CONTINGENT CLAIMS

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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 otherwise
• e.g., lottery tickets, life insurance, my World
  Series tickets, options

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OPTION A Contingent Claim
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CALL OPTION PAYOFF
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PUT OPTION PAYOFF
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PROPERTIES OF CALL OPTION PRICES
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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

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OPTION TIME PREMIUMThe Value of Waiting
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TIME PREMIUM CHARACTERISTICS

• Since the payoff on an option is asymmetric, the
  ability to wait is valuable
• Option time premium is generally positive
• Time premium decreases as an option becomes
  further in-the-money or out-of-the-money
• Time premium is greatest when the option is
  exactly at the money

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OPTIONS AND RISK

• What happens if risk of underlying asset
  increases?
• Greater chance of a larger payoff
• Downside payoffs are limited to zero
• Implication Option values increase with risk of
  underlying asset

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PUT-CALL PARITY(nondividend-paying stocks)

• Buy stock, buy put, sell call (option exercise
  price X)
• Stock Price Call Exer? Put Exer? Wealth
• S gt X yes no X
• S lt X no yes X
• S X ? ? X
• ? S0 P0 - C0 X/(1rf)T

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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

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OPTION ELEMENTS OF EQUITY AND RISKY DEBT

• Equity is like a call option on firm assets
• Equityholders have sold assets to bondholders
• By paying off the debt obligation (exercise
  price) equityholders can buy back assets
• Risky debt contains an embedded put
• Equityholders can put the assets to the
  bondholders and cancel bondholders claim

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MARKET VALUE BALANCE SHEET
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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

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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

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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 - 60 100
• Strategy 2 is a replicating strategy
• Since both strategies have the same payoff, they
  must have the same cost
• Value of call 38.1/10 3.81

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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

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PUT OPTION PRICING

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

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IMPORTANT ASSUMPTIONS

• Nondividend-paying stock
• Constant interest rate
• Stochastic process governing stock price
  movements stays constant