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FIN 685: Risk Management

FIN 685: Risk Management. Topic 5: Simulation Larry Schrenk, Instructor. Topics. Why Simulation? Monte Carlo Simulation Example: European Call. Solution Types. Closed Form FV = PV(1+r) t Numerical Algorithm Binomial Option Pricing Simulation. Definition:.

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FIN 685: Risk Management

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  1. FIN 685: Risk Management Topic 5: Simulation Larry Schrenk, Instructor

  2. Topics • Why Simulation? • Monte Carlo Simulation • Example: European Call

  3. Solution Types • Closed Form • FV = PV(1+r)t • Numerical • Algorithm • Binomial Option Pricing • Simulation

  4. Definition: “Simulation is the process of designing a model of a real system and conducting experiments with this model for the purpose of either understanding the behavior of the system and/or evaluating various strategies for the operation of the system.” - Introduction to Simulation Using SIMAN (2nd Edition)

  5. What is Simulation? • Simulation is the use of a computer to evaluate a systemmodelnumerically, in order to estimate the desired true characteristics of the system. • Simulation is useful when a real-world system is too complex to allow realistic models to be evaluated analytically.

  6. Why Simulation • Complexity/Flexibility • Real World Applications • Dependencies • Descriptive Model • Distributional Assumptions • Distributions not Tractable • Empirically Based Distributions

  7. Basics • System: The physical process of interest • Model: Mathematical representation of the system • Models are a fundamental tool of science, engineering, business, etc. • Abstraction of reality • Models always have limits of credibility • Simulation: A type of model where the computer is used to imitate the behavior of the system • Monte Carlo Simulation: Simulation that makes use of internally generated (pseudo) random numbers

  8. Classification • Static vs. dynamic • Static: E.g., Simulation solution to integral • Dynamic: Systems that evolve over time; simulation of traffic system over morning or evening rush period • Deterministic vs. stochastic • Deterministic: No randomness; solution of complex differential equation in aerodynamics • Stochastic (Monte Carlo): Operations of store with randomly modeled arrivals (customers) and purchases • Continuous vs. discrete • Continuous: Differential equations; “smooth” motion of object • Discrete: Events occur at discrete times; queuing networks

  9. Ways to Study System System Experiment w/ actual system Experiment w/ model Physical Model Mathematical Model Analytical Model Simulation Model

  10. Monte Carlo Simulation • The process of generating a sequence of random values from a probability distribution • Formal Distribution • Empirical Distribution

  11. Uses • General Motors, Proctor and Gamble, Pfizer, Bristol-Myers Squibb, and Eli Lilly use simulation to estimate both the average return and the risk factor of new product • Sears uses simulation to determine how many units of each product line should be ordered from suppliers. • Financial planners use Monte Carlo simulation to determine optimal investment strategies for their clients’ retirement.

  12. Advantages • It is relatively straightforward and flexible • Recent advances in computer software make simulation models very easy to develop • Can be used to analyze large and complex real-world situations • Allows “what-if?” type questions • Does not interfere with the real-world system • Enables study of interactions between components • Enables time compression • Enables the inclusion of real-world complications

  13. Disadvantages • It is often expensive as it may require a long, complicated process to develop the model • Does not generate optimal solutions, it is a trial-and-error approach • Requires managers to generate all conditions and constraints of real-world problem • Each model is unique and the solutions and inferences are not usually transferable to other problems

  14. Simulation Steps • Define a problem • Introduce the variables associated with the problem • Construct a numerical model • Set up possible courses of action for testing • Run the experiment • Consider the results • Decide what courses of action to take

  15. Monte Carlo Simulation • Determine • Probability Distribution • Dependencies • Generate Random Variables • Find Terminal Values • Discount • Average

  16. 1. Probability Distributions

  17. Determine Distributions and Dependencies • Sources • Historical Data • Surveys • Judgment • Theory • Misc • Goodness-of-Fit Software

  18. 2. Generate Random Numbers

  19. Empirical Distribution

  20. PSEUDO RANDOM NUMBERS • Statistical Qualities • Excel: RAND() • Returns an evenly distributed random real number greater than or equal to 0 and less than 1 • RAND()*(b-a)+a

  21. Data Analysis Pack • Data > Data Analysis (Add-In)

  22. Theoretical Distribution

  23. 3. Find (Terminal) Value

  24. Terminal Value of Stock • What is the Stock Price for each Trial?

  25. Terminal Value of Stock • St

  26. Terminal Value of Call • MAX[St – X, 0]

  27. 4. Discount

  28. Present Value • MAX[St – X, 0]e-rt

  29. 5. Average

  30. Average

  31. Convergence

  32. Verification and Validation • Verification • Whether software correctly implements specified model • Validation • Whether the simulation model (perfectly coded) is acceptable representation

  33. Advanced Techniques • Antithetic Variables • Control Variate Technique • Quasi-Random Sequences

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