Essentials of Investments: Chapter 16 - Option Valuation to discuss factors that affect option prices and to present quantitative option pricing models. It includes Factors influencing option values, BlacScholes option valuation, Using the BlacScholes formula, Binomial Option Pricing.
Bodie Kane Marcus Perrakis Ryan Chapter 16 INVESTMENTS, Fourth Canadian Edition Option Valuation Slide 18-1 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Chapter Summary Objective: To discuss factors that affect option prices and to present quantitative option pricing models Factors influencing option values Black-Scholes option valuation Using the Black-Scholes formula Binomial Option Pricing Slide 18-2 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Option Values Intrinsic value - profit that could be made if the option was immediately exercised Call: stock price - exercise price Put: exercise price - stock price Time value - the difference between the option price and the intrinsic value Slide 18-3 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Time Value of Options: Call Option value Value of Call Intrinsic Value Time value X Slide 18-4 Stock Price Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Factors Influencing Option Values: Calls Factor Stock price Exercise price Volatility of stock price Time to expiration Interest rate Dividend Rate Slide 18-5 Effect on value increases decreases increases increases increases decreases Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Restrictions on Option Value: Call Value cannot be negative Value cannot exceed the stock value Value of the call must be greater than the value of levered equity C > S0 - ( X + D ) / ( + Rf )T C > S0 - PV ( X ) - PV ( D ) Slide 18-6 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Allowable Range for Call Call Value Lower Bound = S0 - PV (X) - PV (D) S0 PV (X) + PV (D) Slide 18-7 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Summary Reminder Objective: To discuss factors that affect option prices and to present quantitative option pricing models Factors influencing option values Black-Scholes option valuation Using the Black-Scholes formula Binomial Option Pricing Slide 18-8 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Black-Scholes Option Valuation Co = SoN(d1) - Xe-rTN(d2) d1 = [ln(So/X) + (r + 2/2)T] / (T1/2) d2 = d1 + (T1/2) where, Co = Current call option value So = Current stock price N(d) = probability that a random draw from a normal distribution will be less than d Slide 18-9 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Black-Scholes Option Valuation (cont’d) X = Exercise price e = 2.71828, the base of the natural log r = Risk-free interest rate (annualizes continuously compounded with the same maturity as the option) T = time to maturity of the option in years ln = Natural log function Standard deviation of annualized continuously compounded rate of return on the stock Slide 18-10 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Hedging and Delta The 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 Change in the value of the option Delta change in the value of the stock Slide 18-22 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Mispriced Option: Text Example Implied volatility = 33% Investor believes volatility should = 35% Option maturity = 60 days Put price P = $4.495 Exercise price and stock price = $90 Risk-free rate r = 4% Delta = -.453 Slide 18-23 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Hedged Put Portfolio Cost to establish the hedged position 1000 put options at $4.495 / option $ 4,495 453 shares at $90 / share 40,770 Total outlay 45,265 Slide 18-24 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Profit Position on Hedged Put Portfolio Value of put as function of stock price: implied volatility = 35% Stock Price 89 90 Put Price $5.254 $4.785 Profit/loss per put 759 290 91 $4.347 (.148) Value of and profit on hedged portfolio Stock Price 89 90 91 Value of 1,000 puts $ 5,254 $ 4,785 $ 4,347 Value of 453 shares 40,317 40,770 41,223 Total 45,571 45,555 45,570 Profit 306 290 305 Slide 18-25 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Summary Reminder Objective: To discuss factors that affect option prices and to present quantitative option pricing models Factors influencing option values Black-Scholes option valuation Using the Black-Scholes formula Binomial Option Pricing Slide 18-26 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Binomial Option Pricing: Text Example 200 100 75 C 50 Stock Price Slide 18-27 Call Option Value X = 125 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Binomial Option Pricing: Text Example Alternative Portfolio Buy share of stock at $100 Borrow $46.30 (8% Rate) 53.70 Net outlay $53.70 Payoff Value of Stock 50 200 Repay loan - 50 -50 Net Payoff 150 Slide 18-28 150 Payoff Structure is exactly times the Call Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Binomial Option Pricing: Text Example 150 53.70 75 C 0 2C = $53.70 C = $26.85 Slide 18-29 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Another View of Replication of Payoffs and Option Values Alternative Portfolio - one share of stock and calls written (X = 125) Portfolio is perfectly hedged Stock Value Call Obligation Net payoff 50 50 200 -150 50 Hence 100 - 2C = 46.30 or C = 26.85 Slide 18-30 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Generalizing the Two-State Approach Assume 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) Slide 18-31 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Generalizing the Two-State Approach 121 110 104.50 100 95 90.25 Slide 18-32 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Expanding to Consider Three Intervals Assume 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 Slide 18-33 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Expanding to Consider Three Intervals S+++ S++ S++- S+ S+- S S+-SS Slide 18-34 S - Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Possible Outcomes with Three Intervals Event Probability Stock Price up 1/8 100 (1.05)3 up down 3/8 100 (1.05)2 (.97) =106.94 up down 3/8 100 (1.05) (.97)2 = 98.79 down 1/8 100 (.97)3 Slide 18-35 =115.76 = 91.27 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Multinomial Option Pricing Incomplete markets If the stock return has more than two possible outcomes it is not possible to replicate the option with a portfolio containing the stock and the riskless asset Markets are incomplete when there are fewer assets than there are states of the world (here possible stock outcomes) No single option price can be then derived by arbitrage methods alone Only upper and lower bounds exist on option prices, within which the true option price lies An appropriate pair of such bounds converges to the Black-Scholes price at the limit Slide 18-36 Copyright © McGraw-Hill Ryerson Limited, 2003 ... Put-Call Parity P = C + PV (X) - So = C + Xe-rT - So Using the example data C = 13.70 X = 95 S = 100 r = 10 T = 25 P = 13.70 + 95 e -. 10 x 25 - 100 P = 6.35 Slide 18 -1 6 Copyright © McGraw-Hill... quantitative option pricing models Factors influencing option values Black-Scholes option valuation Using the Black-Scholes formula Binomial Option Pricing Slide 1 8-2 6 Copyright © McGraw-Hill... affect option prices and to present quantitative option pricing models Factors influencing option values Black-Scholes option valuation Using the Black-Scholes formula Binomial Option