Tài chính doanh nghiệp - Chapter 21: Option valuation

Chapter 21Option ValuationIntrinsic value - profit that could be made if the option was immediately exercised.Call: stock price - exercise pricePut: exercise price - stock price Time value - the difference between the option price and the intrinsic value.Option ValuesTime Value of Options: CallOption valueXStock PriceValue of Call Intrinsic ValueTime valueFactor Effect on valueStock price increasesExercise price decreasesVolatility of stock price increasesTime to expiration increasesInterest rate increasesDividend Rate decreasesFactors Influencing Option Values: CallsRestrictions on Option Value: CallValue cannot be negativeValue cannot exceed the stock valueValue 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 hedged Stock Value 50 200 Call Obligation 0 -150 Net payoff 50 50Hence 100 - 2C = 46.30 or C = 26.85Replication of Payoffs and Option ValuesGeneralizing the Two-State ApproachAssume that we can break the year into two six-month segments.In each six-month segment the stock could increase by 10% or decrease by 5%.Assume the stock is initially selling at 100.Possible outcomes: Increase by 10% twice Decrease by 5% twice Increase once and decrease once (2 paths).Generalizing the Two-State Approach1001101219590.25104.50Assume that we can break the year into three intervals.For each interval the stock could increase by 5% or decrease by 3%.Assume the stock is initially selling at 100.Expanding to Consider Three IntervalsSS +S + +S -S - -S + -S + + +S + + -S + - -S - - -Expanding to Consider Three IntervalsPossible Outcomes with Three IntervalsEvent Probability Stock Price3 up 1/8 100 (1.05)3 =115.762 up 1 down 3/8 100 (1.05)2 (.97) =106.941 up 2 down 3/8 100 (1.05) (.97)2 = 98.793 down 1/8 100 (.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 pricee = 2.71828, the base of the natural logr = 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 functionStandard deviation of annualized cont. compounded rate of return on the stockBlack-Scholes Option ValuationSo = 100 X = 95r = .10 T = .25 (quarter)= .50d1 = [ln(100/95) + (.10+(5 2/2))] / (5.251/2) = .43 d2 = .43 + ((5.251/2) = .18Call Option ExampleN (.43) = .6664Table 17.2 d N(d) .42 .6628 .43 .6664 Interpolation .44 .6700Probabilities from Normal DistN (.18) = .5714Table 17.2 d 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 ValuePut Value Using Black-ScholesP = Xe-rT [1-N(d2)] - S0 [1-N(d1)]Using the sample call dataS = 100 r = .10 X = 95 g = .5 T = .2595e-10x.25(1-.5714)-100(1-.6664) = 6.35P = C + PV (X) - So = C + Xe-rT - SoUsing the example dataC = 13.70 X = 95 S = 100r = .10 T = .25P = 13.70 + 95 e -.10 X .25 - 100P = 6.35Put Option Valuation: Using Put-Call ParityBlack-Scholes Model with DividendsThe call option formula applies to stocks that pay dividends.One approach is to replace the stock price with a dividend adjusted stock price.Replace S0 with S0 - PV (Dividends)Hedging: Hedge ratio or delta The number of stocks required to hedge against the price risk of holding one option. Call = N (d1) Put = N (d1) – 1Option Elasticity Percentage change in the option’s value given a 1% change in the value of the underlying stock.Using the Black-Scholes FormulaBuying Puts - results in downside protection with unlimited upside potential.Limitations Tracking errors if indexes are used for the puts.Maturity of puts may be too short.Hedge ratios or deltas change as stock values change.Portfolio Insurance Hedging On Mispriced OptionsOption value is positively related to volatility:If an investor believes that the volatility that is implied in an option’s price is too low, a profitable trade is possible.Profit must be hedged against a decline in the value of the stock.Performance depends on option price relative to the implied volatility.Hedging and DeltaThe appropriate hedge will depend on the delta.Recall the delta is the change in the value of the option relative to the change in the value of the stock.Delta = Change in the value of the optionChange of the value of the stockMispriced Option: Text ExampleImplied volatility = 33% Investor believes volatility should = 35%Option maturity = 60 daysPut price P = $4.495Exercise price and stock price = $90Risk-free rate r = 4%Delta = -.453Hedged Put PortfolioCost to establish the hedged position1000 put options at $4.495 / option $ 4,495453 shares at $90 / share 40,770Total outlay 45,265Profit Position on Hedged Put PortfolioValue of put option: implied vol. = 35% Stock Price 89 90 91Put Price $5.254 $4.785 $4.347Profit (loss) for each put .759 .290 (.148)Value of and profit on hedged portfolio Stock Price 89 90 91Value of 1,000 puts $ 5,254 $ 4,785 $ 4,347Value of 453 shares 40,317 40,770 41,223Total 45,571 45,555 5,570Profit 306 290 305