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:“Simulation is the process of designinga model of a real system and conductingexperiments with this model for thepurpose of either understanding the behavior of the system and/or evaluating various strategies for theoperation of the system.” - Introduction to Simulation Using SIMAN (2nd Edition)

5 of 50

WHAT IS SIMULATION?

• 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.

WHY SIMULATION

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

7 of 50

BASICSSystem: 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

CLASSIFICATIONStatic 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

WAYS TO STUDY SYSTEM

System

Experiment w/ actual system

Experiment w/ model

Physical Model

MathematicalModel

Analytical Model

SimulationModel

MONTE CARLO SIMULATION The process of generating a

sequence of random values from a probability distribution

– Formal Distribution

– Empirical Distribution

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.

ADVANTAGES

1. It is relatively straightforward and flexible2. 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 questions5. Does not interfere with the real-world

system6. Enables study of interactions between

components7. Enables time compression8. Enables the inclusion of real-world

complications

DISADVANTAGES

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

SIMULATION STEPS

1. Define a problem2. Introduce the variables associated with

the problem3. Construct a numerical model4. Set up possible courses of action for

testing5. Run the experiment6. Consider the results7. Decide what courses of action to take

MONTE CARLO SIMULATION1. Determine

1. Probability Distribution 2. Dependencies

2. Generate Random Variables3. Find Terminal Values4. Discount5. Average

1. Probability Distributions

DETERMINE DISTRIBUTIONS AND DEPENDENCIES Sources– Historical Data– Surveys– Judgment– Theory

Misc– Goodness-of-Fit Software

2. Generate Random Numbers

EMPIRICAL DISTRIBUTION

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

DATA ANALYSIS PACK

Data > Data Analysis (Add-In)

THEORETICAL DISTRIBUTION

3. Find (Terminal) Value

TERMINAL VALUE OF STOCK What is the Stock Price for each

Trial?

0 0 0fS S r t S tRV

TERMINAL VALUE OF STOCK St

TERMINAL VALUE OF CALL MAX[St – X, 0]

4. Discount

PRESENT VALUE

MAX[St – X, 0]e-rt

5. Average

AVERAGE

CONVERGENCE

1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071727374757677787980818283848586878889909192939495969798991001011021031041051061071081091101111121131141151161171181191201211221231241251261271281291301311321331341351361371381391401411421431441451461471481491501511521531541551561571581591601611621631641651661671681691701711721731741751761771781791801811821831841851861871881891901911921931941951961971981992002012022032042052062072082092102112122132142152162172182192202212222232242252262272282292302312322332342352362372382392402412422432442452462472482492500.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

Option Value for Increasing Number of Runs

Number of Runs

Op

tio

n V

alu

e

VERIFICATION AND VALIDATIONVerification –Whether software correctly implements specified model

Validation –Whether the simulation model (perfectly coded) is acceptable representation

ADVANCED TECHNIQUES

Antithetic Variables

Control Variate Technique

Quasi-Random Sequences