1. From Binomial to Black-Scholes-Merton
Finance 30233, Fall 2017
The Neeley School
S. Mann
S.Mann, 2017

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, 2017

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-F2017_Binomial Tutor.xls” (available on Mann’s course website)

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6. VBA code for binomial European Call (for later reference)
Function Binom(n, X)
Binom = Application.Combin(n, X) * 0.5 ^ X * 0.5 ^ (n - X)
End Function
Function scm_bin_eur_call(S, X, rf, sigma, t, n)
dt = t / n
up = Exp((rf - (sigma ^ 2) / 2) * dt + sigma * Sqr(dt))
down = Exp((rf - (sigma ^ 2) / 2) * dt - sigma * Sqr(dt))
r = Exp(rf * dt)
p = (r - down) / (up - down)
q = 1 - p
scm_bin_eur_call = 0
For Index = 0 To n
scm_bin_eur_call = scm_bin_eur_call + Exp(-rf * t) *
Application.Combin(n, Index) * p ^ Index * q ^ (n - Index) *
Application.Max((S * up ^ Index) * (down ^ (n - Index)) - X, 0)
Next Index
Note: code in red has to be on one line for vba to compile the code

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8. 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, 2017

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

10. Simulated lognormally distributed asset prices
Lognormal price simulation: visual basic subroutine : lognormalprice)
S.Mann, 2017

11. Black-Scholes-Merton Model (no dividends)
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, 2017

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