Option Valuation

CHAPTER 16 16.1 OPTION VALUATION: INTRODUCTION Option Valuation Option Values Intrinsic value - profit that could be made if the option was immediately exercised Figure 16.1 Call Option Value Before Expiration – Call: stock price - exercise price – Put: exercise price - stock price Time value - the difference between the option price and the intrinsic value Determinants of Call Option Values Stock price Exercise price Volatility of the stock price Time to expiration Interest rate Dividend rate of the stock Table 16.1 Determinants of Call Option Values Binomial Option Pricing: Text Example 120 16.2 BINOMIAL OPTION PRICING 100 10 C 90 0 Call Option Value if X = 110 Stock Price Binomial Option Pricing: Text Example (cont.) Alternative Portfolio Buy 1 share of stock at $100 18.18 Borrow $81.82 (10% Rate) Net outlay $18.18 Payoff Value of Stock 90 120 Repay loan - 90 -90 Net Payoff 0 30 Binomial Option Pricing: Text Example (cont.) 30 30 18.18 C 0 0 Payoff Structure is exactly 3 times the Call Another View of Replication of Payoffs and Option Values Alternative Portfolio - one share of stock and 3 calls written (X = 110) Portfolio is perfectly hedged Stock Value 90 120 Call Obligation 0 - 30 Net payoff 90 90 Hence 100 - 3C = 81.82 or C = 6.06 10 0 3C = $18.18 C = $ 6.06 Generalizing the Two-State Approach 121 110 100 104.50 95 90.25 Generalizing the Two-State Approach Figure 16.2 Probability Distributions Cuu Cu C Cud Cd C dd Black-Scholes Option Valuation 16.3 BLACK-SCHOLES OPTION VALUATION Co = Soe-δTN(d1) - Xe-rTN(d2) d1 = [ln(So/X) + (r – δ + σ2/2)T] / (σ (σ T1/2) 1/2 d2 = d1 - (σ T ) where Co = Current call option value. So = Current stock price N(d) = probability that a random draw from a normal dist. will be less than d. Black-Scholes Option Valuation X = Exercise price. δ = Annual dividend yield of underlying stock e = 2.71828, the base of the natural log r = RiskRisk-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 σ = Standard deviation of annualized cont. compounded rate of return on the stock Figure 16.3 A Standard Normal Curve Call Option Example So = 100 X = 95 r = .10 T = .25 (quarter) σ = .50 δ = 0 d1 = [ln(100/95)+(.10[ln(100/95)+(.10-0+(. 0+(.5 2/2))]/(. /2))]/(.5 .251/2) = .43 d2 = .43 - ((. ((.5)( .25)1/2 = .18 Probabilities from Normal Distribution N (.43) = .6664 Table 17.2 d N(d) .42 .6628 .43 .6664 Interpolation .44 .6700 Call Option Value Probabilities from Normal Distribution N (.18) = .5714 Table 17.2 d N(d) .16 .5636 .18 .5714 .20 .5793 Figure 16.5 Implied Volatility of the S&P 500 (VIX Index) Co = Soe-δTN(d1) - Xe-rTN(d2) Co = 100 X .6664 - 95 e- .10 X .25 X .5714 Co = 13.70 Implied Volatility Using BlackBlack-Scholes and the actual price of the option, solve for volatility. Is the implied volatility consistent with the stock? Put-Call Parity Relationship Payoff for Call Owned Payoff for Put Written Total Payoff ST < X ST > X 0 ST - X -( X -ST) ST - X 0 ST - X Figure 16.6 The Payoff Pattern of a Long Call – Short Put Position Arbitrage & Put Call Parity Since 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)T If the prices are not equal arbitrage will be possible Put Call Parity - Disequilibrium Example Example 16.3 Put-Call Parity Put Option Valuation Put Option Valuation: Using Put-Call Parity Stock Price = 110 Call Price = 17 Put Price = 5 Risk Free = 5% Maturity = .5 yr Exercise (X) = 105 C - P > S0 - X / (1 + rf)T 1414- 5 > 110 - (105 e (-.05 x .5)) 9 > 7.59 Since the leveraged equity is less expensive; acquire the low cost alternative and sell the high cost alternative P=Xe-rT [1[1-N(d2)] - S0e-δT [1[1-N(d1)] Using the sample data P = $95e(-.10X.25)(1(1-.5714) - $100 (1(1-.6664) P = $6.35 P = C + PV (X) - So + PV (Div) = C + Xe-rT - So + 0 Using the example data C = 13.70 X = 95 S = 100 r = .10 T = .25 P = 13.70 + 95 e -.10 X .25 – 100 + 0 P = 6.35 Black-Scholes Formula 16.4 USING THE BLACK-SCHOLES FORMULA 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) - 1 Option Elasticity Percentage change in the option’ option’s value given a 1% change in the value of the underlying security Figure 16.7 Call Option Value and Hedge Ratio Portfolio Insurance - Protecting Against Declines in Stock Value Buying 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 Figure 16.8 Profit on a Protective Put Strategy Figure 16.9 Hedge-Ratios Change as the Stock Price Fluctuates Empirical Tests of Black-Scholes Option Pricing 16.5 EMPIRICAL EVIDENCE Figure 16.10 Implied Volatility as a Function of Exercise Price Implied volatility varies with exercise price – If the model were accurate, implied volatility should be the same for all options with the same expiration date – Implied volatility steadily falls as the exercise price rises