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Consider a continuous time log-normal model for a security price. S, with parameters L and o. (i) Write down formulae for: the log-return of the process (b) the expected value of an investment at a specified future time the variance of the value of an investment at a specified future time [3] (ii) Explain what the model implies 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(A) 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 opportunities, the following relation holds true, for K, S K, . VIE [o,1] AC(Kj)+(1-2.)C(K2) = CCK|+(1-2)K2) [10] In a situation where the zero-coupon bond market is arbitrage-free and complete. consider the following Vasicck model for the short-rate process: dr(t) = a(b- r())at +odw, where (W,;7 2 0) is a standard Brownian motion with respect to the risk-neutral probability measure Q. (i) State the general expression r() of the solution of this stochastic differential equation. [2] (ii) Derive an expression for [ r(u)du , where r and Tare given. Hint: consider the stochastic differential equation of r(e), for u 21. [6] (iii) State the distribution of [ ru)du. (iv) Derive the price of a zero-coupon bond at time : with maturity I 27 related to the distribution of [ ru)du. [61 [Total 15]