Chapter 8 — Multi-Dimensional Itô Calculus

Guiding question

Most real financial systems involve many correlated assets. How does Itô calculus extend to vector-valued processes, and what role does the correlation matrix play? When does correlation create new drift terms?

Every multi-asset model — baskets, spreads, rainbow options, cross-currency products — requires understanding how correlated Brownian motions interact. The key new object is the cross-variation $d[W^i, W^j] = \rho_{ij}\,dt$, which encodes how randomness in one dimension influences another.

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8.1 Multi-dimensional Brownian motion

A $d$-dimensional standard Brownian motion is a vector process $\mathbf W_t = (W_t^1,\ldots,W_t^d)$ where:

1. Each $W_t^i$ is a one-dimensional Brownian motion.
2. The components are independent: $W_t^i - W_s^i$ is independent of $W_t^j - W_s^j$ for $i\ne j$.
3. $\mathbf W_0 = \mathbf 0$.

For a correlated $d$-dimensional Brownian motion with correlation matrix $\rho = (\rho_{ij})_{i,j=1}^d$, write

$$\mathbf W_t = \mathbf B_t\Sigma^{1/2},$$

where $\mathbf B_t$ is a standard $d$-dimensional BM and $\Sigma^{1/2}$ is the Cholesky factor of the correlation matrix $\rho$ (so $(\Sigma^{1/2})(\Sigma^{1/2})^\top = \rho$).

The defining properties of correlated BM are:

$$\text{Cov}(W_t^i, W_t^j) = \rho_{ij}\,t, \qquad d[W^i,W^j]_t = \rho_{ij}\,dt.$$

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8.2 Constructing correlated BM from independent BM

Given independent BMs $Z^1,\ldots,Z^d$ and a correlation matrix $\rho$ with Cholesky decomposition $\rho = LL^\top$, set:

$$W_t^i = \sum_j L_{ij}Z_t^j.$$

Then

$$d[W^i,W^j] = \sum_k L_{ik}\,dZ^k\,\sum_\ell L_{j\ell}\,dZ^\ell = \sum_k L_{ik}L_{jk}\,dt = (LL^\top)_{ij}\,dt = \rho_{ij}\,dt.$$

Correlated Brownian motions: dW₁dW₂ = ρ dt

Correlation ρ = 0.70

Cross-variation ⟨W¹,W²⟩_T ≈0.7012 (theory: 0.70)

Covariation ruledW¹dW² = ρ dt

When ρ = 1 the two paths are identical; ρ = 0 gives independent paths; ρ = −1 gives mirror paths. The scatter ellipse tilts with ρ, geometrically encoding the covariance. This bivariate structure underpins basket options, spreads, and correlated rate models.

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8.3 The multi-dimensional Itô formula

Let $\mathbf X_t = (X_t^1,\ldots,X_t^n)$ be a vector of Itô processes:

$$dX_t^i = b_t^i\,dt + \sum_{j=1}^d \sigma_t^{ij}\,dW_t^j, \qquad i=1,\ldots,n.$$

The quadratic (co-)variation is:

$$d[X^i,X^j]_t = \sum_{k=1}^d\sigma_t^{ik}\sigma_t^{jk}\,dt = (\sigma\sigma^\top)_{ij}\,dt =: c_t^{ij}\,dt.$$

Multi-dimensional Itô formula

For $f\in C^{1,2}([0,T]\times\mathbb R^n)$,

$$df(t,\mathbf X_t) = \partial_t f\,dt + \sum_i \partial_i f\,dX_t^i + \tfrac12\sum_{i,j}\partial_{ij}f\,d[X^i,X^j]_t.$$

Expanded:

$$df(t,\mathbf X_t) = \left[\partial_t f + \sum_i b_t^i\partial_i f + \tfrac12\sum_{i,j}c_t^{ij}\partial_{ij}f\right]dt + \sum_{i,k}\sigma_t^{ik}\partial_i f\,dW_t^k.$$

The Hessian and the diffusion matrix

The second-order term $\frac12\sum_{i,j}c^{ij}\partial_{ij}f$ is the contraction of the Hessian $\nabla^2 f$ with the diffusion covariance matrix $c = \sigma\sigma^\top$. In geometric terms, this is the trace of the product $c\cdot\nabla^2 f$. The Laplacian $\Delta f$ corresponds to $c = I$ (isotropic diffusion). Anisotropic diffusion — different volatilities and correlations — creates a non-isotropic "heat" operator.

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8.4 The Itô multiplication table in $d$ dimensions

The extended multiplication table for correlated BMs $(dW^i_t\,dW^j_t = \rho_{ij}\,dt)$:

	$dW^i$	$dt$
$dW^j$	$\rho_{ij}\,dt$	$0$
$dt$	$0$	$0$

For uncorrelated components ($i\ne j$, $\rho_{ij}=0$): $dW^i\,dW^j = 0$. Cross terms vanish, and Itô's formula simplifies to the sum of independent one-dimensional formulas.

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8.5 Key example: the log of a basket

Let $S_t^1,\ldots,S_t^n$ follow correlated GBMs:

$$dS_t^i = \mu_i S_t^i\,dt + \sigma_i S_t^i\,dW_t^i, \qquad d[W^i,W^j] = \rho_{ij}\,dt.$$

The log of the basket $B_t = \sum_i w_i S_t^i$ (equal weights $w_i = 1/n$) satisfies:

$$dB_t = \sum_i w_i\,dS_t^i = \left(\sum_i w_i\mu_i S_t^i\right)dt + \sum_i w_i\sigma_i S_t^i\,dW_t^i.$$

The variance of $dB_t/B_t$ is (locally):

$$\sigma_{\text{basket}}^2\,dt = \frac{1}{B_t^2}\sum_{i,j}w_i w_j\sigma_i\sigma_j\rho_{ij}S_t^i S_t^j\,dt.$$

This formula shows that correlation reduces basket volatility: when $\rho_{ij} < 1$, the basket is less volatile than any single component.

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8.6 Correlated Black–Scholes: two assets

For two correlated assets $S^1, S^2$ with correlation $\rho$, the joint risk-neutral dynamics are:

$$dS_t^1 = rS_t^1\,dt + \sigma_1 S_t^1\,dW_t^1, \qquad dS_t^2 = rS_t^2\,dt + \sigma_2 S_t^2(\rho\,dW_t^1 + \sqrt{1-\rho^2}\,dW_t^2),$$

where $W^1,W^2$ are independent BMs under $\mathbb Q$. The correlation structure enters through the $\rho\,dW^1$ term in $S^2$'s dynamics.

The covariance matrix of the log-returns over $[0,T]$ is:

$$\Sigma = T\begin{pmatrix}\sigma_1^2 & \rho\sigma_1\sigma_2\\ \rho\sigma_1\sigma_2 & \sigma_2^2\end{pmatrix}.$$

Correlation in options: dispersion trading

When correlation $\rho$ decreases, basket options become cheaper relative to individual options. Dispersion traders sell basket variance and buy individual variance, profiting when realized correlation is lower than implied correlation.

The key formula: $\sigma_{\text{basket}}^2 = \sum_{i,j}w_iw_j\rho_{ij}\sigma_i\sigma_j$. Lower correlation $\to$ lower basket vol $\to$ basket options are cheaper. This is the mathematical basis of the diversification benefit.

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8.7 Itô's product rule in $d$ dimensions

For two Itô processes $X$ and $Y$ with possibly non-zero cross-variation:

$$d(XY) = X\,dY + Y\,dX + d[X,Y].$$

For $X = S^1$, $Y = S^2$ with $d[S^1,S^2] = \rho\sigma_1\sigma_2 S^1 S^2\,dt$:

$$d(S^1 S^2) = S^2\,dS^1 + S^1\,dS^2 + \rho\sigma_1\sigma_2 S^1 S^2\,dt.$$

The extra term $\rho\sigma_1\sigma_2 S^1 S^2\,dt$ is the stochastic correction from the cross-variation. It modifies the drift of the product process.

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8.8 Multi-dimensional Girsanov

For a $d$-dimensional BM $\mathbf W$ under $\mathbb P$ and adapted process $\boldsymbol\theta = (\theta^1,\ldots,\theta^d)$, the stochastic exponential is:

$$Z_T = \exp\!\left(\sum_i\int_0^T\theta_s^i\,dW_s^i - \tfrac12\sum_i\int_0^T(\theta_s^i)^2\,ds\right).$$

Under $\mathbb Q$ with $dQ/dP = Z_T$:

$$\widetilde W_t^i = W_t^i - \int_0^t\theta_s^i\,ds$$

is a $d$-dimensional $\mathbb Q$-BM.

In the multi-asset Black–Scholes model, the market prices of risk $\boldsymbol\theta = ((\mu_1-r)/\sigma_1, \ldots, (\mu_d-r)/\sigma_d)$ are used to transform $\mathbb P$ to $\mathbb Q$.

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8.9 Worked example: spread option

A spread option pays $(S_T^1 - S_T^2 - K)^+$. Under $\mathbb Q$:

$$\log(S_T^i/S_0^i) \sim N\!\left((r-\tfrac12\sigma_i^2)T, \sigma_i^2 T\right),$$

and the correlation is $\text{Corr}(\log S_T^1,\log S_T^2) = \rho$.

No closed-form Black–Scholes formula exists for the spread. The Margrabe formula gives the price of an exchange option $(S_T^1 - S_T^2)^+$ with $K=0$:

$$\text{Margrabe} = S_0^1 N(d_1) - S_0^2 N(d_2),$$ $$d_1 = \frac{\log(S_0^1/S_0^2) + \frac12\sigma^2 T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T}, \quad \sigma^2 = \sigma_1^2 + \sigma_2^2 - 2\rho\sigma_1\sigma_2.$$

The effective volatility $\sigma$ is the volatility of the ratio $S^1/S^2$.

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8.10 Problems

Problem 1: Quadratic variation of a linear combination

Let $X_t = aW_t^1 + bW_t^2$ with $d[W^1,W^2] = \rho\,dt$. Find $[X]_t$.

Problem 2: Itô for $f(X,Y) = XY$

Let $dX = \sigma_X\,dW^1$ and $dY = \sigma_Y(\rho\,dW^1 + \sqrt{1-\rho^2}\,dW^2)$. Compute $d(XY)$ and identify the extra drift term.

Problem 3: Log of correlated GBM

For two correlated GBMs as above, compute $d(\log S^1/S^2)$. This is the dynamics of the log spread. What does it mean for $\rho=1$?

Problem 4: Margrabe check

Verify that the Margrabe formula satisfies put-call parity: $(S_T^1 - S_T^2)^+ - (S_T^2 - S_T^1)^+ = S_T^1 - S_T^2$.

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8.11 Selected solutions

Solution 1

$[X]_t = a^2[W^1]_t + 2ab[W^1,W^2]_t + b^2[W^2]_t = a^2 t + 2ab\rho t + b^2 t = (a^2 + 2ab\rho + b^2)t$.

Solution 2

Using Itô's formula for the product:

$$d(XY) = X\,dY + Y\,dX + d[X,Y],$$

where $d[X,Y] = \sigma_X\sigma_Y\rho\,dt$ (only the $dW^1$ components contribute to cross-variation). Therefore:

$$d(XY) = X\sigma_Y(\rho\,dW^1+\sqrt{1-\rho^2}\,dW^2) + Y\sigma_X\,dW^1 + \sigma_X\sigma_Y\rho\,dt.$$

The extra drift $\sigma_X\sigma_Y\rho\,dt$ appears only because of correlation.

Solution 3

Let $R_t = S_t^1/S_t^2$. By Itô's quotient rule (which follows from the multi-dimensional formula):

$$d\log R_t = d\log S_t^1 - d\log S_t^2 = (\mu_1-\mu_2)\,dt + (\sigma_1^2-\rho\sigma_1\sigma_2)\,dt + \sigma_1\,dW^1 - \sigma_2\rho\,dW^1 - \sigma_2\sqrt{1-\rho^2}\,dW^2.$$

More cleanly: $d\log R = (\mu_1-\mu_2-\frac12\sigma_{\text{eff}}^2)\,dt + \sigma_{\text{eff}}\,d\hat W$ where $\sigma_{\text{eff}}^2 = \sigma_1^2-2\rho\sigma_1\sigma_2+\sigma_2^2$. For $\rho=1$ and $\sigma_1=\sigma_2$: $\sigma_{\text{eff}}=0$, and $\log R$ is deterministic — the ratio $S^1/S^2$ is non-random.

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8.12 What this chapter gives you

Concept	Formula
Correlated BM	$d[W^i,W^j] = \rho_{ij}\,dt$
Multi-dimensional Itô	$df = \partial_t f\,dt + \sum\partial_i f\,dX^i + \frac12\sum\partial_{ij}f\,c^{ij}\,dt$
Product rule	$d(XY) = X\,dY + Y\,dX + d[X,Y]$
Basket vol	$\sigma_{\rm basket}^2 = \sum_{ij}w_iw_j\rho_{ij}\sigma_i\sigma_j$
Margrabe exchange	$(S^1 - S^2)^+$ priced by replacing $\sigma^2$ with $\sigma_1^2+\sigma_2^2-2\rho\sigma_1\sigma_2$

The next chapter develops the infinitesimal generator, which packages all of Itô calculus into an operator acting on functions — the bridge to partial differential equations.