Market incompleteness refers to the situation where there are not enough tradable assets to replicate all possible payoffs. In other words, some payoffs cannot be created by combining existing assets (or their derivatives) in a portfolio. This poses challenges for derivative pricing and risk management because traditional methods based on no-arbitrage arguments may not be applicable.
One way to address market incompleteness is by using model-based pricing. This involves specifying a stochastic model for the underlying asset(s) and deriving the prices of derivatives as expected values under a risk-neutral measure. The risk-neutral measure allows for different probabilities of future events than the actual (physical) probabilities, in order to account for the possibility of arbitrage opportunities.
However, model-based pricing also requires making assumptions about the dynamics of the underlying asset(s) and the behavior of market participants, which can be challenging in the presence of market incompleteness. For example, the model may not capture all relevant risk factors or may underestimate the impact of large market moves.
Another approach is to use a combination of model-based pricing and hedging. This involves identifying a set of tradable assets that can be used to replicate some (but not all) of the payoffs of the derivative, and then adjusting the hedging strategy based on the model’s predictions of market movements. This can help reduce the risk of the derivative position, but may also result in higher costs due to the need to trade in multiple markets.
A third approach is to incorporate some degree of subjective judgment or market views into the pricing and risk management process. This recognizes that market incompleteness may reflect differences in beliefs or preferences among investors, and that these differences can be exploited through active trading strategies. For example, an investor may use a combination of derivatives and underlying assets to express a particular view on the direction of a market, and then adjust the position as new information becomes available.
In summary, dealing with market incompleteness in derivative pricing and risk management requires a combination of quantitative modeling, hedging, and subjective judgment. The optimal approach will depend on the specific market conditions, the types of derivatives involved, and the investor’s risk tolerance and investment objectives.