# Bridge Copula Model for Option Pricing

**Authors:** Giuseppe Campolieti, Roman N. Makarov, Andrey Vasiliev  
**arXiv:** [1110.4669](https://arxiv.org/abs/1110.4669)  
**Date:** 2011-10-21  
**Summary page:** https://www.summarizepaper.com/en/paper/1110.4669v1/

## Key points

- The paper presents a novel multi-asset pricing model built upon families of solvable multi-parameter single-asset diffusions with nonlinear smile-shaped volatility and affine drift.
- Copula methods are employed to develop the model, where all discounted single-asset price processes are modeled as martingale diffusions under a risk-neutral measure.
- UOU diffusions are used to generate the single-asset dynamics by combining an Ito transformation with a measure change performed on an underlying Ornstein-Uhlenbeck process.
- A normal bridge copula is used to couple the single-asset dynamics and reduce the distribution of the multi-asset price process to a multivariate normal distribution, allowing for precise and efficient simulation of multidimensional price paths.
- The proposed model has significant implications for pricing path-dependent financial derivatives such as Asian-style and Bermudan options, making it valuable for practitioners in finance.

## Summary

In their paper titled "Bridge Copula Model for Option Pricing," Giuseppe Campolieti, Roman N. Makarov, and Andrey Vasiliev present a novel multi-asset pricing model that is built upon newly developed families of solvable multi-parameter single-asset diffusions with a nonlinear smile-shaped volatility and an affine drift. The authors employ copula methods to develop this model, where all discounted single-asset price processes are modeled as martingale diffusions under a risk-neutral measure. These price processes are known as UOU diffusions, which are generated by combining a variable (Ito) transformation with a measure change performed on an underlying Ornstein-Uhlenbeck (Gaussian) process. To couple the single-asset dynamics and reduce the distribution of the multi-asset price process to a multivariate normal distribution, the authors exploit the use of a normal bridge copula. This approach allows them to simulate multidimensional price paths in a precise and efficient manner, making it possible to accurately price path-dependent financial derivatives such as Asian-style and Bermudan options using the Monte Carlo method. The authors also demonstrate how to successfully calibrate their multi-asset pricing model by fitting respective equity option and asset market prices to the single-asset models and their return correlations (i.e., the copula function) using least square and maximum likelihood estimation methods. Overall, this paper presents an innovative approach for developing multi-asset pricing models that can precisely simulate multidimensional price paths while reducing distribution complexities through copula methods. The proposed model has significant implications for pricing path dependent financial derivatives such as Asian style and Bermudan options, making it valuable for practitioners in finance.