Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
Nonlinear Option Pricing (MATHGR5400)
Homework III
Re-insurance Deals
In this assignment, we study the Uncertain Mortality Model for the pricing of reinsurance deals. For the sake of simplicity, in this assignment we only consider one type of default: the risk of death.
We consider the following reinsurance product:
● At maturity, if the insurance subscriber is alive, the issuer delivers a put on the underlying X
● At the time of death, if it is before the maturity, the issuer delivers an exit payoff, typically another put on the underlying X.
● The subscriber pays a constant fee aAt at every time step until death or the maturity or the product
● The insurance sells a large number of these contracts to subscribers. We assume that the times of death of the subscribers 
are independent,land identically distributed, and also independent of the underlying's stock price.
● We assume that the underlying's risk neutral price dynamics is the Black-Scholes model with zero interest rate/repo/dividend yield
The Insurers' Approach and Risk-Neutral Pricing
This contract shows two types of risk: the times of death of the subscribers and the changes in the price of the undertying.
In this case, the issuer can apply the insurer's approach to the risk of death times, i.e., the law of large numbers. The more people buy the contract, the less risk.
Choosing a risk-neutral measure under which the death times D have the same distribution as under the historical probability measure is equivalent to applying the arbitrage-pricing approach to the financial risk insurer's rule on the risk of death. The price of the contract is then
Deterministic Death Rate
If the death intensity is a deterministic function 
(i.e. 
has an exponential distribution with time-dependent intensity ), then we have seen that u is the solution to the linear PDE