Quantitative Finance with Python

A Practical Guide to Investment Management, Trading, and Financial Engineering

Chris Kelliher

• Publisher: Chapman and Hall/CRC
• Year: 2022
• Print ISBN:
  9781032014432
• Edition: 1
• eBook ISBN:
  9781000582376
• Topic: Business
• Subtopic: Financial Engineering
• Index: Business

Quantitative Finance with Python: A Practical Guide to Investment Management, Trading and Financial Engineering bridges the gap between the theory of mathematical finance and the practical applications of these concepts for derivative pricing and portfolio management. The book provides students with a very hands-on, rigorous introduction to foundational topics in quant finance, such as options pricing, portfolio optimization and machine learning. Simultaneously, the reader benefits from a strong emphasis on the practical applications of these concepts for institutional investors.

Features

• Useful as both a teaching resource and as a practical tool for professional investors.
• Ideal textbook for first year graduate students in quantitative finance programs, such as those in master's programs in Mathematical Finance, Quant Finance or Financial Engineering.
• Includes a perspective on the future of quant finance techniques, and in particular covers some introductory concepts of Machine Learning.
• Free-to-access repository with Python codes available at www.routledge.com/ 9781032014432 and on https://github.com/lingyixu/Quant-Finance-With-Python-Code.

II_________________Options Modeling

Chapter 8 Stochastic Models

DOI: 10.1201/9781003180975-8

In this chapter we explore the most common stochastic models that are used in quant finance applications. We begin by reviewing the simplest class of models including the Black-Scholes model/Geometric Brownian Motion and the Bachelier/Normal model. These models are ubiquitous in quant finance in spite of their simplicity and serve as quoting conventions in many markets. We then turn our attention to different approaches to modeling volatility skew, starting with stochastic volatility, then proceeding to jump-diffusion models before finally turning to local volatility models. The goal of this chapter is to orient the reader to the features and limitations of these models and to highlight how to work with the models in practice.

As a note to the reader, in this chapter, as we introduce new models we often introduce the characteristic functions associated with the model. Characteristic functions will turn out to be a key building block for efficient pricing via FFT, as we explore in chapter 9. More background on characteristic functions and their importance in solving pricing problems is provided in that chapter as well.

8.1 SIMPLE MODELS

8.1.1 Black-Scholes Model

Perhaps the most famous and commonly used stochastic model in finance is the so-called Black-Scholes model. In this model, we assume that an asset price follows the dynamics below:

$$\begin{array}{l}d{S}_t=r{S}_{t}dt+\sigma S_{t}dW\hfill \end{array}$$(8.1)

These dynamics are often referred to as Geometric Brownian motion, and it can be shown that specifying this model is equivalent to assuming that prices are log-normally distributed, and that returns are subsequently normally distributed. The model is characterized by a single parameter, σ which defines the volatility of the asset.

Importantly, volatility in the model is a function of the current asset price, and the σ parameter is interpreted as a relative or percentage volatility. As the price process increases, so will the magnitude of the dW term. That is, the volatility of the price of the underlying asset depends on the level of the underlying asset. The volatility of the return of the underlying asset, $\frac{d{S}_t}{S_t}$, however, is constant. This is a defining feature of the Black-Scholes model.

This model is perhaps still the most fundamental model in all of quant finance, and quants will find themselves constantly working with these dynamics. Even in the case where more complex models are used, these models are generally translated back to a Black-Scholes model in order to gain intuition and present results to market participants. Consequently, both technical and conceptual mastery of this model is of paramount importance for readers aspiring to be quants.

Under Black-Scholes dynamics, European option prices can be shown to have closed form solutions that are commonly known as the Black-Scholes formula.

$$\begin{array}{ll}{C}_0\hfill & =\Phi (d_1)S_0-\Phi (d_2)K{e}^{-rT}\hfill \end{array}$$(8.2)

$$\begin{array}{ll}{P}_0\hfill & =\Phi (-d_2)K{e}^{-rT}-\Phi (-d_1)S_0\hfill \end{array}$$(8.3)

Here, S₀ is the current asset price, K is the strike, r is risk-free rate and T is the option expiry time. Further, Φ is the CDF of the standard normal distribution d₁ and d₂ are defined as:

$$\begin{array}{ll}{d}_1\hfill & =\frac{1}{\sigma \sqrt{T}}\left(\ln \frac{S_0}{K}+\left(r+\frac{{\sigma }^2}{2}\right)T\right),\hfill \\ d_2\hfill & =d_1-\sigma \sqrt{T}\hfill \end{array}$$(8.4)

There are multiple ways to derive the Black-Scholes formula from the underlying dynamics. Perhaps the simplest approach is to leverage our knowledge of the distribution of asset prices and log returns in the model to calculate the appropriate integrals from first principles. To see this, recall that the price of a call option can be written as:

$$\begin{array}{ll}{C}_0\hfill & =\exp (-rT){\displaystyle {\int }_{-\infty }^{\infty }{(S-K)}^{+}}\varphi (S_T)d{S}_T\hfill \end{array}$$(8.5)

As the payoff is equal ...

Table of contents

• Cover Page
• Half-Title Page
• Series Page
• Title Page
• Copyright Page
• Dedication Page
• Contents
• Foreword
• Author
• Contributors
• Acknowledgments
• Section I Foundations of Quant Modeling
• Section II Options Modeling
• Section III Quant Modeling in Different Markets
• Section IV Portfolio Construction & Risk Management
• Bibliography
• Index