Correlated Brownian motions and multi-asset models

Lesson 15 of 171 min read

Everything so far used one Brownian motion at a time. Real models rarely do. A stock and its volatility move together, a basket option depends on several assets at once, and a portfolio is exactly the situation of "Diversification and correlation". This lesson builds two Brownian motions with a chosen correlation, from the properties in "Properties of Brownian motion", shows how the same construction extends to many assets, and states the two-dimensional version of the rule in "Itô's lemma". It also says where correlation enters pricing models and how it is simulated with the scheme of "Simulating SDEs: the Euler–Maruyama scheme".

Two Brownian motions with correlation ρ

Two standard Brownian motions $W_1$ and $W_2$ have correlation $\rho \in [-1, 1]$ if their increments over a short interval $dt$ have correlation $\rho$ . In the language of quadratic variation this is a single rule for the cross term:

d\langle W_1, W_2 \rangle_t = \rho\,dt, \qquad d\langle W_i \rangle_t = dt.

It is easy to build such a pair from two independent Brownian motions $W_1$ and $W_\perp$ . Keep $W_1$ and mix the other in:

W_2 = \rho\,W_1 + \sqrt{1 - \rho^2}\;W_\perp.

Then $W_2$ has independent normal increments with variance $(\rho^2 + 1 - \rho^2)\,dt = dt$ , so it is a standard Brownian motion in its own right, and its increment covaries with that of $W_1$ by $\rho\,dt$ . At $\rho = 0$ the two are independent. At $\rho = \pm 1$ they are the same path or its mirror image. The figure draws both paths from one set of random shocks and lets you move $\rho$ . Start at the default $\rho = -0.7$ and watch the paths mostly mirror each other; the sample correlation of the increments is noisy for a path with only 250 steps, but the construction fixes the exact value of $\langle W_1, W_2 \rangle_T = \rho\,T$ , which is $-0.70$ for $T = 1$ .

Figure: Correlated Brownian motions and multi-asset modelsOpen in Heston

Correlation ρ-0.70

-0.950.95

W₁W₂

Sample correlation of increments: -0.64
Cholesky L₂₂ = √(1 − ρ²): 0.714
⟨W₁, W₂⟩ at T = ρT: -0.70

The Cholesky factor

The construction above is a matrix product. Write the independent shocks of one step as a vector $\varepsilon = (\varepsilon_1, \varepsilon_2)$ of standard normals and the correlated ones as $Z = L\,\varepsilon$ with

L = \begin{pmatrix} 1 & 0 \\ \rho & \sqrt{1 - \rho^2} \end{pmatrix}, \qquad L L^{\top} = \begin{pmatrix} 1 & \rho \\ \rho & 1 \end{pmatrix}.

This $L$ is the Cholesky factor of the correlation matrix. For $n$ assets with correlation matrix $C$ , the same idea works: find a lower-triangular $L$ with $L L^{\top} = C$ and set $Z = L\,\varepsilon$ for independent $\varepsilon$ . Such an $L$ exists exactly when $C$ is positive definite, which is a real restriction. A table of pairwise correlations chosen independently, say $\rho_{12} = \rho_{13} = 0.9$ and $\rho_{23} = -0.9$ , can fail it, and then no set of random variables has those correlations. The algorithm reports the failure when a square root of a non-positive number would be needed; the figure's readout $\sqrt{1 - \rho^2}$ is that square root for two assets, and it vanishes at $\rho = \pm 1$ .

The two-dimensional Itô rule

Let $dX = \mu_1\,dt + \sigma_1\,dW_1$ and $dY = \mu_2\,dt + \sigma_2\,dW_2$ with $d\langle W_1, W_2 \rangle = \rho\,dt$ . For a smooth function $f(t, x, y)$ , "Itô's lemma" extends by one extra term, the cross term, because $dX\,dY = \rho\,\sigma_1 \sigma_2\,dt$ :

df = f_t\,dt + f_x\,dX + f_y\,dY + \tfrac{1}{2} f_{xx}\,\sigma_1^2\,dt + \tfrac{1}{2} f_{yy}\,\sigma_2^2\,dt + f_{xy}\,\rho\,\sigma_1 \sigma_2\,dt.

With $f(x, y) = x\,y$ this is the product rule for correlated processes:

d(XY) = X\,dY + Y\,dX + \rho\,\sigma_1 \sigma_2\,dt.

For two Brownian motions themselves ( $X = W_1$ , $Y = W_2$ , $\sigma_1 = \sigma_2 = 1$ ) the extra term is $\rho\,dt$ . The product $W_1 W_2$ is therefore not a martingale; its drift is $\rho$ , and taking expectations gives $E[W_1(T)\,W_2(T)] = \rho\,T$ .

Where correlation enters

In the Heston model of "Stochastic volatility and the Heston model", the price and the variance are driven by Brownian motions with correlation $\rho$ :

dS = \mu S\,dt + \sqrt{v}\,S\,dW_1, \qquad dv = \kappa(\theta - v)\,dt + \xi\sqrt{v}\,dW_2, \qquad d\langle W_1, W_2 \rangle = \rho\,dt.

Equity indices fall when volatility rises, so the calibrated $\rho$ is negative, typically between $-0.5$ and $-0.9$ . That negative sign is what tilts the implied volatility smile into a skew. Correlation also drives basket and spread options, whose payoff depends on several assets: the variance of a weighted sum includes $2\,w_i w_j\,\rho_{ij}\,\sigma_i \sigma_j$ terms, exactly as the portfolio variance does. A copula would let the dependence differ from this linear one; we do not need it here.

Simulating correlated assets

In the Euler scheme the only change is the shock. At each step draw independent standard normals $\varepsilon_1, \varepsilon_2$ and use $Z_1 = \varepsilon_1$ , $Z_2 = \rho\,\varepsilon_1 + \sqrt{1 - \rho^2}\,\varepsilon_2$ in place of the two independent shocks. Everything else, including the pricing average of "Monte Carlo option pricing", stays as it was.

Worked example

Correlated shocks for ρ = −0.7, and the product rule

The weight of the independent shock is the second Cholesky entry.
$\displaystyle L_{22} = \sqrt{1 - (-0.7)^2}$ 0.7141
Draw $\varepsilon_1 = 0.8$ and $\varepsilon_2 = -0.3$ . The first correlated shock is $Z_1 = \varepsilon_1 = 0.8$ ; the second mixes both.
$\displaystyle Z_2 = \rho\,\varepsilon_1 + L_{22}\,\varepsilon_2 = -0.7 \times 0.8 + 0.7141 \times (-0.3)$ -0.7742
The product rule gives $d(W_1 W_2) = W_2\,dW_1 + W_1\,dW_2 + \rho\,dt$ . The two stochastic integrals have mean zero, so the expected product after $T = 1$ is $\rho T$ .
$\displaystyle E[W_1(1)\,W_2(1)] = \rho \cdot 1$ -0.7

Summary

Two Brownian motions have correlation $\rho$ when $d\langle W_1, W_2 \rangle = \rho\,dt$ ; build them as $W_2 = \rho W_1 + \sqrt{1 - \rho^2}\,W_\perp$ .
For $n$ assets, correlated shocks are $Z = L\varepsilon$ , where $L L^{\top} = C$ is the Cholesky factor, which exists exactly when $C$ is positive definite.
The two-dimensional Itô rule adds the cross term $f_{xy}\,\rho\,\sigma_1 \sigma_2\,dt$ , which gives $d(XY) = X\,dY + Y\,dX + \rho\,\sigma_1 \sigma_2\,dt$ .
Correlation drives the skew in Heston (with $\rho < 0$ for equities) and the price of basket and spread options.

The next lesson, "Mean reversion: Ornstein–Uhlenbeck and Vasicek", returns to one-dimensional models and adds a pull towards a long-run level.

2 questions on this lesson at the end of the series

Correlated Brownian motions and multi-asset models

Lesson 15 of 171 min read

Two Brownian motions with correlation ρ

d\langle W_1, W_2 \rangle_t = \rho\,dt, \qquad d\langle W_i \rangle_t = dt.

It is easy to build such a pair from two independent Brownian motions $W_1$ and $W_\perp$ . Keep $W_1$ and mix the other in:

W_2 = \rho\,W_1 + \sqrt{1 - \rho^2}\;W_\perp.

Figure: Correlated Brownian motions and multi-asset modelsOpen in Heston

Correlation ρ-0.70

-0.950.95

W₁W₂

Sample correlation of increments: -0.64
Cholesky L₂₂ = √(1 − ρ²): 0.714
⟨W₁, W₂⟩ at T = ρT: -0.70

The Cholesky factor

L = \begin{pmatrix} 1 & 0 \\ \rho & \sqrt{1 - \rho^2} \end{pmatrix}, \qquad L L^{\top} = \begin{pmatrix} 1 & \rho \\ \rho & 1 \end{pmatrix}.

The two-dimensional Itô rule

df = f_t\,dt + f_x\,dX + f_y\,dY + \tfrac{1}{2} f_{xx}\,\sigma_1^2\,dt + \tfrac{1}{2} f_{yy}\,\sigma_2^2\,dt + f_{xy}\,\rho\,\sigma_1 \sigma_2\,dt.

With $f(x, y) = x\,y$ this is the product rule for correlated processes:

d(XY) = X\,dY + Y\,dX + \rho\,\sigma_1 \sigma_2\,dt.

Where correlation enters

In the Heston model of "Stochastic volatility and the Heston model", the price and the variance are driven by Brownian motions with correlation $\rho$ :

dS = \mu S\,dt + \sqrt{v}\,S\,dW_1, \qquad dv = \kappa(\theta - v)\,dt + \xi\sqrt{v}\,dW_2, \qquad d\langle W_1, W_2 \rangle = \rho\,dt.

Simulating correlated assets

Worked example

Correlated shocks for ρ = −0.7, and the product rule

The weight of the independent shock is the second Cholesky entry.
$\displaystyle L_{22} = \sqrt{1 - (-0.7)^2}$ 0.7141
Draw $\varepsilon_1 = 0.8$ and $\varepsilon_2 = -0.3$ . The first correlated shock is $Z_1 = \varepsilon_1 = 0.8$ ; the second mixes both.
$\displaystyle Z_2 = \rho\,\varepsilon_1 + L_{22}\,\varepsilon_2 = -0.7 \times 0.8 + 0.7141 \times (-0.3)$ -0.7742
The product rule gives $d(W_1 W_2) = W_2\,dW_1 + W_1\,dW_2 + \rho\,dt$ . The two stochastic integrals have mean zero, so the expected product after $T = 1$ is $\rho T$ .
$\displaystyle E[W_1(1)\,W_2(1)] = \rho \cdot 1$ -0.7

Summary

Two Brownian motions have correlation $\rho$ when $d\langle W_1, W_2 \rangle = \rho\,dt$ ; build them as $W_2 = \rho W_1 + \sqrt{1 - \rho^2}\,W_\perp$ .
For $n$ assets, correlated shocks are $Z = L\varepsilon$ , where $L L^{\top} = C$ is the Cholesky factor, which exists exactly when $C$ is positive definite.
The two-dimensional Itô rule adds the cross term $f_{xy}\,\rho\,\sigma_1 \sigma_2\,dt$ , which gives $d(XY) = X\,dY + Y\,dX + \rho\,\sigma_1 \sigma_2\,dt$ .
Correlation drives the skew in Heston (with $\rho < 0$ for equities) and the price of basket and spread options.

The next lesson, "Mean reversion: Ornstein–Uhlenbeck and Vasicek", returns to one-dimensional models and adds a pull towards a long-run level.

2 questions on this lesson at the end of the series