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Chapter 16
Option Valuation
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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
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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
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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
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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
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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.
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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
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
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Probabilities from Normal Dist.
N (.43) = .6664
Table 17.2
d
N(d)
.42
.6628
.43
.6664
.44
.6700
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Probabilities from Normal Dist.
N (.18) = .5714
Table 17.2
d
N(d)
.16
.5636
.18
.5714
.20
.5793
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Call Option Value
Co = Soe-dTN(d1) - Xe-rTN(d2)
Co = 100 X .6664 - 95 e- .10 X .25 X .5714
Co = 13.70
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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
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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<delta<1, for put -1<delta<0
• In Black-Schole model, hedge ratio for
call is N(d1), for put is N(d1)-1
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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
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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.
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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)
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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
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Important measurements in trading
• Calculate option price change
Stock
Option
Now(Time 0)
AAPL
Option
implied volatility0(%)
Delta
Vega
AAPL
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
5/10/2010
200
Implied volatility1
volatility change
Option Price change due to increased volatility
Total Option Price change
Option Price 1
Gain per put contract wirte
$
60
-
0
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Important measurements in trading
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
Importance in
Trading
Delta, Gamma
Theta
Very
Very
+
+
-
+
+
Vega
Rho
Very