Summary - Stochastic Processes: The Fundamentals (Master Finance)

Summary

Joya da Silva Patricio Gomes

Stochastic Processes: the Fundamentals

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Student Number: 2806884




December 13, 2024

,Contents

Lecture 1 1
Learning Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
No-Arbitrage Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Risk-Neutral Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Deriving the Risk-Neutral Measure (Q) . . . . . . . . . . . . . . . . . . . . . . . . 3

Lecture 2 4
Basic Derivative Pricing: Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
The Binomial Tree Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Derivative Pricing in the Binomial Tree Model . . . . . . . . . . . . . . . . . . . . 5
Risk-Neutral Pricing and the First Fundamental Theorem . . . . . . . . . . . . . . 5
Multi-Period Binomial Tree Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Lecture 3 7
Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Probability Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Finite Probability Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Algebras and Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Random Variables and Probability Distributions . . . . . . . . . . . . . . . . . . . 9
Expectation and Conditional Expectation . . . . . . . . . . . . . . . . . . . . . . . 9
Stochastic Processes and Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Risk-Neutral Pricing Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Lecture 4 11
Finite Probability Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Continuous Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Sigma-Algebras and Probability Measures . . . . . . . . . . . . . . . . . . . . . . . 12
Almost Sure Events and Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Borel Sigma-Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
General Definition of a Random Variable . . . . . . . . . . . . . . . . . . . . . . . 13
Measure and Probability Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Expectation and Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Lecture 5 15
Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Properties of the Symmetric Random Walk . . . . . . . . . . . . . . . . . . . . . . 15
Scaled Symmetric Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2

,SP: the Fundamentals Summary

Limiting Distribution of the Scaled Symmetric Random Walk . . . . . . . . . . . . 16
Brownian Motion vs. Scaled Symmetric Random Walk . . . . . . . . . . . . . . . 17
Stochastic Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Building Stochastic Processes from Brownian Motion . . . . . . . . . . . . . . . . 18
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Lecture 6 19
Properties of Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Quadratic Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Stochastic Calculus vs. Ordinary Calculus . . . . . . . . . . . . . . . . . . . . . . . 20
Itô’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Lecture 7 22
More Properties of Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Application to Stock Price Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Introduction to the Black-Scholes World . . . . . . . . . . . . . . . . . . . . . . . . 24
Deriving the Black-Scholes Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Solving the Black-Scholes Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Lecture 8 26
Black-Scholes Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Risk-Neutral Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
No-Arbitrage Pricing and Linear Pricing Rules . . . . . . . . . . . . . . . . . . . . 27
Change of Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Girsanov’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
First Fundamental Theorem of Asset Pricing in Continuous Time . . . . . . . . . 28
Illustration in the Black-Scholes World . . . . . . . . . . . . . . . . . . . . . . . . . 29

Lecture 9 30
Black-Scholes Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Risk-Neutral Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
The Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Put-Call Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Second Fundamental Theorem of Asset Pricing . . . . . . . . . . . . . . . . . . . . 31
Beyond Black-Scholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Lecture 10 33
Term Structure of Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Short Rate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Pricing Caps and Caplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35




CONTENTS 3

, Lecture 1

Learning Goals
• No-Arbitrage Principle: Students should understand and apply the concept of no-
arbitrage, which ensures there are no opportunities to make riskless profits in the
market.
• Continuous-Time Models: Gain familiarity with continuous-time financial models,
including concepts like Brownian Motion and stochastic processes.
• Itô’s Lemma: Learn to apply Itô’s Lemma, a fundamental result in stochastic calculus,
used in modeling financial derivatives.
• Black-Scholes Model: Develop a deep understanding of the Black-Scholes option
pricing formula, a cornerstone of modern financial theory.
• Risk-Neutral Pricing: Comprehend the risk-neutral pricing method, which finds
the fair value of derivatives by discounting the expected payoff under a risk-neutral
measure.


No-Arbitrage Pricing
Definition
The no-arbitrage pricing approach dictates that financial markets are structured in such a
way that there are no ”free lunch” opportunities, i.e., there are no riskless profit opportuni-
ties.

Arbitrage Opportunity
An arbitrage opportunity is a scenario where a portfolio has an initial value of zero, cannot
lose value, and has a positive probability of generating a profit.

Simple Example of Arbitrage
Suppose a coin toss pays EUR 10 for heads and EUR 20 for tails. If the cost to enter this
game is EUR 5, an arbitrage opportunity exists since the expected payoff (EUR 15) exceeds
the cost.

Generalizing to Markets
In financial markets with various traded instruments, arbitrage-free conditions mean that if
two assets or portfolios have identical future payoffs, they should be priced the same.

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