CONTINGENT CLAIMS

Any risky security has a payoff that is contingent on the “state of the world”e.g., equity and debt in our asset substitution

and underinvestment problem examples

Some securities make a payoff in a range of states or in one state, but zero otherwisee.g., lottery tickets, life insurance, my World

Series tickets, options

OPTION: A Contingent Claim

Not obligation call put underlying asset strike (exercise) price

Right to buy (sell) specified asset at specified priceon (or before) specified date

European American maturity (expiration) date

CALL OPTION PAYOFF

Gross Payoff

ST - X

0 X ST

PUT OPTION PAYOFF

Gross Payoff

X - ST

0 X ST

PROPERTIES OF CALL OPTION PRICES

XSr1

XS

)X(V)S(V)XS(V

)]0,XS(max[VC

0f

0

TT

T0

IMPLICATIONS FOR CALL OPTION PRICES (nondividend-paying stocks)

The price of a European call option is at least the maximum of zero or the stock price minus the present value of the exercise price

An American call option will never be exercised early

American call option must have same value as equivalent European call option

OPTION TIME PREMIUM:The Value of Waiting

$

Value of Call

Time Premium

Exercise Value

X Value of Stock

TIME PREMIUM CHARACTERISTICS

Since the payoff on an option is asymmetric, the ability to wait is valuable:Option time premium is generally positiveTime premium decreases as an option becomes

further in-the-money or out-of-the-moneyTime premium is greatest when the option is

exactly at the money

OPTIONS AND RISK

What happens if risk of underlying asset increases?Greater chance of a larger payoffDownside payoffs are limited to zero

Implication: Option values increase with risk of underlying asset

PUT-CALL PARITY(nondividend-paying stocks)

Buy stock, buy put, sell call (option exercise price = X)

Stock Price Call Exer? Put Exer? Wealth

S > X yes no X

S < X no yes X

S = X ? ? X

S0 + P0 - C0 = X/(1+rf)T

OPTIONS ARE EVERYWHERE

Explicit• Traded options• Executive stock

options• Call provisions on

bonds• Convertible bonds• Warrants

Hidden• Capital budgeting:

(postponement, abandonment)

• Tax timing• Common stock and

risky debt

OPTION ELEMENTS OF EQUITY AND RISKY DEBT

Equity is like a call option on firm assetsEquityholders have “sold” assets to

bondholdersBy paying off the debt obligation (exercise

price) equityholders can buy back assets

Risky debt contains an embedded putEquityholders can put the assets to the

bondholders and cancel bondholders’ claim

MARKET VALUE BALANCE SHEET

Value of riskless debt Minus default option

Assets Equity

Equity = Assets –(Value of riskless debt – default option)

Assets + default option – Equity = Value of riskless debt

SIMPLE CASE: TWO POSSIBLE FUTURE STATES OF THE WORLD

• Stock price now is 100/1.05 = 95.24

• Stock pays no dividend

• Next year, stock price goes up ( by factor u =1.68) to 160 or down (by factor d =.63) to 60

uS=160

S=95.24

dS=60

PRICING A EUROPEAN CALL OPTION

• Suppose current stock price is $95.24, and we know that, one year from now, stock price will be either $60 or $160

• Consider two investment strategies:1. Buy ten call options on stock with exercise

price of $150. Cost = ?

2. Buy one share of stock and borrow 60/1.05 at 5% int. rate. Total cost = 95.24 - 57.14 = 38.1

PRICING A EUROPEAN CALL OPTION

Payoff from two strategies after one year:

Strategy If S = 60 If S = 160

1 0 10x10 = 100

2 60 - 60 = 0 160 - 60 = 100Strategy 2 is a replicating strategy Since both strategies have the same payoff, they

must have the same costValue of call = $38.1/10 = 3.81

BLACK-SCHOLES MODEL

• C = call price• S = stock price• X = exercise price

• rf = risk-free rate

• T = time to expiration = std. dev. stock

price changes• N( ) = cumulative std.

Normal probabilities

Tdd

T2

1

T

Tr)X/Sln(d

)d(NXe)d(SNC

12

f1

2Tr

1f

PUT OPTION PRICING

• We can use put-call parity to derive a Black-Scholes put option pricing model

))d(N1(S

))d(N1(XeP

SCXeP

1

2Tr

Tr

f

f

IMPORTANT ASSUMPTIONS

Nondividend-paying stock

Constant interest rate

Stochastic process governing stock price movements stays constant