Tomoyuki ICHIBA

Associate Professor, Department of Statistics & Applied Probability and Center for Financial Mathematics and Actuarial Research, and Mathematics in College of Creative Studies

at University of California Santa Barbara

Teaching at UCSB

223 B Financial Modeling (Winter 2015)

This course is an introduction to stochastic models in finance with applications to fixed income and credit markets. It consists of three parts:

Part I. Brownian Motion and Stochastic Calculus (3 weeks)

We discuss Brownian motion, martingales, Stochastic Differential Equations and connections to Partial Differential Equations. Topics are constructions of Brownian motions, quadratic variations, martingale inequalities, local martingales, first-passage time distribution, change-of-variables formulae, stochastic integration, \textsc{Girsanov}'s theorem, stochastic differential equations and \textsc{Feynman-Kac} formula. These mathematical tools are used to develop the next two parts.

Part II. Equity and Interest Rate Market Models (4 weeks)

This part covers financial products and derivatives. The topics are financial options, \textsc{Black-Scholes} model, portfolios, interest rates, coupon Bonds, swaps and yields. We shall see how prices of these products are determined under some nice conditions (e.g., no arbitrage condition) in continuous-time financial model. We also discuss advantage and limitation of mathematical modeling for financial market, in particular, some situations of arbitrage.

Part III. Stochastic Optimal Control (3 weeks)

The aim of this final part is to solve some stochastic optimization problems in continuous time. We discuss various techniques such as \textsc{Hamilton-Jacobi-Bellman} equation and the {\it maximum principle}. The main topics are optimal stopping problem for pricing American options and stochastic control problem for optimal portfolio selections. This last part leads us various applications not only to Finance but also Applied Mathematics, Economics, Engineering, Physics and other fields.

Prerequisites: PSTAT 223A or equivalent. If you haven't taken PSTAT 223A, then please consult and get an approval from the instructor.

Textbook: `` Arbitrage Theory in Continuous Time'' by Tomas Bjork (2009, 3rd edition)

References: ``Stochastic Differential Equations: An Introduction with Applications'' by Bernt Oksendal
``Stochastic Calculus for Finance II'' by Steven E. Shreve
``Applied Stochastic Control of Jump Diffusions'' by Bernt Oksendal and Anes Sulem
``Brownian Motion and Stochastic Calculus'' by Ioannis Karatzas and Steven E. Shreve

Class Meeting Time: TR 12:30-1:45 at HSSB 4202
Office hour: tentatively MW 10:00-11:00 at South Hall 5508.