Chapter 15 – Arbitrage and Option Pricing Theory u Arbitrage pricing theory is an alternate to CAPM u Option pricing theory applies to pricing of contingent claims u Both have applications for capital budgeting and financing u Both are based on the arbitrage pricing principle

Arbitrage pricing principle u Principle: In an efficient market identical sets of benefits sell at identical prices u One way to define the set of benefits is in terms of identical probability distributions of returns u 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 u Foundation for the theory is the arbitrage pricing principle u Based on a much less restrictive set of assumptions than CAPM u Arbitrage principle holds u Markets are efficient

Arbitrage Pricing Theory Standard (Ross) APT E(R s ) = R f + [E(R 1 ) - R f ] s,1 + [E(R 2 ) - R f ] s,2 +…+ [E(R n ) - R f ] s,n Where E(R s ) = Expected return for an asset E(R i ) = 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 u Unitary sensitivity portfolios can be created from publicly available securities u Some suggested factors u Industrial production or return on the market portfolio u Changes in the risk premium between high-grade and lower-grade bonds u Slope of the yield curve u Unanticipated inflation

Application of the Ross APT u The difficulty is in identifying factors and economic surrogates for those factors u This difficulty has limited application

State-based Arbitrage Analysis of Capital Investments u Often identify various possible states of the world and compute the cash benefits for a capital investment in each of these states u We might also estimate returns for a variety of stock investment portfolios in each state

State-based Arbitrage Analysis of Capital Investments u We can replicate the cash flows of the proposed capital investment with publicly traded securities u The NPV of the capital investment = cost of replicating the cash flows - cost of the capital investment u A key advantage is that state-based arbitrage analysis does not require the identification of probabilities

Option Pricing Models u Option pricing models are also based on the arbitrage pricing principle u 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 u In equilibrium, the price of the option must be such that this risk-free investment pays the risk- free rate of return

Option terminology u Call option: an option to buy an asset. u Put option: an option to sell an asset. u Exercise of an option: the buying or selling of the asset as provided for in the option contract. u Exercise price (striking price): the price at which the asset can be bought or sold, as stated in the option contract.

Option terminology u Expiration date: the last day on which the option may be exercised. u European option: an option that may be exercised only on the expiration date. u American option: an option that may be exercised at any time prior to its expiration date.

Option Terminology u 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 u Value of a call option: C = [S o – S d /(1+R f )](S u – E)/(S u – S d ) Where C = Value of a call option So = current price of the underlying stock S u, S d = higher and lower of two possible prices for the stock at the end of the period E = Exercise price of the option R f = Risk-free rate

Black-Scholes Model C = S o N(d 1 ) - [E ÷ e R f T ]N(d 2 ) Where S o = current price of the stock E = exercise price of the option R f = risk-free rate, continuously compounded N(d i ) = the value from the table of the normal distribution representing the probability of an outcome less than d i d 1 = [ln(S o /E) + (R f +.5 S 2 )T]/( S T ) d 2 = d 1 - ( 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 = , the base of the natural logarithm

Real Options u Many capital investments are real options u R & D investment creates the option to invest in production u Can value using a two-state model or Black-Scholes

Financing Choices as Options u Stock u 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. u Debt u Creditors essentially own the company, and have written an option which can be exercised by the stockholders.

Financing Choices as Options u 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 u For the levered firm, increased standard deviation may increase the wealth of the stockholders at the expense of the bondholders u This leads to agency costs

Motivating Managers to Take Risks u Managers often receive a combination of a fixed salary and stock options u The fixed salary is similar to holding debt instruments in that the return is realized as long as the firm is solvent u 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 u 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 u 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 u Currency future: contract to exchange a specific amount of one currency for a specific amount of another currency at a designated future date u Currency swap: spot transaction in one direction offset by futures contract in the opposite direction; often used when direct futures are not available u Options: the right but not the obligation to exchange one currency for another at a specified future date