Corporate finance chapter 015 option and contingent claims

Chapter 15: Options & Contingent Claims 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 Copyright © Prentice Hall Inc 2000 Author: Nick Bagley, bdellaSoft, Inc Chapter 15 Contents 15.1 How Options Work 15.7 The Black-Scholes Model 15.2 Investing with Options 15.8 Implied Volatility 15.3 The Put-Call Parity Relationship 15.9 Contingent Claims Analysis of Corporate Debt and Equity 15.4 Volatility & Option Prices 15.10 Credit Guarantees 15.5 Two-State Option Pricing 15.11 Other Applications of Option-Pricing Methodology 15.6 Dynamic Replication & the Binomial Model Objectives • To show how the Law of One Price can be used to derive prices of options • To show how to infer implied volatility form option prices Table 15.1 List of IBM Option Prices (Source: Wall Street Journal Interactive Edition, May 29, 1998) IBM (IBM) Strike 115 115 115 120 120 120 125 125 125 Underlying stock price 120 1/16 Call Put Expiration Volume Jun Oct Jan Jun Oct Jan Jun Oct Jan Last 1372 … … … … 2377 121 91 1564 91 87 1/2 5/16 12 1/2 1/2 1/2 10 1/2 Open Interest 4483 2584 15 8049 2561 8842 9764 2360 124 Volume Last 756 10 53 873 45 … 3/16 3/4 7/8 1/8 … 17 … … 3/4 … … Open Interest 9692 967 40 9849 1993 5259 5900 731 70 Table 15.2 List of Index Option Prices (Source: Wall Street Journal Interactive Edition, June 6, 1998) S & P 500 INDEX -AM Underlying S&P500 (SPX) Jun Jun Jul Jul Jun Jun Jul Jul High Low 1113.88 1084.28 Strike 1110 call 1110 put 1110 call 1110 put 1120 call 1120 put 1120 call 1120 put Chicago Exchange Close 1113.86 Net Change 19.03 Volume 2,081 1,077 1,278 152 80 211 67 10 Last 17 1/4 10 33 1/2 23 3/8 12 17 27 1/4 27 1/2 From 31-Dec 143.43 Net Change 1/2 -11 1/2 -12 1/8 -11 1/4 -11 % Change 14.8 Open Interest 15,754 17,104 3,712 1,040 16,585 9,947 5,546 4,033 Terninal or Boundary Conditions for Call and Put Options 120 100 Dollars 80 Call Put 60 40 20 0 20 40 60 80 100 120 140 -20 Underlying Price 160 180 200 Terminal Conditions of a Call and a Put Option with Strike = 100 Strike 100 Share 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 Call 0 0 0 0 0 10 20 30 40 50 60 70 80 90 100 Put Share_Put 100 100 90 100 80 100 70 100 60 100 50 100 40 100 30 100 20 100 10 100 100 110 120 130 140 150 160 170 180 190 200 Bond Call_Bond 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 110 100 120 100 130 100 140 100 150 100 160 100 170 100 180 100 190 100 200 Stock, Call, Put, Bond 200 Call Put 160 Share_Put 140 Bond Stock, Call, Put, Bond, Put+Stock, Call+Bond 180 Call_Bond 120 Share 100 80 60 40 20 0 20 40 60 80 100 120 140 Stock Price 160 180 200 Put-Call Parity Equation Call ( Strike, Maturity ) + Strike = Put ( Strike, Maturity ) + Share Maturity (1 + rf ) 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 10 Debtco Security Payoff Table ($’000,000) Security Payoff State a Payoff State b Firm 140 70 Bond 80 70 Stock 60 38 Debtco’s Replicating Portfolio • Let – 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 60 = 140 x − 1.04Y ; = 70 x − 1.04Y ⇒ x = ; Y = $57,692,308 39 Debtco’s Replicating Portfolio ($’000) Position Immediate Case a Case b 6/7 assets -85,714 120,000 60,000 Bond (rf) 57,692 -60,000 -60,000 Total 28,022 60,000 40 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 $71,978,022 • The yield on this debt is (80…/71…) - = 11.14% 41 Another View of Debtco’s Replicating Portfolio (‘$000) Security Bonds Stock Bonds + Stock Total Equivalent Equivalent market Amount Amount Value of Firm of Rf Debt 71,978 14,286 57,692 28,022 85,714 -57,692 100,000 100,000 42 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) E + Y 20,000,000 + 57,692,308 E = xV − Y ; ⇒ V = = = $90,641,026 x is then – The implied value of the bonds $90,641,026 - $20,000,000 = $70,641,026 & the yield is (80.00-70.64)/70.64 = 13.25% 43 Replication Portfolio Position Purchase x of firm Purchase Y RF Bond Total Portfolio Immediate Scenario a Scenario b Cash Flow V1 = 70 V1 = 140 - x* V 70 x 140 x -Y Y (1.04) Y (1.04) -x*V-Y 70 80 44 Determining the Weight of Firm Invested in Bond, x, and the Value of the R.F.-Bond, Y 70 = 70 x + 1.04Y  ⇒ 80 = 140 x + 1.04Y  x = ; Y = $57,692,308 45 Valuing Stock – We can replicate the bond by purchasing 1/7 of the company, and $57,692,308 of defaultfree 1-year bonds – The market value of the bonds is $909.0909 * 80,000 = $72,727,273 D = xV + Y ; ⇒ V = E − Y 72,727,273 − 57,692,308 = = $105,244,753 x – The value of the stock is therefore E=V -D = $105,244,753 - $72,727,273= $32,517,480 46 Convertible Bond Value of Stock and Bond Issue 140 ConvertibleBondValue 120 DilultedStockValue 100 80 60 40 20 0 20 40 60 80 100 120 Value of the Firm 47 140 160 180 200 Month Month Month 12 Outline Decision Tree Node-D $140MM Node-B $115MM Node-F $110MM Node-A $100MM Node-E $90MM Node-C $90MM 48 Node-G $70MM Valuing Pure State-Contingent Securities Security Payoff Scenario a Payoff Scenario b Firm $70,000,000 $140,000,000 Contingent Security #1 $0 $1 Contingent Security #2 $1 $0 49 State-Contingent Security #1 S C S #1 70,000,000 x + 1.04Y = 0 1 ⇒ x = ; Y = −  140,000,000 x + 1.04Y =  70,000,000 1.04 100,000,000 P1 = 1,000,000 x + Y = − = 0.467 032 967 70,000,000 1.04 S C S #2 70,000,000 x + 1.04Y =  ; Y= ⇒ x = − 140,000,000 x + 1.04Y = 0 70,000,000 1.04 100,000,000 P2 = 1,000,000 x + Y = − + = 0.494 505 495 70,000,000 1.04 $1 P1 + P2 = $0.467 033 + $0.494 505 = $0.961 538 = 1.04 50 Payoff for Debtco’s Bond Guarantee Security Scenario a Firm $70,000,000 $140,000,000 Bonds Guarantee Scenario b $1,000 $875 $0 $125 51 SCS Conformation of Guarantee’s Price • Guarantee’s price is 125 * $0.494505 = $61.81 52 [...]...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 Call ( Strike, Maturity ) + Strike Forward = Put ( Strike , Maturity ) + (1 + rf ) Maturity (1 + rf ) Maturity 11 Implications for European Options • If (F > E) then (C > P) • If (F = E) then (C =... price 12 Call and Put as a Function of Forward Call = Put 16 call put asy_call_1 asy_put_1 14 Put, Call Values 12 10 8 6 4 2 0 90 92 94 96 98 100 102 104 106 108 Forward Strike = Forward 13 110 Put and Call as Function of Share Price 60 call Put and Call Prices 50 put asy_call_1 40 asy_call_2 asy_put_1 30 asy_put_2 20 10 0 50 60 70 80 90 100 110 -10 Share Price 14 120 130 140 150 Put and Call as Function... Despair • The 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-scaled 34 Strategy 1-Week Later 2.0 Strategy Value 1.5 Portfolio PortfolioLater 1.0 0.5 0.0 90 95 100 105 110 -0.5 Share Price 35 115 120 Payoffs for Bond and Stock Issues Value of Bond and Stock (Millions) 120 100 80 60 40 BondValue StockValue 20 0... 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 y • the fraction x is called the hedge ratio C = xS − y 17 Binary Model: Call • Specification: – We have an equation, and given the value of the terminal share price, we know the terminal option value for two cases: 20 = x120 − y 0 = x80 − y – By inspection, the... 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 y • the fraction x is called the hedge ratio P = − xS + y 20 Binary Model: Put • Specification: – We have an equation, and given the value of the terminal share price, we know the terminal option value for two cases: 20 = x120 − y 0 = x80 − y – By inspection, the solution... N ( d 2 ) ) S C=P≈ σ T ≈ 0.39886 Sσ T 2π 27 Determinants of Option Prices Increases in: Stock Price, S Exercise Price, E Volatility, sigma Time to Expiration, T Interest Rate, r Cash Dividends, d Call Put Increase Decrease Increase Ambiguous Increase Decrease Decrease Increase Increase Ambiguous Decrease Increase 28 Value of a Call and Put Options with Strike = Current Stock Price 11 10 put 8 7 6 5... Value of a Call and Put Options with Strike = Current Stock Price 11 10 put 8 7 6 5 4 3 2 1 0 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 Time-to-Maturity 29 0.2 0.1 0.0 Call and Put Price call 9 Call and Put Prices as a Function of Volatility 6 Call and Put Prices 5 call put 4 3 2 1 0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Volatility 30 0.14 0.16 0.18 0.20 Computing Implied Volatility volatility call strike share... asy_put_2 20 10 0 50 60 70 80 90 100 110 -10 Share Price 14 120 130 140 150 Put and Call as Function of Share Price 20 call Put and Call Prices put asy_call_1 15 asy_call_2 asy_put_1 asy_put_2 10 5 0 80 85 90 95 100 105 110 115 Share Price PV Strike 15 Strik e 120 Volatility and Option Prices, P0 = $100, Strike = $100 Stock Price Call Payoff Put Payoff Low Volatility Case Rise Fall Expectation 120 80 100... rate_for maturity 0.3154 10.0000 100.0000 105.0000 0.0500 0.0000 0.2500 factor 0.0249 d_1 d_2 0.4675 0.3098 n_d_1 n_d_2 0.6799 0.6217 call_part_1 call_part_2 error Insert any number to start Formula for option value minus the actual call value 71.3934 -61.3934 0.0000 31 Computing Implied Volatility volatility 0.315378127101852 call strike share rate_dom rate_for maturity 10 100 105 0.05 0 0.25 factor... Model: Put • Solution: – We now substitute the value of the parameters x=1/2, y = 60 into the equation P = − xS + y – to obtain: 1 P = − 100 + 60 = $10 2 22 Decision Tree for Dynamic Replication of a Call Option < -0 Months > < 6 Months > 12 Months StockPrice x y CallPrice x y CallPrice $120.00 $110.00 $100.00 $90.00 $80.00 $20.00 $10.00 50.00% 100.00% -$100.00 -$45.00 $0.00 $0.00