PPlease solve the following assignment with precision. Thank you

(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 $,, the price of a non-dividend paying security at times r =0, 1 and 2, with real world dynamics: Sel =5, u with probability p =S, d with probability I- p and # > 1 3 0. There also exists a risk-free instrument that offers a continuously compounded rate of return of 5% per period. The state price deflator in this model after one period is: A =0.7510 when S, = Squ =1.5220 when S, = Sod The price of a derivative at time 0 that pays 1 at time 2 if Sy So using the risk-neutral probability measure derived in (ii). 121 [Total 9]The stem and leat plot below gives the surrender values (to the nearest (1,000) of 40 endowment policies issued in France and recently purchased by a dealer in such policies in Paris. The stem unit is (10,000 and the leaf unit is (1,000. 5779 122344 55(677899 1123444 567778 024 6 Determine the median surrender value for this batch of policies. [2] In a certain large population 45% of people have blood youup A. A random sample of 300 individuals is chosen from this population. Calculate an approximate value for the probability that more than 115 of the sample have blood group A. [3] A random sample of size 10 is taken from a normal distribution with mean pt = 20 and variance G = = 1. Find the probability that the sample variance exceeds 1, that is find P($- > 1). [3] In a one-way analysis of variance, in which samples of 10 claim amounts (f) from each of three different policy types are being compared, the following means were calculated: VI. = 276.7, y2. =254.6, 53. =296.3 with residual sum of squares SS given by 15 SSR = EXO-V. ) = 15,508.6 Calculate estimates for each of the parameters in the usual mathematical model, that is. calculate pi tot2. i,. and A. [4]