (slide 09).
Over short timescales the random term $\sigma\phi\sqrt{\delta t}$ dominates the deterministic term $\mu\delta t$, because $\sqrt{\delta t} \gg \delta t$ as $\delta t \to 0$. Randomness wins on short horizons. Over long horizons the linear drift catches up.
$\mu$ is the expected return per unit time. For equities it is broadly positive (long-run equity premium); it varies by asset class and over time. Typical equity $\mu \approx 5$–$10\%$/year (nominal).
To pin $\mu$ down to within $\pm 1\%$/year you need decades of data — and by then $\mu$ has changed. The "expected return" is the least-well-known parameter in finance.
Estimator: $\hat\mu = \frac{1}{T}\sum R_i$ on disjoint windows of length $\delta t$, then $\hat\mu / \delta t$.
Black–Scholes pricing does not need $\mu$. The hedging argument of Deck 03 made the real probability $p$ irrelevant; in the continuous limit it makes $\mu$ irrelevant in the same way. Option prices depend on $\sigma$, not on $\mu$.
For a forward (Deck 01) we already knew $\mathbb{E}[S_T] = S_0 e^{rT}$ under the risk-neutral measure, not $S_0 e^{\mu T}$. The same will be true here: the relevant drift for pricing is $r$, the riskless rate, not the actual drift $\mu$. We need $\mu$ only to simulate the real-world distribution.
For risk (VaR, P&L attribution, scenario analysis) you need $\mu$. For pricing, you can ignore it. This is why a pure pricing desk talks almost exclusively about volatility surfaces, never about expected returns.
$\sigma$ measures the uncertainty about the return per unit time. It has units of $1/\sqrt{\text{time}}$ — typically annualised, with the convention that "20% vol" means $\sigma = 0.20\,\text{yr}^{-1/2}$.
To convert across timescales (assuming iid increments):
$$\sigma_T = \sigma_1 \sqrt{T}.$$
Daily vol (252 trading days) $\approx$ annual vol $/\sqrt{252}$ $\approx$ annual vol $/15.87$.
| Equities, large-cap | 15–25% |
| Single stocks, biotech | 40–80% |
| FX, G10 pairs | 6–12% |
| Interest rates (short) | 0.5–2% (abs) |
| Crude oil | 30–60% |
| VIX itself | ~80% |
Because pricing doesn't see $\mu$, $\sigma$ is the input that does matter, and it is hard to measure. Two flavours:
If returns really were $\mathcal{N}(\mu \delta t, \sigma^2 \delta t)$ with constant $\sigma$, implied vol would be the same number for every strike. It isn't — Deck 08.
Given daily closes $S_0, S_1, \ldots, S_n$, the standard estimator of annualised volatility is the sample standard deviation of log-returns:
$$r_i = \log\!\frac{S_i}{S_{i-1}},\qquad \hat\sigma_{\text{daily}} = \sqrt{\frac{1}{n-1}\sum_{i=1}^n (r_i - \bar r)^2}.$$
$$\hat\sigma_{\text{annual}} = \hat\sigma_{\text{daily}}\, \sqrt{252}.$$
| Issue | Effect |
|---|---|
| Vol clusters | Recent windows mis-estimate future vol after a regime change. |
| Choice of window | Short window = noisy estimate; long window = stale. |
| Open-close vs intraday | Close-to-close ignores overnight gaps; high-frequency improves the estimate at the cost of microstructure noise. |
| EWMA / GARCH | Down-weight old data exponentially — standard for risk management. |
Historic vol tells you what already happened. Implied vol tells you what the market thinks will happen. They are routinely different — the gap is itself a tradable spread (variance swaps, Deck 08).
The discrete return model
$$\delta S = \mu S\, \delta t + \sigma S\, \phi\, \sqrt{\delta t},\qquad \phi \sim \mathcal{N}(0,1)$$
can be implemented in fewer than five spreadsheet cells. This is Wilmott's "random walk on a spreadsheet" — the most useful sanity check in the book.
=NORM.S.INV(RAND())).Jagged, scale-free, locally noisy. Recalculate $F9$ in Excel and the entire path changes — you've just drawn a fresh sample.
Overlay 100 paths and you get a "cone" widening like $\sqrt{t}$. The terminal distribution is approximately lognormal — positive skew, fat right tail.
If you use $S_i = S_{i-1}(1 + \mu\delta t + \sigma\phi\sqrt{\delta t})$ the price can go negative when $\phi$ is very negative. The fix used in continuous time is the multiplicative form $S_i = S_{i-1} \exp((\mu - \tfrac12 \sigma^2)\delta t + \sigma\phi\sqrt{\delta t})$, which guarantees $S > 0$ and is exact for GBM.
Take $\delta t \to 0$ in the discrete model. The random part $\sigma\phi\sqrt{\delta t}$ becomes $\sigma\, dW_t$ where $dW_t$ is the increment of a Wiener process (a.k.a. standard Brownian motion).
Treat $dW$ as a random variable of size $\sqrt{dt}$:
$$dW_t \approx \phi\, \sqrt{dt},\quad \phi \sim \mathcal{N}(0,1),$$
independent for non-overlapping intervals. Then $(dW)^2 = \phi^2 dt$ has mean $dt$, variance $2(dt)^2$. The mean term survives to leading order; the variance is lower order. In the limit $(dW)^2 = dt$, exactly — not as a random variable but as a deterministic identity in stochastic calculus.
The identity $(dW)^2 = dt$ is what creates Itô's lemma in Deck 05 — an extra $\tfrac12 \sigma^2 S^2 \partial^2/\partial S^2$ term that has no analogue in ordinary calculus. It is the entire reason Black–Scholes contains a $\tfrac12 \sigma^2$ rather than the naive zero.
Putting everything together, the standard continuous-time model for an asset price is
$$\boxed{\;dS_t \;=\; \mu\, S_t\, dt \;+\; \sigma\, S_t\, dW_t.\;}$$
This is geometric Brownian motion (GBM). It is the workhorse for the rest of the book.
A 1% move in a £100 stock and in a £1,000 stock should both be "1% moves." The natural noise is multiplicative, not additive. Empirically, the SD of returns is roughly stable across price levels — not the SD of price changes.
Also: a multiplicative process can't go negative.
Applying Itô's lemma to $\log S$ (Deck 05) gives
$$\log S_T = \log S_0 + (\mu - \tfrac12 \sigma^2)T + \sigma W_T,$$
so $\log S_T \sim \mathcal{N}\bigl(\log S_0 + (\mu - \tfrac12\sigma^2)T,\; \sigma^2 T\bigr)$ — $S_T$ is lognormal.
$$\mathbb{E}[S_T] = S_0\, e^{\mu T},\qquad \mathrm{Var}(S_T) = S_0^2\, e^{2\mu T}\bigl(e^{\sigma^2 T} - 1\bigr).$$
GBM says: log-returns are Gaussian, vol is constant, no jumps, no fat tails. It is good enough for a first pass and for closed-form pricing (Black–Scholes, Deck 06) but the next several decks chip away at each assumption: stochastic vol, jumps, regime-switching, and the volatility surface. The model is the start of the story, not the end.
Drag the sliders to set $S_0, \mu, \sigma, T, N$. The left panel plots ~50 sample GBM paths; the right panel shows the histogram of terminal $S_T$ with the theoretical lognormal density overlaid. Hit Resample to draw a fresh set.
Push the vol up and the paths fan out; push $T$ out and the lognormal skew becomes obvious. Notice the empirical mean and theoretical $S_0 e^{\mu T}$ agree only on average — with $50$ paths there's noise. That noise is what Monte Carlo pricing has to fight.