Binomial Tree for Stock Price & American Option Pricing

The assignment requires the construction of a binomial tree to price an American option with a payoff function [max(S^2 - 100^2, 0)]^0.5 expiring in 1 year, given initial stock price $100, volatility range (20%-40%), and riskfree rate range (1%-10%). Additionally, the assignment involves calculating the delta of the option as a function of the stock price, plotting the cash in the replicating portfolio as a function of the stock price, and constructing a binomial tree for a convertible bond.

1. Construct a binomial tree for the price of a stock where the initial price is equal to 100, the
volatility  = 30% per year, the length of the period is one day (h=1/252), the riskfree rate r = 1%
per year, compounded continuously, u = exp[rh +  (h^0.5)], d = exp[rh -  (h^0.5)]. The stock
does not pay dividends.
a. Use this framework to calculate the price (at the beginning of the tree) of an American
option whose payoff is equal to [max(S^2 – 100^2, 0)]^0.5 and expiring in 1 year.
b. Repeat part a. with volatilities ranging from 20% to 40%, in increments of 5%, and plot
the option price as a function of volatility.
c. Repeat part a. for a range of riskfree rates from 1% to 10%, in increments of 1%, and
plot the option price as a function of the riskfree rate.
d. Plot the Delta of the option as a function of the stock price on the path of the tree
where the stock price goes only up.
e. Plot the cash in the replicating portfolio as function of the stock price on the path in the
binomial tree on which the stock price only goes up.
Make your excel spreadsheet very clear.
2. Construct a binomial tree with three periods, such that the riskfree rate is zero, u=1.1, d=0.9 and
the initial stock price is $100. Calculate the price of the following convertible bond: you have the
choice between holding your security as a bond tha pays you $200 at maturity or converting
(irreversibly) that security in two shares of stock.