Session 1 - Introduction and Arbitrage Pricing

8/12/2019 Session 1 - Introduction and Arbitrage Pricing

1/36

1

Advanced Decision ModelsOPMG-GB. 60.2351

Session 1

Introduction and Arbitrage Pricing

Professor Jiawei Zhang


2/36

2

About Me

Education: Ph.D. Management Science, Stanford Univ., 2004

M.S. Operations Research, Tsinghua Univ., 1999

B.A. Applied Mathematics, Tsinghua Univ., 1996

Research: Deterministic and Stochastic OptimizationHealth Care Operations

Supply Chain Network Design

Production Planning and Scheduling

Inventory Management

Teaching: Operations Management (Undergraduate)

Advanced Optimization (PhD)

Supply Chain Optimization (PhD)

Decision Models (MBA, Undergraduate)

Advanced Decision Models (MBA)


3/36

3

Agenda

Introduction of the course

Arbitrage pricing

Option pricing

Capital budgeting

Lognormal distribution and Black-Scholes

Valuation of strategic flexibility (real options)


4/36

4

Decision Models

Deterministic Optimization (Math Programming)

Linear Programming

(Binary) Integer Linear Programming

Nonlinear Programming

Network Flow

Stochastic Models: Simulation

Focusing on evaluating outcomes of (given) decisions

Not searching for optimal decision

Decision Analysis:

Decision: # of alternatives is small

Uncertainty: # of states is small


5/36

5

B60:2351: Advanced Decision Models

Designed for students who have taken Decision Models

Focus on decision making under uncertainty

Optimization incorporates uncertain parameters

Emphasis on

Model formulation

Interpretation of results

Not mathematical algorithms


6/36

6

Modeling Tools: Stochastic Programming

Static Stochastic Optimization Decisions made beforeuncertainty is resolved

Stochastic Optimization with Chance Constraints Constraints are satisfied with probability

Two-Stage Stochastic Optimization with Recourse Some decisions are made beforerandom variables are realized

Other decisions may wait until afterrandom variables are realized

Dynamic Stochastic Programming Uncertainty resolved over time

Decisions made over time


7/36

7

A Production Planning Example

Sailco must determine how many sailboats to produce during each of the next four quarters.At the beginning of the first quarter, Sailco has an inventory of 10 sailboats.

Sailco must meet demand on time. The demand during each of the next four quarters is asfollows:

1stQtr 2ndQtr 3rdQtr 4thQtr

40 60 75 25

For simplicity, assume that sailboats made during a quarter can be used to meet demand forthat quarter. During each quarter, Sailco can produce up to 50 sailboats with regular-timeemployees, at a labor cost of $400 per sailboat. By having employees work overtime during a

quarter, Sailco can produce unlimited additional sailboats with overtime labor at a cost of$450 per sailboat.

At the end of each quarter (after production has occurred and the current quarters demandhas been satisfied,), a holding cost of $20 per sailboat is incurred.


8/36

8

A Production Planning Example

Demands are often uncertain

Scenario 1: Production quantities for the next four quartershave to be determined now

Scenario 2: Decide now the production quantity for thenext quarter only, wait until quarter 2 to decide its

production quantity, etc.


9/36

9

Finance Examples

The Miller-Orr Cash Management Model

Retirement Planning: The Kelly Criteria

Optimal Hedging of Dell Computer Investment

Hedging Foreign Exchange Risk, Hedging with Futures

Value a Compound Option Capital Budgeting with Uncertain Resource Usage

Pricing an American Option

Valuing an Option to Purchase a Company

Valuing an Option to Purchase with an Abandonment Option


10/36

10

Operations Example

Inventory Management with Diversion

Optimal Sampling in Quality Control

Truck Loading

Animal Feed Formulation at Agway

Product Mix Multi-period Production Planning

Manpower Scheduling Under Uncertainty

Optimal Selection of Employees

Optimal Ordering Policies for Style Goods

Optimal Plant Capacities and Transportation Plan

Modeling Managerial Flexibility

Capacity Planning for an Electric Utility

Agricultural Planning Under Uncertainty

Optimal Strategy of Gold Mine


11/36

11

Marketing Examples

Targeted Marketing

Flexibility in Capacity Decision for New Product

Test-Marketing a New Product

Timing Market Entry

Pricing Airline Revenue Management


12/36

12

Valuation of Strategic Flexibility

Valuing an Internet Start-up

Valuing a Pioneer Option: Web TV

Valuing an R&D Project

Options to Postpone, Expand, and Contract

Value an Option to Develop Vacant Land Value a Licensing Agreement


13/36

13

Software: Risk Solver Platform

Add-in for Excel

Powerful tool for

Optimization & Stochastic Optimization Simulation & Risk Analysis

Decision Tree

Sensitivity Analysis


14/36

14

Recommended Books (by Wayne Winston) Financial Models using Simulation and Optimization II

Decision Making Under Uncertainty with RISKOptimizer

Learning by doing

Teaching by example

In-class exercises

Real-world cases

Course Materials


15/36

15

Four assignments, 20% each

Class Participation 20%

Grading


16/36

16

A European call option gives the owner the right to buya share ofstock for a particular price (the exercise/strike price) ona particulardate (the exercise/expiration date).

A European put option gives the owner the right tosella share ofstock for the exercise price onthe exercise date.

An American call/put option allows you buy/sell the stock at anydatebetween the present and the exercise date.

Option: Basic Definitions


17/36

17

Many actual investment opportunities (not just those involving stocks)may be viewed as combinations of puts and calls.

If we know how to value puts/calls we can value many actual

investment opportunities.

Example: The option to purchase an airplane 3 years from now for$20 million. The value of an airplane 3 years from now is uncertainand would depend on the economic cycle, fuel prices etc.

Example: We are undertaking an R&D project and five years fromnow we can sell what we have accomplished so far for $80 million.

Strategic Flexibility/Real Options


18/36

18

Expansion Option: Three years from now we have an option todouble the size of a project.

Contraction Option: Three years from now we have the option to cut

the scale of a project in half.

Postponement Option: We are thinking of developing a new SUV-minivan hybrid. In two years we will know more about the size of themarket. We have the option to wait two years before deciding todevelop the car.

Licensing Option: During any year in which profit from drug exceeds$50 million, we pay 20% of all profits to developer of drug.

More Examples on Strategic Flexibility


19/36

19

Arbitrage pricing: If an investment has no risk it should yield therisk-free rate of return.

Arbitrage opportunity:we can spend $0 today and ensure we have

no chance of losing money and a positive chance of making money.

Example: A stock is currently selling for $40. One period from nowthe stock will either increase to $50 or decrease in price to $32. Therisk free rate of interest is 1/9. What is a fair price for a European calloption with an exercise price of $40.

Any model in which a stock price can only increase or decrease by acertain amount during a period is called a binomial model.

Arbitrage Pricing Approach


20/36

20

The key is to construct a portfolio that has no risk. Why is thispossible?

Increase in the stock price benefits the stock owned, and the cash flow

of the European option.

What happens if we have a portfolio consisting ofxshares of stock andshortone call option?

If next period is in the good state, the value of portfolio is 50x-10. In abad state, the value of portfolio is 32x-0.

If we choosexsuch that 50x-10=32x, then the portfolio is risk-free.We can solve forxand getx=5/9.

Arbitrage Pricing: Example


21/36

21

Now we have a risk-freeportfolio withxshares of stock and short onecall option. It should grow at the risk-free rate 1/9.

Wed like to find out a fair pricep for this call option.

At present, the portfolio is worth $40x-p=200/9p

Next period, regardless of the stock price, the portfolio is worth$50x-10=250/9 -10

The growth is (250/9-10)/(200/9-p), which should be equal to 1+1/9.

We getp=56/9.

Arbitrage Pricing: Example


22/36

22

If p56/9(the call is overpriced), we can

Short one call

Buy 5/9shares of stock

Borrowp-200/9 dollars

Arbitrage Opportunity


23/36

23

In this example, the factors that influence the option price are

Current stock price ($40)

Two values of stock price in next period ($50, $32)

Risk free interest rate (1/9)

Exercise price of the call ($40)

How about the probabilitythat the stock will go up or down?

The average growth rate of the stock does notaffect the calls value!

If stock has more chance of increasing in price, shouldnt the call sellfor more because the call pays off for high stock prices?

The current price of the stock incorporates information about the

stocks growth rate!

Stocks Growth Rate


24/36

24

Discounted Cash Flow (DCF) approach

The value of a project is defined as the future expected cash flowsdiscounted at rate that reflects the riskiness of the cash flow.

Typically, these discount rates are defined as the equilibrium expectedrate of return on securities equivalent in risk to the project beingvalued.

Two commonly used discount rates Risk-adjusted discount rate

Weighted average cost of capital

The Capital Budgeting Example: DCF


25/36

25

Risk-adjusted discount rate

Rf + * ( Km- Rf)

Where Rf is the risk free rate

describes the relation of the return (of the project/stock) with amarket benchmark

=cov(return of a stock, return of market)/var(return of market)

Km is the expected rate of return of the market benchmark

Risk-Adjusted Discount Rate


26/36

26

Weighted Average Cost of Capital

Re = cost of equityRd = cost of debtE = market value of the firm's equity

D = market value of the firm's debtV = E + DTc = corporate tax rate


27/36

27

The Lognormalor Geometric Brownian Motion random variable isoften used to model the evolution of stock prices or project values.

It assumes that in a small time t, the stock price changes (difference)

by an amount that is normally distributed withMean=St

Standard Deviation =

Here

S: current stock price: instantaneous rate of return

: the percentage volatility in the annual return (not the sdev ofstock price).

(In 1999, Microsoft 47%, AOL 65%, Amazon.Com 120%)

Lognormal Model of Stock Prices

tS


28/36

28

During a small period of time, the standard deviation of the stocksmovement can greatly exceed the mean.

The Lognormalmodel leads to really jumpy changes in stock

prices.

Assume the current price is S0, then the stock price at time t is given by

Or

The natural logarithm follows a normal distribution

Continuously compounded rate of return.


)]1,0(Normal)5.0exp[( 20 ttSSt

)1,0(Normal)5.0()()( 20 ttSLnSLn t


29/36

29


E S S e

S S e e

T

T

T

T T

( )

( ) ( )

0

0

2 2 2 1

var


30/36

30

We can estimate volatility byimplied volatility or historicalvolatility.

Historical estimation of mean and volatility of stock return

1. Compute Ln(St/St-1) for t=1,2,,T2. Average the values we obtain an estimate of

3. Take the standard deviation, we obtain an estimate of .

4. Convert the daily/monthly estimate to annual estimate.

Lognormal: Estimation of Volatility

)5.0( 2


31/36

31

There is a lot of evidence that changes in stock prices have fattertails than a Lognormal random variable.

Lognormal model is still widely used to model changes in stock prices.

Sudden jumps in exchange rates and stock prices could happen, andcan be modeled with ajump diffusionprocess which combined a

jump process with a Lognormal process.

The jump diffusion usually assumes that the number of jumps per unittime follows a Poisson random variable and the size of each jump (as apercentage of the current price) follows a normal distribution.

Lognormal: Remarks

32


32/36

32

Black-Scholes Option Pricing Model

Consider a European put/call option with

S: todays stock price

t: duration of the option

X: Exercise or strike price

r: Risk free rate. (continuously compounded; if r=0.05, then growat exp(0.05)

: Annual volatility of stock

y: percentage of stock value paid annually in dividends.

LetN(x)be the cumulative normal probability for a normal randomvariable having mean 0 and standard deviation 1. In Excel, it is given

by Normsdist() function. For example,N(0)=0.5, N(1)=0.84,N(1.96)=0.975.

33


33/36

33

Black-Scholes Option Pricing Model

Let

Then the European call price is given by

The European put price is given by

These formulas are based on the arbitrage pricing approach. Noticethat does notplay a role in these formulas.

American options are usually modeled using binomial trees (to be

discussed in Dynamic Programming part of the course).

tddt

tyrLnd X

S

12

2

1 ,)5.0()(

)()exp()()exp( 21 dNrtXdNytS

]1)()[exp(]1)()[exp( 21 dNrtXdNytS

34


34/36

34

Suppose the current stock price of MSFT is $100, and we own a 7-yearEuropean call option with an exercise price of $95. Assume risk freerate of 5% and the annual volatility of 47%. What is a fair price of thisoption? (see Black Scholes Pricing MSFT template.xlsx)

Keep the solution file as a template for future use.

Sensitivity analysis:

Increase in todays stock price

Increase in the exercise price Increase in the duration

Increase in volatility

Increase in risk-free rate (assuming they do not affect currentstock prices but they do).

Black-Scholes: An Example

35


35/36

35

Consider a traded option and its actual price

Assume that the price matches the predicted Black-Scholes price

Then we can compute the volatility of the stock. This is called impliedvolatility.

What if there are several traded options on the same stock?

Example: At the end of trading on February 8, 2000 MSFT sold for$106.61. A put option expiring on March 25th, 2000 with exercise price$100 sold for $3.75. Risk free rate is 5.33%. Whats the impliedvolatility of MSFT at this point in time?

Implied Volatility

36


36/36

36

Risk Neutral Pricing and Real Options by Simulation

Read the following examples (pages 5-9) before class

Valuing an R&D Project

Options to Postpone, Expand, and Contract A Pioneer Option: Merck

Develop Vacant Land

Value a Licensing Agreement

Read the article Real Options Analysis and Strategic DecisionMaking by Bowman and Moskowitz

Download the template file

Next Class

Date post:	03-Jun-2018
Category:	Documents
Upload:	anand-r-bhat
View:	232 times
Download:	0 times

Session 1 - Introduction and Arbitrage Pricing

Documents