Chapter 16 Option Valuation

Outline Valuation How valuation helps trading (optional) Intrinsic and time values Factors determining option price Black-Scholes Model How valuation helps trading (optional) Hedge ratio (Delta) and option elasticity Other variables

1. 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 However, option price is always higher than or equal to its intrinsic value Time value - the difference between the option price and the intrinsic value 2

Time Value of Options: Call Value of Call Intrinsic Value Time value X Stock Price 3

Factors Influencing Option Values: Calls If this variable increases Value of a call option Stock price increases Exercise price decreases Volatility of stock price increases Time to expiration increases Interest rate increases Dividend Rate decreases Interest affects the PV(x), your obligation to pay in the future. Higher interest, the less you need to pay in today’s value, the higher the value of call Div is a drag on stock price, call holder want stock price to be higher 4

Factors Influencing Option Values: Puts If this variable increases Value of a Put option Stock price decreases Exercise price increases Volatility of stock price increases Time to expiration increases Interest rate decreases Dividend Rate Increases Interest affects the PV(x), your sell price in the future. Higher interest, the less you get paid in today’s value, the lower the value of put Div is a drag on stock price, put holder want stock price be low 4

Black-Scholes Option Valuation Co = SoN(d1) - Xe-rTN(d2) d1 = [ln(So/X) + (r – d + s2/2)T] / (s T1/2) d2 = d1 - (s 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 1. 9

Black-Scholes Option Valuation X = Exercise price. d = Annual dividend yield of underlying stock 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 s = Standard deviation of annualized cont. compounded rate of return on the stock 10

Call Option Example So = 100 X = 95 r = .10 T = .25 (quarter) s = .50 d = 0 d1 = [ln(100/95)+(.10-0+(.5 2/2))]/(.5 .251/2) = .43 d2 = .43 - ((.5)( .251/2) = .18 11

Probabilities from Normal Dist. Table 17.2 d N(d) .42 .6628 .43 .6664 .44 .6700 12

Probabilities from Normal Dist. Table 17.2 d N(d) .16 .5636 .18 .5714 .20 .5793 13

Call Option Value Co = Soe-dTN(d1) - Xe-rTN(d2) Co = 100 X .6664 - 95 e- .10 X .25 X .5714 Co = 13.70 14

Put Option Value: Black-Scholes P=Xe-rT [1-N(d2)] – S0 [1-N(d1)] Using the sample data P = $95e(-.10X.25)(1-.5714) - $100 (1-.6664) P = $6.35

2.HOW VALUATION HELPS TRADING

Hedge ratio Hedge ratio: The change in the price of an option for a $1 increase in stock price. Hedge ratio is also called delta If we graph option value as a function of stock price, hedge ratio is the slope For call, 0<delta<1, for put -1<delta<0 In Black-Schole model, hedge ratio for call is N(d1), for put is N(d1)-1

How to use hedge ratio in trading Hedge ratio (delta) help to understand your potential gain and loss for options positions Leverage Option elasticity: (%change of option price)/(% change of stock price) Option elasticity=(delta/option price)/(1/stock price) Elasticity measures your leverage (with options) vs. investing in stocks My own measurement: delta/option price Measures % change of option value for $1 change of stock price

Important measurements in trading Delta: the change in an option price for one dollar increase in stock price Gamma: the change of Delta for one $ increase in stock price Theta: the change in an option price given a one-day change in time. Always negative, Good for option sellers.

Important measurements in trading Rho: the change in an option price for one % change in risk free rate ( not a big concern in trading. 1% rate is huge change, compared with $1 change of underlying stock price)

Important measurements in trading Vega: sensitivity to volatility. The change in an option price for 1%change in implied volatility Vega declines overtime Example: June 2010 S&P index Put, exercise price: 800 Index now: 1015; option Price/premium: $33 Vega: 2.3;implied volatility 35% If implied volatility increase by 10% from 35% to 45%. (CBOE Volatility Index soars as Wall St slumps) Put price: 2.3*10+33=$56

Important measurements in trading Calculate option price change

Important measurements in trading