Price Two Assets Call Formula

This question considers a European call option written on the product of two underlying assets, each modeled as a correlated geometric Brownian motion. Instead of a standard single-asset option, the payoff depends on whether the product of the two asset prices exceeds a strike at maturity. The candidate must recognize how to reinterpret this exotic payoff in terms of a more familiar structure and translate the two-dimensional price dynamics into a tractable, effectively one-dimensional problem suitable for closed-form valuation, as is often expected in front-office derivatives and exotics pricing roles.

The derivation leans heavily on properties of lognormal distributions, correlated Brownian motions, and the behavior of sums of normal variables. It tests whether the candidate can identify an appropriate change of variables, compute an effective drift and volatility for a transformed underlying, and then correctly embed that into a risk-neutral pricing framework. Interviewers look for comfort with joint distributions, covariance and correlation, and the ability to manipulate stochastic processes algebraically rather than relying only on memorized formulas.

What it tests

When dealing with payoffs that depend on the product (or ratio) of multiple correlated assets following geometric Brownian motions, the key is to recognize that the logarithm of the product is the sum of the individual logarithms. Since each asset's log-price evolves as a Brownian motion with drift and volatility, the sum of their log-prices is itself a Brownian motion with drift equal to the sum of the individual drifts and variance equal to the sum of variances plus twice the covariance. This means the product of two correlated GBMs is itself a GBM, but with an 'effective' volatility that aggregates their individual volatilities and their correlation. The pricing problem then reduces to the classic Black-Scholes framework, but applied to this new synthetic asset whose price is the product and whose volatility is the combined volatility. This structure holds because log-normality is preserved under multiplication, and the joint distribution of correlated normals is still normal when summed.