Session 3: Options III

C15.0008 Corporate Finance Topics

Summer 2006

Outline

• Risk Neutral Valuation

• Options embedded in projects

• Valuation of real options– DCF (decision trees)

Risk neutral valuationWhat is the expected return on this tree?

110

90

100

0.6

0.4

Recall the definition of “expected” return from the first session

Risk neutral valuation

110

90

100

p

1-p

Why 1-p?

What is the value of p if the expected return is to be 2%?

Risk-Neutral Valuation

• Given the prices, is it possible to find implied probabilities such that we can apply the risk-free rate as our discounting rate.

• Risk-neutral valuation is equivalent to solving for the replicating strategy, but it is computationally quicker for large trees.

• Risk-neutral probabilities are not really probabilities!!

A note on how to get u and d

• If the per period volatility of a stock is σ, then u and d are given by:– u = e σ/√2

– d = e -σ/√2

• If we reduce the time period per step of a tree, so that a very large number of steps represent a year of time, we get the Black Scholes formula.

Black Scholes formula

• The Black-Scholes formula for a call option is• C = S N(d1) – PV(K) N(d2)

– S is the current price of the stock– PV(K) is the present value of the exercise price (PV(k)

= Ke-rt)– d1 = (log(S/ PV(K) )/ σ√t) + σ√t/2– d1 = (log(S/ PV(K) )/ σ√t) - σ√t/2– σ is the volatility

• Notice that N(d1) is like “H”. PV(K) N(d2) is like “B”

• Use put-call parity to get the price of a put

Back to Discounted Cash-flows

• Recall that when we compute the NPV of a project, we discount the expected cash flows by the cost of capital.– What exactly are “expected” cash flows?– Are we missing out on something when we

take these average cash flows?

A hint..

“Discounted cash flow is going to look at an average scenario," comments Triantis. "But if you talk to any manager, that's not how they think. They think about contingencies — what's going to happen, how would we react. And even if they don't think that way, once it's presented to them that way, they say, 'Yeah, that's the way we should be thinking.'"

CFO Magazine – “Will Real Options take roots?”, July 2003

Real Options in Projects

Projects have some characteristics

• If things go well, you can expand

• If things go poorly, you can shut down

• Timing options (when to invest, the option to delay investment)

• These options are related to decisions that are to be taken when more is known about the project

Decision Trees

• What does the decision tree for the decision to take C15.008 look like?

• What are the action points?

• What are the information points?

A Project as an Option

Before the decision is made to undertake a project, every project is really an option on a project• NOT accept vs. reject• Accept today vs. revisit decision tomorrow

Note: generally, the option has no value unless uncertainty will be resolved over time, i.e., information will be revealed about the value of the underlying asset

Example: Product Introduction

Baldwin Inc. is considering investment in a new technology to produce colored bowling balls. The primary determinant of cash flows is demand, i.e., consumer acceptance of colored bowling balls: 0 Demand Prob. 1 2 3...-100 High 1/3 24 24 24...

Low 2/3 1.5 1.5 1.5...E(CF)-100 9 9 9…At a discount rate of 10% the NPV is-100 + 9/0.1 = -10

The Abandonment Option

The project can be abandoned at the end of year 1 and the equipment salvaged for 19.5. Should the project be scrapped? If demand is low…

1 2 3... PV(10%) in yr 1

Continue 1.5 1.5... 15

Abandon 19.5 19.5

The modified project

Prob. 0 1 2 3... NPV(10%)

1/3 -100 24 24 24...

2/3 -100 21

E(CF) -100 22 8 8… -7.27

The Option Value

• -7.27 is the value of the project plus the option to abandon (using DCF)

• Recall that the value of the project without the option to abandon was -10

The value of the abandonment option is -7.27 - (-10) = 2.73 or

(19.5-15)(2/3)/1.1 = 2.73

The Expansion Option

Baldwin can add a second factory at the end of the first year, which will also operate at capacity if demand is high.

The modified project

Prob. 0 1 2 3... NPV(10%)

1/3 -100 24 24 24...

-100 24 24…

2/3 -100 21

E(CF) -100 -11.3 16 16… 35.15

Why not open this second factory immediately?

Why not delay the first investment until time 1?

The Option Value

• 35.15 is the value of the project plus the option to expand and the option to abandon (using DCF)

• Recall that the value of the project without the option to expand was -7.27

The value of the expansion option is 35.15 - (-7.27) = 42.42 or

140(1/3)/1.1 = 42.42

The total value of the project is project w/o options + abandonment + expansion = -10 + 2.73 + 42.42 = 35.15

Valuing Real Options

Why not value real options using a DCF approach (i.e., decision trees)?• In principle, a DCF approach will work• In practice, the discount rate may be a problem. The required return on the option is generally not the same as the required return on the underlying asset (project).

Solution:

Use a binomial approach

Example: Product Introduction

Underlying project: 0 Demand Prob. 1 2 3...-100 High 1/3 24 24 24...

Low 2/3 1.5 1.5 1.5...Abandonment option: project can be abandoned at time 1 and the equipment salvaged for 19.5Expansion option: a second factory can be built at time 1 that will also operate at full capacity if demand is high 0 Demand Prob. 1 2 3...

High 1/3 -100 24 24...

The Option to Abandon

• The abandonment option is a put option where the underlying asset is an operating bowling ball factory and the exercise price is the salvage value, i.e., it is the right to sell (salvage) the factory and receive the salvage value.

• Conceptually it is a quasi-American option, i.e., it can be exercised starting in 1 year and at any time thereafter.

If exercised at all, will the option be exercised immediately?

How would you handle a salvage value that varies over time?

Inputs

• exercise price = salvage valueE = 19.5

• underlying asset = operating factoryvalue (time 0) = PV(expected cash flows)0 1 2 3...

9 9 9…S = 9/0.1 = 90

• t = 1, rf = 2%

The Option to Expand

• The expansion option is a call option where the underlying asset is a new (second) operating bowling ball factory and the exercise price is the initial investment required, i.e., it is the right to buy (build) the factory and receive the resulting future cash flows.

• Conceptually it is a quasi-American option, i.e., it can be exercised starting in 1 year and at any time thereafter.

If exercised at all, will the option be exercised immediately?

Inputs

• exercise price = initial investment E = 100

• underlying asset = operating factory (built at time 1)value (time 0) = PV(expected cash flows)0 1 2 3...

9 9…S = (9/0.1)/1.1 = 90/1.1 = 81.82Assumption: the second factory looks just like the first

• T = 1, rf = 2%, = 130%

The Option to Abandon

E=19.5

P

0

4.5

90

24 24 24…

1.5 1.5 1.5…

90

264

16.5

240

15

Option Value

rf = 2% H = -0.018, B = -4.71, P = 3.07

Note: replicate with the cum-dividend values of the underlying asset, calculate the put payoff based on the ex-dividend values

The Option to Expand

E=100

C

140

0

81.82

0 24 24…

0 1.5 1.5…

81.82

240

15

Option Value

rf = 2% H = 0.622, B = 9.15, C = 41.76

Project Value

base abandon expand total

DCF -10 2.73 42.42 35.15

binomial -10 3.07 41.76 34.83

Note:• DCF uses the wrong discount rate• B-S uses the wrong distribution

When is There a Real Option?

• There has to be a clearly defined underlying asset whose value changes over time in a predictable way

• The payoffs on this asset have to be contingent on a specific observable event, i.e, there has to be a resolution of uncertainty about the value of the asset

When Does the Option Have Value?

• For an option to have significant economic value, there has to be a restriction on competition in the event of the contingency (in a perfectly competitive market, no option generates positive NPVs)

• Real options are most valuable when there is exclusivity

Assignments

• Reading– RWJ: Chapters 14, 16.7, 20– Problems: 14.1, 20.16– Problem Set 1 due next class