Covariation, the Product Rule, and Multiple Brownian Motions

A two-asset derivatives desk prices products where the P&L depends on two correlated stock prices

S^1_t

and

S^2_t

simultaneously — spread options (

V = (S^1_T - S^2_T - K)^+

), basket options, and cross-Gamma terms in the hedging PDE. The key tool is the covariation process

[X, Y]_t

, which tracks how two Itô processes co-oscillate:

d[X, Y]_t = dX_t \cdot dY_t \quad \text{(computed via the Itô multiplication table)}.

For $dX_t = A_t \, dW^1_t$ and $dY_t = C_t \, dW^1_t$ (same BM), $d[X,Y]_t = A_t C_t \, dt$ . For independent drivers $dX_t = A_t \, dW^1_t$ and $dY_t = C_t \, dW^2_t$ , $d[X,Y]_t = 0$ (independent BMs have zero covariation).

The product rule $d(XY) = X \, dY + Y \, dX + d[X,Y]$ then extends to all Itô processes, and the multidimensional Itô formula (Theorem 3.7.1) gives the chain rule for functions of several Itô processes — the formula underlying correlated-asset PDEs and the multi-factor interest rate models.

Learning objectives. By the end of this note you should be able to:

Define the covariation process $[X,Y]_t$ from the partition limit and compute it via the multiplication table.
State and apply the stochastic product rule (Theorem 3.6.1).
Work with multiple independent BMs and compute cross-covariations $[X,Y]_t = \sum_i A_t^i C_t^i \, dt$ .
State the final form of Itô's formula (Theorem 3.7.1) in $n$ variables driven by $d$ BMs.
Derive the two-asset Black-Scholes PDE from the multidimensional formula.

In this lesson

Covariation of Two Itô Processes
Stochastic Product Rule
Multiple Independent Brownian Motions
Multidimensional Itô's Formula

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