# FINANCIAL MATHEMATICS ACTIVITIES

July 17, 2024

## Probability Theory and Modern Finance sponsored by October 18-19, 2000

 ACCOMMODATION DIRECTIONS SCHEDULE FACULTY

### Instructors:

Professor Thomas S. Salisbury
Professor Moshe Arye Milevsky

The course has been approved as eligible for 12 units of SOA-approved PD credit.

### Introduction

If you work in the financial engineering field as an analyst, forecaster, portfolio manager, risk manager, asset manager, commodity or derivative trader, valuation actuary or pricing actuary, you are frequently exposed to some very sophisticated mathematical techniques, many of which were only invented in the last few decades. This course puts those techniques in perspective, and shows how they are used for everyday strategic financial decision-making. It is designed to be an interactive program offering personal attention to small groups for a learning-centered experience.

### Topics of Discussion

• Why do option prices depend on volatility but not on expected returns?
• What is the difference between a Stochastic and a Deterministic integral and why does it require new techniques?
• How can you determine whether theoretical arbitrage opportunities are present?
• Why is Brownian motion the basic mathematical object on which much of modern financial theory is based?
• What are Martingales and how do they relate to security prices?
• How does Ito's Lemma let one calculate expected values for functions of Brownian motion?
• What is Girsanov’s theorem and why is it so important?
• How does one distinguish between real-world and risk-neutral probabilities?
• What concepts lie behind the Black-Scholes/Merton formula?
• What economic intuition does the model provide?

### Course Schedule - Day One

Stochastic Processes and Brownian Motion

• Probability Measures & Martingales
• Stochastic Processes and Brownian Motion
• Sigma Fields and Conditional Expectations
• Markov Processes and Diffusions
• Efficient Market Hypothesis and Modeling Financial Markets

Stochastic Calculus in Financial Decision-making

• Stochastic Differential Equations v.s. Ordinary Differential Equations
• Stochastic Integrals and Calculus
• Ito's Formula and Geometric Brownian Motion
• Statistical Estimation of Parameters

Pricing of Options

• Hedging Replication and Arbitrage
• Hitting Probabilities and PDEs
• Simulation of Sample Paths
• Computational Issues
• Financial Versus Actuarial Pricing

Evening Case Study

• Valuation and Analysis of Death-protected Mutual Funds (Variable Annuities) and Segregated Funds with Maturity Guarantees. What are the risks and how can they be hedged?
• Valuation and Analysis of Protected Mutual Funds (Variable Annuities) and Segregated Funds

Course Schedule - Day Two

Risk-Neutral Valuation and Black Scholes-Merton

• Equivalent Martingale Measures
• Fundamental Theorem of Asset Pricing
• Girsanov's Theorem and BSM Option Pricing
• Partial Differential Equation Satisfied by all Derivative Securities
• A Typography of Exotic Options

Interest Rates and Basic Processes

• Models for the Term Structure of Interest Rates
• Vasicek, CIR, Hull-White and HJM Models
• Ornstein-Uhlenbeck and Brownian Bridge Processes
• Feynman Kac Formula and Killing

American Option Pricing

• Stopping Times and Optimal Stopping
• Perpetual American Put Option
• Optimal Exercise Time and Boundary

Real-World Translation

• How is all this used on the "Street"?