Market Viability and Completeness for Multinomial Models

This paper characterizes the set of equivalent martingale measures in two-period finite markets as convex combinations of a finite number of measures via a new algorithm, applying these results to analyze the Korn-Kreer-Lenssen model and demonstrate the limitations of discrete-time approximations for continuous-time markets.

The Gist

Imagine you are the mayor of a small, bustling town called Financia. In this town, people trade goods, but instead of apples and oranges, they trade "future promises" (stocks and bonds). Your job is to make sure the market is fair (no one can cheat to make free money) and complete (everyone can buy insurance against any specific disaster that might happen).

This paper by Nahuel I. Arca is like a rulebook and a construction manual for building these fair markets, specifically when the future is uncertain but limited to a few specific possibilities.

Here is the breakdown of the paper using simple analogies:

1. The Setting: The "Tree of Possibilities"

In the simplest version of this town (the "Binomial Model"), every morning, the price of a stock either goes Up or Down. It's like a fork in the road.

• The Paper's Twist: The author looks at more complex towns where the road can split into three (Up, Down, or Stay the Same) or even more paths. This is called a "Multinomial Model."
• The Problem: When you have too many paths, it gets messy. How do you know if the market is fair? How do you know if you can create a product to hedge against every specific outcome?

2. The "Magic List" (Equivalent Martingale Measures)

To check if a market is fair, mathematicians look for a "Magic List" of probabilities. Let's call this the Fairness Scorecard.

• If you can find a set of probabilities where the expected future price of a stock equals its current price (adjusted for interest), the market is Fair (Arbitrage-Free). No one can rig the game.
• If you cannot find such a list, the market is broken, and a clever person could make infinite money for free (Arbitrage).

The Paper's Big Discovery:
The author realized that finding this "Magic List" isn't about guessing randomly. It's a geometry problem!

• Imagine all possible probability lists are points inside a shape (like a triangle or a pyramid).
• The "Fairness Scorecards" are just the points where a specific line (representing the stock prices) cuts through that shape.
• The Key Insight: You don't need to find every possible scorecard. You only need to find the corners (or "generators") of the shape where the line touches. Once you have those corners, you can mix them together (like mixing paints) to create any valid scorecard you need.

3. The Construction Kit (The Algorithm)

The author built a step-by-step recipe (algorithm) to find these "corners."

• Analogy: Imagine you are trying to find the exact spots where a laser beam hits a complex 3D sculpture. Instead of scanning the whole thing, the algorithm checks the edges, then the corners, and quickly identifies the exact hit points.
• Why it matters: This allows computers to instantly tell you if a market is fair and, if it's not complete, exactly what new assets you need to add to make it complete.

4. Filling the Gaps (Completing the Market)

Sometimes, the town has too many possible futures, but not enough products to cover them all.

• Analogy: Imagine you have insurance for "Rain" and "Fire," but you don't have insurance for "A meteor hitting the town." The market is incomplete.
• The Solution: The paper tells you exactly how to design that new "Meteor Insurance" policy. By looking at the "corners" of the Fairness Scorecard, you can calculate the perfect price for this new product so that the market becomes complete. Now, no matter what happens (Rain, Fire, or Meteor), everyone is covered.

5. The Trap: Discrete vs. Continuous (The "Pixelated" World)

The most fascinating part of the paper is a warning about discrete-time models (checking prices every second) vs. continuous-time models (checking prices every nanosecond).

• The Analogy: Imagine a video game.
  • Discrete: The game updates 60 times a second. It looks smooth, but if you zoom in, you see the pixels.
  • Continuous: The game is perfectly smooth, like real life.
• The Warning: The author shows that you can build a sequence of "perfectly fair" pixelated games (discrete markets) that, when you zoom out and look at the "smooth" version (continuous limit), suddenly become unfair and allow for free money.
• Real-world implication: Just because a complex mathematical model works perfectly in a computer simulation (step-by-step), it doesn't guarantee it will work in the real, continuous flow of time. You have to be careful when translating the math from "pixels" to "smooth video."

Summary

• Goal: Determine when a financial market is fair and how to make it cover every possible risk.
• Method: Use geometry to find the "corner points" of all possible fair pricing rules.
• Tool: A new algorithm to calculate these points and design new financial products to fill the gaps.
• Warning: Be careful when assuming that a market that works in a step-by-step simulation will work perfectly in the real, continuous world. Sometimes, the "smooth" version breaks the rules.

In short, this paper gives us a geometric map to navigate the complex world of financial risk, ensuring we don't get lost in the trees of probability and that we don't build a house of cards that collapses when the wind (time) blows.

Technical Summary

1. Problem Statement

The paper addresses the fundamental problems of market viability (absence of arbitrage) and market completeness in finite, discrete-time financial markets with multinomial structures (where the future branches into $b > 2$ possibilities at each node).

While the binomial model ($b=2$) is well-understood and complete, multinomial models (e.g., trinomial trees, $b=3$) are generally incomplete. The paper seeks to:

1. Characterize the set of Equivalent Martingale Measures (EMM) for these finite markets.
2. Provide a constructive algorithm to find the generators of this set.
3. Determine necessary and sufficient conditions for a market to be arbitrage-free and complete.
4. Analyze how to complete an incomplete market by adding derivative assets.
5. Investigate the convergence of discrete-time multinomial models to continuous-time limits, specifically highlighting limitations in the Korn-Kreer-Lenssen (KKL) model.

2. Methodology

The author employs a geometric approach rooted in convex geometry and linear algebra, moving away from purely probabilistic arguments.

• Geometric Formulation: The set of Martingale Measures (MM) is defined as the intersection of an affine space $A(S)$ (defined by the pricing equations of existing assets) and the standard simplex $\Delta_{b-1}$ (the set of probability vectors).
• Krein-Milman Theorem: The paper utilizes the Krein-Milman theorem (and its finite-dimensional equivalent, the Minkowski-Carathéodory theorem) to characterize the set of MM as the closed convex hull of its extreme points.
• Algorithmic Generation: A novel algorithm (Procedure 2) is proposed to compute these extreme points (generators). The algorithm iteratively checks intersections between the affine space $A(S)$ and the faces of the simplex (0-simplices, 1-simplices, etc.) to identify the vertices of the solution set.
• Market Completion Strategy: Once the generators of the EMM set are found, the paper derives a procedure to add new assets (rows to the pricing matrix) such that the new system has full rank, ensuring market completeness while preserving the no-arbitrage condition.
• Discrete-to-Continuous Analysis: The paper applies these discrete results to a discretized version of the KKL model and constructs a counter-example sequence of complete markets converging to a limit market with arbitrage.

3. Key Contributions

A. Characterization of Martingale Measures

The paper establishes that for any finite market, the set of Martingale Measures $M_a(S)$ is a compact convex set. Crucially, it proves that $M_a(S)$ is the convex hull of a finite number of generators (extreme points).

• Theorem: $M_a(S) = \text{conv}(\{p_1, \dots, p_k\})$.
• Equivalent Measures: The set of Equivalent Martingale Measures $M_e(S)$ is characterized as the subset of these convex combinations where the resulting probability vector has strictly positive components.

B. Computational Algorithm

The paper provides a step-by-step algorithm (implemented in Python in the author's repository) to find the generators:

1. Check if vertices of the simplex lie in the affine space.
2. Check intersections with edges (1-simplices) formed by valid vertices.
3. Proceed to higher-dimensional simplices until the intersection is found or the dimension limit is reached.

• Optimization: The paper proves that if a solution exists within a $k$-simplex, it is unique, allowing the algorithm to stop early if the dimension of the affine space is known.

C. Conditions for Viability and Completeness

• Arbitrage-Free Condition: A market is arbitrage-free if and only if the affine space $A(S)$ intersects the interior of the simplex (i.e., $M_e(S) \neq \emptyset$).
• Completeness Condition: A market is complete if and only if the rank of the asset price matrix equals the number of states $b$.
• Single Asset Case: For a single asset with $b$ branches:
  • The market is arbitrage-free if the current price lies strictly between the minimum and maximum possible future prices.
  • The market is complete if and only if $b=2$ (the binomial case).

D. Market Completion

The paper details how to complete an incomplete market (e.g., $b=3$) by adding derivatives.

• It derives the specific condition for a new derivative with payoffs $c_1, c_2, c_3$ to complete the market: the determinant of the augmented matrix must be non-zero.
• It characterizes the no-arbitrage price interval for the new derivative as the open interval between the expected values of the derivative under the extreme EMMs.

4. Results and Applications

Application to the Korn-Kreer-Lenssen (KKL) Model

The KKL model is a continuous-time model where an asset can jump up, jump down, or stay constant. The paper analyzes its discrete-time approximation:

• Arbitrage Condition: The discrete market is arbitrage-free if and only if the time step $\Delta t$ satisfies $\Delta t < \frac{1}{|r|(S(0) + n - 1)}$.
• Completeness: The standard put option used to complete the continuous KKL model fails to complete the discrete version because the discrete condition for rank (equation 8) is not satisfied for the specific payoff of a put option at maturity.
• Proposition 10: The paper proves that while the standard put fails, one can find a derivative with payoffs arbitrarily close to the put option that does complete the market.

The Convergence Paradox (Example 7)

The paper presents a significant counter-intuitive result regarding the limit of discrete models:

• A sequence of complete binomial markets (with specific parameters $u_n, d_n, p_n$) is constructed.
• As $n \to \infty$, these markets converge (in distribution and uniform convergence of bond prices) to a continuous-time limit market.
• Result: The limit market admits arbitrage opportunities (specifically, a portfolio $S(t) - B(t)$ is an arbitrage).
• Significance: This demonstrates that the properties of "viability" and "completeness" are not preserved under weak convergence of discrete models to continuous ones.

5. Significance

• Theoretical Rigor: The paper bridges the gap between financial economics and convex geometry, providing a rigorous geometric characterization of EMM sets in finite markets.
• Practical Utility: The proposed algorithm offers a practical tool for practitioners and researchers to compute the set of risk-neutral measures and determine the exact price bounds for derivatives in incomplete markets.
• Limitations of Discretization: The most profound contribution is the warning against blindly assuming that discrete multinomial models converge to their continuous counterparts in terms of market properties. The KKL example and the convergence paradox highlight that discrete approximations can mask or create arbitrage opportunities that do not exist (or do exist) in the continuous limit.
• Future Directions: The paper identifies the need for a general theory connecting discrete-time market properties with continuous-time limits, suggesting that current methods of taking limits in financial modeling require more careful scrutiny regarding viability and completeness.