# Optimal Investment and Pricing in the Presence of Defaults

## TL;DR

This paper develops a comprehensive model for optimal investment and pricing in markets with default risk, incorporating a factor-driven asset dynamics and providing explicit PDE characterizations for indifference and default protection prices.

## Contribution

It introduces a new framework for computing indifference and default protection prices using a semi-linear PDE in a factor model with affine processes.

## Key findings

- Explicit PDE characterizations for prices under default risk
- Numerical examples illustrating factor influence on prices
- Protection price not equal to default intensity under the dual measure

## Abstract

We consider the optimal investment problem when the traded asset may default, causing a jump in its price. For an investor with constant absolute risk aversion, we compute indifference prices for defaultable bonds, as well as a price for dynamic protection against default. For the latter problem, our work complements Sircar & Zariphopoulou (2007), where it is implicitly assumed the investor is protected against default. We consider a factor model where the asset's instantaneous return, variance, correlation and default intensity are driven by a time-homogenous diffusion X taking values in an arbitrary region E. We identify the certainty equivalent with a semi-linear degenerate elliptic partial differential equation with quadratic growth in both function and gradient. Under a minimal integrability assumption on the market price of risk, we show the certainty equivalent is a classical solution. In particular, our results cover when X is a one-dimensional affine diffusion and when returns, variances and default intensities are also affine. Numerical examples highlight the relationship between the factor process and both the indifference price and default insurance. Lastly, we show the insurance protection price is not the default intensity under the dual optimal measure.