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(0) Derive an expression for the theta of an option under the Black-Scholes model involving delta and gamma. (4] (1i) Explain why a deep out of the money call option in the Black-Scholes world will experience a rate of return close to the risk-free rate of return [2] [Total 6] Consider a set of risky assets in a mean-variance framework where: V: = variance of the return for asse. i Cy= covariance between the returns of assets i and / where i * j Derive an expression for the variance of a portfolio of / such assets where ; is the relative weight of asset / in the portfolio. Assume that the weights sum to unity and that short selling is prohibited. [3] (ii) Show that the variance of the returns of a very large portfolio of equally weighted allocations to the assets depends mainly on the average covariance between the asset returns. [5] [Total 8] Consider a two period recombining binomial model for $,, the price of a non-dividend paying security at times r =0, 1 and 2, with real world dynamics: Sel =5, u with probability p =S, d with probability I- p and # > 1 3 0. There also exists a risk-free instrument that offers a continuously compounded rate of return of 5% per period. The state price deflator in this model after one period is: A =0.7510 when S, = Squ =1.5220 when S, = Sod The price of a derivative at time 0 that pays 1 at time 2 if Sy So using the risk-neutral probability measure derived in (ii). 121 [Total 9]An actuary has been advised to use the following positively-skewed claim size distribution as a model for a particuku type of claim, with claim sizes measured in units of fl(0. f(x: 0) =. 243 3 exp : 0