1. Chapter 16
Option Valuation

2. 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

3. 1. valuation

4. 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

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

6. 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

7. 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

8. 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

9. Black-Scholes Option Valuation
X = Exercise price.
d = Annual dividend yield of underlying stock
e = , 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

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

11. Probabilities from Normal Dist.
Table 17.2
d N(d) 12

12. Probabilities from Normal Dist.
Table 17.2
d N(d) 13

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

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

15. 2.HOW VALUATION HELPS TRADING

16. 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

17. 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

18. 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.

19. 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)

20. 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

21. Important measurements in trading
Calculate option price change

22. Important measurements in trading