Explain how to get to the answer and difference between european and american. A stock price is currently $$40.$ Over each of the next two three$-$month periods it is expected to go up by $10\%$ or down by $10\%.$ The risk$-$free interest rate is $12\%$ per annum with continuous compounding. Use risk$-$neutral valuation.

a$)$ What is the value of a six$-$month European put option with a strike price of $$42?$

b$)$ What is the value of a six$-$month American put option with a strike price of $$42?$

a$.$ A tree describing the behavior of the stock price is shown below. The risk$-$neutral probability of an up move, $p,$ is given by

$p=\frac{e^{0\mathrm{.}12\frac{3}{12}}-0\mathrm{.}90}{1\mathrm{.}1-0\mathrm{.}9}=0\mathrm{.}6523$

The value of the European option is $2\mathrm{.}118.$ This can be calculated by working back through the tree as shown in figure below. The second number at each node is the value of the European option.

b$.$ The value of the American option is shown as the third number at each node on the tree. It is $2\mathrm{.}537.$ This is greater than the value of the European option because it is optimal to exercise early at node C$.$a$)$ Consider a European call option on a non$-$dividend$-$paying stock where the stock price is $140,$ the strike price is$140,$ the risk$-$free rate is $5$