Transcript Volatility Adjustment (Full) and Weighted Volatility

VOLATILITY
ADJUSTMENT (FULL)
AND
WEIGHTED VOLATILITY
ADJUSTMENT
FOR THE BLACK
SCHOLES MODEL
Michael
Aussieker
Oliver
Grace
Fredrik
Jonsӓll
MODEL INPUTS
 Full Volatility
Adjustment
Implemented in the
Black Scholes




BSM (S 0 - D d , X-D 0 , T,  Adj , r)
 Adj=  S 0 / (S 0 - D d )
D d =D 0 e -rTd
Td=Ex Dividend Date
 Assuming same as pay
date
 Weighted Volatility
Adjustment
Implemented in the
Black Scholes





BSM (S 0 - D d , X-D 0 , T,  Adj , r)
 Adj=  S 0 / (S 0 -KD d )
D d =D 0 e -rTd
K=Td/T
Td=Ex Dividend Date
 Assuming same as pay
date
DIVIDEND MODELS AT MATURIT Y
Method
Calls
Puts
Full Volatility
13.26971
10.31426
Weighted Volatility
13.26971
10.31426
Standard BSM
13.28331
10.32786
S=100, X=100, T=1, Td=1, R=0.03, σ=0.3, D0=5,
 These results are consistent across different variables as long as
the dividend date is the same as the terminal date or very near.
COX ROSS RUBINSTEIN
 Using Standard CRR model for a European call with constant
volatility, where the dividend amount is known in advance, in
which the results prior to the dividend dose not recombine.
 CRR has been calculated with n=500, both with and without a
single discrete dividend. Where the dividend is paid at first,
second, and, third quarter respectively.
 We used this as a standard discounted base case for
comparison a rough comparison.
DIFFERENCE WITH BSM AT $2 DIV
0.25
0.2
FV.25
0.15
FV.5
FV.75
WV.25
0.1
WV.5
WV.75
CRR.25
CRR.5
0.05
CRR.75
0
70
-0.05
80
90
100
Strike
110
120
DIFFERENCE WITH BSM AT $5 DIV
0.6
0.5
0.4
FV.25
FV.5
FV.75
0.3
WV.25
WV.5
WV.75
0.2
CRR.25
CRR.5
0.1
CRR.75
0
70
-0.1
80
90
100
Strike
110
120
DIFFERENCE WITH BSM AT $10 DIV
1.2
1
0.8
FV.25
FV.5
FV.75
0.6
WV.25
WV.5
WV.75
0.4
CRR.25
CRR.5
0.2
CRR.75
0
70
-0.2
80
90
100
Strike
110
120
CONCLUSIONS
 A Full volatility adjustment better approximates the BS than the
Weighted volatility adjustment or CRR.
 For the Black Scholes adjusted models the difference is a
(negative slope) linear function of dividend time. This
dependence is greater for the Weighted model, than the Full
volatility implementation.
 The larger the dividend, the greater the impact on pricing.
 Out of money options, the dividend has greater impact on pricing
differences.
 In the money options, results in less impact for the Full volatility
adjusted model.
 Weighted model is not a viable option when accounting for
dividends.
 The Full volatility model is feasible when the underlying pays
small dividends and the option is at or in the money.
 Pricing improves for all models as the value of the underlying
falls
RAW DATA
Strike
70
80
90
100
110
120
Difference between BSM and Full Volatility Adjustment
Ex @.25
Ex @.5
Ex @.75
0.058316573
0.045220151
0.03221951
0.05505174
0.04368235
0.032394966
0.043237086
0.03396502
0.024758892
0.026509337
0.019373368
0.012287487
0.009744562
0.004523882
-0.000660566
-0.003620258
-0.007278583
-0.010911758
Difference between BSM and Weighted Volatility Adjustment
Ex @.25
Ex @.5
Ex @.75
0.120523292
0.08671919
0.052981061
0.163387609
0.11580473
0.068403046
0.192318966
0.13311066
0.074208246
0.200021157
0.134732407
0.069806858
0.188393566
0.123328183
0.05859128
0.164202311
0.104398851
0.044821488
70
80
90
100
110
120
Difference between BSM and Full Volatility Adjustment
Ex @.25
Ex @.5
Ex @.75
0.147234416
0.114458456
0.081914368
0.139264541
0.110854794
0.082638734
0.109203267
0.086086952
0.063122299
0.066506443
0.048758108
0.031122094
0.023751134
0.010792918
-0.002085857
-0.010274585
-0.019341811
-0.02835493
Difference between BSM and Weighted Volatility Adjustment
Ex @.25
Ex @.5
Ex @.75
0.29846462
0.216635158
0.133685372
0.408372833
0.291696468
0.173778036
0.483184732
0.336732167
0.189106064
0.503351522
0.341316125
0.178063276
0.47369031
0.312315501
0.149453617
0.41186345
0.264023946
0.114293596
70
80
90
100
110
120
Difference between BSM and Full Volatility Adjustment
Ex @.25
Ex @.5
Ex @.75
0.299653839
0.23396319
0.168710808
0.284478721
0.227696791
0.171263492
0.22246822
0.176456468
0.130703005
0.133894529
0.098720449
0.063727279
0.045331228
0.019745967
-0.005717854
-0.02494342
-0.042797806
-0.060573242
Difference between BSM and Weighted Volatility Adjustment
Ex @.25
Ex @.5
Ex @.75
0.5863296
0.432042487
0.271394244
0.815680549
0.590540211
0.357198801
0.974263147
0.687067632
0.390876903
1.0179198
0.698196821
0.368694267
0.956471604
0.638457216
0.309448917
0.828013437
0.538431467
0.236514517
R CODE
 Black Scholes function in R
ADJBlackscholes<-function(S, X, rf, T,td, sigma,D0) {
values <- matrix(3,2)
Dd<-(D0*exp(-rf*td))
#discounted dividend from Ex date#
Sd<-(S-Dd)
#Stock adjusted for present value of dividends#
Xd1=(X-D0)
#Strike adjusted after dividend paid#
K=td/T
#dividend payment to terminal date impact#
SigA=(sigma*S)/(Sd)
#adjusted volatility full#
WSig=(sigma*S)/(S-(Dd*K)) #volatility adjustment impact of payment date#
d1
d2
d3
d4
d5
d6
<<<<<<-
(log(Sd/Xd1)+(rf*T))/(SigA*sqrt(T)) +((SigA*sqrt(T))/2)
d1 - (SigA * sqrt(T))
(log(Sd/Xd1)+(rf*T))/(WSig*sqrt(T)) +((WSig*sqrt(T))/2)
d3 - (WSig * sqrt(T))
(log(S/X)+(rf*T))/(sigma*sqrt(T)) +((sigma*sqrt(T))/2)
d5 - (sigma * sqrt(T))
values[1]
values[4]
values[2]
values[5]
values[3]
values[6]
<<<<<<-
(Sd)*pnorm(d1) - (Xd1)*exp(-rf*T)*pnorm(d2)
(Xd1)*exp(-rf*T)*pnorm(-d2)-(Sd)*pnorm(-d1)
(Sd)*pnorm(d3) - (Xd1)*exp(-rf*T)*pnorm(d4)
(Xd1)*exp(-rf*T)*pnorm(-d4)-(Sd)*pnorm(-d3)
(S)*pnorm(d5) - (X)*exp(-rf*T)*pnorm(d6)
(X)*exp(-rf*T)*pnorm(-d6)-(S)*pnorm(-d5)
print("Calls COL=1, Puts COL=2, Full Vol ADJ ROW=1, Weighted Vol ROW=2, Std BSM ROW=3")
values
}
CRR CODE IN PY THON
CRR.py
''‘
The code is inspired from:
Binomial Tree for America and European options by
Mehdi Bounouar
@author: Oliver Grace
@group team members: Michael Aussieker, Fredrik
Jonsall
'''
import numpy as np
def BinomialTreeCRR(n, Spot, k, r, v, T, D, td):
"""
n: steps
Spot: Spot price
K: Strike price
r: Risk free rate
v: volatility
T: Maturity
td: Dividend payment date
"""
D0 = D*np.exp(r*td)
true_sigma = v*(Spot/(Spot - D0))
dt = T/n
u = np.exp(true_sigma*np.sqrt(dt))
d = 1./u
p = (np.exp(r*dt)-d)/(u-d)
#Binomial price tree
stkval = np.zeros((n+1,n+1))
stkval[0,0] = Spot - D0
for i in range(1,n+1):
stkval[i,0] = stkval[i-1,0]*u
for j in range(1,i+1):
stkval[i,j] = stkval[i-1,j-1]*d
#option value at each final node
optval = np.zeros((n+1,n+1))
for j in xrange(n+1):
optval[n,j] = max(0, stkval[n,j]-k+D)
#backward recursion for option price
for i in xrange(n-1,-1,-1):
for j in xrange(i+1):
optval[i,j] = max(stkval[i,j]-k+D, np.exp(r*dt)*(p*optval[i+1,j]+(1-p)*optval[i+1,j+1]))
return optval[0,0]
if __name__ == "__main__":
Spot = 100.
# Spot Price
k = 100.
# Strike Price
r = .03
# Annual Risk-free rate
v = .3
# Annual Volatility
T = 1.0
# Time in year (days/365)
n = 500
# Number of steps
D = 10.0
# Dividend per share
td = .5
# Time to Dividend Payment
print "European Call on Dividend Paying Stock:
%s"%(BinomialTreeCRR(n, Spot, k, r, v, T, D, td))