LOGISTICS: Teaching Assistant: Textbook: Class Schedule and Location

Transcription

1 FRE 6331: FINANCIAL RISK MANAGEMENT AND OPTIMIZATION Richard Van Slyke LOGISTICS: Office: LC121 Dibner Building (Brooklyn) or Phone: (not very reliable!) Office hours and location TBA Teaching Assistant: Kamyar Neshvadian Class Schedule and Location We meet on Wednesdays from 5:30-8:00 PM at 14 Wall St. on the 20 th floor, Room Meetings: 3/12, No class 3/19 (Spring Vacation), 3/26, 4/2, 4/9, 4/16, 4/23, 4/30 (Final). Textbook: Optimization Methods in Finance, by Gerard Cornuejols and Reha Tütüncü, Cambridge,

2 Reference Materials: We will: pass out and post PowerPoint presentations. post supplementary notes. post links to on-line references. put relevant references on reserve in the Polytechnic Library. COURSE OBJECTIVE: Matching Optimization Techniques to Financial Applications LIMITATIONS Because time is limited (to 7 weeks) we will: 1. Emphasize an intuitive view of the computational techniques that are most important. There will be few proofs. (More details can be found in the text, and in Supplementary Notes.) 2. Similarly for the financial applications we use will be streamlined and simplified. Practical issues will be ignored unless they illuminate the computational technique they are meant to illustrate. But, For Each Technique We Will Try to Cover: The basic class of problems the technique can address directly Some modeling techniques that can be used to extend the problem class Examples of financial engineering applications of the technique References to books, web sites and computer programs that you can use to learn more about the technique if it seems relevant to your application AN ABSTRACT OPTIMIZATION PROBLEM The Generic Problem The generic mathematical optimization problem, (P), is: Terminology The real valued function f is called the objective function, and D the constraint set, or decision set. We choose to minimize cost. (P) Minimize f(x) subject to x D * 2

3 Terminology (cont.) The problem (P) is called feasible if D is nonempty, and a point in D is called a feasible solution. The decision x* D solves (P) or is optimal if x D implies that f(x) f(x*). In this case, x* is an optimal solution or optimizer of (P), and f(x*) is the optimal value. SOME VERY GENERAL TOOLS Even for the very general generic problem (P) that we defined we can establish some very useful results. We will consider three: 1. Solving maximization problems using a minimization tool. 2. Substituting monotone functions of the objective, and 3. Using relaxations * MODELING TRICKS Maximization A simple modeling trick allows us to maximize profit instead of minimizing cost because: Maximum f(x) subject to x D = -Minimum f(x) subject to x D Modeling Tricks This is the first of several tricks we will introduce that make the techniques we discuss much more powerful than they at first seem. These tricks allow us to convert wide varieties of optimization problems into standard forms that have effective algorithms. These tricks, while usually very simple, can be quite powerful. Another Modeling Trick: Monotone Transformations Suppose we modify the objective of (P): (P 0 ) Minimize f 0 (x) subject to x D If always we have: (f 0 (x) < f 0 (y) if and only if f(x) < f(y) whenever x D, and y D), then x* D is an optimizer of (P) if and only if it is an optimizer of (P 0 ). 3

4 Applications f 0 (x) = kf(x), for any positive constant k (for example, if you change from dollars to cents as units). f 0 (x) = f(x) + c, for any constant c; for this reason you rarely see constant terms in objective functions. (Of course the optimal value of the objective depends on the constant!) f 0 (x) = g(f(x)), where g is strictly monotone increasing f 0 (x) = f(x) 2, when f(x) 0 for x of interest. f 0 (x) = f(x) 1/2, when f(x) > 0 for x of interest. f 0 (x) = log f(x), when f(x) > 0 for x of interest. f 0 (x) = e f(x) RELAXATIONS The Generic Problem Minimize f(x) subject to x D A Relaxation: Minimize f (x) subject to x D Where f f, and D contains D. * Consequences Relaxation (cont.) Any optimal solution value for f (x) on D is a lower bound for any solution of (P). If the relaxation is infeasible so is the generic problem. Suppose x* D and f (x*) f (x) for any x D (that is, x* solves (P) with f, D replacing f, D). Then if x* D, and f (x*) = f(x*) then x* solves the original problem (P). * Example (P) f(x) = 2x 2-3x+5, D= {x 0 x 1}, where x is a real variable. The relaxation we consider is replacing D with D where D is the set of all real numbers. We essentially do away with the constraint. Do You Remember Calculus? Set the first derivative of f to 0 to find stationary points: f (x*) = 4x* 3 = 0 => x* = ¾ is the only stationary point. Now to make sure we have a minimum, we evaluate f (x*) = 4 > 0. Since x* satisfies the constraint defining D we are done! * * 4

5 More Applications Later Later we will consider relaxations of integer and convex programming to linear programming. KINDS OF DECISION VARIABLES: Continuous vs. discrete One-dimensional, multidimensional, functionals (e.g., for dynamic problems), stochastic functionals * OPTIMIZATION PROBLEMS: We look at a structured list of optimization problem types that we might consider. Based on classical calculus (review): One-dimensional, continuous, deterministic, decision variable Multi-dimensional, continuous deterministic, decision variables Mathematical Programming Linear Programming (linear objective with linear inequality constraints; continuous variables) Integer Programming (some discrete variables) Non-linear programming (objective and/or constraints non-linear; variables continuous) Dynamic Problems (time enters; multi-stage): Discrete Dynamic Programming Continuous Dynamic Programming 5

6 Stochastic Problems (risk!): Stochastic Programming Stochastic Discrete Dynamic Programming Stochastic Continuous Dynamic Programming Modeling Languages There are software systems supporting modeling languages that try to be closer to natural languages than to programming languages. The software translates your application into a form that can be optimized using a solver and passes the solution back. AMPL and GAMS are common systems of this sort. We will be using GAMS in this course. 6