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Financial EngineeringLecture 3
Option Valuation Methods Genentech call options have an exercise
price of $80 and expire in one year.
Case 1
Stock price falls to $60
Option value = $0
Case 2
Stock price rises to $106.67
Option value = $26.67
Option Valuation Methods If we are risk neutral, the expected return on Genentech
call options is 2.5%. Accordingly, we can determine the price of the option as follows, given equal probabilities of each outcome.
01.13$
)050(.)67.2650(.e
0rise ofy probabilit167.26rise ofy probabilit valueOption
0.1025.
rt
e
Binomial ModelThe price of an option, using the Binomial method, is significantly impacted by the time intervals selected. The Genentech example illustrates this fact.
Binomial Pricing
)()( upy Probabilit
dudap
p1downy Probabilit
yearof % as interval time
theu
ed
ea
h
h
rh
The prior example can be generalized as the binomial model and shown as follows.
ExamplePrice = 36 = .40 t = 90/365 t = 30/365 Strike = 40 r = 10%
a = 1.0083u = 1.1215d = .8917Pu = .5075Pd = .4925
Binomial Pricing
40.37
32.1036
37.401215.13610
UPUP
Binomial Pricing
40.37
32.1036
37.401215.13610
UPUP
10.328917.3610
DPDP
Binomial Pricing
50.78 = price
40.37
32.10
25.52
45.28
36
28.62
40.37
32.1036
1 tt PUP
Binomial Pricing
50.78 = price10.78 = intrinsic value
40.37.37
32.100
25.520
45.28
36
28.62
36
40.37
32.10
Binomial Pricing
50.78 = price10.78 = intrinsic value
40.37.37
32.100
25.520
45.285.60
36
28.62
40.37
32.1036
trdduu ePUPO
The greater of
Binomial Pricing
50.78 = price10.78 = intrinsic value
40.37.37
32.100
25.520
45.285.60
36.19
28.620
40.372.91
32.10.10
36
1.51
trdduu ePUPO
Binomial Pricing
Price Comparisons
Black Scholes price= 1.70
Binomial price = 1.51
Volatility Only non-observable variable Historical volatility Predictive models
◦ ARCH (Robert Engel)◦ GARCH
Weighted Average Historical Volatility Implied Volatility VIX – Exchange traded volatility option
◦ 1993◦ S&P 500 Implied Volatility
Implied Volatility is highest where time premium is highest…usually at the money
Time Decay
Option Price
Stock Price
Days to Expiration906030
Volatility Surface Term Structure of Volatilities
Time to Expiration 80% 90% 100% 110% 120%
15 0.150 0.134 0.130 0.134 0.150 30 0.148 0.139 0.135 0.139 0.148 45 0.145 0.144 0.140 0.144 0.145 60 0.151 0.149 0.145 0.149 0.151 75 0.164 0.156 0.152 0.156 0.164 90 0.170 1.604 1.600 1.604 0.170
Stock Price as % of Strike Price
Volatility Smile
Strike PriceAsset Price
Implied Volatility
Volatility Smirk
Strike PriceAsset Price
Implied Volatility
Volatility Smirk
Strike PriceAsset Price
Implied Volatility
Volatility Calculate the Annualized variance of the
daily relative price change Square root to arrive at standard deviation Standard deviation is the volatility
Volatility Develop Spreadsheet Download data from internet
http://finance.yahoo.com
Implied Volatility All variables in the option price can be
observed, other than volatility. Even the price of the option can be
observed in the secondary markets. Volatility cannot be observed, it can only be
calculated. Given the market price of the option, the
volatility can be “reverse engineered.”
Implied VolatilityUse Numa to calculate implied volatility.Example (same option)P = 41 r = 10% PRICE = 2.67EX = 40 t = 30 days / 365 v = ????
Implied volatility = 42.16%
Implied Volatility CBOE Example
Use Actual option ◦ Calculate historical volatility◦ Calculate implied volatility
http://www.math.columbia.edu/~smirnov/options13.htmlhttp://www.cboe.comhttp://www.numa.com
Expected Returns Given a normal or lognormal distribution of
returns, it is possible to calculate the probability of having an stock price above or below a target price.
Wouldn’t it be nice to know the probability of making a profit or the probability of being “in the money?”
Expected ReturnSteps for Infinite Distribution of Outcomes
tVVt
y volatilitadjusted timeConvert to :1 Step
t
Pq
VNq ln pricebelow ty Probabili:2 Step
qbelow prob -1 price abovety Probabili:3 Step
q
Expected Return
Example
Example (same option)P = 41 r = 10% v = .42EX = 40 t = 30 days / 365
1204.42. :1 Step 36530 tVVt
4187.2051.1204.ln40)( Prob:2 Step 41
40
NN
6299.3316.1204.
ln)67.24( Prob:2 Step 4167.42
NN
Expected ReturnExample (same option)P = 41 r = 10% v = .42EX = 40 t = 30 days / 365
$2.67
40 42.67
37%58%
63%
Option Pricing Project
• See handout for specs
• Walk through sample project