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Unformatted text preview: 132.69
100
75.36 0 0 10.72 0
24.64
1517 Details in the code *
In the program “binomial_euro”, we initialise asset prices at
In
maturity step n
maturity
S(0) = Stock price of all ups
S(0)
= 100*(1.1519^2) = 132.69
100*(1.1519^2)
For I = 1 To n (n=2),
For
S(1) = (S(1  1) / u) * d = (132.69/1.1519)*0.8681 = 100
S(1)
S(2) = (S(2  1) / u) * d = (100/1.1519)*0.8681= 75.36
Now, we can easily compute the value of the option at the
maturity (n)
= MAX( OptionType*(S(I)  K), 0 )
We now have the all option prices in step 2:
price(0)=0, price(1)=0, price(2)=24.64
price(0)=0,
1518 Details in the code ** (1)
In the program “binomial_euro”, we step back the tree
For j = n  1 To 0 Step 1
For
For I = 0 To j
For
price(I) = pv * (Pu * price(I) + Pd * price(I + 1))
price(I)
Next I
Next
Next j
For j = n – 1 = 1
For I = 0 to 1, we will have the following option price in j=1:
For
price(0) = pv*(Pu*price(0)+Pd*price(1))
= 0.9753*(0.5539*0 + (10.5539)*0) = 0
(Note that price(0) was step 2 option price but is now
replaced by the updated value!! price(0) is now the option
1519
price in step 1)
price Details in the code ** (2)
price(1) = pv*(Pu*price(1)+Pd*price(2))
= 0.9753*(0.5539*0 + (10.5539)*24.64) = 10.72
(Note that price(1) is now replaced by the option price
in step 1)
in
For j = n – 2 = 0
For
For I = 0 to 0, we will have the following option price in j=0:
For
price(0) = pv*(Pu*price(0)+Pd*price(1))
= 0.9753*(0.5539*0 + (10.5539)*10.72) = 4.66
(Note that price(0) is now replaced by the updated
value!! So, price(0) was step 1 option price, but is now
step 0 option price)
step
At the end, price(0) gives us the option price for today (step
At
0)
0)
1520 Cox, Ross, Rubinstein’s (CRR)
binomial trees – American options 1521 Using binomial tress to price American
options (1)
In the final step, the payoffs of American
In
options are the same as the payoffs of
same
European options
European
So, pricing American options is similar to what
So,
we have done for European options as we all
start from the final step’s payoffs
final
In any step before the final step, the riskneutral
In
investor makes the decision between (a)
exercise the option now and (b) wait for the
next step
next
So, investor compare (a) the payoff from
So,
exercise and (b) the expected payoff for waiting
exercise
1522 Using binomial tress to price American
options (2)
Check the “option” tab for American options
Take the bottom triangle of step 8 (J57) as the
Take
example, we find the following
example, p=0.5177 $14.05 1p= 0.4823 $17.29 $?
As shown earlier, the claim in this node is
As
15.54, i.e., if the investor choose not to
exercise the option now, the expected payoff is
15.54 (present value)
15.54
1523 Using...
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 Spring '09

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