Problem C$1$: Option Valuation and Properties of Options

In this question, you need to price options with different valuation approaches and comment on your results. You will consider puts and calls on a share with a spot price of $$44\mathrm{.}21.$ Strike price is $$42\mathrm{.}5.$

The risk$-$free interest rate is $4\mathrm{.}34\%$ per annum with continuous compounding.

Binomial trees:

Furthermore, assume that over each of the next two three$-$month periods, the share price is expected to go up by $7\%$ or down by $7\%.$

a$.$ Draw a two$-$step binomial tree using the template provided and populate the individual nodes with the share price values at each node.

b$.$ Use the two$-$step binomial tree from a$.$ to calculate the value of a six$-$month European call option using risk$-$neutral valuation.

c$.$ Use the two$-$step binomial tree from a$.$ to calculate the value of a six$-$month European put option using risk$-$neutral valuation.

d$.$ Use the two$-$step binomial tree from a$.$ to calculate the value of a six$-$month European call option using the no$-$arbitrage approach.

e$.$ Use the two$-$step binomial tree from a$.$ to calculate the value of a six$-$month European put option using the no$-$arbitrage approach.

f$.$ Use the two$-$step binomial tree from a$.$ to calculate the value of a six$-$month American put option.