Chapter 21 Options Valuation

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

Time Value of Options: Call X Stock Price Value of Call Intrinsic Value Time value

Factors Influencing Option Values: Calls Factor Effect on value Stock price increases Exercise price decreases Volatility of stock price increases Time to expiration increases Interest rate increases Dividend Rate decreases

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 ) / ( 1 + Rf )T C > S0 - PV ( X ) - PV ( D )

Allowable Range for Call Call Value Upper bound = S0 Lower Bound = S0 - PV (X) - PV (D) S0 PV (X) + PV (D)

Binomial Option Pricing: Text Example 200 C 75 Call Option Value X = 125 100 50 Stock Price

Binomial Option Pricing: Text Example 150 Alternative Portfolio Buy 1 share of stock at $100 Borrow $46.30 (8% Rate) Net outlay $53.70 Payoff Value of Stock 50 200 Repay loan - 50 -50 Net Payoff 0 150 53.70 Payoff Structure is exactly 2 times the Call

Binomial Option Pricing: Text Example 150 C 75 53.70 2C = $53.70 C = $26.85

Another View of Replication of Payoffs and Option Values Alternative 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 50 Hence 100 - 2C = 46.30 or C = 26.85

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)

Generalizing the Two-State Approach 100 110 121 95 90.25 104.50

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

Expanding to Consider Three Intervals

Possible Outcomes with Three Intervals Event Probability Stock Price 3 up 1/8 100 (1.05)3 =115.76 2 up 1 down 3/8 100 (1.05)2 (.97) =106.94 1 up 2 down 3/8 100 (1.05) (.97)2 = 98.79 3 down 1/8 100 (.97)3 = 91.27

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 dist. will be less than d.

Black-Scholes Option Valuation X = 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 years ln = Natural log function Standard deviation of annualized cont. compounded rate of return on the stock

Call Option Example So = 100 X = 95 r = .10 T = .25 (quarter) = .50 = .50 d1 = [ln(100/95) + (.10+(5 2/2))] / (5.251/2) = .43 d2 = .43 + ((5.251/2) = .18

Probabilities from Normal Dist Table 17.2 d N(d) .42 .6628 .43 .6664 Interpolation .44 .6700

Probabilities from Normal Dist. Table 17.2 d N(d) .16 .5636 .18 .5714 .20 .5793

Call Option Value Co = SoN(d1) - Xe-rTN(d2) Co = 100 X .6664 - 95 e- .10 X .25 X .5714 Co = 13.70 Implied Volatility Using Black-Scholes and the actual price of the option, solve for volatility. Is the implied volatility consistent with the stock?

Put Option Valuation: Using 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

Adjusting the Black-Scholes Model for Dividends The 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)

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’s value given a 1% change in the value of the underlying stock

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