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A binomial lattice is used to model the price of a non-dividend-paying share up to time 7. The interval (0,7) is subdivided into a large number of intervals of length St = T. It is assumed that, at each node in the lattice, the share price is equally likely to increase by a factor # or decrease by a factor d, where u ="ortoo and d= 4di-0.6: The movements at each step are assumed to be independent. (1) Show that, if the share price makes a total of X, "up jumps", the share price at time 7 will be: Sy = Soexp HT toVT 24, -1 Vn where So denotes the initial share price. [4] (ii) Write down the distribution of X,, and state how this distribution can be approximated when a is large. [2] (iii) Hence determine the asymptotic distribution of ST for large n . [4] So [Total 10](i) Sketch rough graphs on the same diagram showing the price of a call option on a non-interest bearing stock as a function of the volatility parameter o, in the cases where the option is: (a) deep in-the-money (b) at-the-money (c) deep out-of-the-money. [4] (ii) The following table relates to the prices for 6-month call options on Grade A Copper on a particular day: Strike price Option Price Implied Volatility 1,350 ? 0.255 1,400 102.5 0.260 1,450 85.2 ? The current spot price is 1,370 and the risk-free interest rate is 0.05 (expressed as a continuously-compounded annual rate). Assuming that the Black-Scholes approach is valid here, complete the table by calculating the option price for a contract with a strike price of 1,350 and the implied volatility for a contract with a strike price of 1,450. [8] [Total 12]