What factors affect LRAS.

A bank has been selling a "home equity release" product for several years. The product is aimed at homeowners without a mortgage, usually married couples, who want to receive an enhanced pension in return for giving up some of the value of their property upon death. The terms of the product are as follows: The bank lends the homeowners a fixed sum (typically a proportion of the property value), which they are free to spend as they wish. The homeowners pay no interest directly on the loan, but interest accumulates at a fixed rate f% per annum over the rest of their lives. They live rent free in the property until the last survivor dies, when the property is sold and the loan repaid from the proceeds: - If the proceeds exceed the loan, then the difference passes to the homeowners' estate. - If the proceeds are less than the outstanding loan, the bank waives the shortfall (referred to as a no negative equity guarantee or NNEG). The bank is considering how to value the NNEGs on its balance sheet and has concluded that it should use a valuation formula based on the Garman-Kohlagen model: P = Ke-"IN(-d,) -Spe " N(-d) where d = : dy = d -ovT and I is the time of death of ONT the last survivor, assumed to be fixed. (1) Identify precisely the option that the bank is writing, and state what K, So, r, q and o represent in the formula. [4] (ii) Discuss how the values for So, q and o could be determined, and outline any assumptions that would need to be made. [5] (iii) Describe briefly any difficulties the bank might experience in trying to hedge its NNEG exposure. [3]7 Consider a non-dividend paying asset X of current value S and volatility o, in a market in which the risk-free rate is constant for all maturities. Consider also a European Call option Y based on \\" with maturity ' and an "at-the-money" strike equal to the risk-neutral forward price of X. (D) (a ) Write down the Black-Scholes formula for the price of Y. (b) Demonstrate, by using a Taylor expansion of your formula in (a), or otherwise, that the theoretical price of Y is approximately equal to SOVT [Hint: You may wish to use the fact that N'(0) = 1 (c) Using Put-Call parity, derive a similar approximation for the equivalent "at-the-money" Put option on X. [6] A dealer wishes to put together a trade where she buys 1-year "at-the-money" straddles (Calls + Puts in equal quantities with identical strikes) on X, and sells 3- month "at-the-money" straddles on X, so that her position is funding neutral. (mi) (a) Calculate the approximate hedge ratio for her trade. (b) Describe under what future conditions the strategy might be successful or, indeed, unsuccessful. (c) Sketch a rough graph of the value of the position against the price of X in three months' time, assuming no other parameters have changed. [7] During the period that the trade is in place, the risk-free rate drops sharply across all maturities, although this does not appear to affect the price of asset X. (iii) Describe any impact the rate change might have on the option strategy. [2] [Total 15]