Correlation matrices, Cholesky factorisation, the multi-dimensional BS PDE, baskets and Margrabe — and why correlation is the hardest parameter in the book.
The natural generalisation of Black–Scholes to $n$ underlyings. Each $S_i$ follows its own GBM, with correlated drivers:
$$dS_i = \mu_i S_i\, dt + \sigma_i S_i\, dW_i, \qquad i = 1, \dots, n,$$
$$\mathrm{corr}(dW_i, dW_j) = \rho_{ij}.$$
Three sets of parameters:
One per asset. Under risk-neutral measure each is replaced by $r - q_i$ where $q_i$ is the dividend / convenience yield of asset $i$.
One per asset. The marginal scale of fluctuation. Usually calibrated from each asset's own vanilla market.
$n(n-1)/2$ pairwise numbers. Couples the assets. The painful parameter — observable only indirectly, time-varying, and the one the model is most sensitive to.
Each $S_i$ marginally is still a lognormal; what's new is the joint distribution. The terminal vector $(S_1^T, \dots, S_n^T)$ is multivariate lognormal — lognormal margins glued together by a Gaussian copula in log-space.
Stack the pairwise correlations into a matrix:
$$\rho = \begin{pmatrix} 1 & \rho_{12} & \cdots & \rho_{1n} \\ \rho_{12} & 1 & \cdots & \rho_{2n} \\ \vdots & & \ddots & \vdots \\ \rho_{1n} & \rho_{2n} & \cdots & 1 \end{pmatrix}.$$
$\rho_{ij} = \rho_{ji}$. Correlation is a property of the pair, not of order. The matrix is determined by its upper triangle.
$\rho_{ii} = 1$. Every asset is perfectly correlated with itself by definition.
$x^\top \rho\, x \ge 0$ for all $x$. Equivalent to: every eigenvalue is non-negative. The variance of any linear combination must be non-negative — if it weren't, that combination would have a negative variance, which is impossible.
The PSD constraint is non-trivial when $n$ is large. A naive estimate (pairwise sample correlations on different histories) often yields a matrix with a small negative eigenvalue — an "almost-PSD" matrix that has to be projected back. Higham's algorithm finds the nearest correlation matrix in Frobenius norm.
If $\rho_{12} = 0.9$, $\rho_{13} = 0.9$, $\rho_{23} = -0.5$ — this matrix is not PSD. Two assets each highly correlated with a third cannot be negatively correlated with each other. The PSD condition is the matrix-level expression of these triangle-inequality-style constraints.
How do you generate $n$ correlated normals on a computer that has only independent ones? The Cholesky decomposition.
For any symmetric positive-definite $\rho$, there is a unique lower-triangular $L$ with positive diagonal such that
$$\rho = LL^\top.$$
Given $\mathbf{Z} = (Z_1, \dots, Z_n)^\top$ with i.i.d. standard normal entries, the vector
$$\mathbf{X} = L\, \mathbf{Z}$$
has $\mathrm{Cov}(\mathbf{X}) = L\, I\, L^\top = LL^\top = \rho$. So $\mathbf{X}$ has the desired correlation structure.
For two assets with correlation $\rho$:
$$L = \begin{pmatrix} 1 & 0 \\ \rho & \sqrt{1-\rho^2} \end{pmatrix},\quad \mathbf{X} = L\mathbf{Z} = \begin{pmatrix} Z_1 \\ \rho Z_1 + \sqrt{1-\rho^2}\, Z_2 \end{pmatrix}.$$
So $X_1$ takes the first noise as-is; $X_2$ mixes $\rho$ of that same noise with $\sqrt{1-\rho^2}$ of fresh independent noise. Easy to verify: $\mathrm{Var}(X_2) = \rho^2 + (1-\rho^2) = 1$ and $\mathrm{Cov}(X_1, X_2) = \rho$.
Cholesky is $O(n^3)$ for the factorisation but only $O(n^2)$ per draw thereafter — you factor $\rho$ once and reuse $L$ for every Monte Carlo step. For very large $n$, eigendecomposition with truncation (PCA) is an alternative that also dimension-reduces.
For $V = V(S_1, \dots, S_n, t)$ smooth, applying Itô to each $dS_i$ and collecting the second-order terms gives
$$dV = \frac{\partial V}{\partial t}\,dt + \sum_{i=1}^{n}\frac{\partial V}{\partial S_i}\, dS_i + \tfrac{1}{2}\sum_{i,j=1}^{n} \rho_{ij}\,\sigma_i\sigma_j\, S_i S_j\, \frac{\partial^2 V}{\partial S_i \partial S_j}\, dt.$$
The new piece, compared to single-asset Itô, is the double sum — all pairs $(i,j)$ contribute, including $i \ne j$ off-diagonal terms whose coefficient is $\rho_{ij}\sigma_i\sigma_j S_iS_j$. The mixed second partials $V_{S_iS_j}$ are the cross-gammas, and they pick up explicit correlation dependence.
One way to read the formula: define the "diffusion matrix"
$$\Sigma_{ij} = \rho_{ij}\sigma_i\sigma_j S_i S_j$$
and the Itô correction is $\tfrac{1}{2}\,\mathrm{tr}(\Sigma\, \mathrm{Hess}_S V)\,dt$. Setting up the right $\Sigma$ is half the work of multi-asset pricing.
$n=1$: $\rho_{11} = 1$, no off-diagonal terms, and you recover $dV = V_t\,dt + V_S\,dS + \tfrac{1}{2}\sigma^2 S^2 V_{SS}\,dt$ — the Itô formula of deck 05.
Construct a hedged portfolio $\Pi = V - \sum_i \Delta_i S_i$. Differentiate, choose $\Delta_i = \partial V / \partial S_i$ to cancel all the $dW_i$ terms, equate the remaining deterministic return to the risk-free rate. The result is the multi-asset Black–Scholes equation:
$$\boxed{\;\frac{\partial V}{\partial t} + \tfrac{1}{2}\sum_{i,j=1}^{n}\rho_{ij}\sigma_i\sigma_j S_i S_j \frac{\partial^2 V}{\partial S_i \partial S_j} + \sum_{i=1}^{n}(r - q_i) S_i \frac{\partial V}{\partial S_i} - rV = 0.\;}$$
Same shape as the single-asset PDE, with the second-derivative term replaced by the matrix-style sum. Now solved on $\mathbb{R}_+^n \times [0,T]$, with terminal condition $V(S_1,\dots,S_n,T) = $ payoff.
The PDE is $n+1$ dimensional. Finite-difference grids scale as $G^n$ where $G$ is the grid resolution per dimension. With $G = 50$ this is fine for $n=2$, painful for $n=3$, infeasible for $n \ge 4$. Hence: Monte Carlo is the workhorse for multi-asset pricing as soon as $n > 3$, and tensor-train / PCA-projection methods for very large $n$.
Payoff written on a weighted sum:
$$B_T = \sum_{i=1}^{n} w_i S_i^T, \qquad \text{payoff} = \max(B_T - K, 0).$$
An index option is the natural example — the S&P 500 is a basket of 500 components with capitalisation weights. Index futures and ETF options are baskets in everything but name.
The sum of lognormals is not lognormal. There's no exact Black–Scholes formula even when each leg is itself a lognormal.
Levy (1992) and Turnbull–Wakeman style approximations: choose an effective volatility $\sigma_B$ and an effective drift so that the first two moments of the basket match those of an artificial lognormal:
$$\sigma_B^2 = \frac{1}{T}\ln\!\left(\frac{E[B_T^2]}{(E[B_T])^2}\right),$$
then plug $\sigma_B$ into a single-asset Black–Scholes formula with spot $E[B_T]e^{-rT}$. Surprisingly accurate at the money; worse in the tails. Modern practice: use the lognormal approximation as a control variate for MC.
Basket-option price is a decreasing function of average correlation in most regimes. A basket call with $\rho = 1$ is just a vanilla on a single asset (worth more); with $\rho = 0$ the basket variance is averaged down by diversification (worth less). This relationship is what dispersion trading exploits.
The rare two-asset payoff that does admit a closed form. Margrabe (1978): an option to exchange asset 2 for asset 1 at maturity,
$$\text{payoff} = \max(S_1^T - S_2^T, 0).$$
Divide everything by $S_2$. The ratio $R = S_1 / S_2$ is itself lognormal under the $S_2$-measure, with variance
$$\sigma_R^2 = \sigma_1^2 + \sigma_2^2 - 2\rho\,\sigma_1\sigma_2.$$
(The minus sign is the variance-of-difference identity in log-space; correlation reduces dispersion.) The exchange option then prices like a vanilla call on $R$ with strike $1$, zero rate, and the effective vol $\sigma_R$:
$$V = S_1 N(d_1) - S_2 N(d_2), \qquad d_1 = \frac{\ln(S_1/S_2) + \tfrac{1}{2}\sigma_R^2 T}{\sigma_R\sqrt{T}},\; d_2 = d_1 - \sigma_R\sqrt{T}.$$
No interest rate appears in the formula — the rate has cancelled, because the exchange is between two risky assets. This is the cleanest example in derivative pricing of numéraire choice as a problem-simplification technique.
Set $\sigma_2 = 0$ and $S_2 = K e^{-rT}$: Margrabe reduces to ordinary Black–Scholes. So vanilla BS is an exchange option — exchange the asset for a bond worth $K$.
Multi-asset pricing inherits all the difficulties of single-asset pricing — volatility surfaces, skew, jumps — and adds correlation, which is harder than every one of them.
You can compute a sample correlation from historical returns, but it's noisy, time-varying, and what you actually need is the forward-looking correlation under the pricing measure — which the market doesn't quote explicitly.
Implied correlation must be extracted from basket-vs-component option prices: take dispersion trades' implied vols and back out the correlation that makes them consistent. Limited liquidity outside indices.
The single biggest model failure of 2008: correlations between "diversified" assets all jumped towards $1$ when systemic risk hit. A model fitted in calm times underprices the basket put in a stressed time.
Practitioner fix: stress correlations as part of risk management; price with $\rho \to 1$ as a sanity check; consider regime-switching or stochastic-correlation models for tail products.
Define cega: $\partial V / \partial \rho$. For a basket call it's negative; for a worst-of (or rainbow) it's positive. Hedging cega means trading correlation — via dispersion books, or via single-name vs index variance swaps. A specialist activity.
"If your multi-asset model agrees with the market to four decimal places, you have probably overfitted." The right standard is robustness across plausible correlation regimes, not fit to a single snapshot.
Wilmott closes chapter 12 with a useful distinction. Two time series can be very correlated (their moves agree) without being cointegrated (their levels stay together), and vice versa.
A property of returns. A short-horizon statistic. Two stocks can have correlation $0.9$ in daily returns yet drift apart over a decade — the levels are unbounded random walks even when the increments line up.
Useful for: pricing options whose payoff depends on a snapshot of joint behaviour (baskets, Margrabe, worst-of). Wrong tool for long-horizon dynamics.
A property of levels. Two series are cointegrated if a linear combination is stationary — the spread is mean-reverting even though each leg is non-stationary. Engle–Granger (1987).
Useful for: spread trading, pairs strategies, modelling long-run relationships (e.g. front- vs back-month futures, on-the-run vs off-the-run treasuries).
For derivatives the correlation framework is dominant because option payoffs depend on the joint distribution at a single time $T$, not on the dynamics of the spread over $[0,T]$. For trading and risk management the cointegration framework matters because it tells you when "diverged" prices are likely to mean-revert.
Two assets, correlated via Cholesky. Simulate $N$ paths of
$$dS_i = (r - q_i) S_i\,dt + \sigma_i S_i\,dW_i,\quad \mathrm{corr}(dW_1, dW_2) = \rho,$$
and look at terminal $(S_T^1, S_T^2)$ scatter on the left, the histogram of basket payoffs $\max(0.5\, S_T^1 + 0.5\, S_T^2 - K, 0)$ on the right.
Drag $\rho$ from $-0.9$ to $+0.9$ and watch the scatter morph from a downward-sloping cloud to an upward-sloping one. The basket-payoff histogram simultaneously narrows (high $\rho$, more concentrated payoff) or fattens (low $\rho$, more diversification). The MC price tracks the trade-off: typically decreasing in $\rho$ for an out-of-the-money basket call, because diversification lets more outcomes drift into the money.