please address the following questions with precision. Thanks

Consider a two-period binomial model for a non-dividend paying stock whose current price is So =100. Assume that: over each six-month period, the stock price can either move up by a factor # = 1.2 or down by a factor d = 0.8 . the continuously compounded risk-free rate is r= 5% per six-month period (1) (a) Prove that there is no arbitrage in the market. ( b ) Construct the binomial tree. [2] (ii) Calculate the price of a standard European call option written on the stock $ with strike price A =100 and maturity one year. [5] Consider a special type of call option with strike price K =100 and maturity one year. The underlying asset for this special option is the average price of the stock over one year, calculated as the average of the prices at times 0. 0.5 and 1 measured in years. (iii) Calculate the initial price of this call option assuming it can be exercised only at time 1. [5] [Total 12] Consider the following stochastic differential equation for the instantaneous risk free rate (also referred to as the short-rate): di(n) =a(b-r(!))at + odw, Its solution is given by: r(r)= exp(-ar) +b(1-exp(-ar))+ oexp(-ar ] [exp(as)aw, You may also use the fact that for T > /: ruku =D( I-1) + ( 1(0)-6 )- I-exp(-a (T-1)) =[(1-exp(-a (1 -$ )jaw. (i) Derive the price at time r of a zero-coupon bond with maturity T. [10] (1) (a) State the main drawback of such a model for the short-rate. (b) State the name and stochastic differential equation of an alternative model for the short-rate that is not subject to the drawback. 121 [Total 12]