Deriving the Density of a Two-Asset Basket in a Lognormal Model

Summary

The note presents a way to obtain the probability density of a basket formed by summing two correlated geometric Brownian motion assets. Although each asset is lognormally distributed, their sum is generally not lognormal. The proposed density follows from integrating the joint density along all pairs of asset values whose sum equals a given basket value.

The joint density is expressed using the marginal densities and a Gaussian copula density, which captures dependence between the assets. This provides an integral representation for the basket density and suggests extending the approach to larger baskets. The note asks whether this representation is useful for basket option pricing and observes that higher-dimensional integration may become difficult. It does not provide a worked pricing example, numerical comparison, or evidence that the method is computationally preferable to alternatives, so the proposal is best read as a distributional formulation rather than a validated pricing procedure.

Key ideas

• The sum of two lognormally distributed assets is generally not itself lognormal.
• The basket density can be represented by integrating the joint asset density along values that sum to the basket level.
• A Gaussian copula combines the marginal densities while accounting for dependence.
• Extending the integral approach to many assets may make evaluation computationally difficult.

Tags

# Basket option density in BS model


# Basket option density in BS model












Let X and Y be two GBM’s, they have each a univariate log-normal distribution for some time t, that is $X_t\sim{LnN(µ_x, σ^2_x)}$, $Y_t\sim{LnN(µ_y, σ^2_y})$ and $Z_t=[X_t,Y_t]\sim{ MvLnN(μ, Σ)}$ where $µ_x, σ^2_x, µ_y, σ^2_y , Σ$ have the known expressions as functions of time and initial values..

We take the value of the basket as $B_t=X_t+Y_t$. The distribution of B is not log-normal but its density can be written as :

$f(b)=\int_{-\infty}^{+\infty}f(x,b-x)dx$

We write the multivariate density as the product of univariate densities and the copula density :

$f(x,y)=c(x,y)f(x)g(y)$

So we can write the density of the basket as :

$f(b)=\int_{-\infty}^{+\infty}c(x,b-x)f(x)g(b-x)dx$

Where c(x,y) is the density of the normal copula (since the copula remain the same under increasing transformations of rvs), which has a known form. Moreover we can generalise this to a sum of n variables by looking at the bivariate pairs of combinations.

Question

Is this used in pricing basket options? Since I haven’t seen such a procedure used I was wondering what is the reason for not using it?

Edit I modified the integral expressions above because they weren't totally correct. The question is still up, though I'm less convinced by this approach in the case where B is a sum of many rvs since the last integral above might become quite difficult to evaluate. Still, this would give an exact solution.

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