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# Applied Mathematics & Statistics

AMS 513, Financial Derivatives and Stochastic Calculus
Foundations of stochastic modeling for finance applications, starting with general probability theory leading up to basic results in pricing exotic and American derivatives.  We will cover filtrations and generalized conditional expectation, Girsanov theorem and the Radon-Nikodym process, martingales, Brownian motion, Ito integration and processes, Black-Scholes formula, risk neutral pricing, Feynman-Kac theorem, exotic options such as barrier and lookback, and the perpetual American put.  If time permits we will discuss term structure modeling, volatility estimation, and mortgage backed securities.

Prerequisite:  AMS 511

Spring 2019 Semester:

Required Text:   "Stochastic Calculus for Finance II: Continuous Time Models", by Steven E. Shreve, Springer Finance; 2004; ISBN: 978-0-387-40101-0

Recommended Text: "Introduction to Stochastic Calculus Applied to Finance" by Damien Lamberton and Bernard Lapeyre, 2nd Edition (Chapman and Hall/CRC Financial Mathematics Series), November 30, 2007; ISBN:  978-1584886266

Learning Outcomes:

1) Be able to apply concepts from information and conditioning to derive properties of stochastic processes that are at the foundation of financial derivatives pricing theory.

2) Be able to derive basic properties of Brownian motion.

3) Understand basic theory of stochastic integrals, including the Martingale property, Ito’s lemma, and Ito processes.

4) Be able to solve stochastic differential equations and derive results, including types of Brownian motion and Hull-White and Cox-Ingersoll-Ross interest rate models.

5) Derive binomial no-arbitrage pricing model and understand its relation to the Black-Scholes equation and risk neutral pricing.

6) Derive the Black-Scholes-Merton partial diff. equation and its consequences.

7) Understand the Girsanov Theorem and use it to transform processes into martingales.

8) Understand the origin of risk-neutral pricing and be able to apply it to a variety of financial problems, including risk-free discounting, risk neutral pricing, implied volatility, complete market modeling, and asset pricing.

9) Understand the Feynman-Kac formula and be able to solve SDE problems by transforming them into PDEs.

10) Obtain an in-depth understanding of how to price exotic options from both PDE and risk-neutral pricing perspectives.

11) Understand basic principles for pricing American derivatives.

12) Basic concepts behind Heath-Jarrow-Merton term structure interest rate models. 