金融市场学期权攀 登nBuy - Long nSell - ShortnCallnPut nKey ElementsExercise or Strike PricePremium or PriceMaturity or ExpirationOption TerminologyIn the Money - exercise of the option would be profitableCall: market priceexercise pricePut: exercise pricemarket priceOut of the Money - exercise of the option would not be profitableCall: market priceexercise pricePut: exercise pricemarket priceAt the Money - exercise price and asset price are equalMarket and Exercise Price RelationshipsAmerican - the option can be exercised at any time before expiration or maturityEuropean - the option can only be exercised on the expiration or maturity dateAmerican vs. European OptionsnStock OptionsnIndex OptionsnFutures OptionsnForeign Currency OptionsnInterest Rate OptionsDifferent Types of OptionsNotation Stock Price = ST Exercise Price = XPayoff to Call Holder (ST - X) if ST X 0if ST X 0if ST X(X - ST)if ST X-(X - ST)if ST XProfits to Put WriterPayoff + PremiumPayoffs and Profits at Expiration - PutsPayoff Profiles for Puts0PayoffsStock PricePut WriterPut HolderInvestmentStrategyInvestmentEquity onlyBuy stock 100 100 shares$10,000Options onlyBuy calls 101000 options$10,000LeveragedBuy calls 10100 options $1,000equityBuy T-bills 2% $9,000YieldEquity, Options & Leveraged EquityIBM Stock Price$95$105$115All Stock$9,500$10,500$11,500All Options$0 $5,000$15,000Lev Equity $9,270 $9,770$10,770Equity, Options & Leveraged Equity - PayoffsIBM Stock Price$95$105$115All Stock-5.0%5.0% 15%All Options-100% -50% 50%Lev Equity -7.3%-2.3% 7.7%Equity, Options & Leveraged EquityProtective PutUse - limit lossPosition - long the stock and long the putPayoffST XStock ST STPut X - ST 0Protective Put ProfitSTProfit -PStockProtective Put PortfolioCovered CallUse - Some downside protection at the expense of giving up gain potentialPosition - Own the stock and write a callPayoffST XStock ST STCall 0 - ( ST - X)Covered Call ProfitSTProfit -PStockCovered Call PortfolioStraddle (Same Exercise Price)Long Call and Long PutSpreads - A combination of two or more call options or put options on the same asset with differing exercise prices or times to expirationVertical or money spreadSame maturityDifferent exercise priceHorizontal or time spreadDifferent maturity datesOption StrategiesST XPayoff forCall Owned 0ST - XPayoff forPut Written-( X -ST) 0Total Payoff ST - X ST - XPut-Call Parity RelationshipLong CallShort PutPayoffStock PriceCombined =Leveraged EquityPayoff of Long Call & Short PutSince the payoff on a combination of a long call and a short put are equivalent to leveraged equity, the prices must be equal.C - P = S0 - X / (1 + rf)TIf the prices are not equal arbitrage will be possibleArbitrage & Put Call ParityStock Price = 110 Call Price = 17Put Price = 5 Risk Free = 10.25%Maturity = .5 yr X = 105C - P S0 - X / (1 + rf)T17- 5 110 - (105/1.05) 12 10Since the leveraged equity is less expensive, acquire the low cost alternative and sell the high cost alternativePut Call Parity - Disequilibrium ExamplePut-Call Parity ArbitrageImmediateCashflow in Six MonthsPositionCashflowST 105Buy Stock-110 ST STBorrowX/(1+r)T = 100+100-105-105Sell Call+17 0-(ST-105)Buy Put -5105-ST 0Total 2 0 0Optionlike SecuritiesnCallable BondsnConvertible SecuritiesnWarrantsnCollateralized LoansExotic OptionsnAsian OptionsnBarrier OptionsnLookback OptionsnCurrency Translated OptionsnBinary OptionsnIntrinsic value - profit that could be made if the option was immediately exercisedCall: stock price - exercise pricePut: exercise price - stock price nTime value - the difference between the option price and the intrinsic valueOption ValuesTime Value of Options: CallOption valueXStock PriceValue of Call Intrinsic ValueTime valueFactorEffect on valueStock price increasesExercise price decreasesVolatility of stock priceincreasesTime to expirationincreasesInterest rate increasesDividend RatedecreasesFactors Influencing Option Values: CallsRestrictions on Option Value: CallnValue cannot be negativenValue cannot exceed the stock valuenValue of the call must be greater than the value of levered equityC S0 - ( X + D ) / ( 1 + Rf )TC S0 - PV ( X ) - PV ( D )Allowable Range for CallCall ValueS0PV (X) + PV (D)Upper bound = S0Lower Bound = S0 - PV (X) - PV (D)10020050Stock PriceC750Call Option Value X = 125Binomial Option Pricing:Text ExampleAlternative PortfolioBuy 1 share of stock at $100Borrow $46.30 (8% Rate)Net outlay $53.70PayoffValue of Stock 50 200Repay loan - 50 -50Net Payoff 0 15053.701500Payoff Structureis exactly 2 timesthe CallBinomial Option Pricing:Text Example53.701500C7502C = $53.70C = $26.85Binomial Option Pricing:Text ExampleAlternative Portfolio - one share of stock and 2 calls written (X = 125) Portfolio is perfectly hedgedStock Value50200Call Obligation 0 -150Net payoff50 50Hence 100 - 2C = 46.30 or C = 26.85Another View of Replication of Payoffs and Option ValuesGeneralizing the Two-State ApproachAssume that we can break the year into two six-month segmentsIn each six-month segment the stock could increase by 10% or decrease by 5%Assume the stock is initially selling at 100Possible outcomesIncrease by 10% twiceDecrease by 5% twiceIncrease once and decrease once (2 paths)Generalizing the Two-State Approach1001101219590.25104.50nAssume that we can break the year into three intervalsnFor each interval the stock could increase by 5% or decrease by 3%nAssume the stock is initially selling at 100Expanding to Consider Three IntervalsSS +S + +S -S - -S + -S + + +S + + -S + - -S - - -Expanding to Consider Three IntervalsPossible Outcomes with Three IntervalsEventProbabilityStock Price3 up 1/8100 (1.05)3 =115.762 up 1 down 3/8100 (1.05)2 (.97)=106.941 up 2 down 3/8100 (1.05) (.97)2= 98.793 down 1/8100 (.97)3= 91.27Co = SoN(d1) - Xe-rTN(d2)d1 = ln(So/X) + (r + 2/2)T / (T1/2)d2 = d1 + (T1/2)whereCo = Current call option value.So = Current stock priceN(d) = probability that a random draw from a normal dist. will be less than d.Black-Scholes Option ValuationX = Exercise price.e = 2.71828, the base of the nat. log.r = Risk-free interest rate (annualizes continuously compounded with the same maturity as the option)T = time to maturity of the option in yearsln = Natural log functionStandard deviation of annualized cont. compounded rate of return on the stockBlack-Scholes Option ValuationSo = 100X = 95r = .10T = .25 (quarter)= .50d1 = ln(100/95) + (.10+(5 2/2) / (5.251/2) = .43 d2 = .43 + (5.251/2) = .18Call Option ExampleN (.43) = .6664Table 17.2d N(d) .42 .6628 .43.6664 Interpolation .44.6700Probabilities from Normal DistN (.18) = .5714Table 17.2d N(d) .16 .5636 .18.5714 .20.5793Probabilities from Normal Dist.Co = SoN(d1) - Xe-rTN(d2)Co = 100 X .6664 - 95 e- .10 X .25 X .5714 Co = 13.70Implied VolatilityUsing Black-Scholes and the actual price of the option, solve for volatility.Is the implied volatility consistent with the stock?Call Option ValueP = C + PV (X) - So = C + Xe-rT - SoUsing the example dataC = 13.70X = 95S = 100r = .10T = .25P = 13.70 + 95 e -.10 X .25 - 100P = 6.35Put Option Valuation: Using Put-Call ParityAdjusting the Black-Scholes Model for DividendsnThe call option formula applies to stocks that pay dividendsnOne approach is to replace the stock price with a dividend adjusted stock priceReplace S0 with S0 - PV (Dividends)Hedging: Hedge ratio or delta The number of stocks required to hedge against the price risk of holding one optionCall = N (d1)Put = N (d1) - 1Option ElasticityPercentage change in the options value given a 1% change in the value of the underlying stockUsing the Black-Scholes FormulanBuying Puts - results in downside protection with unlimited upside potentialnLimitations Tracking errors if indexes are used for the putsMaturity of puts may be too shortHedge ratios or deltas change as stock values changePortfolio Insurance - Protecting Against Declines in Stock Value