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

7 Modeling Hybrids: Stochastic Processes

7.1 INTRODUCTION

The value X_t is the outcome of a stochastic process X observed at time t. An example of such a stochastic process is the Brownian motion. Inevitably one tends to encounter this name very early on when studying financial calculus. The description of a Brownian motion and its properties are indeed a logical starting point of many books on derivative securities. A book on hybrids is no exception to this. In this context the story is often told of how Robert Brown (1773–1858), who was an army surgeon of the Fifeshire Regiment based in Ireland, studied pollen suspended in water [182]. Brownian motion is named after the rapid oscillatory movements of these small particles. A rather unknown protagonist when it comes to stochastic processes in finance is Jules Regnault (1834–1894). The latter was a French economist who approached the stock market in a scientific way. He based his findings on observations of share prices and concluded that the price differences of a share over a time interval tended to be proportional to the square root of the elapsed time. With this empirical observation, modern financial history was born in Paris back in 1863, more than a century before Fisher Black, Myron Scholes, and Robert Merton developed their ground-breaking formula [182]. Louis Bachelier (1870–1946) followed in the footsteps of Jules Regnault when he developed his own mathematical theory of share price movements.

7.2 PROBABILITY DENSITY ...