Brownian motion, quadratic variation, the Itô integral, SDEs and Itô's lemma — the machinery you need before Black–Scholes makes sense.
The objects that move financial models — share prices, exchange rates, interest rates — are not smooth. Sampled at any frequency, they look jagged; sampled finer, they look just as jagged. Classical calculus assumes you can write $df = f'(x)\,dx$ and let $dx \to 0$. For Brownian-driven paths that limit doesn't exist in the ordinary sense.
"The most important difference between deterministic and stochastic calculus is the appearance of the second-derivative term in Itô's lemma."
— Wilmott, Ch. 5The fundamental obstruction is that the sample paths of Brownian motion have unbounded variation: between any two times $0 \le s < t$, the total length traced out by the path is infinite. So $\int f(W_s)\,dW_s$ cannot be defined path-by-path as a Riemann–Stieltjes integral. We need a new definition — the Itô integral — and once we have it the consequence is the extra second-derivative term in Itô's lemma.
Brownian motion exists, has continuous paths, and is the limit of a simple random walk under a particular scaling. The book proves none of this rigorously; nor will this deck. What matters is the algebra of $(dW)^2 = dt$ and the consequences for derivative pricing.
Two properties dominate the kinds of processes Wilmott uses for share prices.
The future depends on the past only through the present.
$$\mathbb{E}[\,f(X_t)\mid \mathcal{F}_s\,] = \mathbb{E}[\,f(X_t)\mid X_s\,], \quad s < t.$$
For modelling, this is the assumption that "today's price contains all relevant information." It is the working assumption behind technical analysis being a waste of time.
The best forecast of the future value is the current value.
$$\mathbb{E}[\,X_t \mid \mathcal{F}_s\,] = X_s, \quad s < t.$$
Real share prices are not martingales (they have drift). But under a clever change of measure (Deck 06) the discounted price becomes one — the riskneutral pricing principle.
Brownian motion $W_t$ is both Markov and a martingale. So is $W_t^2 - t$. So is $e^{\sigma W_t - \tfrac{1}{2}\sigma^2 t}$ (the exponential martingale, the workhorse of Girsanov's theorem). These will come up repeatedly.
If $X$ is a martingale and you bet on it, your expected wealth is constant: you can't make money in expectation by trading a martingale. Risk-neutral pricing turns derivative valuation into the calculation of the expected payoff of an artificially constructed martingale.
Partition $[0,T]$ into $N$ equal pieces of width $\delta t = T/N$. Let $\Delta W_k = W_{t_k} - W_{t_{k-1}}$. The quadratic variation of the path is
$$Q_N(T) = \sum_{k=1}^{N} (\Delta W_k)^2.$$
Each $\Delta W_k$ is normal with mean zero and variance $\delta t$, so $\mathbb{E}[(\Delta W_k)^2] = \delta t$ and $\mathrm{Var}[(\Delta W_k)^2] = 2(\delta t)^2$. Summing over $N$ terms:
$$\mathbb{E}[Q_N(T)] = T, \qquad \mathrm{Var}[Q_N(T)] = 2T\,\delta t \;\to\; 0.$$
So $Q_N(T) \to T$ in mean square as $N \to \infty$. The crucial point is that the limit is deterministic — even though every $\Delta W_k$ is random.
We summarise this as $\;(dW)^2 = dt\;$ — an identity that doesn't quite make sense pathwise but is the right book-keeping for every Itô-calculus calculation. Anywhere a $(dW)^2$ appears in a Taylor expansion you may replace it by $dt$. Higher powers ($dW^3, dW\,dt, dt^2$) vanish.
| × | $dt$ | $dW$ |
|---|---|---|
| $dt$ | $0$ | $0$ |
| $dW$ | $0$ | $dt$ |
This is the single fact that distinguishes Itô calculus from Newton–Leibniz.
The hero of the chapter. Standard Brownian motion (a.k.a. the Wiener process) is the unique stochastic process $W_t$ satisfying:
$t \mapsto W_t$ is almost surely continuous, with $W_0 = 0$.
$W_t - W_s \sim \mathcal{N}(0, t-s)$ for all $s < t$.
For $0 \le t_1 < t_2 < t_3 < t_4$ the increments $W_{t_2}-W_{t_1}$ and $W_{t_4}-W_{t_3}$ are independent.
From these three axioms a great deal follows. The paths are nowhere differentiable (almost surely). They have infinite total variation on every interval. They achieve every real value infinitely often. They are self-similar: $W_{ct} \stackrel{d}{=} \sqrt{c}\,W_t$.
Heuristically a typical increment over $\delta t$ has size $\sqrt{\delta t}$, not $\delta t$. That mismatch is exactly why $(\Delta W)^2 / \delta t \to 1$ and is the seed of every stochastic-calculus subtlety.
To simulate a path on $[0,T]$ with $N$ steps: draw $Z_k \sim \mathcal{N}(0,1)$ i.i.d., set $\Delta W_k = \sqrt{\delta t}\,Z_k$, then $W_{t_k} = \sum_{j \le k}\Delta W_j$. The interactive widget on slide 10 does exactly this.
For a smooth function $f$ and a smooth path $g$ the Riemann–Stieltjes integral $\int_0^T f\,dg$ is defined by partition sums. That definition fails for $g = W$, because $W$ has unbounded variation.
Itô's fix: fix the evaluation point of the integrand to the left endpoint of each partition cell. Define
$$\int_0^T X_s\, dW_s \;:=\; \lim_{N \to \infty} \sum_{k=1}^{N} X_{t_{k-1}}\,(W_{t_k} - W_{t_{k-1}})$$
where convergence is in mean-square. The choice of left endpoint is essential: it makes the integral a martingale and respects the non-anticipating nature of finance — you decide how much stock to hold at $t_{k-1}$ before you see the move $\Delta W_k$.
$$\mathbb{E}\!\left[\,\Big(\int_0^T X_s\,dW_s\Big)^{\!2}\,\right] = \mathbb{E}\!\left[\,\int_0^T X_s^2\,ds\,\right]$$
The "variance of the integral equals the integral of the variance." This is the only formula you need to compute second moments.
If $X$ is reasonable (square-integrable, adapted), then
$$M_t = \int_0^t X_s\,dW_s$$
is a martingale with $\mathbb{E}[M_t] = 0$.
Evaluating the integrand at the midpoint instead of the left gives the Stratonovich integral, denoted $\int X \circ dW$. It obeys ordinary calculus — no second-derivative term — but loses the martingale property. Finance uses Itô.
The shorthand
$$dX_t \;=\; a(X_t, t)\,dt \;+\; b(X_t, t)\,dW_t$$
is precisely the integral equation
$$X_t \;=\; X_0 + \int_0^t a(X_s, s)\,ds + \int_0^t b(X_s, s)\,dW_s.$$
The first integral is an ordinary Riemann integral; the second is an Itô integral. The function $a$ is the drift, $b$ is the diffusion coefficient (sometimes called the volatility).
| Process | SDE | Used for |
|---|---|---|
| BM with drift | $dX = \mu\,dt + \sigma\,dW$ | arithmetic walks; toy models |
| Geometric BM | $dS = \mu S\,dt + \sigma S\,dW$ | share prices; Black–Scholes |
| Ornstein–Uhlenbeck | $dX = -\theta X\,dt + \sigma\,dW$ | mean-reverting rates |
| Cox–Ingersoll–Ross | $dr = \kappa(\theta-r)\,dt + \sigma\sqrt{r}\,dW$ | short rate (positive) |
If $a$ and $b$ are Lipschitz in $X$ and grow at most linearly, the SDE has a unique strong solution. The book skips proofs; we should know they exist for the cases above.
The fundamental theorem of the chapter. Suppose $X_t$ satisfies $dX = a\,dt + b\,dW$, and $f(x,t)$ is twice differentiable in $x$ and once in $t$. Then
$$\boxed{\; df \;=\; \Big(f_t + a\,f_x + \tfrac{1}{2}\,b^2\,f_{xx}\Big)\,dt \;+\; b\,f_x\,dW \;}$$
The shape is the same as the deterministic chain rule, except for the new $\tfrac{1}{2} b^2 f_{xx}$ piece in the drift — the Itô correction.
Taylor-expand $f(X_{t+dt}, t+dt) - f(X_t, t)$ to second order:
$$df = f_t\,dt + f_x\,dX + \tfrac{1}{2}f_{xx}(dX)^2 + \cdots$$
and use $dX = a\,dt + b\,dW$ together with the multiplication table $(dW)^2 = dt$, $dt\,dW = 0$, $(dt)^2 = 0$:
$$(dX)^2 = b^2 (dW)^2 + 2ab\,dt\,dW + a^2 (dt)^2 = b^2\,dt.$$
The $(dX)^2$ term, which Newton would throw away as "second order," now contributes a first-order $dt$ piece. That's the entire story.
For Black–Scholes, $S$ is geometric BM ($a = \mu S$, $b = \sigma S$) and $V$ is a function of $S$ and $t$. Then $dV = (V_t + \mu S V_S + \tfrac{1}{2}\sigma^2 S^2 V_{SS})\,dt + \sigma S\,V_S\,dW$ — this is the formula Wilmott applies to derive the BS PDE in chapter 6.
Side by side, deterministic and stochastic chain rules:
If $x(t)$ is smooth, $f = f(x,t)$:
$$df = f_t\,dt + f_x\,\dot x\,dt.$$
The Taylor expansion has only first-order terms. $(dx)^2 \sim (dt)^2$, which is negligible.
If $dX = a\,dt + b\,dW$:
$$df = (f_t + a f_x + \tfrac{1}{2} b^2 f_{xx})\,dt + b\,f_x\,dW.$$
The Taylor expansion must include the $f_{xx}(dX)^2$ piece. $(dW)^2 \sim dt$ is not negligible.
Let $f(x) = x^2$ and $X_t = W_t$. Then $df = 2W_t\,dW_t + dt$. Integrate:
$$W_T^2 = 2\int_0^T W_s\,dW_s + T.$$
Rearranging gives a useful identity that classical calculus would never produce:
$$\int_0^T W_s\,dW_s = \tfrac{1}{2} W_T^2 - \tfrac{1}{2}\,T.$$
The extra $-T/2$ is the Itô correction. It's also why $W_t^2 - t$ is a martingale (slide 02).
Whenever you expand a function of $W$ to second order, replace $(dW)^2$ by $dt$ and drop $dW\,dt$ and $(dt)^2$. Then re-collect into $dt$ and $dW$ parts. That's Itô's lemma in one sentence.
Three SDEs that recur in the rest of the book, each solved by an application of Itô's lemma.
$dX = \mu\,dt + \sigma\,dW \implies X_t = X_0 + \mu t + \sigma W_t$. Gaussian at every $t$, with $\mathbb{E}[X_t] = X_0 + \mu t$ and $\mathrm{Var}[X_t] = \sigma^2 t$. Can go negative — not a stock-price model.
$dS = \mu S\,dt + \sigma S\,dW$. Apply Itô to $f = \log S$ ($f_S = 1/S$, $f_{SS} = -1/S^2$, $f_t = 0$):
$$d(\log S) = \Big(\mu - \tfrac{1}{2}\sigma^2\Big)\,dt + \sigma\,dW.$$
So $\log S_t$ is Brownian motion with drift, hence Gaussian. Exponentiating:
$$S_t = S_0\,\exp\!\Big(\,(\mu - \tfrac{1}{2}\sigma^2) t + \sigma W_t\,\Big).$$
Always positive. $\mathbb{E}[S_t] = S_0 e^{\mu t}$ (the $-\tfrac{1}{2}\sigma^2$ is absorbed). This is the model Black–Scholes uses.
$dX = -\theta(X - \mu)\,dt + \sigma\,dW$. The drift pulls $X$ back toward $\mu$ at rate $\theta$. Closed form via integrating factor $e^{\theta t}$:
$$X_t = \mu + (X_0 - \mu)\,e^{-\theta t} + \sigma\int_0^t e^{-\theta(t-s)}\,dW_s.$$
Gaussian, with stationary variance $\sigma^2/(2\theta)$ as $t \to \infty$. Used for short rates (Vasicek) and for any quantity that "knows its own equilibrium."
Slide $T$ and the partition size $N$. The left panel plots a sample path of $W_t$; the middle panel plots the running quadratic variation $Q_t = \sum_{k\le t/\delta t}(\Delta W_k)^2$ together with the deterministic limit line $y=t$; the right panel is a histogram of $\Delta W_k$ over the path, with the theoretical $\mathcal{N}(0,\delta t)$ density overlaid.
Push $N$ up: the empirical sum tightens around $T$, even as the path you're sampling has rougher and rougher visual texture. That's the quadratic variation theorem in motion — the path becomes more wiggly but its quadratic content settles to $T$. Resample with the button to see how stable the limit is despite path-by-path randomness.