CHAPTER 21 Option Valuation

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 Option Values

Figure 21.1 Call Option Value before Expiration

Table 21.1 Determinants of Call Option Values

Stock Price C 10 0 Call Option Value X = 110 Binomial Option Pricing: Text Example

Alternative Portfolio Buy 1 share of stock at $100 Borrow $81.82 (10% Rate) Net outlay $18.18 Payoff Value of Stock Repay loan Net Payoff Payoff Structure is exactly 3 times the Call Binomial Option Pricing: Text Example Continued

C C = $18.18 C = $6.06 Binomial Option Pricing: Text Example Continued

Alternative Portfolio - one share of stock and 3 calls written (X = 110) Portfolio is perfectly hedged Stock Value90120 Call Obligation Net payoff90 90 Hence C = 90/(1 + r f ) = 90/(1.1) = C = 100 – = Thus C = 6.06 Replication of Payoffs and Option Values

Number of stocks per option (H) H = (C + - C - )/(S + - S - ) = (10 – 0)/(120 – 90) =1/3 Number of options per stock = 1/H = 3 Hedge Ratio (H)

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 Continued

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

S S + S + + S - S - - S + - S S S S Expanding to Consider Three Intervals Continued

Possible Outcomes with Three Intervals EventProbability Final Stock Price 3 up 1/8100 (1.05) 3 = up 1 down 3/8100 (1.05) 2 (.97)= up 2 down 3/8100 (1.05) (.97) 2 = down 1/8100 (.97) 3 = 91.27

Figure 21.5 Probability Distributions

C o = S o N(d 1 ) - Xe -rT N(d 2 ) d 1 = [ln(S o /X) + (r +  2 /2)T] / (  T 1/2 ) d 2 = d 1 + (  T 1/2 ) where C o = Current call option value S o = Current stock price N(d) = probability that a random draw from a normal distribution will be less than d Black-Scholes Option Valuation

X = Exercise price e = , 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 Black-Scholes Option Valuation Continued

Figure 21.6 A Standard Normal Curve

S o = 100X = 95 r =.10T =.25 (quarter)  =.50 d 1 = [ln(100/95) + (.10+(  5 2 /2))(0.25)]/(  5 .25 1/2 ) =.43 d 2 =.43 + ((  5 .25 1/2 ) =.18 Call Option Example

N (.43) =.6664 Table 21.2 d N(d) Interpolation Probabilities from Normal Distribution

N (.18) =.5714 Table 21.2 d N(d) Probabilities from Normal Distribution Continued

Table 21.2 Cumulative Normal Distribution

C o = S o N(d 1 ) - Xe -rT N(d 2 ) C o = 100 X e -.10 X.25 X.5714 C o = Implied Volatility Using Black-Scholes and the actual price of the option, solve for volatility. Is the implied volatility consistent with the stock? Call Option Value

Spreadsheet 21.1 Spreadsheet to Calculate Black- Scholes Option Values

Figure 21.7 Using Goal Seek to Find Implied Volatility

Figure 21.8 Implied Volatility of the S&P 500 (VIX Index)

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 S 0 with S 0 - PV (Dividends)

Put Value Using Black-Scholes P = Xe -rT [1-N(d 2 )] - S 0 [1-N(d 1 )] Using the sample call data S = 100 r =.10 X = 95 g =.5 T =.25 95e -10x.25 ( )-100( ) = 6.35

P = C + PV (X) - S o = C + Xe -rT - S o Using the example data C = 13.70X = 95S = 100 r =.10T =.25 P = e -.10 X P = 6.35 Put Option Valuation: Using Put-Call Parity

Hedging: Hedge ratio or delta The number of stocks required to hedge against the price risk of holding one option Call = N (d 1 ) Put = N (d 1 ) - 1 Option Elasticity Percentage change in the option’s value given a 1% change in the value of the underlying stock Using the Black-Scholes Formula

Figure 21.9 Call Option Value and Hedge Ratio

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 Portfolio Insurance

Figure Profit on a Protective Put Strategy

Figure Hedge Ratios Change as the Stock Price Fluctuates

Figure S&P 500 Cash-to-Futures Spread in Points at 15 Minute Intervals

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 Delta = Change in the value of the option Change of the value of the stock

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

Table 21.3 Profit on a Hedged Put Portfolio

Table 21.4 Profits on Delta-Neutral Options Portfolio

Figure Implied Volatility of the S&P 500 Index as a Function of Exercise Price