Incomplete Markets: Trading and Key Concepts

Market Incompleteness

Esscher Transform

The Esscher transform is a mathematical tool used to change one probability distribution into another. In finance, its primary use is to transform the real-world probability measure (P) into a risk-neutral pricing measure (Q).

It works by "tilting" the original density function using an exponential function. This new measure, Q, is chosen to ensure the discounted price process of the underlying asset becomes a martingale, which is the fundamental condition for risk-neutral pricing. It is particularly useful for pricing derivatives on assets whose log-returns are not normally distributed.

Minimal Martingale Measure

In a complete market, there is only one unique risk-neutral (martingale) measure. However, in an incomplete market, there are infinitely many. This raises the question: which one should be used for pricing?

The minimal martingale measure is a specific choice of martingale measure that is "closest" to the original real-world measure P. The "closeness" is typically defined in a way that minimizes the relative entropy or variance. It's a popular choice because it preserves certain statistical properties of the original market model and has convenient mathematical features.

f-Divergences

An f-divergence is a broad class of mathematical functions that measure the "distance" or difference between two probability distributions. It's a way to generalize the concept of distance.

Notable examples of f-divergences include:

• Kullback-Leibler (KL) Divergence: Measures the information lost when one distribution is used to approximate another.

• Hellinger Distance: Another measure of similarity between distributions.

In finance, f-divergences are used to select a specific martingale measure (Q) from the infinite set of possibilities in an incomplete market. The goal is often to find the measure Q that has the minimal f-divergence from the real-world measure P. This provides a systematic way to choose a pricing measure based on a chosen distance metric.

Portfolio Optimization in Incomplete Markets

This is the problem of finding the best investment strategy for an investor in a market where perfect hedging is impossible. Since risks cannot be completely eliminated, the goal is typically to maximize the expected utility of the investor's wealth at a future time.

Unlike in complete markets, the solution depends heavily on the investor's individual risk aversion and preferences, which are captured by their utility function. This leads directly to the concept of indifference pricing.

Indifference Pricing

Indifference pricing is a method for valuing a derivative based on an investor's utility function. The indifference price is defined as the amount of money that leaves the investor equally satisfied (i.e., with the same maximum expected utility) in two scenarios:

1. They do not trade the derivative.

2. They do trade the derivative at that price and re-optimize their portfolio accordingly.

Essentially, it answers the question: "What price for this contract would make me indifferent between having it and not having it?" This price is subjective and depends on the investor's wealth and risk tolerance. It's a powerful concept for pricing non-replicable or illiquid assets.

Good-Deal Bounds

In an incomplete market, the absence of arbitrage only provides a very wide range of possible prices for a derivative. Good-deal bounds are a way to narrow this range to a more economically meaningful interval.

The core idea is to exclude any prices that would create "too good to be true" investment opportunities, even if they aren't risk-free arbitrages. A "good deal" is typically defined as a strategy with an unusually high Sharpe ratio (return per unit of risk). By imposing a limit on the maximum allowable Sharpe ratio in the market, one can derive much tighter upper and lower bounds on the derivative's price, effectively ruling out prices that are economically unreasonable.