Use the binomial method to determine the value of an American Call option at time $t=0.$ The option expires at time $t=T=1$ and has exercise price $E=100.$ The current value of the underlying is $S(0)=100$ with the underlying paying continuous dividends at the rate $D=0\mathrm{.}2.$ The interest rate is $r=0\mathrm{.}05.$ Use a time step of $t=\frac{1}{3}.$ Consider the case of $p=\frac{1}{2}$ and suppose the volatility is $=0\mathrm{.}4.$ Perform all calculations using a minimum of $4$ decimal places of accuracy.

Use the binomial method to determine the value of compound Call on a Put option, where both the Call and the Put are of European style. The current time is $t=0$ and the current value of the underlying, which does not pay dividends, is $S(0)=50.$ The Put option expires at $T_2=3$ with exercise price $E_2=50,$ and the Call option expires at time $T_1=1$ with $E_1=10.$ The interest rate is $r=0\mathrm{.}1.$ Use a time step of $t=1.$ Consider the case of $ud=1$ and suppose the volatility is $=0\mathrm{.}3.$ Perform all calculations using a minimum of $4$ decimal places of accuracy.