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Assume the Black-Scholes model applies. (i) State on expression for the price of a derivative security with payoff D at maturity date 7 in terms of the risk-neutral measure. 121 An at the money European call option on a stock has an exercise date one year away and a strike price of f1 18.57. The option is priced at f10. The continuously compounded risk-free rate is 1% per annum. (ii) (a) Estimate the implied volatility to within 1% per annum, (b) Calculate the corresponding hedging portfolio in shares and cash for 1000 options on the share, quoting any results that you use. (C) Calculate the option's Vega. [10] (iii) Price a put on the some stock with the same expiry date and a strike price of Ello. (2] The hedging portfolio of the call option has the same value. the same Delta and the same Vega as the option. The Delia of the put option is -0.29975 and its Vega is 39.435. (iv) Determine the hedging portfolio of the call option in terms of shares, cash and the put option 14] [Total 18] In an extension of the Merton model, a very highly geared company has two tiers of debt, a senior debt and a junior debt. Both consist of zero coupon bonds payable in three years time. The senior debt is paid before the junior debt. Let /, be the value of the company at time . L, the nominal of the senior debt and Ly the nominal of the junior debt. (i) (a) State the value of the senior debt at maturity. (b) Deduce the value of the junior debt at maturity. The current gross value of the company is $3.2m. The nominal of the senior debt is f1.2m and that of the junior debt is f2m. The continuously compounded risk-free rate is 4% per annum, the volatility of the value of the company is 30% per annum and the price of C100 nominal of the senior bond is $88.26. (ii) Calculate the theoretical price of f100 nominal of the junior debt. [6] [Total 10]