Real Options

“Life wasn’t 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

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Options Are Everywhere!

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

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Real versus Financial Options

• Transparency

• Information

• Contract details

• Valuation?

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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?

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Current State of Real Options

• Gaining traction in valuing corporate investments (but the slope is slippery)– About a third of the CFO’s “always” or “almost

always” use real options (Graham & Harvey)– 10% to 15% of CFO’s use real options “always” or

“often” (Ryan & Ryan)– 9% of senior executives use real options but about a

third had stopped using (Bain & Co.)

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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. R&D, 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

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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!

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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

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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

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Call Option Value Basics

• Option Premium

• Option Value– Intrinsic

• At the expiration date:

• ST > X value =

• ST = X value =0

• ST < X value =0

– Time: At expiration date=0

• Premium=Intrinsic value+Time Value

• Limited Liability

ST - X

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Long (Buying) a Call Payoff Diagram

With Premium (P)

$-P

$X

Value of underlying asset (ST) at expiration

Value of Option at expiration

$0

$X+P

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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 > X value =

• ST = X value =0

• ST < X value =0

• Unlimited liability

-(ST - X)

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Short Call Payoff Diagram

With Premium (P)

$P

$X

Value of underlying asset (ST) at expiration

Value of Option at expiration

$0

$X+P

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Buying Versus Selling

• Liability

• Obligation versus Commitment

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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 < X

– Value = 0 if ST X

• Limited liability

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Long (Buying) a Put Payoff Diagram

With Premium (P)

$-P

$X-P

Value of underlying asset (ST) at expiration

Value of Option at expiration

$0

$X

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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 < X– Value = 0 if ST X

• Unlimited liability

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Short (Selling/Writing) a Put Payoff Diagram

With Premium (P)

$P

$X-P

Value of underlying asset (ST) at expiration

Value of Option at expiration

$0

$X

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Four Option Positions

Payoff Diagram

-20

-10

0

10

20

30 40 50 60 70

Underlying Price

Pro

fit/L

oss Buy Call

Sell Put

Buy Put

Sell Call

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Valuation Basics

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How Do We Value?• Option Value=Intrinsic+Time

• 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)

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Value of a Call

Exercise price Share price

Lower bound

Upper bound

A

B

C

Option Value

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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 asset…Volatility (S)

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Option Pricing Model• Characterize the underlying asset price

• Simple Binomial World:– 1 period (time t)

– Two possible prices: uS0 and dS0

St=uS0=150

St=dS0=50

S0=100

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St=uS0=150

Ct=50

St=dS0=50

Ct=0

S0=100

C0=?

Call Option Intrinsic Value• Call Option (C)

• Strike Price (X) 100

• Expires at t

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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

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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 state=payoff in down state• 150h-50=50h-0 h=0.5• Generalize h for one period model:

huS0=h150

1Ct=-50

hdS0=h50

1Ct=0

S0=100

00 dSuS

CCh du

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huS0=0.5(150)=75

1Ct=-50

Pt=25

hdS0=0.5(50)=25

1Ct=0

Pt=25

S0=100

C0=?

P0=?

Portfolio’s Value• Self-financing portfolio

• Total cash flows must equal zero

• What’s the valuation equation?

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Call Price

• Assume r=10%

• hS0-C=PV(payoff)

• hS0-C=[huS0-Cu]/(1+r)

• (0.5)100-C=[75-50]/1.1

• (0.5)100-C=25/1.1

• C=50-22.73

• C=27.27

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Generalization of the One Step Model

Substitute in the following for h:

00 dSuS

CCh du

Implications:

du

drp

)1(

])1([)1(

1du CppC

rC

r

ChuSChS u

1

00

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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!

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Risk Neutral Valuation• Investors are risk-neutral (the probability of underlying

asset moves don’t matter), they require 10% • The share price can increase/decrease by 50%• What is the probability of an increase/decrease?

– p=[(1+r)-d]/(u-d)=[1.1-0.5]/(1.5-0.5)=0.6– Intuitively: – 1+0.1 = 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

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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!).

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Two Step Binomial ModelAssume: u=1+0.5, d=1-0.5, r=0.10, t=1, S=100 and X=100

First: Build the stock prices from today forward by u and d.

uS0=150

C=?

dS0=50

C=?

S0=100

C=?

uuS0=225

C=?

udS0=75

C=?

ddS0=25

C=?

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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).

uS0=150

C=68.18

dS0=50

C=0

S0=100

C=37.19

uuS0=225

C=125

udS0=75

C=0

ddS0=25

C=0

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Valuing a Put

• Remember: Intrinsic value of a put is Max[0,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 Max[0,ST-X]

– Recursive solution• Risk neutral probability (p)?

– No difference

• Put computation (C=)?– No difference

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Two Step Binomial Model: PutSame assumptions as the call: u=1+0.5, d=1-0.5, r=0.10, t=1, S=100 and X=100

First: Build the stock prices from today forward by u and d. NO DIFFERENCE FROM THE CALL!

uS0=150

P=?

dS0=50

P=?

S0=100

P=?

uuS0=225

P=?

udS0=75

P=?

ddS0=25

P=?

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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).

uS0=150

P=9.09

dS0=50

P=40.91

S0=100

P=19.83

uuS0=225

P=0

udS0=75

P=25

ddS0=25

P=75

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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 (C=S-X, P=X-S)

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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 t is the length of one period in the binomial model , is the historical volatility over that same period.

ud

eu t

/1

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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

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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

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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)

tddt

trX

S

d

dSNdXNeP

dXNedSNCrt

rt

12

2

1

12

21

2ln

)()(

)()(

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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

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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?

)(

)(

1

1

dN

dN

P

C

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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.

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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 ()?– 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.

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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