Chapter 15 –
Arbitrage and Option Pricing Theory
Arbitrage pricing theory is an alternate to CAPM
 Option pricing theory applies to pricing of
contingent claims
 Both have applications for capital budgeting and
financing
 Both are based on the arbitrage pricing principle

Arbitrage pricing principle
Principle: In an efficient market identical sets of
benefits sell at identical prices
 One way to define the set of benefits is in terms of
identical probability distributions of returns
 Two ways of creating the same probability
distribution of future cash flows should have the
same current price and therefore the same
expected return

Arbitrage Pricing Theory
Foundation for the theory is the arbitrage pricing
principle
 Based on a much less restrictive set of
assumptions than CAPM

 Arbitrage
principle holds
 Markets are efficient
Arbitrage Pricing Theory
Standard (Ross) APT
E(Rs) = Rf+ [E(R1) - Rf] s,1 + [E(R2) - Rf] s,2
+…+ [E(Rn) - Rf] s,n
 Where
E(Rs) = Expected return for an asset
E(Ri) = Expected return on a portfolio with
unitary sensitivity to factor n and zero
sensitivity to other factors
s,i = Beta of the asset with regard to factor i

Application of the Ross APT
Unitary sensitivity portfolios can be created from
publicly available securities
 Some suggested factors

 Industrial
production or return on the market portfolio
 Changes in the risk premium between high-grade and
lower-grade bonds
 Slope of the yield curve
 Unanticipated inflation
Application of the Ross APT
The difficulty is in identifying factors and
economic surrogates for those factors
 This difficulty has limited application

State-based Arbitrage Analysis of
Capital Investments
Often identify various possible states of the world
and compute the cash benefits for a capital
investment in each of these states
 We might also estimate returns for a variety of
stock investment portfolios in each state

State-based Arbitrage Analysis of
Capital Investments
We can replicate the cash flows of the proposed
capital investment with publicly traded securities
 The NPV of the capital investment = cost of
replicating the cash flows - cost of the capital
investment
 A key advantage is that state-based arbitrage
analysis does not require the identification of
probabilities

Option Pricing Models
Option pricing models are also based on the
arbitrage pricing principle
 Option pricing models are based on the fact that
options can generally be combined with purchase
or sale of the underlying asset to create a risk-free
investment
 In equilibrium, the price of the option must be
such that this risk-free investment pays the riskfree rate of return

Option terminology
Call option: an option to buy an asset.
 Put option: an option to sell an asset.
 Exercise of an option: the buying or selling of the
asset as provided for in the option contract.
 Exercise price (striking price): the price at which
the asset can be bought or sold, as stated in the
option contract.

Option terminology
Expiration date: the last day on which the option
may be exercised.
 European option: an option that may be
exercised only on the expiration date.
 American option: an option that may be
exercised at any time prior to its expiration date.

Option Terminology

Writer: the person who sells an option contract to
another, thereby granting the buyer an option to
buy or sell the asset at the exercise price under the
terms specified in the contract.
Two-state Option Valuation
Value of a call option:
C = [So – Sd/(1+Rf)](Su – E)/(Su – Sd)
Where

C = Value of a call option
So = current price of the underlying stock
Su, Sd = higher and lower of two possible prices for the
stock at the end of the period
E = Exercise price of the option
Rf = Risk-free rate
Black-Scholes Model
C = SoN(d1) - [E ÷ eRfT]N(d2)
Where
So = current price of the stock
E = exercise price of the option
Rf = risk-free rate, continuously compounded
N(di) = the value from the table of the normal distribution
representing the probability of an outcome less than di
d1 = [ln(So/E) + (Rf + .5S2)T]/(ST )
d2 = d1 - (ST )
S = standard deviation of the continuously compounded
annual rate of return for the stock
T = time in years or fractions of years until expiration of option
e = 2.71828 ..., the base of the natural logarithm
Real Options

Many capital investments are real options
R
& D investment creates the option to invest in
production
 Can value using a two-state model or Black-Scholes
Financing Choices as Options

Stock
 Stockholders
of a leveraged firm have the option of
paying the creditors and “buying” the company or
turning the company over to the creditors. The exercise
price of the option is thus the amount owed to creditors.

Debt
 Creditors
essentially own the company, and have
written an option which can be exercised by the
stockholders.
Financing Choices as Options
For an unlevered firm, increasing standard
deviation of the probability of asset returns
without increasing the expected return will
decrease value as long as any of that risk is
systematic
 For the levered firm, increased standard deviation
may increase the wealth of the stockholders at the
expense of the bondholders
 This leads to agency costs

Motivating Managers to Take Risks
Managers often receive a combination of a fixed
salary and stock options
 The fixed salary is similar to holding debt
instruments in that the return is realized as long as
the firm is solvent
 Increased standard deviation of asset returns may
increase or decrease the wealth of managers
depending on the mix of salary and options in
their compensation package

Exchange Rate Risk
Translation risk is the risk that the company will
report lower income because unfavorable
exchange rate movements decrease the U.S. dollar
value of foreign income
 Transaction risk is the risk that actual dollar
amount of cash flows coming from international
activities will be smaller because of changes in
exchange rates

Managing Exchange Rate Risk
Currency future: contract to exchange a specific
amount of one currency for a specific amount of
another currency at a designated future date
 Currency swap: spot transaction in one direction
offset by futures contract in the opposite direction;
often used when direct futures are not available
 Options: the right but not the obligation to exchange
one currency for another at a specified future date
