2-period binomial tree
data to enter: rf= 1.09
u= 1.196 K= 55
d= 0.836
S0= 50
Payoff of:
price of stock: S0 S1 S2 Call Put
71.50 16.50 0.00
59.79
50.00 50.00 0.00 5.00
41.81
34.96 0.00 20.04
p= 0.706
price of call: C Cu/d Cuu/ud/dd price of put: P Pu/d Puu/ud/dd
16.50 0.00
10.68 1.35
6.92 0.00 3.21 5.00
0.00 8.65
0.00 20.04
replicating plio: D0 Du/d replicating plio: D0 Du/d
0.767 -0.23
0.594 -0.41
0 -1
L0 Lu/d L0 Lu/d
35.21 -15.3
22.79 -23.5
0 -50.5
positive L means borrowing
negative L means lending
3-period binomial tree
data to enter: rf= 1.014
u= 1.13 K= 65
d= 0.885
S0= 60
Payoff of:
price of stock: S0 S1 S2 S3 Call Put
86.64 21.64 0.00
76.65
67.82 67.82 2.82 0.00
60.00 60.00
53.08 53.08 0.00 11.92
46.96
41.55 0.00 23.45
p= 0.526
price of call: C Cu/d Cuu/ud/dd price of put: P Pu/d Puu/ud/dd
21.64 0.00
12.54 0.00
7.19 2.82 2.60 0.00
4.09 1.46 6.45 5.57
0.76 0.00 10.91 11.92
0.00 17.15
0.00 23.45
replicating plio: D0 Du/d Duu/ud/dd replicating plio: D0 Du/d Duu/ud/dd
1 0
0.665 -0.33
0.437 0.191 -0.56 -0.81
0.112 -0.89
0 -1
L0 Lu/d Luu/ud/dd L0 Lu/d Luu/ud/dd
64.11 0
37.94 -25.3
22.11 10 -40.3 -54.1
5.19 -58
0 -64.1
positive L means borrowing
negative L means lending
Black-Scholes formula
data to enter: S= 50 K= 60
yearly s= 0.3
ann. r= 0.09
years! T= 0.2493
d1= -0.992 N(d1)= 0.160 N'(d1)= 0.2438
d2= -1.142 N(d2)= 0.127
C= 0.593 Dcall= 0.1605 Lcall= 7.432 (if positive then borrow)
P= 9.261 Dput= -0.8395 Lput= -51.237 (if negative then lend)
Gcall= 0.0326
relation to binomial model: Gput= 0.0326
n= 3 Vcall= 6.0866
u= 1.090 Vput= 6.0866
d= 0.917
rf= 1.007 Qcall= -4.3308
Qput= 0.9493
rcall= 1.8529
rput= -12.774
sitive then borrow)
gative then lend)