Pricing Vulnerable Exchange Options with Correlated Skew Brownian Motions

We study the valuation of European exchange options subject to the issuer’s terminal default risk under correlated skew Brownian motions. The two underlying assets and the issuer’s asset value are exposed to correlated Gaussian risks and a common reflected Brownian factor, with different skewness parameters. The contract allows full payment when the issuer is solvent and proportional recovery otherwise. Conditioning on the reflected factor gives a joint Gaussian distribution for the log asset values. Using Gaussian exponential tilting, we derive a pricing formula consisting of four truncated moments, each expressed as a one-dimensional integral involving the bivariate normal distribution function. The formula treats the exercise and default conditions exactly. Its mathematical contribution is the joint evaluation of the exchange and default regions, including the asset–issuer products arising in recovery, without a payoff approximation. This retains a one-dimensional integration structure despite the three-asset dependence. Numerical experiments compare the resulting prices with direct Monte Carlo estimates and illustrate the effects of the issuer’s asset value, bankruptcy costs, skewness parameters, and credit-related correlations.