The Handbook of Hybrid Securities: Convertible Bonds, CoCo Bonds and Bail-In

11 Constant Elasticity of Variance

11.1 FROM BLACK–SCHOLES TO CEV

11.1.1 Introduction

In Section 9.2 we left the Black–Scholes trail when introducing the jump-diffusion model. The equity prices became defaultable since a jump to zero was allowed. The probability of such a jump taking place was based on the reduced-form approach such as in [81]. The probability of “jumping” to default is now determined through the default intensity parameter λ. The share price process turns from a geometric Brownian motion into a jump-diffusion process:

(11.1)

with

The volatility of the diffusion component of the process described in Equation (11.1) is σ_JD and is different from σ_S. The use of this latter volatility is constrained to the geometric Brownian motion.

The jump-diffusion process allows, in sharp contrast with the geometric Brownian motion, the share price level S to collapse. Here, we work with the setting that in case of default, S_t = 0. Such a jump to default takes place as soon as dN_t = 1 and hence dS_t = −S_t.

There are different alternatives allowing us to step away from the Black–Scholes framework. Jump-diffusion is only one of the many possible choices. The most accepted extensions to Black–Scholes are those solutions which allow for a closed-form or semi-closed-form formula ...