1. Black-Scholes-Merton model assumptions
Implied & Historical Volatility
Finance 30233, Fall 2009
The Neeley School
S. Mann
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(1+r ) ~ N (m , s)
reflect limited liability
-100% is lowest possible
stable return variance over option life
S.Mann, 2009

2. 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 = -ln[B(0,T)]/T
e.g.
T = 0.5
B(0,.5) =
r = -ln(.975)/0.5 = /.5 =
S.Mann, 2009

3. Code for Mann’s Black-Scholes-Merton VBA functions
Function scm_d1(S, X, t, r, sigma)
scm_d1 = (Log(S / X) + r * t) / (sigma * Sqr(t)) * 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))
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
To enter code:
tools/macro/visual basic editor
at editor:
insert/module
type code, then compile by:
debug/compile VBAproject
S.Mann, 2009

4. 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
S.Mann, 2009

5. Implied volatility (implied standard deviation)
annualized standard deviation of
asset rate of return, or volatility.
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 > market price, choose lower volatility, go to 2)
model value < market price, choose higher volatility, go to 2)
eventually,
if model value  market price,
volatility is the implied volatility
S.Mann, 2009

6. 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
S.Mann, 2009