An “Ameropean” option is one which acts like an American option for the first half of the option’s life and then acts as a European option for the second half of the option’s life. Under this circumstance, we need to analyse this option in 2 parts separately, one for European option, and another for American option. According to the European option pricing rule, we calculate the option’s value for the second half backwards from time period 6 to time period 4 and then calculate it for the first half backwards from time period 3 to time period 1.
1) European Option Part
Because the option is a European option from period 4 to period 6, the value of the option at Node 11 can be calculated directly by the formula as below.
The formula:
At time period 4, n=4, j=4, P=0.7367, 1-P=0.2633, r=0.05, f(6,6)=f(6,5)=f(6,4)=0, and f(6,3)=5.1111
Then we can derive f(4,4)=0
In the same way, f(4,3)=0.35, f(4,2)=2.7, f(4,1)=7.93, and f(4,0)=13.24
2) American Option Part
The American Option Part is needed to be checked node by node to decide whether we need to early exercise it or not at each node.
At Node 7:
If exercise the option, the value=max{0, X-St}=max{0, 99-106.12}=0
If not exercise it, the value==0.09
So we do not exercise it at Node(3,3) and the value of f(3,3)=0.09
Implying the same rule to the rest of nodes in the American Option Part,
At Node 8:
If exercise the option, the value=0
If not exercise it, the value=0.97
At Node 9:
If exercise the option, the value= max{0, X-St}=max{0, 99-94}=5
If not exercise it, the value==4.06
So we exercise the option at Node(3,1) and the value f(3,1)=5
At Node 10:
If exercise the option, the value= max{0, X-St}=max{0, 99-88.47}=10.53
If not exercise it, the value==9.3
So we exercise the option at Node(3,0) and the value f(3,0)=10.53
At Node 4:
If exercise the option, the value=0
If not exercise it, the value ==0.3206
So at Node(2,2), the option will not be exercised and the value equals to 0.3206
At Node 5:
If exercise the option, the value=1.08
If not exercise it, the value==2.02
At this node, we know that f(3,2)=0.97 and f(3,1)=5 which was decided in the Node3.
The option still not be exercised and the value f(2,1)=2.02
At Node 6:
If exercise the option, the value= 6.84
If not exercise it, the value ==6.43, given f(3,1)=5 and f(3,0)=10.53
So we exercise the option and the value f(2,0)=6.84.
Node 3:
If exercise the option, the value=0
If not exercise it, the value ==0.77, given f(2,2)=0.32 and f(2,1)=2.02
So we exercise the option and the value f(1,1)=0.77
Node 2:
If not exercise the option, the value= 3
If not exercise the value==3.28, given f(2,1)=2.02 and f(2,0)=6.84
So we do not exercise the option and the value f(1,0)=3.28
At Node 1:
If not exercise the option, the value= 0
If not exercise the value==1.42, given f(1,1)=0.77 and f(1,0)=3.28
We do not exercise it at Node(0,0), therefore the six period Ameropean option's value is $1.42
Question 2
TABLE 1
PORTFOLIO
NOW
AT MATURITY
T=0
St≤4
4< St≤10
10< St≤14
14< St≤20
20< St≤27
St >27
BUY 1 BC1
-12.38
0
St -4
St -4
St -4
St -4
St -4
SELL 2 BC2
13.96
0
0
2(10- St)
2(10- St)
2(10- St)
2(10- St)
BUY 3 BC3
-9.00
0
0
0
3(St-14)
3(St-14)
3(St-14)
SELL 4 BC4
4.20
0
0
0
0
4(20-St)
4(20-St)
BUY 2 BC5
-0.32
0
0
0
0
0
2(St-27)
NET
-3.54
0
St-4
16-St
2St-26
54-2St
0
A investor decides to enter into a “Twin-Peaks” trading strategy. We put those 5 options into a portfolio and calculate the payoff in different maturity time as the table1 shown below:
Then we can draw this table into a diagram as below:
Payoff Diagram
(ii) The stock price finishes ABOVE the current stock price, which means the stock price is greater than $16, according
