session 1 - introduction and arbitrage pricing
TRANSCRIPT
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Advanced Decision ModelsOPMG-GB. 60.2351
Session 1
Introduction and Arbitrage Pricing
Professor Jiawei Zhang
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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)
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Agenda
Introduction of the course
Arbitrage pricing
Option pricing
Capital budgeting
Lognormal distribution and Black-Scholes
Valuation of strategic flexibility (real options)
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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
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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
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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
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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.
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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.
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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
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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
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Marketing Examples
Targeted Marketing
Flexibility in Capacity Decision for New Product
Test-Marketing a New Product
Timing Market Entry
Pricing Airline Revenue Management
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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
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Software: Risk Solver Platform
Add-in for Excel
Powerful tool for
Optimization & Stochastic Optimization Simulation & Risk Analysis
Decision Tree
Sensitivity Analysis
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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
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Four assignments, 20% each
Class Participation 20%
Grading
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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
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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
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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
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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
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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
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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
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If p56/9(the call is overpriced), we can
Short one call
Buy 5/9shares of stock
Borrowp-200/9 dollars
Arbitrage Opportunity
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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
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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
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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
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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
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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
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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.
Lognormal Model of Stock Prices
)]1,0(Normal)5.0exp[( 20 ttSSt
)1,0(Normal)5.0()()( 20 ttSLnSLn t
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Lognormal Model of Stock Prices
E S S e
S S e e
T
T
T
T T
( )
( ) ( )
0
0
2 2 2 1
var
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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
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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
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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.
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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
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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
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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
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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