1. From Binomial to Black-Scholes-Merton
Option Sensitivities (Delta, Gamma, Vega , etc)
Implied Volatility
Finance 30233, Fall 2016
The Neeley School
S. Mann
S.Mann, 2010

2. Historical Volatility Computation
annualized standard deviation of asset rate of return
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
take square root
Or:
1) Compute monthly returns
3) Multiply monthly variance by 12
4) take square root
S.Mann, 2010

3. Calibrating Binomial Tree:
1) Set “up” and “down” factors to reflect volatility:
U = 1 + s√T
D = 1 - s√T
Adjust so expected future price is forward price:
E[S(T)] = S0 ( 1 + (r – d)T )
Simple form:
U = 1 + (r – d)T + s√T
D = 1 + (r – d)T - s√T
Or, in continuous time (McDonald):
U = exp((r – d)T + s√T)
D = exp((r – d)T - s√T)
See “Fin30233-F2016_Binomial Tutor.xls” (available on Mann’s course website)

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7. 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, 2010

8. Simulated lognormal returns
Lognormal simulation: visual basic subroutine : logreturns

9. Simulated lognormally distributed asset prices
Lognormal price simulation: visual basic subroutine : lognormalprice)
S. Mann, 2000.

10. Black-Scholes-Merton Model
C = S N(d1 ) - KZ(0,T) N(d2 )
ln (S/K) + (r + s2/2 )T
d1 =
s T
d2 = d1 - s T
N( x) = Standard Normal [~N(0,1)] Cumulative density function:
N(x) = area under curve left of x; e.g., 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, 2010

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16. 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, 2010

17. VBA Code for Black-Scholes-Merton functions (no dividends)
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
VBA code for implied volatility:
Function scm_BS_call_ISD(S, X, t, r, C)
high = 1
low = 0
Do While (high - low) >
If scm_BS_call(S, X, t, r, (high + low) / 2) > C Then
high = (high + low) / 2
Else: low = (high + low) / 2
End If
Loop
scm_BS_call_ISD = (high + low) / 2
End Function
To enter code:
tools/macro/visual basic editor
at editor:
insert/module
type code, then compile by:
debug/compile VBAproject
S.Mann, 2010

18. Call Theta:
Time decay
S.Mann, 2010