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

8 Modeling Hybrids: Risk Neutrality

8.1 INTRODUCTION

The geometric Brownian motion is the starting point for most equity derivative models:

(8.1)

The equation above describes the changes dS_t in the value of the share price and illustrates the dependence of those changes on the expected return μ of the stock at its volatility σ. From Equation (7.31) we know that the possible values for S_t at a future time t, starting from the current value S_s at the current time s, are given by:

(8.2)

The random component driving the stock price from S_s to S_t is the Brownian motion W = {W_t, t ≥ 0}. One factor in the equation above is investor-dependent. Investors will indeed have different values for the drift component μ in the equation above. An investor with a bullish mindset will have a very positive value for μ opposed to more moderate values for the investors taking a bearish stance on the stock. We now want to calculate the price of a derivative security P whose value depends on the future price path followed by the underlying share price S. How can we eliminate the different assumptions regarding μ? This is where a risk-neutral setting enters on stage, which allows derivative pricing independent from the individual investor’s views. We start illustrating the possibility to hedge away ...