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Course Code  :           AMAT 32343

Title                :           Mathematics for Finance II

Pre-requisites :           AMAT 31303

Learning outcomes:

On successfully completion of the course the student will be able to

• derive the payoff/profit diagrams for given trading strategy
• calculate the option price on various underlying assets using Binomial tree method
• solve Black-Scholes equation numerically
• identify the Greeks and their use
• define appropriate Swap strategies.

 Course Contents:

Trading Strategies: Single option and stock, Spreads: Bull spread, Bear Spread, Box spreads, Butterfly spreads and Combinations: Straddle, Strips and Straps.

Option Pricing using Binomial Trees: A one-step binomial model and a no-arbitrage argument, Risk-neutral valuation, Two-step binomial trees, Put and Call options, American options, Delta, Matching volatility with u and d, binomial tree formulas, increasing the number of steps, create spreadsheet application.

The Black-Scholes Formula: Brownian motion, martingales, stochastic calculus, Ito processes, stochastic models of security prices, Black-Scholes Merton Model, Black-Scholes Pricing formula on call and put options, Applying formula to other assets.

Numerical Solutions to Black-Scholes Equation: Converting to parabolic type, Finite difference methods, FTCS, BTCS and Crank-Nicholson Schemes for Black-Scholes Equation, implement the various numerical schemes using an appropriate software.

Option Greeks: Definition of Greeks, Greek Measures for Portfolios.

Swaps: swap, swap term, prepaid swap, notional amount, swap spread, deferred swap, simple commodity swap, interest rate swap

 Method of Teaching and Learning : A combination of lectures and tutorial discussions

 Assessment     :           Based on tutorials, tests and end of course examination

 Recommended Readings:

1. Hull, J.C. (10th Ed., 2018). Options, Futures and Other Derivatives, Pearson.
2. McDonald, R.L. (3rd Ed., 2013). Derivatives Markets, Addison Wesley.
3. Kosowski, R. & Neftci, S.N. (3rd Ed., 2015). Principles of Financial Engineering, Academic Press.

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Course Unit Code     : AMAT 32323

Course Title               : Mathematical Modeling

Pre-requisites             : PMAT 22282

Learning outcomes:

Upon successful completion of the course unit the student will be able to:

• explain how the general principles arise in the context of mathematical modeling
• analyze existing mathematical models using ordinary differential equations
• formulate simple ODE models for real world problems
• solve system of ordinary differential equations
• analyze the qualitative behavior of mathematical models
•  identify the solutions of difference equations
• solve system of linear difference equations using Putzer algorithm and Jordan form.

 Course Content:

Introduction to Mathematical Modeling: Philosophy of modeling, Modeling Methodology, Problem formulation, Mathematical Description, Analysis, Interpretation.

Mathematical Modeling Using Ordinary Differential Equations: Classification of ODE, Equilibrium points. First order Differential Equations: Mixing, chemical reactions, Population models: Logistic growth model, Harvesting models, Traffic Dynamic models: Microscopic and macroscopic models. System of Differential equations: Interacting population models (Predator–Prey models, Competition models), Compartment models (Dynamic of infectious disease, Age structured models, Reaction kinetics), Qualitative analysis of models.

Mathematical Modeling Using Difference Equations: First order difference equations, Equilibrium points, asymptotic stability of equilibrium points, System of linear difference equations: Autonomous systems, Discrete analogue of Putzer algorithm, Jordan form, linear periodic systems.

 Method of Teaching and Learning: A combination of lectures and tutorial discussions.

Assessment : Based on tutorials, tests and end of course examination.

Recommended Textbook:

1. Kapur, J.N. (2015). Mathematical Modeling, New Age International.

2. Bender, A. (2012). An introduction to Mathematical Modeling, Courier Corporation.

3. Haberman, R. (1998). Mathematical Models: Mechanical Vibrations, Population Dynamics and Traffic Flow. SIAM.

4. Allen, L. (2006). An Introduction to Mathematical Biology, Pearson.

5. Elaydi, S. (2005). An Introduction to Difference Equation, Springer.

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Course Code  :           AMAT 32333

Title                :           Introduction to Fluid Dynamics

Pre-requisites :           PMAT 22293

Learning Outcomes   :

At the end of this course the student will be able to

• identify fluid flow motions and their properties
• formulate equations of motions based on three conservation laws
• simplify equations of motions considering flow characteristics and apply them in real world problems
• identify appropriate boundary conditions
• make use of complex analysis for two-dimensional fluid motions
• distinguish the dominant terms through dimensional analysis.

 Course Content         :

Vector Analysis Review: Orthogonal curvilinear coordinates, Gradient, Divergence and curl.

Basic Principles of Fluid Dynamics: Fluids and fluid flow variables, Streamlines and path lines, Lagrangian and Euler approaches for describing fluid motions, Reynold’s Transport Theorem, conservation of mass (equation of continuity), momentum and energy

Newtonian fluid: Inviscid and viscous fluids, Euler’s equation of Motion, Vorticity, irrotational motion under conservative forces, Bernoulli’s equation

Boundary condition: Inlet and outlet conditions, no slip condition, pressure boundary conditions, radial and axisymmetric boundary conditions.

Flow in Pipes: Laminar flow in pipes, Pressure drop and head loss, flows in non-circular and inclined pipes.

Two-Dimensional Motion: Stream function and plotting streamlines, Complex potential, Sources and sinks, Vortices, Doublets and image systems, Milne-Thompson theorem.

Axi-symmetric Motion: Stokes’ stream function in three dimensional flows.

Dimensional Analysis and modeling: Nondimensionalization of equations

 Method of Teaching and Learning : A combination of lectures and tutorial discussions

 Assessment     :           Based on tutorials, tests and end of course examination.

 Recommended Reading       :

1. Ruban, A.I. & Gajjar, J.S.B. (1 st Ed., 2014). Fluid Dynamics (classical fluid dynamics), Oxford.

2. Cengel, Y.A. & Cimbala, J.M. (2006). Fluid Mechanics (Fundamentals and Applications), McGraw Hill.

3. Feistauer, M. (1993). Mathematical Methods in Fluid Dynamics, Chapman and Hall/CRC.

4. Chorin, A.J. & Marsden, J.E. (2012). A Mathematical Introduction to Fluid Mechanics, Springer Science & Business Media.

5. Henningson, D.H. & Berggren, B. (2005). Fluid Dynamics Theory and Computation, Stockholm.

6. Chorlton, F. (2005). Textbook of Fluid Dynamics, CBS Publishers & Distributors.

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Course Unit Code     : AMAT 31313

Course Title               : Computational Mathematics

Pre-requisites             : AMAT 22292

 Learning outcomes:

Upon successful completion of the course unit the student will be able to:

• classify Partial Differential Equations (PDE)
• identify initial and boundary conditions of PDE
• calculate finite difference operators to approximate derivatives and corresponding truncation errors
• apply finite difference methods to obtain the approximate solution of PDEs together with prescribed boundary and/or initial conditions
• analyze the stability, consistency and convergence of numerical schemes
• compare the accuracy of the approximate solution obtained by finite difference scheme using simulation results
• solve boundary value problems using basic finite elements methods
• solve one dimensional PDEs using finite element method by using appropriate software.

 Course Content:

Finite Difference Methods: Introduction, Classification of Partial Differential Equations (PDE): parabolic, hyperbolic and elliptic, Taylor series expansion: analysis of truncation error. Initial and boundary conditions: Dirichlet and Neumann boundary conditions. Finite difference methods: Forward, Backward, Centered and Crank-Nicholson schemes, Implicit and Explicit methods. Stability and Convergence analysis of numerical schemes: Von Neumann Analysis, Consistency and Stability, Lax Equivalent Theorem, Comparison of Numerical Schemes.

Finite Element Methods: Introduction, Weak Formulation. Solving one dimensional PDEs using finite element method: Weighted residual methods: Collocation method, least square method, Galerkin method.

Practical: Simulate the Finite Difference solutions using appropriate programming language

 Method of Teaching and Learning:  A combination of lectures, classroom discussions and computer laboratory sessions

 Assessment : Based on assignments, group projects and Final examination.

 Recommended Readings:

1. Burden, R.L., Faires, J.D, Burden, M.L. (10th Ed., 2016). Numerical Analysis, Cengage Learning.
2. Smith, G. D. (3 rd Ed., 1986). Numerical Solution of Partial Differential Equations: Finite Difference Methods, Clarendon press.
3. Evans, J., Blackledge, J., & Yardley, P. (2000). Numerical Methods for Partial Differential Equation, Springer.
4. Davies, A.J. (2 nd Ed., 2011). Finite Element Method: An Introduction to Partial Differential Equations, OUP Oxford.
5. Desai, Y.M. (2011). Finite Element Method with Applications in Engineering, Pearson Education India.
6. Ŝolín, P. (2013). Partial Differential Equations and the Finite Element Method, Wiley.