Nonlinear Portfolio Risk in Closed Form

We extend the Hermite-COS framework from linear portfolios (weighted sums of correlated lognormals) to portfolios containing options, structured products, and arbitrary derivative payoffs. The key observation is that the Gauss-Hermite quadrature grid — which evaluates the underlying Gaussian variables at \(Q^n\) deterministic nodes — computes the joint realization of ALL underlying assets simultaneously. Any payoff function \(g(S_1, \ldots, S_n)\) can be evaluated on these same nodes without recomputing the grid. The COS inversion then produces the full distribution, VaR, and ES of the derivative portfolio.

The deeper consequence is exact cross-correlation between arbitrary derivative payoffs. Given two options \(V_1 = g_1(S)\) and \(V_2 = g_2(S)\), their correlation is:

\[\rho(V_1, V_2) = \frac{\sum_\ell w_\ell g_1(S_\ell) g_2(S_\ell) - (\sum_\ell w_\ell g_1(S_\ell))(\sum_\ell w_\ell g_2(S_\ell))}{\sqrt{\text{Var}_w(g_1) \cdot \text{Var}_w(g_2)}}\]

This gives practitioners the exact correlation matrix of an option book — something previously available only through Monte Carlo simulation. The entire computation is deterministic, sub-second, and requires no additional infrastructure beyond the linear portfolio framework.

We demonstrate on a mixed portfolio of spot positions, call options, put options, and a collar, computing the joint distribution and the full correlation matrix of all positions.

Keywords: derivative portfolio risk, option correlation, nonlinear payoff, closed-form distribution, Gauss-Hermite quadrature.