Real Option Pricing

what is a call option?

right to buy an asset in the future at a predetermined price

what is a put option?

right to sell an asset in the future at a predetermined price

what is the strike price?

price at which the underlying can be bought or sold

what is the underlying?

the asset on which the option is written

what is the exercise value of a call?

Max(0, S - X)

when is an American option exercisable?

at any time until expiration

when is a European option exercisable?

only at expiration

a call is in-the-money when:

S > X

a call is out-of-money when:

S < X

option pride refers to:

market price of the option contract

payoff of a long call at maturity

Max(0, S - X)

payoff of a short call

−Max(0, S − X)

payoff of a long put

Max(0, X − S)

payoff of a short put

−Max(0, X − S)

put-call parity formula

S + P - C = PV(X)

rearranged version

C - P = S - B

what does C - P represent?

leveraged stock position

why must put-call parity hold?

to avoid arbitrage

in the binomial model, stock prices:

move up or down each period

number of states per period

two

stock evolution assumption

multiplicative process

trading assumption

occurs at discrete times

upward stock price

uS

downward stock price

dS

call payoff in up state

Max(0, uS − X)

call payoff in down state

Max(0, dS − X)

purpose of hedge portfolio

create risk-free position

hedge portfolio composition

stock + short m options

condition for risk-free portfolio

payoffs equal in both states

hedge ratio m equals

(cu − cd) / (uS − dS)

why is hedge portfolio discounted at rf?

it is risk-free

risk-neutral probability p

(1 + rf − d) / (u − d)

call price formula

c = [p·cu + (1−p)·cd] / (1 + rf)

meaning of risk-neutral probability

adjusted probability for pricing under no arbitrage

a recombining tree means

paths lead to same nodes

two step up-up price

u²S

middle node price

udS

backward induction is used to

price options recursively

stock dynamics assumption

geometric dynamics motion

returns are

normally distributed

volatility assumption

constant

market assumption

no arbitrage

dividend assumption

no dividends

Black-Scholes formula gives price of

European call

key inputs

S, X, r, T, σ

d2 equals

d1 − σ√T

discounting factor in BSM

e^(−rfT)

equity is modeled as

call option on firm value

underlying

firm value V

strike

debt value D

equity payoff

Max(0, V − D)

default occurs when

V < D

NPV rule

accept project if NPV > 0

key limitation of NPV

ignores flexibility

NPV assumes

deterministic decisions

expansion option resembles

American call

contraction option resembles

American put

abandonment option

right to sell asset

deferral option

right to delay project

switching option

turn project on/offc

compound option

option on option

DTA uses

physical probabilities

discounting in DTA

WACC

problem with DTA

same WACC across branches

why problematic?

risk changes with flexibility

ROA uses

risk-neutral probabilities

discounting in ROA

risk-free rate

advantage of ROA

arbitrage-free valuation

main requirement

observable market value

drilling right is equivalent to

call option

exercise price

investment cost

NPV without flexibility

negative

NPV with flexibility

positive

value of option increases because

ability to delay decision

DTA overvalues project because

uses wrong discount rate

ROA adjusts risk via

probabilities

DTA adjusts risk via

discount rate

staging creates

real options

advantage of staging

flexibility

disadvantage

delay cost

best first stage in

most risky

also preferable

least capital intensive

also preferable

longest duration

failure cost index used for

optimal ordering

real options are about

flexibility and future rights

real options most relevant in

high uncertainty projects

typical application

R&D investments

why is the NPV method problematic?

it assumes no flexibility after investment

what does flexibility create?

additional project value

why does flexibility increase value?

it allows adaptation to new information

which method violates the law of one price?

decision tree analysis (DTA)

which method is arbitrage-free?

Real Option Approach (ROA)

why is DTA theoretically incorrect?

it uses a constant WACC despite changing risk

what changes when flexibility is introduced?

the risk profile of the project

what probabilities are used in ROA?

risk-neutral probabilities

what probabilities are used in DTA?

physical (objective) probabilities

how does ROA adjust for risk?

through probabilities

how does DTA adjust for risk?

through discount rates

what discount rate is used in ROA?

risk-free rate

why construct a hedge portfolio?

to eliminate risk