# Mathematical Models and Financial Derivatives

Objectives and Learning Outcomes

The goal of the course is to develop firm understanding of the principal ideas and models that underpin modern financial practice and theory and to build hands-on experience in valuation, hedging and trading of financial derivatives. At the end of the course, students understand the institutional aspects and practical uses and methods of valuation and hedging of derivative securities in discrete and continuous time and effectively utilize data on financial derivatives. They shall also test their derivative-based investment strategies using a realistic trading simulator.

Lectures

Types, uses and risks of financial derivative instruments. Non-arbitrage pricing, complete markets, and replicating portfolio valuation method in discrete time. Risk-neutral expectation (martingale) approach in discrete time. Pricing of forwards, futures and swaps. Use of linear contracts for hedging and speculation. The binomial model and option pricing in discrete time. Brownian motion and Ito processes. Ito’s Lemma and Stochastic Differential Equations. Girsanov Theorem and risk neutral pricing in continuous time. Black-Scholes formula for options pricing. Option Greeks. Hedging equity portfolio using options. Trading strategies with options. Pricing options on indices and currency options. Implied volatility and volatility skew. Introduction to Monte Carlo option pricing. Feynman-Kac theorem and Black-Scholes partial differential equation for derivatives pricing. Pricing of early exercise and exotic options. Volatility models and their trading.

Exercises

Hedging and speculating with forwards, futures and swap contracts. Writing functions for European and American option pricing using binomial tree. Convergence of binomial to the corresponding Black-Scholes values. Working with options data and studying key empirical properties of these data. Back-testing equity portfolio with out-of-the-money put and comparison with long-only strategy. Using Greeks. Constructing volatility smiles and volatility surfaces using options data. Numerically solving Black-Scholes partial differential equations for option pricing in Python implementing finite different method. Monte Carlo experiments in Python for pricing European and American options. Replicating VIX in Python and volatility trading. Group competition in options trading using paper trading accounts in Interactive Broker trading platform.