Real Options

1
Real Options

• Life wasnt designed to be risk-free. The key
  is not to eliminate, but to estimate it
  accurately and manage it wisely.
• -William Schreyer, former Chair and CEO of
  Merrill Lynch

2
Options Are Everywhere!

• Callable Bonds
• Convertible Securities
• Warrants
• Secure Loans
• Firm with Debt
• Exotics
• Compensation
• Real Options

3
Real versus Financial Options

• Transparency
• Information
• Contract details
• Valuation?

4
What is a real option?

• The holder of an option has the right, but
  not the obligation
  to do something
• Some typical capital investment real options
• Timing Can the project occur sometime besides
  now or never?
• Growth Can you alter capacity?
• Abandonment Can you stop production before the
  end of economic life of the asset?
• Flexibility Can you alter operations?

5
Current State of Real Options

• Gaining traction in valuing corporate investments
  (but the slope is slippery)
• About a third of the CFOs always or almost
  always use real options (Graham Harvey)
• 10 to 15 of CFOs use real options always or
  often (Ryan Ryan)
• 9 of senior executives use real options but
  about a third had stopped using (Bain Co.)

6
Advantages of the Option Framework

• Creates a different way to think of how a firm
  can create shareholder value
• Ties strategic decisions directly to maximizing
  shareholder wealth
• May use capital market data
• More closely models actual decision process
• Encourages optimal growth opportunity investing,
  i.e. RD, or infrastructure
• Creates metric to better monitor and reward
  managers
• Think less of the most likely cash flow and
  more of the distribution of cash flows

7
How do we deal with real options?

• Use NPV and assume option value is zero
• Estimate expected cash flow
• Each cash flow is usually the most likely case or
  the probability weighted average of the cash
  flows
• Incorporate the riskiness of cash flows into r
• Assume the project is a now or never
  opportunity
• Use NPV and qualitatively recognize option
• Same as above except recognize options exist
• Without computing a value make a qualitative
  appraisal
• Financially engineer a project specific model
• Sky is the limit!

8
Another Method Financial Option Models

• Assumptions which may be problematic for real
  options
• Prices have various distributions
• Least restrictive assumption Brownian motion
• No arbitrage A replicating portfolio must exist
• Marketed Asset Disclaimer (MAD) assumes the
  project without the option is the replicating
  portfolio
• Binomial option pricing model (discrete time)
• More flexible, can be difficult to model
• Typically better economic description of real
  options
• Black-Scholes variations (continuous time)
• Requires stricter assumptions
• Better economic description of certain financial
  options
• Simple method for first pass valuation

9
Call Option Basics

• The buyer of an option has the right, but
  not the obligation to
  buy an asset
• The seller has a commitment to sell an asset
• Underlying Asset Price (S)
• Strike Price (X)
• Expiration Date (T)
• American Exercise at any time
• European Only exercise at maturity

10
Call Option Value Basics

• Option Premium
• Option Value
• Intrinsic
• At the expiration date
• ST gt X value
• ST X value 0
• ST lt X value 0
• Time At expiration date0
• PremiumIntrinsic valueTime Value
• Limited Liability

ST - X
11
Long (Buying) a Call Payoff Diagram
Value of Option at expiration
With Premium (P)
0
-P
Value of underlying asset (ST) at expiration
X
XP
12
Selling (Writing/Short) a Call

• The seller has a commitment to sell an agreed
  amount of an asset at an agreed price on or
  before a specified future date.
• Intrinsic value
• At the expiration date
• ST gt X value
• ST X value 0
• ST lt X value 0
• Unlimited liability

-(ST - X)
13
Short Call Payoff Diagram
Value of Option at expiration
P
With Premium (P)
0
Value of underlying asset (ST) at expiration
X
XP
14
Buying Versus Selling

• Liability
• Obligation versus Commitment

15
Put Option Basics

• The buyer of a put option has the right, but not
  the obligation to sell an agreed
  amount of an asset at an
  agreed price on or before a specified future
  date.
• Intrinsic Value
• At the expiration date
• Value Max 0 , X-ST
• Value X-ST if ST lt X
• Value 0 if ST ? X
• Limited liability

16
Long (Buying) a Put Payoff Diagram
Value of Option at expiration
With Premium (P)
0
-P
Value of underlying asset (ST) at expiration
X-P
X
17
Selling a Put

• The buyer of a put option has the right, but not
  the obligation to sell an asset
• The seller has a commitment to buy an asset .
• Intrinsic value
• At the expiration date
• Value -(X-ST ) if ST lt X
• Value 0 if ST ? X
• Unlimited liability

18
Short (Selling/Writing) a Put Payoff Diagram
Value of Option at expiration
P
0
With Premium (P)
Value of underlying asset (ST) at expiration
X-P
X
19
Four Option Positions
20
Valuation Basics
21
How Do We Value?

• Option ValueIntrinsicTime
• Focus on Time Component
• Time till Maturity (T)
• What if the call option is out of the money?
• Near the money?
• In the money?
• Exercise before expiration?
• S0-PV(X)

22
Option Value
Value of a Call
B
Upper bound
Lower bound
C
Exercise price
A
Share price
23
What are the determinants of value?

• Exercise price (X)
• Time till expiration (T)
• Underlying asset (S)
• Interest rate (rf)
• Variability of the value of the underlying
  assetVolatility (sS)

24
Option Pricing Model

• Characterize the underlying asset price
• Simple Binomial World
• 1 period (time t)
• Two possible prices uS0 and dS0

StuS0150
S0100
StdS050
25
Call Option Intrinsic Value

• Call Option (C)
• Strike Price (X) 100
• Expires at t

StuS0150 Ct50
S0100 C0?
StdS050 Ct0
26
What is the value of the call today?

• Assumption No Arbitrage Opportunities
• Establish an arbitrage portfolio of the option
  and underlying asset
• This portfolio has a constant return
• What is that return?
• Portfolio
• Short Call
• Long stock

27
Portfolio Composition

• Portfolio Short call and long stock
• What proportions?
• Say we short ONE call, now how many stocks do we
  buy (h)?
• Remember the goal is constant payoff
• Payoff in up statepayoff in down state
• 150h-5050h-0 h0.5
• Generalize h for one period model

huS0h150 1Ct-50
S0100
hdS0h50 1Ct0
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Portfolios Value

• Self-financing portfolio
• Total cash flows must equal zero
• Whats the valuation equation?

huS00.5(150)75 1Ct-50 Pt25
S0100 C0? P0?
hdS00.5(50)25 1Ct0 Pt25
29
Call Price

• Assume r10
• hS0-CPV(payoff)
• hS0-ChuS0-Cu/(1r)
• (0.5)100-C75-50/1.1
• (0.5)100-C25/1.1
• C50-22.73
• C27.27

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Generalization of the One Step Model
Substitute in the following for h
Implications
31
Important Characteristics

• Backward induction method
• Rollback values one period at a time
• Arbitrage portfolio guarantees that the payoff is
  identical in all states of nature
• No longer concerned which path the price on the
  underlying asset takes
• Implies that you are not concerned about state
  dependent risk
• Implies that all investors are risk neutral
• Used in all derivatives pricing
• Simplifying assumption
• What is the appropriate r?
• The risk free rate!

32
Risk Neutral Valuation

• Investors are risk-neutral (the probability of
  underlying asset moves dont matter), they
  require 10
• The share price can increase/decrease by 50
• What is the probability of an increase/decrease?
• p(1r)-d/(u-d)1.1-0.5/(1.5-0.5)0.6
• Intuitively
• 10.1 P(increase) 1.5 P(decrease) 0.5
• 1.1 P(increase) 1.5 P(1- increase) 0.5
• P(increase) 60 and P(decrease)40
  (Risk-neutral probabilities)
• Given the call prices for the two scenarios
• 1/1.1 x pCu(1-p)Cd
• 1/1.1 x 0.60 50 0.40 0 27.27

33
Two Ways to Value

• No-Arbitrage Valuation To value an option, you
  can take a levered position in the underlying
  asset that replicates the payoffs of the option.
  So you have to estimate the price of the
  replicating portfolio and the option.
• Risk-Neutral Valuation Assume investors do not
  care about risk, so that the expected return on
  the underlying asset is equal to the risk-free
  interest rate. Calculate the expected future
  value of the option then discount it to time 0.
  Only have to estimate stock and option prices (no
  probabilities!).

34
Two Step Binomial Model
Assume u10.5, d1-0.5, r0.10, t1, S100 and
X100 First Build the stock prices from today
forward by u and d.
uuS0225 C?
uS0150 C?
udS075 C?
S0100 C?
dS050 C?
ddS025 C?
35
Second
At expiration, compute the intrinsic value of the
option then one period at a time, recursively
solve to time 0 (today) using the risk neutral
probability (p).
uuS0225 C125
uS0150 C68.18
S0100 C37.19
udS075 C0
dS050 C0
ddS025 C0
36
Valuing a Put

• Remember Intrinsic value of a put is
  Max0,X-ST
• Intuitively, what is different for a put
• Underlying stock price?
• No difference
• At maturity payoff (intrinsic value)?
• Max 0,X-ST instead of Max0,ST-X
• Recursive solution
• Risk neutral probability (p)?
• No difference
• Put computation (C)?
• No difference

37
Two Step Binomial Model Put
Same assumptions as the call u10.5, d1-0.5,
r0.10, t1, S100 and X100 First Build the
stock prices from today forward by u and d. NO
DIFFERENCE FROM THE CALL!
uuS0225 P?
uS0150 P?
udS075 P?
S0100 P?
dS050 P?
ddS025 P?
38
Second Put
At expiration, compute the intrinsic value of the
option then one period at a time, recursively
solve to time 0 (today) using the risk neutral
probability (p).
uuS0225 P0
uS0150 P9.09
S0100 P19.83
udS075 P25
dS050 P40.91
ddS025 P75
39
Valuing an American Option

• Remember an American option can be exercised any
  time before (or at) maturity
• European option can only be exercised at maturity
• Intuitively, what is the difference?
• Underlying stock price?
• No difference
• At maturity payoff?
• No difference
• Backward induction
• Risk neutral probability (p)?
• No difference
• Put or Call computation
• Partial difference The same except at each
  node, take the higher of the option value or the
  exercise price (CS-X, PX-S)

40
Real Data

• Usually has many periods
• Estimating r, t, S0 and X is relatively easy.
• Estimating u and d are more difficult
• Usually based on the volatility of the underlying
  asset.
• Typical estimation problems period, data
  frequency, etc
• Assume Dt is the length of one period in the
  binomial model , s is the historical volatility
  over that same period.

41
Option Delta

• How many shares are needed to replicate an
  option?
• Option Delta Spread of Possible Option Prices /
  Spread of Possible Share Prices
• Delta (125 - 0) / (225 - 25) 5/8
• Delta of a Put Delta of a Call with the same
  exercise price minus 1

42
Black-Scholes versus Binomial Model

• Continuous versus Discrete Time
• Assume stock price can be characterized by a
  continuous process
• Special limiting case of Binomial as the length
  of the period approaches zero
• Computationally easier
• More restrictive assumptions
• Several variations to account for different types
  of underlying assets

43
Black-Scholes Model

• European option on non-dividend paying asset
• N(d) cumulative normal density function
• X exercise price
• t number of periods to the expiration date
• S current stock price (underlying asset)
• ? standard deviation per period of the rate of
  return on the underlying asset (continuously
  compounded)

44
BS Example

• Value a 100 strike price call on ABC
• 3 months (0.25 years) till expiration
• 0.5 standard deviation
• risk free rate/year is 0.04
• ABC is trading at 101
• Value a 100 strike price put on ABC

45
Option Delta

• First derivative of the option value with respect
  to the price of the asset (S)
• Interpretation If the price of the asset
  increases by 1 how much will the value of the
  option change?

46
Black-Scholes Variations

• American options on non-dividend paying assets
• European options on assets that have
• discrete (point in time before expiration)
  dividend
• continuous proportion dividend such as index
  futures
• European options on foreign currency
• European options on futures and forwards
• Etc.

47
DCF to Real Options

• Exercise Price (X)?
• Initial investment (investment required to
  acquire the assets)
• Underlying asset (S)?
• PV (CF) excluding initial investment (value of
  the operating assets to be acquired)
• Time to expiration (t)?
• Length of time the choice is available
• Volatility (s)?
• Riskiness of underlying operating assets
• Risk free rate?
• Time value of money If replicating portfolio
  exists the risk free rate. If not, the risk free
  rate will provide the upper bound.

48
DCF versus Real Options

• One method does not completely dominate
• Both methods used correctly will give identical
  values
• Each method frames the question differently
• DCF
• Works well for assets in place type of projects
• Lower uncertainty, less decision nodes
• Real Options
• Works well for growth option type of projects
• Greater span of possible cash flows, many
  decision nodes