Actuarial Finance

13Market incompleteness and one-period trinomial tree models

Market incompleteness is everywhere in actuarial science, as typical insurable risks embedded in life insurance or homeowner’s insurance cannot be (exactly) replicated using tradable assets. Public policy and tight regulations in the insurance industry prevent insurers from trading their insurance policies to exactly offset a liability. It may also be impossible to (exactly) replicate certain financial risks. For example, prior to the creation of credit default swaps (see Chapter 4), it was nearly impossible to hedge against losses resulting from credit risk.

The risk management implications of an incomplete market are rather important. It basically means that some risks cannot be synthetically replicated. We illustrate this in the next example.

Example 13.0.1 Impact of default risk on replicating portfolios¹

Suppose that on the financial market, a share of ABC inc. currently trades for $40 and it is assumed that, in a year, its value will be either $50 or $30. You also know that $100 invested at the risk-free rate will accumulate to $104 in a year. Your company sells a 1-year at-the-money investment guarantee on ABC inc.; this is marketed as an investment that provides the upside on the stock return with a protection against the downside. We know from Chapter 6 that the investment guarantee can be replicated by combining the stock and a put option on this stock. As the risk manager, you decide to replicate this ...