Analytical Option Pricing Models:

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• Analytical Option Pricing Models
• introduction and general concepts

Finance 30233, Fall 2007 The Neeley School S. Mann
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• Black-Scholes-Merton model assumptions

Asset pays no dividends European call No taxes or
transaction costs Constant interest rate over
option life Lognormal returns ln(1r ) N
(m , s) reflect limited liability -100 is
lowest possible stable return variance over
option life
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• Black-Scholes-Merton Model

C S N(d1 ) - KB(0,t) N(d2 )
ln (S/K) (r s2/2 )t
d1
s? t
d2 d1 - s? t
Note that B(0,T) present value of 1 to be
received at T define r
continuously compounded risk-free rate find r by
exp(-rT) B(0,T) so that r
-lnB(0,T)/T e.g. T 0.5 B(0,.5) 0.975
r -ln(.975)/0.5 0.02532/.5 0.05064

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Code for Manns Black-Scholes-Merton VBA
functions
Function scm_d1(S, X, t, r, sigma) scm_d1
(Log(S / X) r t) / (sigma Sqr(t)) 0.5
sigma Sqr(t) End Function Function
scm_BS_call(S, X, t, r, sigma) scm_BS_call S
Application.NormSDist(scm_d1(S, X, t, r,
sigma)) - X Exp(-r t) Application.NormSDist
(scm_d1(S, X, t, r, sigma) - sigma Sqr(t)) End
Function Function scm_BS_put(S, X, t, r, sigma)
scm_BS_put scm_BS_call(S, X, t, r, sigma) X
Exp(-r t) - S End Function
To enter code tools/macro/visual basic editor at
editor insert/module type code, then compile
by debug/compile VBAproject
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• Black-Scholes-Merton Model Delta

C S N(d1 ) - KB(0,t) N(d2 )
N( x) Standard Normal (N(0,1))
Cumulative density function N(x) area
under curve left of x N(0) .5
coding (excel) N(x) NormSdist(x)
N(d1 ) Call Delta (D) call hedge ratio
change in call value for small change in asset
value slope of call first derivative of
call with respect to asset price
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Call and Delta over time
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Call gamma (curvature)
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• Implied volatility (implied standard deviation)

annualized standard deviation of asset rate
of return, or volatility.
s
Use observed option prices to back out the
volatility implied by the price. Trial and error
method 1) choose initial volatility, e.g.
25. 2) use initial volatility to generate model
(theoretical) value 3) compare theoretical value
with observed (market) price. 4) if model
value gt market price, choose lower volatility, go
to 2) model value lt market price, choose higher
volatility, go to 2) eventually, if model value
? market price, volatility is the implied
volatility
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Discrete-time lognormal evolution
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• Historical annualized Volatility Computation

annualized standard deviation of asset rate of
return
s
1) compute daily returns 2) calculate variance
of daily returns 3) multiply daily variance by
252 to get annualized variance s 2
4) take square root to get s
or 1) compute weekly returns 2) calculate
variance 3) multiply weekly variance by 52 4)
take square root
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Call Theta Time decay