Exchange Call Option Pricing

This question focuses on pricing an option whose payoff depends on the relative performance of two underlying assets instead of a single stock versus cash. The setup is an exchange option that pays off if one asset finishes above another, making it a classic example of multi-asset option pricing in continuous time. Candidates must think carefully about modeling two correlated geometric Brownian motions and about how to treat "exchanging" one risky asset for another at maturity, rather than exercising into a risk-free numeraire. This style of question is common in more theoretical derivatives roles and in interviews where comfort with advanced option pricing frameworks is expected.

Conceptually, the problem leans on using a change of numeraire to reduce a two-dimensional pricing problem to a one-dimensional one. It uses the structure of equivalent martingale measures, lognormal distributions, and the behavior of ratios of correlated diffusions. Interviewers look for a clear explanation of why a particular numeraire is natural, a correct derivation of the dynamics under the new measure, and recognition of the connection to standard single-asset option pricing results. They also watch how candidates handle correlation, volatilities, and measure changes rigorously but efficiently.

What it tests

The core structure of this problem class is the use of a change of numeraire to simplify multi-asset derivative pricing. When a payoff depends on the difference or ratio of two tradable assets, switching to a numeraire that is one of those assets can reduce the dimensionality and complexity of the problem. This works because, under the new measure where the chosen asset is the numeraire, the relative price process of the other asset becomes a martingale, and the payoff can often be recast as a standard option on this relative price. The principle holds because the fundamental theorem of asset pricing guarantees that, under an equivalent martingale measure for any tradable numeraire, discounted asset prices are martingales, allowing us to leverage familiar single-asset option pricing machinery in a transformed space. This approach is especially powerful for options whose payoffs are invariant to the scale of the numeraire, such as exchange options, quanto options, and spread options.