r-bottom: solid #7ba0cd 0.5pt; border-right: solid #7ba0cd 0.5pt;">
97.47
95.00
95.00
92.60
92.60
90.25
87.97
0.01644
0.0329
0.0493
0.0658
0.0822
Step 3: Starting with the final node we work backwards to determine the option price or value at time 0. The option prices at the final nodes of the tree are the intrinsic values of the option.
At earlier nodes we will first need to calculate a value assuming that the option is held for a further time step. We compare this value with the value if the option is exercised early. If the former value exceeds the latter the earlier exercise is not optimal. If the opposite is true the early exercise value will be selected for further stages in the calculation.
This is illustrated for our example below:
B 13.68 | |||||
A | 113.68 | ||||
10.80 | |||||
D | 110.80 | C | |||
8.00 | 10.82 | 8.00 | |||
108.00 | 108.00 | ||||
5.26 | 8.04 | 5.26 | |||
105.26 | 105.26 | ||||
2.60 | 5.65 | 2.60 | 5.29 | 2.60 | |
102.60 | 102.60 | 102.60 | |||
0.00 | 3.80 | 0.00 | 3.29 | 0.00 | |
100.00 | 100.00 | 100.00 | |||
2.47 | 0.00 | 1.96 | 0.00 | 1.30 | 0.00 |
97.47 | 97.47 | 97.47 | |||
1.14 | 0.00 | 0.65 | 0.00 | ||
95.00 | 95.00 | ||||
0.32 | 0.00 | 0.00 | 0.00 | ||
92.60 | 92.60 | ||||
0.00 | 0.00 | ||||
90.25 | |||||
0.00 | 0.00 | ||||
87.97 | |||||
Node Time: 0 | 0.01644 | 0.0329 | 0.0493 | 0.0658 | 0.0822 |
The numbers above the prices (given in the boxes) are the value of the call option, i.e. max [Price –Exercise Price, 0] let’s call this O_{1}.
For nodes 0 to 4 the values at the bottom represent the value of holding the option for a further time step, lets call this O_{2}.
O_{2 }at A is (13.68*p +8*(1-p))/0.10 .The option value at the particular nodes will be the maximum of O_{1 }and O_{2}.
These are the values outside the boxes that are highlighted in the diagram above.
At A this is 10.82. O_{2} at D is (10.82*p +5.29*(1-p))/0.10 and so on. Following this process the price of the call option is 2.47.