Consider a stock St and a plain vanilla, at-the-money, put option written on this stock. The option expires at time t + , where

Consider a stock St and a plain vanilla, at-the-money, put option written on this stock. The option expires at time t + Δ, where Δ denotes a small interval. At time t, there are only two possible ways the St can move. It can either go up to Sut+Δ, of go down to Sut+Δ. Also available to traders is risk-free borrowing and lending at annual rate r.
(a) Using the arbitrage theorem, write down a three-equation system with two states that gives the arbitrage-free values of Sr and Ct.
(b) Now plot a two-step binomial tree for St. suppose at every node of the tree the markets are arbitrage-free. How many three-equation systems similar to the preceding case could then be written for entire tree?
(c) Can you find a three-equation system with 4 states that corresponds to the same tree?
(d) How do we know that all the implied state prices are internally consistent?