$1.$ Explain in detail the derivation of the BOPM for a call option. Include

a$.$ Derivation of the single$-$period model $($use parameters, not numbers, to explain$)$

b$.$ Include a single$-$period example where: u $=1\mathrm{.}10,$ d $=0\mathrm{.}95,$ Rf $=0\mathrm{.}05,$ S$0=$ $$100,$ X $=$ $$100.$ Show an arbitrage example when the call is either overpriced or underpriced by $$1\mathrm{.}00.$ In your example, include closing the positions at expiration.

c$.$ Explain the mechanics for pricing a call with the multiple$-$period model.

d$.$ Explain why subdividing the binomial model adds realism to the model

e$.$ Describe the methodology for estimating u and d$.$ Include verbal statement, graphical picture, and math explanation.