1 Chapter 15: Options & Contingent Claims Copyright © Prentice Hall Inc Author: Nick Bagley, bdellaSoft, Inc. Objective To show how the law of one price may be used to derive prices of options To show how to infer implied volatility from option prices

2 Chapter 15 Contents 15.1 How Options Work 15.2 Investing with Options 15.3 The Put-Call Parity Relationship 15.4 Volatility & Option Prices 15.5 Two-State Option Pricing 15.6 Dynamic Replication & the Binomial Model 15.7 The Black-Scholes Model 15.8 Implied Volatility 15.9 Contingent Claims Analysis of Corporate Debt and Equity Credit Guarantees Other Applications of Option-Pricing Methodology

3 Objectives To show how the Law of One Price can be used to derive prices of optionsTo show how the Law of One Price can be used to derive prices of options To show how to infer implied volatility form option pricesTo show how to infer implied volatility form option prices

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9 Put-Call Parity Equation

10 Synthetic Securities The put-call parity relationship may be solved for any of the four security variables to create 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 C=S+P-B S=C-P+B S=C-P+B P=C-S+B P=C-S+B B=S+P-C B=S+P-C

11 Options and Forwards We saw in the last chapter that the discounted value of the forward was equal to the current spotWe saw in the last chapter that the discounted value of the forward was equal to the current spot The relationship becomesThe relationship becomes

12 Implications for European Options If (F > E) then (C > P)If (F > E) then (C > P) If (F = E) then (C = P)If (F = E) then (C = P) If (F < E) then (C < P)If (F < E) then (C < P) E is the common strike priceE is the common strike price F is the forward price of underlying shareF is the forward price of underlying share C is the call priceC is the call price P is the put priceP is the put price

13 Strike = Forward Call = Put

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15 PV Strike Strik e

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17 Binary Model: Call Implementation:Implementation: –the synthetic call, C, is created by buying a fraction x of shares, of the stock, S, and simultaneously selling short risk free bonds with a market value ybuying a fraction x of shares, of the stock, S, and simultaneously selling short risk free bonds with a market value y the fraction x is called the hedge ratiothe fraction x is called the hedge ratio

18 Binary Model: Call Specification:Specification: –We have an equation, and given 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

19 Binary Model: Call Solution:Solution: –We now substitute the value of the parameters x=1/2, y = 40 into the equation –to obtain:

20 Binary Model: Put Implementation:Implementation: –the synthetic put, P, is created by sell short a fraction x of shares, of the stock, S, and simultaneously buy risk free bonds with a market value ysell short a fraction x of shares, of the stock, S, and simultaneously buy risk free bonds with a market value y the fraction x is called the hedge ratiothe fraction x is called the hedge ratio

21 Binary Model: Put Specification:Specification: –We have an equation, and given 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

22 Binary Model: Put Solution:Solution: –We now substitute the value of the parameters x=1/2, y = 60 into the equation –to obtain:

23 Decision Tree for Dynamic Replication of a Call Option ($120*100%) + (-$100) = $20

24 The Black-Scholes Model: Notation C = price of callC = price of call P = price of putP = price of put S = price of stockS = price of stock E = exercise priceE = exercise price T = time to maturityT = time to maturity ln(.) = natural logarithmln(.) = natural logarithm e = e = 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

25 The Black-Scholes Model: Equations

26 The Black-Scholes Model: Equations (Forward Form)

27 The Black-Scholes Model: Equations (Simplified)

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31 Insert any number to start Formula for option value minus the actual call value

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34 Pat in Despair The next diagram shows the the value of the portfolio today and one week henceThe next diagram shows the the value of the portfolio today and one week hence The construction lines have been removed, and the graph has been re- scaledThe construction lines have been removed, and the graph has been re- scaled

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38 Debtco Security Payoff Table ($’000,000)

39 Debtco’s Replicating Portfolio LetLet –x be the fraction of the firm in replicator –Y be the borrowings at the risk-free rate in the replicator –In $’000,000 the following equations must be satisfied

40 Debtco’s Replicating Portfolio ($’000)

41 Debtco’s Replicating Portfolio We know 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 $71,978,022We know 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…) - 1 = 11.14%The yield on this debt is (80…/71…) - 1 = 11.14%

42 Another View of Debtco’s Replicating Portfolio (‘$000)

43 Valuing Bonds –We can replicate the firm’s 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 ( )/70.64 = 13.25%

44 Replication Portfolio

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

46 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 $ * 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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48 Outline Decision Tree Node-B $115MM Node-C $90MM Node-D $140MM Node-F $110MM Node-E $90MM Node-G $70MM Node-A $100MM Month 0Month 6Month 12

49 Valuing Pure State-Contingent Securities

50 State-Contingent Security #1

51 Payoff for Debtco’s Bond Guarantee

52 SCS Conformation of Guarantee’s Price Guarantee’s price is 125 * $ = $61.81Guarantee’s price is 125 * $ = $61.81