Assume that the spot price of a risky stock is $72,$ and its volatility is $35\%$ p$.$a$.$

The risk free rate of interest is $5\%$ p$.$a$.$ continuously compounded. A $1$ year

American put option on the risky stock struck at $75$ is also traded.

Use the binomial model to price the given American put option, by assuming

that the tree parameters are u $=$eVat and d $=$ ul$,$ where o is the stock

volatility, At$=$T$/$N$,$ T is the maturity of the option and N is the number of

steps forming the binomial tree.

$1.$ Calculate the price of the contract for a $3$ step tree using risk neutral

valuation. Provide a justification for the methodology you adopt.

$2.$ Calculate the price of an American call option on the same underlying,

with the same strike and maturity as the American put option considered

above. Provide a justification for the methodology you adopt.

$3.$ For the case of a non$-$dividend paying underlying security, develop a

routine which caleulates the price of both American call and put options

using the binomial tree deseribed above for N$=1,2,..,100.$ You can use

Matlab, Python or Excel. Discuss and justify the results you obtain.

$4.$ Compare the results you obtain from your code with the Black$-$Scholes

prices of the European counterpart contracts. Discuss and justify the

results you obtain by increasing further the umber of steps in the tree.