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## 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**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 Chapter 15 Contents**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**Table 15.1 List of IBM Option Prices**(Source: Wall Street Journal Interactive Edition, May 29, 1998) IBM (IBM) Underlying stock price 120 1/16 Put Call Strike Expiration Volume Last Open Volume Last Open Interest Interest 115 Jun 1372 7 4483 756 1 3/16 9692 115 Oct … … 2584 10 5 967 115 Jan … … 15 53 6 3/4 40 120 Jun 2377 3 1/2 8049 873 2 7/8 9849 120 Oct 121 9 5/16 2561 45 7 1/8 1993 120 Jan 91 12 1/2 8842 … … 5259 125 Jun 1564 1 1/2 9764 17 5 3/4 5900 125 Oct 91 7 1/2 2360 … … 731 125 Jan 87 10 1/2 124 … … 70**Table 15.2List of Index Option Prices**(Source: Wall Street Journal Interactive Edition, June 6, 1998) S&P500 INDEX -AM Chicago Exchange Underlying High Low Close Net From % Change 31-Dec Change S&P500 1113.88 1084.28 1113.86 19.03 143.43 14.8 (SPX) Net Open Strike Volume Last Change Interest Jun 1110 call 2,081 17 1/4 8 1/2 15,754 Jun 1110 put 1,077 10 -11 17,104 Jul 1110 call 1,278 33 1/2 9 1/2 3,712 Jul 1110 put 152 23 3/8 -12 1/8 1,040 Jun 1120 call 80 12 7 16,585 Jun 1120 put 211 17 -11 9,947 Jul 1120 call 67 27 1/4 8 1/4 5,546 Jul 1120 put 10 27 1/2 -11 4,033**Call**Put 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**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 option’s exercise price and simultaneously buying a call option**Terminal Conditions of a Call and a Put Option with Strike =**100 Share Call Put Share_Put Bond Call_Bond 0 0 100 100 100 100 10 0 90 100 100 100 20 0 80 100 100 100 30 0 70 100 100 100 40 0 60 100 100 100 50 0 50 100 100 100 60 0 40 100 100 100 70 0 30 100 100 100 80 0 20 100 100 100 90 0 10 100 100 100 100 0 0 100 100 100 110 10 0 110 100 110 120 20 0 120 100 120 130 30 0 130 100 130 140 40 0 140 100 140 150 50 0 150 100 150 160 60 0 160 100 160 170 70 0 170 100 170 180 80 0 180 100 180 190 90 0 190 100 190 200 100 0 200 100 200**Call_Bond**Share_Put Bond Share Call Put 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**B: bond**Put-Call Parity Equation**Synthetic Securities**• The put-call parity relationship may be solved for any of the four security variables to create synthetic securities • C=S+P-B • S=C-P+B • P=C-S+B • B=S+P-C**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**C=S+P-B**Put-Call Arbitrage**or**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**Implications for European Options**• If (F > E) then (C > P) • If (F = E) then (C = P) • If (F < E) then (C < 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**Call = Put**Strike = Forward**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/(1+r)**Strike**PV Strike**Volatility and Option Prices**P0 = $100, Strike Price = $100 Stock Price Call Payoff Put Payoff Low Volatility Case Rise 120 20 0 Fall 80 20 0 Expectation 100 10 10 High Volatility Case 0 Rise 140 40 Fall 60 0 40 Expectation 100 20 20 The prices of options increase with the volatility of the stock**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 option’s price depends only on the volatility and the time to maturity • The interest rate is assumed to be zero**Binary Model: Call**• The synthetic call, C, is created by • buying a fraction x (which is called the hedgeratio) of shares, of the stock, S • selling short risk-free bonds with a market value y**Binary Model: Creating the Synthetic Call**S = $100, E = $100, T = 1 year, d = 0, r = 0**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 x=1/2, y = 40. The Law of One Price**Binary Model: Call**• Solution: • We now substitute the value of the parameters x=1/2, y = 40 into the equation • to obtain**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**Binary Model: Creating the Synthetic Put**S = $100, E = $100, T = 1 year, d = 0, r = 0**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**Binary Model: Put**• Solution: • We now substitute the value of the parameters x=1/2, y = 60 into the equation • to obtain:**D**$120 B $110 E $100 A $100 C $90 F $80 Decision Tree for Dynamic Replication of a Call Option**D**$120 B $110 $20 E $100 C11 0 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 x=1, y = $100 • Thus, C11= 1*$110 − $100 = $10**E**$100 C $90 0 F $80 C12 0 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 x=0, y = 0 • Thus, C12 = 0*$90 − $0 = $0**B**$110 A $100 $10 C $90 C0 0 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 x=1/2, y = $45 • Thus, C0= (1/2)*$100 − $45 = $5**D**$120 B $110 E $100 A $100 C $90 F $80 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**($120*100%) + (-$100) = $20**Decision Tree for Dynamic Replication of a Call Option**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 modelapproaches the Black-Scholes option price The Black-Scholes Model: The Limiting Case of Binomial Model**Bond**Shares of Stock The Black-Scholes Model**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. dist’n The following are annual, compounded continuously: r = domestic risk free rate of interest d = foreign risk free rate or constant dividend yield σ = volatility The Black-Scholes Model: Notation**The Black-Scholes Model :Dividend-adjusted Form (Simplified)****Increases in:**Call Put Stock Price, S Increase Decrease Exercise Price, E Decrease Increase Volatility, sigma Increase Increase Time to Expiration, T Ambiguous Ambiguous Interest Rate, r Increase Decrease Cash Dividends, d Decrease Increase Determinants of Option Prices**11**10 9 call put 8 7 6 Call and Put Price 5 4 3 2 1 0 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Time-to-Maturity Value of a Call and Put Options with Strike = Current Stock Price**The value of σ that makes the observed market price of the**option equal to its Black-Scholes formula value Approximation: S E r T d C σ 100 108.33 0.08 1 0 7.97 0.2 Implied Volatility**Insert any number to start**Formula for option value minus the actual call value**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**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