Pricing Vulnerable Exchange Options with Correlated Skew Brownian Motions
- 1Department of Applied Mathematics, Kyung Hee University, Yongin 17104, Republic of Korea
- 2School of Natural Sciences, Seoul National University of Science and Technology, Seoul 01811, Republic of Korea
Abstract
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.
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BibTeX
@article{jeon2026pricing,
title = {{Pricing Vulnerable Exchange Options with Correlated Skew Brownian Motions}},
author = {Junkee Jeon and Geonwoo Kim},
journal = {Axioms},
year = {2026},
month = oct,
doi = {10.3390/axioms15100777},
url = {https://www.mdpi.com/2075-1680/15/10/777},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Pricing Vulnerable Exchange Options with Correlated Skew Brownian Motions
AU - Junkee Jeon
AU - Geonwoo Kim
PY - 2026
DA - 2026/10/16
JO - Axioms
DO - 10.3390/axioms15100777
UR - https://www.mdpi.com/2075-1680/15/10/777
AB - 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.
SN - 2075-1680
ER -