Question

2. Consider a two-period binomial tree with u = 1.16, d = 0.96, So = 49, r = 0.07. Find the price of an American put with T = 2 and K = 50. (15 points)

Answer 1

We are having the following binomial tree

65.9344 56.84 54.5664 45.1584 4.8416

The value of the option can be calculated using the following formula

$\large f_{uu} = Max(K -S_T,0)$

When price = $ 65.9344

f = Max(50 - 65.9344,0) = 0

fud = Max(50 – 54.5664,0) = 0

$\large f_{dd} = Max(50 -45.1584,0) = 4.8416$

We have used the Pay off of Put option that is K - St because a put option is excercised when K is greater than St.

Calculate the probability of upward movement

$\large P = \frac{e^{rT}-d}{u-d}$

$\large P = \frac{e^{0.07\times 1}-0.96}{1.16-0.96} = 0.5625$

Probabilaity of downward movement

1 - P = 1 - 0.5625 = 0.4375

Here we have to determine the Value of put option node wise. As we can see from thje binomial tree only the node with a value of 47.04 lead to a positive value thus we have to calculate the value for this only.

$\large f_{d} = [Pf_{ud}+(1-P)f_{dd}]e^{-rT}$

$\large \implies f_{d} = [0.5625\times 0+0.4375\times 4.8416]e^{-0.07\times 1}$

$\large \implies f_{d} = 1.9748$

Other node that is the node with a value of 56.84 leads to a option value of 0. Hence we need not to calculate the value here.

The value of the put option for this intermediate node can be determined as

$\large f_{d} = Max(K-S_T,1.9748)$

$\large f_{d} = Max(50-47.04,1.9748) = 2.96$

Now calculating the value of the option

$\large f = [Pf_{u}+(1-P)f_{d}]e^{-rT}$

$\large \implies f_{d} = [0.5625\times 0+0.4375\times 2.96]e^{-0.07\times 1}$

$\large \implies f_{d} = \$ \ 1.20733$

+65.934 一 .84 54.5664 49 1.2073 47.00 2.96 45.1584 14.8416

Here the maturity of option is 2 years so i have divided each time step of 1 year.

Please contact if still having any query will be obliged to you for your generous support, please help. Thank you