Preface

Welcome to Stochastic Calculus: An Idiosyncratic Approach! This book provides a discussion regarding special features of the mathematics behind random processes like the Brownian Motion and their applications in finance.

What You’ll Learn

This book covers:

  • Random walks and Brownian motion
  • Itô’s lemma and stochastic integration
  • Stochastic differential equations (SDEs)
  • The Feynman-Kac connection between SDEs and PDEs
  • Risk-neutral pricing and martingales
  • Black-Scholes theory and the Greeks
  • Jump-diffusion models
  • Lévy processes

Prerequisites

Readers should be comfortable with:

  • Calculus (derivatives, integrals)
  • Basic probability theory (random variables, expectations, variance)
  • Linear algebra fundamentals

About This Book

This book grew out of years of instruction at the PhD and MBA levels in IIMs Bangalore and Kozhikode. It reflects my idiosyncratic perspective on topics in mathematical finance and its pedagogy. The emphasis is on understanding why things work, not what the formulas are.

Each chapter builds on previous concepts, so I recommend reading sequentially, especially if you’re new to the subject.

Book Overview

This book is organized into eight chapters that build progressively from foundational concepts to advanced generalizations.

Part I: Foundations

  1. Random Walks and Brownian Motion

    From discrete coin flips to continuous paths—establishing why \((dB)^2 = dt\)

    We begin with the simple random walk and watch it transform into Brownian motion through careful scaling. The central revelation: stochastic infinitesimals behave differently from classical ones, setting the stage for everything that follows.

  2. Itô Integral and Itô’s Lemma

    The fundamental theorem for stochastic processes

    Classical integration fails for Brownian paths. We construct the Itô integral and derive Itô’s lemma—the chain rule of stochastic calculus, complete with its surprising correction term.

  3. Stochastic Differential Equations

    Geometric Brownian Motion, Ornstein-Uhlenbeck, and solution techniques

    The workhorse models of mathematical finance. We solve key SDEs and develop intuition for when closed-form solutions exist.

Part II: The PDE Connection

  1. Feynman-Kac and PDEs

    The bridge between probability and partial differential equations

    A remarkable duality: expectations over random paths solve deterministic PDEs. This connection underlies both theoretical insights and computational methods.

Part III: Mathematical Finance

  1. Risk-Neutral Pricing and Martingales

    No-arbitrage, Girsanov’s theorem, and the fundamental theorem of asset pricing

    The conceptual foundation of derivative pricing. We explore why “fair games” (martingales) are central to finance and how changing probability measures eliminates arbitrage.

  2. Black-Scholes Theory

    Option pricing, the Greeks, and delta hedging

    The celebrated formula and its practical toolkit. We derive the Greeks, understand hedging, and confront the model’s limitations through implied volatility.

Part IV: Beyond Diffusions

  1. Jump-Diffusion Models

    When continuous paths aren’t enough—Merton’s model and the PIDE

    Markets jump. We extend our framework to incorporate discontinuous price movements, leading to partial integro-differential equations.

  2. Lévy Processes

    The ultimate generalization: infinite divisibility and the Lévy-Khintchine formula

    The grand unification. Every process with independent, stationary increments—from Brownian motion to pure jump processes—falls under this umbrella.