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

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Why Simulation? Monte Carlo Simulation Example: European Call

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Closed Form – FV = PV(1+r) t Numerical – Algorithm – Binomial Option Pricing Simulation

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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)

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5 Simulation is the use of a computer to evaluate a system model numerically, 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.

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Complexity/Flexibility Real World Applications Dependencies Descriptive Model Distributional Assumptions – Distributions not Tractable – Empirically Based Distributions

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7 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

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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

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System Experiment w/ actual system Experiment w/ model Physical Model Mathematical Model Analytical Model Simulation Model

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The process of generating a sequence of random values from a probability distribution – Formal Distribution – Empirical Distribution

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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.

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1.It is relatively straightforward and flexible 2.Recent advances in computer software make simulation models very easy to develop 3.Can be used to analyze large and complex real- world situations 4.Allows “what-if?” type questions 5.Does not interfere with the real-world system 6.Enables study of interactions between components 7.Enables time compression 8.Enables the inclusion of real-world complications

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1.It is often expensive as it may require a long, complicated process to develop the model 2.Does not generate optimal solutions, it is a trial- and-error approach 3.Requires managers to generate all conditions and constraints of real-world problem 4.Each model is unique and the solutions and inferences are not usually transferable to other problems

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1.Define a problem 2.Introduce the variables associated with the problem 3.Construct a numerical model 4.Set up possible courses of action for testing 5.Run the experiment 6.Consider the results 7.Decide what courses of action to take

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1. Determine 1. Probability Distribution 2. Dependencies 2. Generate Random Variables 3. Find Terminal Values 4. Discount 5. Average

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1. Probability Distributions

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Sources – Historical Data – Surveys – Judgment – Theory Misc – Goodness-of-Fit Software

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2. Generate Random Numbers

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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

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Data > Data Analysis (Add-In)

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3. Find (Terminal) Value

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What is the Stock Price for each Trial?

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StSt

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MAX[S t – X, 0]

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4. Discount

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MAX[S t – X, 0]e -rt

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5. Average

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Verification – Whether software correctly implements specified model Validation – Whether the simulation model (perfectly coded) is acceptable representation

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Antithetic Variables Control Variate Technique Quasi-Random Sequences

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