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Chapter 16
Option Valuation
Fourth Edition
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Outline
• Valuation – –
Intrinsic and time values Factors determining option price
– Black-Scholes Model • How valuation helps trading (optional) – Hedge ratio (Delta) and option elasticity – Other variables
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1. VALUATION
Fourth Edition
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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
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Time Value of Options: Call
Option value Value of Call Intrinsic Value Time value X Stock Price
Fourth Edition
6 Factors Influencing Option Values: Calls If this variable increases Value of a call option Stock price increases Exercise price Volatility of stock price decreases increases Time to expiration Interest rate increases 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
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Factors Influencing Option Values: Puts
If this variable increases Value of a Put option Stock price decreases Exercise price Volatility of stock price increases increases Time to expiration Interest rate increases 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
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Black-Scholes Option Valuation
C
o
d d
1 2
= S
o
N(d
1
) - Xe -
rT
N(d
2
) = [ln(S
o
/X) + (r – d + s
2
/2)T] / ( s T
1/2
) = d 1 - ( s 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 dist. will be less than 1.
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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
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Call Option Example
S o = 100 X = 95 r = .10
T = .25 (quarter) s = .50
d 1 d = 0 = [ln(100/95)+(.10-0+( .
5 2 /2))]/( .
5 .25
1/2 ) d 2 = .43
= .43 - (( .
5 )( .25
1/2 ) = .18
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Probabilities from Normal Dist.
N (.43) = .6664
Table 17.2
d N(d) .42
.43
.44
.6628
.6664
.6700
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Probabilities from Normal Dist.
N (.18) = .5714
Table 17.2
d N(d) .16
.18
.20
.5636
.5714
.5793
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Call Option Value
C
o
= S
o
e d
T
N(d
1
) - Xe -
rT
N(d
2
) C o = 100 X .6664 - 95 e - .10 X .25
X .5714 C o = 13.70
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Put Option Value: Black-Scholes
P=Xe -
rT
[1-N(d
2
)] – S
0
[1-N(d
1
)] Using the sample data P = $95e
(-.10X.25)
(1-.5714) - $100 (1-.6664) P = $6.35
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2.HOW VALUATION HELPS TRADING
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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 1 ), for put is N(d 1 )-1 Fourth Edition 17 • 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 Fourth Edition 18 Fourth Edition • 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 Fourth Edition • 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 • 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 Fourth Edition 21 • Calculate option price change Stock AAPL Option Now(Time 0) AAPL Option implied volatility0(%) Delta Vega 2012 Jan $200 Put 5/7/10 $ 235.86 $ 34.55 46 -0.2732 1.0232 Next trading day(Time 1) Stock price stock price change Option price change due to stock price Implied volatility1 volatility change Option Price change due to increased volatility 5/10/2010 $ 200 60 Total Option Price change Option Price 1 Gain per put contract wirte 0 Fourth Edition 22 Fourth Edition Variables Exercise Price Stock Price Time to Maturity Volatility Risk Free Rate Dividend Yield + + - Relationship with Call Option Value - + + Relationship with put Option Value + - + + - + Sensitivity variables Delta, Gamma Theta Vega Rho Importance in Trading Very Very Very How to use hedge ratio in trading
Important measurements in trading
Important measurements in trading
Important measurements in trading
Important measurements in trading
Important measurements in trading