Suppose we use the standard deviation of possible future returns on a stock as a measure of its volatility. Is it reasonable to take that volatility as a constant over time? I think not.

—Fischer Black

It is widely accepted today that an assumption of a constant volatility fails to explain the existence of the volatility smile as well as the leptokurtic character (fat tails) of the stock distribution. The Fischer Black quote, made shortly after the famous constant-volatility Black-Scholes model was developed, proves the point.

In this chapter, we will start by describing the concept of Brownian Motion for the Stock Price Return, as well as the concept of historic volatility.

We will then discuss the derivatives market and the ideas of hedging and risk neutrality. We will briefly describe the Black-Scholes Partial Derivatives Equation (PDE) in this section.

Next, we will talk about jumps and level-dependent volatility models. We will first mention the jump-diffusion process and introduce the concept of leverage. We will then refer to two popular level-dependent approaches: the Constant Elasticity Variance (CEV) model and the Bensoussan-Crouhy-Galai (BCG) model.

At this point, we will mention local volatility models developed in the recent past by Dupire and Derman-Kani and we will discuss their stability.

Following this, we will tackle the subject of stochastic volatility where we will mention a few popular models such as the Square-Root ...

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