The DerivaGem Application Builder functions enable you to investigate how the prices of options calculated from a binomial tree converge to the correct value as the number of time steps increases. (See Figure 21.4 and Sample Application A in DerivaGem.) Consider a put option on a stock index where the index level is 900 , the strike price is 900 , the risk-free rate is $5 \%$, the dividend yield is $2 \%$, and the time to maturity is 2 years.
(a) Produce results similar to Sample Application A on convergence for the situation where the option is European and the volatility of the index is $20 \%$.
(b) Produce results similar to Sample Application A on convergence for the situation where the option is American and the volatility of the index is $20 \%$.
(c) Produce a chart showing the pricing of the American option when the volatility is $20 \%$ as a function of the number of time steps when the control variate technique is used.
(d) Suppose that the price of the American option in the market is 85.0. Produce a chart showing the implied volatility estimate as a function of the number of time steps.