Explicit Pricing Formulas for European Option with Asset Exposed to Double Defaults Risk

Taoshun He

doi:10.1155/2018/8362912

Abstract

We derive analytical formulas for European call and put options on underlying assets that are exposed to double defaults risks which include exogenous counterparty default risk and endogenous default risk. The endogenous default risk leads the asset price to drop to zero and the exogenous counterparty default risk induces a drop in the asset price, but the asset can still be traded after this default time. A novel technique is developed to evaluate the European call and put options by first conditioning on the predefault and the postdefault time and then obtaining the unconditional analytic formulas for their price. We also compare the pricing results of our model with default-free option model and counterparty default risk option model.

Highlights

Over the past few decades, academic researchers and market practitioners have developed and adopted different models and techniques for pricing European option
The pathbreaking work on option valuation was done by Black and Scholes [1] and Merton [2]; their works assumed that the absence of arbitrage opportunities and the asset price dynamics are governed by a Geometric Brownian Motion (GBM)
We study the impact of the double defaults risk, which includes exogenous counterparty default risk and endogenous default risk, on option pricing problem

Summary

Introduction

Over the past few decades, academic researchers and market practitioners have developed and adopted different models and techniques for pricing European option. The derivation of the analytic formula for pricing European call and put options under the double defaults risk model has not been done in the previous literature. The main difficulty lies in the derivation of the distribution of the stock price at expire time T under the double defaults and the continuous trading of the underlying asset after the exogenous counterparty default time. We use the conditional density approach of default, which is suitable to study what goes on after the default and was adopted by Jiao and Pham [6] for the optimal investment problem, to derive the explicit distribution of the stock price at expire time T and obtain the analytic formulas for valuation of the European call and put options.

Model Setting and Change of Measure

Analytic Formula for Pricing European Options

Conclusions

Conflicts of Interest