Chapter 15: Options and Contingent Claims

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Chapter 15 Options and Contingent Claims

• Objective
• To show how the law of one price may
• be used to derive prices of options
• To explore the range of financial decisions
• that can be fruitfully analyzed
• in terms of options

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Chapter 15 Contents

• How Options Work
• Investing with Options
• The Put-Call Parity Relationship
• Volatility Option Prices
• Two-State Option Pricing
• Dynamic Replication the Binomial Model

• The Black-Scholes Model
• Implied Volatility
• Contingent Claims Analysis of Corporate Debt and
  Equity
• Convertible Bonds
• Valuing Pure State-Contingent Securities

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Terms

• A option is the right (not the obligation) to
  purchase or sell something at a specified price
  (the exercise price) in the future
• Underlying Asset, Call, Put, Strike (Exercise)
  Price, Expiration (Maturity) Date, American /
  European Option
• Out-of-the-money, In-the-money, At-the-money
• Tangible (Intrinsic) value, Time Value

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Terminal or Boundary Conditions for Call and Put
Options
120
100
80
60
Dollars
40
20
0
0
20
40
60
80
100
120
140
160
180
200
-20
Underlying Price
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The Put-Call Parity Relation

• Two ways of creating a stock investment that is
  insured against downside price risk

• Buying a share of stock and a put option (a
  protective-put strategy)

• Buying a pure discount bond with a face value
  equal to the options exercise price and
  simultaneously buying a call option

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200
180
160
140
120
Payoffs
100
80
60
40
20
0
0
20
40
60
80
100
120
140
160
180
200
Stock Price
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Payoff Structure for Protective-Put Strategy
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Payoff Structure for a Pure Discount Bond Plus a
Call
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Put-Call Parity Equation
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Synthetic Securities

• The put-call parity relationship may be solved
  for any of the four security variables to create
  synthetic securities
• CSP-B
• SC-PB
• PC-SB
• BSP-C

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Converting a Put into a Call

• S 100, E 100, T 1 year, r 8, P 10
• C 100 100/1.08 10
  17.41
• If C 18, the arbitrageur would sell calls at a
  price of 18, and synthesize a synthetic call at
  a cost of 17.41, and pocket the 0.59 difference
  between the proceed and the cost

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Put-Call Arbitrage
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Options and Forwards

• We saw in the last chapter that the discounted
  value of the forward was equal to the current
  spot
• The relationship becomes

If the exercise price is equal to the forward
price of the underlying stock, then the put and
call have the same price
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Implications for European Options

• If (F gt E) then (C gt P)
• If (F E) then (C P)
• If (F lt E) then (C lt P)
• E is the common exercise price
• F is the forward price of underlying share
• C is the call price
• P is the put price

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Put and Call as Function of Share Price
60
call
50
put
asy_call_1
40
asy_call_2
asy_put_1
30
asy_put_2
Put and Call Prices
20
10
0
50
60
70
80
90
100
110
120
130
140
150
-10
Share Price
100/(1r)
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The prices of options increase with the
volatility of the stock
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Two-State Option Pricing Simplification

• The stock price can take only one of two possible
  values at the expiration date of the option
  either rise or fall by 20 during the year
• The options price depends only on the volatility
  and the time to maturity
• The interest rate is assumed to be zero

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Binary Model Call

• The synthetic call, C, is created by
• buying a fraction x (which is called the hedge
  ratio) of shares, of the stock, S
• selling short risk-free bonds with a market value
  y

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Binary Model Creating the Synthetic Call
S 100, E 100, T 1 year, d 0, r 0
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Binary Model Call

• Specification
• We have an equation, and give the value of the
  terminal share price, we know the terminal option
  value for two cases
• By inspection, the solution is x1/2, y 40.

The Law of One Price
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Binary Model Call

• Solution
• We now substitute the value of the parameters
  x1/2, y 40 into the equation
• to obtain

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Binary Model Put

• The synthetic put, P, is created by
• selling short a fraction x of shares, of the
  stock, S
• buying risk free bonds with a market value y

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Binary Model Creating the Synthetic Put
S 100, E 100, T 1 year, d 0, r 0
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Binary Model Put

• Specification
• We have an equation, and give the value of the
  terminal share price, we know the terminal option
  value for two cases
• By inspection, the solution is x 1/2, y 60

The Law of One Price
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Binary Model Put

• Solution
• We now substitute the value of the parameters
  x1/2, y 60 into the equation
• to obtain

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Decision Tree for Dynamic Replication of a Call
Option
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Decision Tree for Dynamic Replication of a Call
Option

• The terminal option value for two cases
• 120x y 20
• 100x y 0
• By inspection, the solution is x1, y 100
• Thus, C11 1110 - 100 10

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Decision Tree for Dynamic Replication of a Call
Option

• The terminal option value for two cases
• 90x y 0
• 80x y 0
• By inspection, the solution is x0, y 0
• Thus, C12 090 - 0 0

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Decision Tree for Dynamic Replication of a Call
Option

• The terminal option value for two cases
• 110x y 10
• 90x y 0
• By inspection, the solution is x1/2, y 45
• Thus, C0 (1/2)100 - 45 5

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Decision Tree for Dynamic Replication of a Call
Option
Sell shares 120 Pay off debt -100 Total 20
Buy another half share of stock Increase
borrowing to 100
Sell shares 100 Pay off debt -100 Total 0
Buy 1/2 share of stock Borrow 45 Total
investment 5
Sell stock and pay off debt
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Decision Tree for Dynamic Replication of a Call
Option
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The Black-Scholes Model The Limiting Case of
Binomial Model

• One can continuously and costlessly adjust the
  replicating portfolio over time
• As the decision intervals in the binomial model
  become shorter, the resulting option price from
  the binomial model approaches the Black-Scholes
  option price

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The Black-Scholes Model
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The Black-Scholes Model Notation

• C price of call
• P price of put
• S price of stock
• E exercise price
• T time to maturity
• ln() natural logarithm
• e 2.71828...
• N() cum. norm. distn

• The following are annual, compounded
  continuously
• r domestic risk free rate of interest
• d foreign risk free rate or constant dividend
  yield
• s volatility

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The Black-Scholes Model Dividend-adjusted Form
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The Black-Scholes Model Dividend-adjusted Form
(Simplified)
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Determinants of Option Prices
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Value of a Call and Put Options with Strike
Current Stock Price
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Implied Volatility

• The value of s that makes the observed market
  price of the option equal to its Black-Scholes
  formula value
• Approximation

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Implied Volatility
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Valuation of Uncertain Cash Flows CCA / DCF

• The DCF approach discounts the expected cash
  flows using a risk-adjusted discount rate
• The Contingent-Claims Analysis (CCA) uses
  knowledge of the prices of one or more related
  assets and their volatilities

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An Example Debtco Corp.

• Debtco is in the real-estate business
• It issues two types of securities
• common stock (1 million shares)
• corporate bonds with an aggregate face value of
  80 million (80,000 bonds, each with a face value
  of 1,000) and maturity of 1 year
• risk-free interest rate is 4
• The total market value of Debtco is 100 million

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Debtco Notation

• V be the current market value of Debtcos assets
  (100 million)
• V1 be the market value of Debtcos assets a year
  from now
• E be the market value of Debtcos stocks
• D be the market value of Debtcos bonds

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Two Ways to Think about the Debtcos Market Value

• To think of the assets of the firm, real estates
  in Debtcos case, as having a market value of
  100 million
• To imagine another firm that has the same assets
  as Debtco but is financed entirely with equity,
  and the market value of this all-equity-financed
  twin of Debtco is 100 million

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Value of the Bonds
Value of the Stock
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The payoff is identical to a call option in which
the underlying asset is the firm itself, and the
exercise price is the face value of its debt
Value of the Stock
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• The value of the firms equity

• The value of the debt

DV-E
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Debtco Security Payoff Table (000,000)


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Debtcos Replicating Portfolio

• Let
• x be the fraction of the firm in the replication
• Y be the borrowings at the risk-free rate in the
  replication
• The following equations must be satisfied

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Debtcos Replicating Portfolio (000)
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Debtcos Replicating Portfolio

• We know the value of the firm is 1,000,000, and
  the value of the total equity is 28,021,978, so
  the market value of the debt with a face of
  80,000,000 is 71,978,022
• The yield on this debt is (80/71) -111.14

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Another View of Debtcos Replicating Portfolio
(000)


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Given the Price of the Stock

• Suppose
• 1 million shares of Debtcos stock outstanding,
  and the market price is 20 per share
• two possible future value of for Debtco, 70
  million and 140 million
• the face value of Debtco bonds is 80 million
• risk-free interest rate is 4

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Valuing Bonds

• We can replicate the firms equity using x 6/7
  of the firm, and about Y 58 million riskless
  borrowing (earlier analysis)
• The implied value of the bonds is then
  90,641,026 - 20,000,000 70,641,026 the
  yield is (80.00 - 70.64)/70.64 13.25

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Given the Price of the Bonds

• Suppose
• the face value of Debtco bonds is 80 million,
  the yield-to-maturity on the bonds is 10 (i.e.,
  the price of Debtco bonds is 909.09)
• two possible future value of for Debtco, 70
  million and 140 million
• risk-free interest rate is 4

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Replication Portfolio

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Determining the Weight of Firm Invested in Bond,
x, and the Value of the R.F.-Bond, Y
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Valuing Stock

• We can replicate the bond by purchasing 1/7 of
  the company, and 57,692,308 of default-free
  1-year bonds
• The market value of the bonds is 909.0909
  80,000 72,727,273
• The value of the stock is therefore E V -D
  105,244,753 - 72,727,273 32,517,480

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Convertible Bonds

• A convertible bond obligates the issuing firm
  either to redeem the bond at par value upon
  maturity or to allow the bondholder to convert
  the bond into a prespecified number of shares of
  common stock

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An Example Convertidett Corp.

• Convertidett has assets identical to those of
  Debtco
• Its capital structure consists of
• 1 million shares of common stock
• one-year zero-coupon bonds with a face value of
  80 million (80,000 bonds, each with a face value
  of 1,000), that are convertible into 20 shares
  of Convertidett stock at maturity
• risk-free interest rate is 4
• The total market value of Debtco is 100 million

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Critical value of Convertident for Conversion

• Upon convertion, the total shares of stock will
  be 2.6 million

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Payoff for Convertidetts Stocks and Bonds
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Convertidetts Replicating Portfolio

• Let
• x be the fraction of the firm in the replication
• Y be the borrowings at the risk-free rate in the
  replication
• The following equations must be satisfied

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Values of Convertidetts Stocks and Bonds
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Decomposition of Convertidetts Stocks and Bonds

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Pure State-Contingent Securities

• Securities that pay 1 in one of the states and
  nothing in the others
• For Debtco and Convertidett, if we know the
  prices of the two pure state-contingent
  securities, then we are able to price any
  securities issued by the firmsstocks, bonds,
  convertible bonds, or other securities

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Valuing Pure State-Contingent Securities


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State-Contingent Security 1
80
State-Contingent Security 2
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Valuing Debtcos Securities

• Price of a Debtco stock 60P1 60.4670329
  28.02
• Price of a Debtco bond 1,000P1 875P2
• 1,000.4670329
  875.494505 899.73

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Valuing Convertidetts Securities

• Price of a Convertidett stock 53.86415P1
• 53.86415.4670329 25.15
• Price of a Convertidett bond 1,076.923P1
  875P2
• 1,076.923.4670329
  875.494505 935.65

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Payoff for Debtcos Bond Guarantee

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SCS Conformation of Guarantees Price

• Guarantees price 125P2 125 0.494505 61.81

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Credit Guarantees

• Guarantees against credit risk pervade the
  financial system and play an important role in
  corporate and public finance
• Parent corporations routinely guarantee the debt
  obligations of their subsidiaries
• Commercial banks and insurance companies offer
  guarantees in return for fees on a broad spectrum
  of financial instruments ranging from traditional
  letters of credit to interest rate and currency
  swaps
• The largest providers of financial guarantees are
  almost surely governments and governmental
  agencies

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Credit Guarantees

• Fundamental identity
• Risky loan loan guaranteedefault-free loan
• Risky loan default-free loan-loan guarantee
• The credit guarantee is equivalent to writing a
  put option
• on the firm's assets
• with a strike price equal to the face value of
  the debt. The guarantee's value can, therefore,
  be computed using the adjusted put-option-pricing
  formula