Kindly solve this question precisely

Consider a continuous time log-normal model for a security price. S, with parameters and o. Write down formulae for: (a) the log-return of the process (b) the expected value of an investment at a specified future time (C) the variance of the value of an investment at a specified future time [3] (ii) Explain what the model implics about market efficiency. [2] (iii) Outline the empirical evidence for and against the model. [5] [Total 10] Consider two call options, which are identical (same maturity. same underlying asset] except for the strike price. Denote by C(X) the price at time 0 of the call option with strike price K. Stating the key arguments required, prove that, if there are no arbitrage cpportunities, the following relation holds true, for K, SK, . Vie[o,1] AC(Kj)+ (1-2)C(K,)=CAK, +(1-2) K2) [10] In a situation where the zero-coupon bond market is arbitrage-free and complete. consider the following Vasicek model for the short-rate process: di(t) = a(b-r(!))at + odw, where (W,; 2 0) is a standard Brownian motion with respect to the risk-neutral probability measure Q- State the general expression r() of the solution of this stochastic differential equation. [2] (ii) Derive an expression for [ r(u)du , where , and Tare given. Hint: consider the stochastic differential equation of r(w), for u 21. [6] (iii) State the distribution of ( ru)due (iv) Derive the price of a zero-coupon bond at time / with maturity 7 27 related to the distribution of [ ru)du . [6] [Total 15]