The final exam required us to price longevity bonds and incorporated long-run risks into pricing. It is said no paper does this very successfully so far. One of the biggest problems is that there are many uncertainties should be considered.
To simplify the analysis and make the pricing tractable, I consider only two uncertain sources: interest rates and life expectancy. Since the coupon payment will not be realized until a period of time after the holder purchases the bonds, it is no doubt we need a stochastic discount factor to calculate its price today. This is the same for the mortality rate which is expected to be declined with time. Mimicking the idea of forward rate, I create something called forward mortality rate. Then assume both of them follow Ito process.
Assigning an arbitrary life length, T, I can calculate the present value of coupon for each period starting from the day of realization to T. The price of longevity bonds is just the sum of present values.
Of course it is impossible to get the closed-form present value or price unless we know the exact form of Ito process and T.
The take-home exam is as hard as hell, but it's indeed a good practice for modeling.
To simplify the analysis and make the pricing tractable, I consider only two uncertain sources: interest rates and life expectancy. Since the coupon payment will not be realized until a period of time after the holder purchases the bonds, it is no doubt we need a stochastic discount factor to calculate its price today. This is the same for the mortality rate which is expected to be declined with time. Mimicking the idea of forward rate, I create something called forward mortality rate. Then assume both of them follow Ito process.
Assigning an arbitrary life length, T, I can calculate the present value of coupon for each period starting from the day of realization to T. The price of longevity bonds is just the sum of present values.
Of course it is impossible to get the closed-form present value or price unless we know the exact form of Ito process and T.
The take-home exam is as hard as hell, but it's indeed a good practice for modeling.
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