Option Valuation Chapter 21

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 S0 PV (X) + PV (D) Upper bound = S0 Lower Bound = S0 - PV (X) - PV (D)

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

Binomial Option Pricing: Text Example 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 53.70 150 Payoff Structure is exactly 2 times the Call

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

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 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 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 Value Using Black-Scholes P = Xe-rT [1-N(d2)] - S0 [1-N(d1)] Using the sample call data S = 100 r = .10 X = 95 g = .5 T = .25 95e-10x.25(1-.5714)-100(1-.6664) = 6.35

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

Black-Scholes Model with 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 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.

Hedging On Mispriced Options Option 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 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 Change of the value of the stock Delta =

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

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

Profit Position on Hedged Put Portfolio Value of put option: implied vol. = 35% Stock Price 89 90 91 Put Price $5.254 $4.785 $4.347 Profit (loss) for each put .759 .290 (.148) Value of and profit on hedged portfolio 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 5,570 Profit 306 290 305