DEPARTMENT OF APPLIED MATHEMATICS FECULTY OF TECHNOLOGY AND ENGINEERING

THE M.S.UNIVERSITY OF BARODA

TWO YEAR MASTER OF SCIENCE IN FINANCIAL MATHEMATICS

Semester- I

Subject Code Subject Scheme of Working Scheme of Examination

L P T Th Pr/Viva Total

AMT 2111 Multivariate Calculus

and Mathematical Analysis

4 - 4 100 - 100

AMT 2112 Fundamentals of

Statistics 4 2 6 100 50 150

AMT 2121 Modeling and Simulation

4 - 4 100 - 100

AMT 2114 Differential Equations 4 - 4 100 - 100

AMT 2122 Mathematical Methods in Finance

4 2 6 100 50 150

Total 20 4 24 500 100 600

Semester- II

Subject Code Subject Scheme of Working Scheme of Examination

L P T Th Pr/Viva Total

AMT 2207 Numerical

Techniques in Finance

4 2 6 100 50 150

AMT 2216 Numerical and combinatorial Optimization

4 2 6 100 50 150

AMT 2217

Economic Analysis and Principals of

Financial Management

4 - 4 100 - 100

AMT 2218 Stochastic Calculus and Black-Scholes

Theory 4 2 6 100 50 150

AMT 2219 C programming /

Mathematical softwares

4 2 6 100 50 150

Total 20 8 28 500 200 700

Semester- III

Subject Code Subject Scheme of Working Scheme of Examination

L P T Th Pr/Viva Total

AMT 2315 Discrete Time Modeling

and Derivative securities

4 0 4 100 - 100

AMT 2316 Portfolio Theory and Asset Allocation

4 2 4 100 50 100

AMT 2317 Financial Risk Management

4 0 4 100 - 100

AMT 2318

Modeling of Bonds, Term Structure, and

Interest Rate Derivatives

4 2 4 100 50 100

AMT 2319 Time Series Analysis 4 2 6 100 50 150

Total 20 6 26 500 150 650

Semester- IV

Subject Code Subject Scheme of Working Scheme of Examination

L P T Th Pr/Viva Total

AMT 2408L Financial Mathematics Project - 3 3 - 300 300

** Project work to be continue in any industry during this semester and report to be submitted after the end of fourth semester

Semester I

AMT 2111 Multivariate Calculus and Mathematical Analysis

1. Vectors

Vectors in two and three dimensions

• R2, R3, vector notation, scalars • Vector addition, zero vector, scalar multiplication, and their properties • Geometric interpretation of vectors, parallelogram law for addition Vectors and equations

• Standard basis vectors i, j, k • Parametric equations for lines • Symmetric form equations for a line in R3 • Parametric equations for curves r(t): R->R2 Dot products

• Dot products of vectors, lengths of vectors, law of cosines and angles • Projections of vectors • Normalization of vectors Cross products

• Cross products of pairs of vectors in R3 • Areas of parallelograms and triangles • Matrices, determinants • Triple scalar product, volume of a parallelepiped • Rotation, angular velocity Equations and distance

• Equations of planes, parametric equations of planes • Distance between a point and a line • Distance between parallel planes • Distance between skew lines Summary of geometry in Rn

• Cauchy inequality, triangle inequality • Standard basis • Linear functions correspond to matrices, and composition of functions corresponds to matrix

multiplication • Hyperplanes • Determinants, minors, and cofactors Coordinate systems

• Rectangular coordinates • Polar coordinates of planes • Cylindrical and spherical coordinates for space

2. Differentiation in several variables

Functions of several variables

• Function f: X->Y, domain X, codomain Y, range R(f) • Onto (surjective and one-to-one (injective) functions • One-to-one correspondences (bijective functions) and their inverses • Scalar-valued functions f: Rn->R, also called scalar fields • Vector-valued functions f:Rn->Rm, and their component functions R->Rm • Graphs of functions R2->R as surfaces in R3, and their level curves • Surfaces in R3, hypersurfaces in Rn • [Quadric surfaces] Limits of vector-valued functions

• Intuitive concept and formal definition of limits for multivariate functions • Topological concepts: open and closes subsets, boundaries of subsets, neighborhoods of

points • Properties of limits • Multivariate polynomials • Continuous functions Derivatives for multivariate functions

• Partial derivatives for scalar-valued functions • Differentiability and planes tangent to surface graphs of functions R2->R • Differentiability and hyperplanes tangent to hypersurface graphs of functions Rn->R • Differentiability for vector-valued functions • Gradient vectors for scalar-valued functions • Derivative matrix for vector-valued functions Properties of derivatives, higher-order partial derivatives

• Linearity: sum, difference, and constant multiple rules for vector-valued functions Rn->Rm • Product and quotient rules for scalar-valued functions Rn->R • Partial derivatives of higher order The chain rule in several variables

• The chain rule for composition fog where g:R->Rn and f:Rn->R • The chain rule for the composition fog where g:Rn->Rm and f:Rm->Rp • Polar-rectangular conversions Directional derivatives and gradients, implicit & inverse function theorems

• The gradient vector field for a scalar field • Directional derivatives, definition and evaluation in terms of the gradient • Steepest ascent • Tangent planes and hyperplanes

3. Vector-valued functions

Parameterized curves; Kepler's laws of planetary motion

• Functions f:Rn->R as paths or parameterized curves • Velocity, speed, and acceleration • Kepler's laws of planetary motion Arclength, curvature

• Length of a path as an integral

• Unit tangent vector, curvature of a path or curve Vector fields

• Functions f:Rn->R as scalar fields • Functions F:Rn->Rn as vector fields • The gradient field (a vector field) associated to a potential function (a scalar field) • Equipotential sets • Flow lines of vector fields Gradient, divergence, curl, the del operator

• The del operators and gradients • The del operator and divergence of a vector field • The curl of a vector field in R3 • Gradient fields are irrotational, that is, curl(grad f) = 0 • Div(curl F)=0

4. Maxima and minima in several variables

Differentials and Taylor's theorem

• Taylor's theorem for single variable functions as an extension of the mean value theorem • Taylor polynomials, remainder term • The first-order formula for the multivariate Taylor's theorem • Total differentials • The second-order formula and the Hessian • Higher-order Taylor polynomials Maxima and minima (extrema) of functions

• Local minima and maxima of scalar-valued functions • Critical points and the Hessian criterion • Quadratic forms, positive definite forms • The second derivative test for extrema of scalar-valued functions • Compact sets, the Extreme Value Theorem (EVT) Lagrange multipliers

• Equational constraints • Lagrange multipliers for extrema subject to constraints

5. Multiple integration

Areas and volumes

• Volumes over rectangles as double integrals Double integrals

• Double integrals over rectangles defined as Riemann sums, integrability • Conditions for integrability • Fubini's theorem, linearity of integrals, other basic properties • Double integrals over general regions • Changing order of integration

Triple integrals

• Triple integrals over boxes • Properties of triple integrals • Triple integrals over general regions Change of variables

• Transformations of the plane R2->R2 • Linear transformations and their expansion factors • Change of variables in definite integrals of one variable • Change of variables for double integrals, the Jacobian • Double integrals in polar coordinates • Change of variables for triple integrals Applications of multiple integration

• Mean value (average value) of a scalar-valued function • Center of gravity

6. Line integrals

Scalar and vector line integrals

• Scalar line integrals • Vector line integrals • Reparametrization Green's theorem

• Green's theorem • Divergence theorem in the plane

Conservative vector fields

• Vector fields with path-independent line integrals • Gradient fields and line integrals, conservative vector fields

7. Surface integrals

Parameterized surfaces

• Coordinate curves, normal vectors, tangent planes • Smooth and piecewise smooth surfaces • Areas of surfaces Surface integrals

• Scalar surface integrals • Vector surface integrals • Reparametrization of surfaces Stoke's and Gauss's theorems

• Stoke's theorem • Gauss's theorem

• Divergence and curl

References:

1. Rogawski, Calculus Early Transcendental, W. H. Freeman, 2008, ISBN-10: 0-7167- 7267-1

2. Vector Calculus, Second Edition, by Susan Jane Colley, Prentice-Hall, 2002

AMT 2112 Fundamentals of Statistics

• Illustrating the importance of statisticians in a variety of fields, including medicine and the biological, physical, and social sciences.

• Structure of data sets, histograms, means, and standard deviations. • Correlation and regression. • Elementary concepts of probability and sampling; binomial and normal distributions. • Basic concepts of hypothesis testing, estimation, and confidence intervals; t-tests and chi-

square tests. • Linear regression theory and the analysis of variance. • Introduction to the analysis of data from planned experiments. Analysis of variance for

multiple factors and applications of orthogonal arrays and linear graphs for fractional factorial designs to product and process design optimization.

Reference:

1. Statistics Freedman, Pisani, and Purves. 3rd Edition. New York: W. W. Norton

and Co., Inc. 1998

AMT 2121 Modeling and Simulation • Random numbers: • Generating uniform random variables: Pseudo – random numbers, congruential generators

and their properties, alternative approaches • Testing random numbers: empirical and theoretical tests or sequences of uniform random

numbers • General methods: Inversion method, acceptance-rejection, composition methods • Particular methods for non–uniform random variables, Box–Muller methods to generate

normal variates. • Exponential, Binomial and Poisson variates • Discrete Event simulation modeling: • Introduction to simulation modeling • Queuing models • Simulation of queueing systems • Output analysis of short and long term performance • Verification and validation • Statistical simulations: • Monte-Carlo simulation • Variance reduction technique: Arithmetic variables, conditioning, control variates, importance

sampling and common random numbers • Simulation inference • Quantitative modeling paradigms: queueing networks, stochastic process

algebras and stochastic Petri nets • Input and output analysis: random numbers, generating and analyzing

random numbers, sample generation, trace- and execution-driven simulation, point and interval estimation. Process-oriented and parallel and component simulation and modeling

MATHEMATICAL MODELING

• Needs and Techniques of mathematical modeling: Idea of mathematical modeling, need for mathematical modeling, steps in mathematical modeling, Characteristics of mathematical modeling ,Interpretation

• Models in mechanical vibration :Spring mass system, pendulum problems • Models in population dynamics: One species model, logistic model, growth model in time

delays ,Predator-Prey models, Volterra-Lotka models • Models of Financial Markets

COMPUTATIONAL MODELING

• Modeling dynamical systems: differential equations and their numerical solution, linear and non–linear dynamics, stability, convergence, attractors.

• Physical systems: System types and characteristics behavior, Continuous-time, discrete – time and discrete -event systems, linear and non linear systems

• Exploration of behavior through simulation: developing simulations of dynamical systems using Matlab : representation and visualization of simulation experiments, analyzing behavioral characteristics for a range of classes of physical and computational systems eg. Predictor- prey models, evolutionary systems and cellular systems

References:

1. J.N.Kapur: Mathematical modeling ,Wiley eastern Ltd.,1994. 2. M.M. Gibbons : A concrete approach to Mathematical modeling , John Wiley and sons, 1995. 3. H. Neunzert and A.H. Siddiqui: Topics in Industrial Mathematics, Kluwer Academic

Publishers, London, 2000 4. P. E. Wellstead : Introduction to Physical system modeling, Academic Press, 1979. 5. Richard Haberman: Mathematical Models, Practice- Hall Inc., NJ, 1979. 6. Jery Banks, John S., Carson II, Barry Nelson and David M.Nicol: Discrete – Event system

simulation , Prentice hall, 2001

AMT 2114 Differential Equations

1. First Order Differential Equations

• Direction fields, integral curves. • Explicit methods for solving separable, homogeneous, linear exact equations and some

special types such as Bernoulli. • Second order equations which can be reduced to first order. • Elementary applications of first order equations; exponential growth and decay, population,

mixing problems.

2. Linear Second Order Differential Equations

• Linear dependence and independence of solutions; the Wronskian. • Linear homogeneous equations with constant coefficients. • Nonhomogeneous linear equations and the method of reduction of order. • Undetermined coefficients, variation of parameters.

3. Applications of Linear Second Order Differential Equations

• Harmonic linear oscillators. • Electrical circuits. • Forced vibrations.

4. Laplace Transforms

• Laplace transforms. • Second order equations. • Delta functions and impulse forcing. • Convolutions.

5. Systems of Linear Differential Equations

• Preliminary theory. • Homogeneous linear systems with constant coefficients. • Variation of parameters..

6. Introduction to PDE:

• Mathematical Modeling through Hyperbolic, Parabolic and Elliptic PDE, Dirichlet, Neumann and Robin Condition, Well - posed ness of problem,

• Elliptic equation: Analytical Solution using Greens functions and separable variable approach, Parabolic equation and Hyperbolic equation: Analytical Solution using Fourier Transform, Laplace Transform and separable variable approach

References:

1. Elementary differential equations and boundary value problems, Eighth edition, by William E. Boyce and Richard C. DiPrima

AMT 2122 Mathematical Methods in Finance • Fundamentals of probability: probability space and measure, algebras and sigma-algebras,

random variables, probability distribution, expectation, variance, covariance, correlation. • Conditional expectation, conditional probability, dependence and independence. • Stochastic processes in discrete time; random walk and the Poisson process. • Discrete time martingales, submartingales, supermartingales. • Central Limit Theorem and Brownian motion as a limit of a symmetric random walk,

properties of Brownian motion. • Informal overview of Ito calculus: stochastic integrals, Ito formula, stochastic differential

equations. • Applications of calculus in finance: Black-Scholes equation and Black-Scholes formula.

References:

1. P. Willmott, Derivatives, Wiley 1997. 2. M. Capinski and T. Zastawniak, Mathematics for Finance: An Introduction to

Financial Engineering, Springer 2003. 3. M. Capinski and T. Zastawniak, Probability Through Problems, Springer 2001. 4. Z. Brzezniak and T. Zastawniak, Basic Stochastic Processes, Springer 1999.

Semester II

AMT 2207 NUMERICAL TECHNIQUES IN FINANCE Lattice Methods

• One Period Binomial Model

o Portfolios and Arbitrage

o Contingent Claims

o Completeness

o Risk Neutral Valuation

• The Binomial Tree

o Recombining and non-recombining trees

o The Cox-Rubenstein-Ross (CRR)

o The Jarrow-Rudd (JR) case

o Pricing vanilla European options on a tree

o Pricing American options on a tree

Review of Continuous Time Finance • Motivation to Stochastic Differential Equations • Brownian Motion • Information Flow • Stochastic Integrals • Conditional Expectations • Martingales • Ito's Formula • Examples of SDE's

o Geometric Brownian Motion

o Linear SDE's

• Change of Measure and Girsanov Theorem • Arbitrage • Risk Neutral Valuation Monte Carlo Methods

• Introduction to Monte Carlo Integration • Pseudo Random Number Generators • Generating random variables with a given distribution • Setting the number of replications • Variance Reduction Techniques • Simulation of SDE's Sample Paths • Pricing vanilla european options • Pricing path dependent options

o Barrier Options

o Asian Options

o Lookback Options

Numerical Solutions of Parabolic PDE's

• Introduction and Classification of 2nd Order Partial Differential Equations

o Characteristic Curves

o Examples

• The PDE Approach to Finance

o The Cauchy Problem & Probability Theory

o The Feymann-Kac Formula

o The Black-Scholes Equation

• Reduction of the Black-Scholes Equation to a Diffusion Equation • Finite Difference Methods

o Explicit

o Fully Implicit

o Crank-Nicholson

o Examples in Option Valuation

• Free Boundary Conditions

o Linear Complementarity, PSOR

o American Options

o Convergence and Stability

References:

1. TE, Paolo, Numerical Methods in Finance: a MatLab-based introduction, Wiley Series in Probability and Statistics, 2002.

2. SHAW, Willian, Modeling Financial Derivatives with Mathematica, Cambridge University Press, 1998.

3. BURDEN, Richard L. and FAIRES, J. Douglas, Numerical Analysis, Brooks/Cole, 2001. 4. HEATH, Michael T., Scientific Computing: an introductory survey, McGraw-Hill, 2002. 5. BJORK, Thomas, Arbitrage Theory in Continuous Time, Oxford University Press, 1998. 6. BAXTER, Martin and RENNIE, Andrew, Financial Calculus: an introduction to

derivative pricing, Cambridge University Press, 1998. 7. GOODMAN, Victor and STAMPFLI, Joseph The Mathematics of Finance: modeling and

hedging, The Brooks/Cole Series in Advanced Mathematics, 2001. 8. HULL, John, Options, Futures and Other Derivatives, Prentice Hall, 2000. 9. WILMOTT, Paul, HOWISON, Sam and DEWYNE, Jeff, The Mathematics of Financial

Derivatives: a student introduction, Cambridge University Press, 1998. 10. OKSENDAL, Bernt, Stochastic Differential Equations: an introduction with applications,

Springer-Verlag, 1998. 11. DUFFIE, Darrel, Dynamic Asset Pricing Theory, Princeton University Press, 1996.

AMT 2216 Numerical and Combinatorial Optimization

• Dynamic programming and allocating investments. • Markov chains and sequential decision making. • Networks and graph theory. • Linear programming and the simplex method. • The theory of games. • Scheduling problems.

References:

1. Jorge Nocedal and Stephen J.: Wright Numerical Optimization, Springer-Verlag, New York 1999

2. R. Fletcher: Practical Methods of Optimization, Wiley, Second edition, 1987 3. K. Deb: Optimization for Engineering Design Algorithms and Examples, Prentice Hall of India, 2000.

4. S. S. Rao: Engineering Optimization Theory and Practice, Third Edition, John Wiley and Sons, 1996.

5. R. K. Sundram: First Course in Optimization Theory, Cambridge University Press, 1996.

6. Kantiswarup, P.K.Gupta and Manmohan: Operations Research ,Sultan chand and Sons. 7. S.D. Sharma: Operations Research, Kedar Nath, Ram Nath & Co.

AMT 2217 Economic Analysis and Principals of Financial Management

Economic Analysis

Intermediate Economics • Introduction to the economic analysis of consumer and producer behavior. • Role of the market in the allocation of resources and the distribution of income, and how

these outcomes are affected by imperfections in the market system and by government policy. Introduction to Economics • Noon-economics majors and minors to basic ideas in economic analysis. • price theory, market analysis, national income accounts, growth, and international trade.. Asset Pricing Models • The Mean Variance Model • Capital Asset Pricing Model (CAPM) • Empirical Evaluation of Asset Pricing Models Financial Instruments and Financial Market Analysis • Options Markets and Prices • Forwards and Futures Markets and Prices • The Term Structure of Interest Rates and the Yield Curve

• Stock Price Determination

Principals of Financial Management

• Describe the main classes of financial assets; • Use of mathematical and statistical methods to construct diversified equity portfolios; • explain the nature of risk and risk-reduction in financial markets; describe attitudes to risk and

explain how they influence portfolio choices and the market prices of assets; • explain the concepts of equilibrium and arbitrage and use them to value securities; • demonstrate an understanding of capital market efficiency and explain the significance of

market efficiency for resource allocation References:

1. Bodie, Kane, Marcus, Investments, 5th edition, Irwin McGraw-Hill, 2002, ISBN: 0072339160.

AMT 2218 Stochastic Calculus and Black-Scholes Theory

Stochastic Calculus

• Brownian Motion and its properties. • Development of the Ito integral and its extension to wider classes of integrands;

isometry and martingale properties of the integral. • Ito calculus, Ito formula and its application in calculating stochastic integrals. • Stochastic differential equations. • Risk-neutral pricing: Girsanov"s theorem and equivalent measure change in a

martingale setting; representation of Brownian martingales. • Feynman-Kac formula.

Black-Scholes Theory

• The Black-Scholes model: assumptions and scope; delta-hedging; derivations of the Black-Scholes PDE and its solution via the heat equation and Brownian motion; role of the ‚¬ËœGreeks' as measures of model parameters; the Black-Scholes formula and simple extensions of the model.

• Application of Girsanov's theorem to Black-Scholes dynamics; self-financing strategies and model completeness; buyer's and seller's prices; derivation of the Black-Scholes formula via expectations; wealth processes and minimal hedges.

• (optional, time allowing) A choice of: Barrier and lookback options in the Black-Scholes model; reflexion principle for Brownian motion; comparison of European and American options; dividend-paying stocks; path-dependence.

References:

1. R.A. Dana and M. Jeanblanc, Financial Markets in Continuous Time, Springer 2001. 2. T. Bjork, Arbitrage theory in continuous time, Oxford University Press 1999. 3. M. Baxter and A. Rennie, Financial Calculus, Cambridge University Press 1996. 4. R.J. Elliott and P.E. Kopp, Mathematics of Financial Markets, Springer 1999.

5. P. Wilmott, Derivatives, Wiley 1997. 6. R. Korn and E. Korn, Option Pricing and Portfolio Optimization, Graduate Studies in

Mathematics, vol. 31, American Mathematical Society, 2001. 7. D. Lamberton and B. Lapeyre, Introduction to Stochastic Calculus Applied to Finance,

Chapmans & Hall/CRC, 2000. AMT 2219 C Programming/Mathematical soft wares

Algorithms:

Definition and properties, developing well-known algorithms, flow-charting. Programming languages: machine language, assembly language, High-level languages, assemblers, compilers and interpreters.

C language preliminaries:

Structure of a C program, the function main, header files, C preprocessor. Built - in data types: int, float, char, double, Constants and variables, variable declarations, Input/Output of basic data types.

Arithmetic operators, relational operators, logical operators, expressions, precedence and order of execution, the assignment operator.

Control structures, if…else, else if, switch, while loop, for loop, do…while loop, break and continue statements.

Arrays-one dimensional and two-dimensional arrays, their internal representation, benefits of using arrays, enumerators, structures and unions.

Pointers and pointer arithmetic.

Input/Output operations on files.

User defined functions:

Defining and calling functions, returning data from function, the type void, default arguments, recursive calls. Storage classes and scope: internal, external, automatic, static, register. Command line arguments- passing arguments to functions main.

Mathematical Software

Matlab: Essentials of MATLAB, Vectors, Matrices, and the colon operator, M files: Scripts and Functions, Input and Output, Graphics, Control Structures and Logical Tests, Symbolic Math Toolbox, Advanced MATLAB features

Mathematica: Essentials of Mathematica, Types of arithmetic, Programming Paradigms

Internet: The internet as a source of information about mathematical software

Options/J Java Options Pricing and Analysis software

References:

1. Duane Hanselman and Bruce Littlefield: Mastering Matlab 6, A Comprehensive Tutorial and Reference, Prentice Hall, 2001

2. Stephen Wolfram: The Mathematica Book, Cambridge University Press, Cambridge, Fourth edition, 1999

3. Gottfried: Programming with C, Mcgraw Hill, Schaum’s outline series. 4 Kernighan and Ritchie: The C Programming Language, 2nd edition, Prentice

Hall of India ltd.

For Financial Mathematics Software refer : http://windale.com

SEMESTER III AMT 2315 Discrete Time Modeling and Derivative Securities

Discrete Time Modeling

• Basic assumptions and definitions for discrete-time market models. • The 1-step binary model with 1 stock and bond: viability, completeness, replicating

portfolios, risk-neutral probabilities, pricing derivatives • Viability and incompleteness for more than binary branching.

General discrete-time models – many time steps, several stocks, one bond. • General notions of trading strategies, self-financing strategies, value and gains

processes, arbitrage. • Detailed theory for 1 stock, several time steps and general branching. Viability, risk

neutral probabilities. • Fundamental Theorem of Asset Pricing in discrete time.

Brief consideration of models with several stocks – extension of the Fundamental Theorem.

Derivative Securities

• Derivative securities – definitions and examples especially European options. • Pricing of attainable derivatives. • Specialisation to the 1 stock binomial (CRR) model; the Black-Scholes model. • Black-Scholes formula derived as a limit of the CRR pricing formula. • American options and pricing. • Pricing of European derivatives in incomplete markets (time allowing).

References:

1. A. Etheridge, A Course in Financial Calculus, Cambridge University Press, 2002. 2. N.H. Bingham and R. Kiesel, Risk Neutral Valuation, Springer, 1998. 3. M. Baxter and A. Rennie, Financial Calculus, Cambridge University Press, 1996.

4. J.C. Hull, Introduction to Futures and Options Markets, Prentice- Hal, 1998. 5. M. Capinski and T. Zastawniak, Mathematics for Finance: An Introduction to Financial

Engineering, Springer 2003. 6. S.E. Pliska, Introduction to Mathematical Finance: Discrete Time Models, Blackwell 1997.

AMT 2316 Portfolio Theory and Asset Allocation

• Value at Risk (VaR). • Mean and variance as a measure of return and risk. • Risk and return of a portfolio of two assets, diversification. Construction of the

feasible set. • Risk minimization for two assets. Indifference curves, optimization of portfolio

selection based on individual preferences. Inclusion of risk free asset. Finding the market portfolio in two-assets market. Discussion of the separation principle (single fund theorem). Market imperfections: different rates for borrowing and lending.

• Non-linear optimization: Lagrange Multipliers and Kuhn-Tucker theorem. • General case of many assets, risk-minimization, efficient frontier and its

characterization (reduction to the two-assets case: Two-fund theorem). The role of risk-free asset – Capital Market Line, market portfolio.

• Market imperfections: no short-selling. • Capital Asset Pricing Model (theorem on linear dependence of cost of capital on beta

–Security Market Line), practical applications – equilibrium theory. Certainty equivalent form of CAPM.

• Arbitrage Pricing Theory (if possible, time allowing).

References:

1. D.G. Luenberger, Investment Science, Oxford University Press 1998. 2. D. Duffie Dynamic Asset Pricing Theory, Princeton University Press 2001. 3. T.E. Copeland and J.F. Weston, Financial Theory and Corporate Policy, Addison Wesley

1992. 4. M. Capinski and T. Zastawniak, Mathematics for Finance: An Introduction to Financial

Engineering, Springer 2003. 5. E.K.P. Chong and S.H. Zak, An Introduction to Optimisation, Wiley 1996.

AMT 2317 Financial Risk Management

• Basic concepts of probability, Introduction to measure theory, Riemann integral, Lebesgue Integral, conditional expectation, Borel-Cantelli lemma, notions of convergence, characteristic functions, Limit theorems: law of large number, central limit theorem, Markov chains: discrete and continuous time Markov chains, Random walks, Gaussian stochastic processes, Martingales, Binomial model of asset pricing, Lognormal model of asset prices, Black-Scholes Model, derivation of the Black-Scholes equation, Black-Scholes formula for European option prices, Hedging parameters: the Greeks (Delta, Gamma, Vega, Theta) , American options. Early exercise and time-optionality

• Discrete and continuous time models, Applications of the discrete versions of the stochastic integrals and Ito’s lemma to risk analysis and management. Arbitrage based pricing of derivative securities. Risk neutral valuation, Binomial trees and American options, Black-Scholes and extensions, Some exotic options

• Continuous time risk analysis and management. Arbitrage pricing theory (Harrison-Kreps style) in continuous time with appropriate mathematical precision. Exotic options such as compound options, quantos, basket and barrier options, etc. Interest rate models (Heath-Jarrow-Morton, Hull-White, etc), yield curves, and pricing of interest rate derivatives (swaps, swaptions, caps, etc).

AMT 2318 Modeling of Bonds, Term Structure, and Interest Rate Derivatives

Bonds and Term Structure

• The concept of the term structure of interest rates. Presentation of theories explaining the shapes of the term structure encountered in practice. Methods of constructing long horizon term structure (bootstrapping, presentation of STRIPS).

• Tools describing dynamics of bond prices: yields, forward rates, short (instantaneous) rates. Money market account.

• Risk management in case of parallel shift. Applications of duration and convexity for immunization of bond portfolios. Problems with non-parallel shifts of term structure and tools necessary in this case.

• Necessity of developing a theory of random interest rates. Construction of Binomial trees for bond prices, yields and forward rates. No arbitrage principle and its consequences concerning admissible models. Risk neutral probabilities and a theorem on their dependence on maturity.

• Presentation of a variety of short term models (Vasicek, CIR, Dothan, Brennan-Schwartz). Single factor and multi-factor models. Model calibration. Discrete and continuous time versions.

• Outline of term structure models (Ho-Lee, Black-Derman-Toy, Hull-White) in discrete and continuous time framework. Forward rate model of Heath-Jarrow-Morton. Some other recent models (Brace-Gatarek-Musiela)

Interest Rate Derivatives

• Derivative securities in models with random interest rates. Pricing options. • Pricing and hedging interest rate futures, caps, floors, collars, swaps, caplets, florlets,

swaptions.

References:

1. M. Capiński and T. Zastawniak, Mathematics for Finance, Chapters 10, 11. Springer-Verlag, London 2003.

2. R. Jarrow, Modelling Fixed Income Securities and Interest Rate Options, McGraw-Hill, New York 1996.

3. T. Bjork, Arbitrage Theory in Continuous Time, Oxford University Press, Oxford 1998.

AMT 2319 Time Series Analysis

• Basic idea of time series analysis in both the time and frequency domains. • Autocorrelation • Partial ACF, Box and Jerkins ARIMA modeling • Spectrum and periodogram, order selection, diagnostic and forecasting. • Real life examples.

Reference:

1. P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods

SEMESTER IV

MATHEMATICAL FINANCE DISSERTATION/PROJECT

• Construction of martingale pricing measures by maximizing entropy • Continuous time limit of the binomial model • Estimating volatility using ARCH models • Optimal investments using utility functions • Real options • Mean-VaR portfolio theory • Liquidity risk by means of VaR • Valuation of companies using real options • Coherent risk measures • Capital structure • Optimal portfolios in Heath-Jarrow-Morton model • Conditional Value at Risk • Applications of change of numeraire for option pricing • Copulas in finance • Single factor short rate models • Modeling credit risk – structural approach • Credit risk – reduced form approach • Credit risk – probabilities of default • Stochastic backward equations in finance • Computer simulations of interest rate models • Stochastic differential delay equations in finance • Methods of designing pension schemes • Fundamental theorem of asset pricing and its extensions • Implied volatility, volatility smile, stochastic volatility • Complete market models implied by call options • Monte Carlo valuation of American type derivative securities • Microscopic simulation of the stock market • Pricing weather derivatives by utility maximization • Arbitrage pricing of mortgage-backed securities • Computer implementation of finite-difference option pricing schemes