A trader is considering purchasing several ArrowDebreu securities which all have the same maturity time, but provide a payoff of $1 at all possible states of the maturity time. The trader thinks that this is a very clever investment plan, and is possibly an arbitrage opportunity because it ensures there is always a payoff of $1. Here we investigate the traders investment plan using a two-step binomial pricing model. Assume that the return on an investment over one time-step is constant R and the risk-neutral probability of the upstate is .

(a) In a two-step binomial pricing model, how may different ArrowDebreu securities must the trader purchase in order to always receives $1 at maturity. Explain your answer. (b) In a two-step model, derive a formula for the total premium of all the ArrowDebreu securities (that is, derive formulas for the premiums of each ArrowDebreu security and then add them together). (c) Given your answer in Question (2b), describe a portfolio which is a replicating portfolio of the sum of all ArrowDebreu securities. Confirm that this portfolio is a replicating portfolio by showing that at each node it equals the sum of all ArrowDebreu securities. (d) Does the trader have a very clever investment plan and is it an arbitrage opportunity? Explain why or why not. In your answer, discuss whether or not your conclusions would change if we considered a general N-step binomial pricing model.