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The Handbook of Hybrid Securities: Convertible Bonds, CoCo Bonds and Bail-In by Jan De Spiegeleer, Wim Schoutens, Cynthia Van Hulle

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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:

with

image

The volatility of the diffusion component of the process described in Equation (11.1) is σJD and is different from σS. 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, St = 0. Such a jump to default takes place as soon as dNt = 1 and hence dSt = −St.

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 ...

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