Nassim Nicholas Taleb (Scholar Investor) – How to Price an Election (Oct 2020)

Chapters

Abstract

“Reevaluating Forecast Accuracy: The Importance of Skin in the Game and Consistent Updating in High-Volatility Scenarios”

In a critical analysis of forecast accuracy, particularly in the field of financial predictions and binary event forecasting, recent discussions have illuminated significant flaws in traditional methods. Key figures like Nate Silver have been challenged by Nassim Nicholas Taleb’s argument, which underscores the inherent uncertainty in assigning probabilities to binary events. The discourse pivots on the need for a probabilistic approach that accounts for high volatility and the principle of ‘skin in the game’. This article delves into the nuances of this debate, exploring the implications of Taleb’s perspective for the pricing of out-of-the-money options, the coherence of arbitrage rules, and the concept of bounded martingales. It further examines the practical implications for evaluating forecasts in uncertain scenarios, emphasizing the evolving nature of forecast evaluation methodologies and the critical role of consistent updating schemes.

Main Body:

1. The Problem with Election Forecasts and Binary Event Predictions:

The 2016 US election outcome, which defied many forecasts including those by Nate Silver, highlighted significant discrepancies in the prediction of binary events. This has raised concerns about the fundamental accuracy of such predictions, especially in situations characterized by high uncertainty.

2. Taleb’s Argument on Uncertainty and Probabilities:

Nassim Nicholas Taleb argues that uncertainty in binary events, such as elections, should logically lead to a probability of 0.5 rather than values close to 0 or 1. This perspective challenges the traditional approach of assigning low probabilities to seemingly unlikely outcomes.

Uncertainty and Probability:

Taleb explained that uncertainty increases the probability of outcomes when they are convex and lowers them when they are concave. In a binary event, maximum uncertainty results in a 50-50 probability for both outcomes.

Pricing and Arbitrage:

Taleb pointed out that out-of-the-money options increase in price as uncertainty increases, and vice versa. This pricing behavior is fundamental for arbitrage strategies.

3. Pricing of Out-of-the-Money Options:

The relationship between uncertainty and the pricing of out-of-the-money options is critical. Taleb points out that with increased uncertainty, the prices of these options rise due to the convexity of the payoff function, and vice versa.

4. Scanning the Game: Implications for Decision Making:

Understanding the dynamics between uncertainty and option pricing is essential for making informed decisions in complex scenarios, a concept referred to as “Scanning the Game.” This approach is crucial for identifying opportunities in volatile markets.

5. Arbitrage Rules and Coherent Pricing:

The coherence in pricing is essential to avoid incoherent outcomes that can lead to losing bets and mean reversion. Probabilities must be tradable and not too volatile or extreme.

Finetti’s Argument and the Dutch Book:

Coherent forecasting requires following arbitrage rules to avoid incoherence, which can lead to mean reversion and betting losses. Forecasts must be tradable items, and probabilities cannot be too volatile or deviate significantly from 0.5 due to uncertainty.

6. Maximum Entropy and the Role of Ignorance:

Taleb emphasizes that maximum ignorance, or uncertainty, should lead to maximum entropy, where probabilities are equally distributed (P = 0.5). This principle counters the idea of assigning highly volatile probabilities to uncertain events.

Maximum Ignorance and Maximum Entropy:

Maximum ignorance about the future leads to maximum entropy, resulting in a probability of 0.5 for any event. Volatility in probabilities requires significant information to justify changes, and certainty eliminates the pull towards 0.5.

7. Martingales and Bayesian Updating:

Martingales are processes where expected future prices equal current prices. Taleb aligns consistent Bayesian updating with martingale forecasts, suggesting that irrational updating schemes deviate from this principle.

The Arbitrage Argument and Martingale Processes:

The arbitrage argument treats every forecast as a tradable security. Martingale processes are used to price options, but creating a martingale between 0 and 1 is challenging. A pseudoprocess is constructed to translate a non-martingale forecast into a martingale process for option pricing.

Transformation and Mean Reversion:

A logistic function or error function can be used to transform a non-martingale forecast into a martingale process. To create a martingale process with mean reversion, the Norstein-Lubeck process is employed.

8. Pricing Options and Pseudoprocess:

The concept of using a pseudoprocess to translate probabilities into a martingale process is introduced for pricing options. This approach overcomes the absence of an established process for handling probabilities between 0 and 1.

9. Transformation and Mean Reversion:

Taleb uses transformation functions, like the error function, to map probabilities onto the martingale process. This ensures mean reversion, employing processes like the Norstein-Lubeck process.

10. Shadow Process and Expectation:

The shadow process concept implies that each process mirrors another, with expectations in one process of no change and in the other, a pullback due to convexity. This concept challenges the construction of a martingale between zero and one.

Constructing the Shadow Process and the Limits of Martingales:

A shadow process is created by mapping an inverse probability distribution to a price. The expectation of the shadow process is zero, while the expectation of its counterpart process exhibits pullback due to its convexity. Building a martingale between zero and one is impossible, necessitating the inclusion of drift in the process to enable option pricing.

11. Nate Silver’s Counterargument and Taleb’s Response:

Silver and Tetlock’s criticism of the idea that option prices converge to one-half with infinite volatility is countered by Taleb and Dhruv, who assert the argument’s fallacy. Their research further explores arbitrage and incorporates insights from Bruno Dupier’s exam question.

Challenges in Pricing Binary Options:

When pricing binary options, volatility tends to infinity, causing the option to approach one-half. This concept was initially met with skepticism from individuals like Nate Silver and Tetlock, who lacked a comprehensive understanding of the underlying mechanisms.

Responding to Criticisms:

A critique of the pricing method, claiming a mistake, was dismissed due to its faulty arguments. The authors continued to present additional arguments based on arbitrage and insights from Bruno Dupier, a quantitative analyst.

12. Bounded Martingale and Volatility Constraints:

Dupier’s concept of a bounded martingale, where volatility is capped, is crucial for understanding the limits of forecast variability. This concept is likened to a box with limited size, offering a tangible illustration of bounded variance.

Bruno Dupier’s Perspective:

Dupier framed the issue using the concept of bounded martingales, where volatility cannot exceed a certain point due to the constraints of the bounded box. This analogy illustrates the relationship between boundedness and volatility.

13. Statistical Concepts and Arbitrage:

Taleb distinguishes between the approaches of statisticians and quants to arbitrage, highlighting that while statisticians focus on existing numbers, quants deal with dynamic processes. The bounded martingale is seen as a deep mapping of volatility constraints.

Statistical vs. Dynamic Approaches to Arbitrage:

Statisticians typically deal with existing numbers and sampling, while quants focus on dynamic processes and their properties under arbitrage constraints. The boundedness of binary options, as prices between 0 and 1, introduces constraints on volatility.

14. Tradability of Forecasts and Real-World Implications:

Every forecast about the future is tradable, and Taleb emphasizes that forecasts must be consistent with the underlying volatility. This perspective challenges traditional expectations of forecasting.

Conditions for Martingality:

If an arithmetic Brownian motion is bounded between low and high, its variance is bounded. The formula for this boundary involves variations over time, initial and final points, and sigma squared.

Implications for Forecasting:

Every statement about the future is a tradable price, requiring consideration of the volatility of the underlying. Forecasts should not be viewed as expectations but as processes that incorporate volatility. Convexity of the forecast and its payoff introduces paradoxes and inconsistencies in pricing.

15. Convexity of Forecasts and Paradoxes:

The convexity of forecasts introduces paradoxes, particularly when low-probability events are erroneously priced at zero, disregarding the chance of unexpected occurrences.



The exploration of forecast accuracy in high-volatility scenarios sheds light on the critical importance of reevaluating traditional methods. The insights from Taleb and others highlight the need for a probabilistic approach that considers the principles of ‘skin in the game’ and consistent updating. As the field of forecast evaluation continues to evolve, these principles provide a robust framework for distinguishing reliable forecasts from unreliable ones, offering a pathway to more informed decision-making in uncertain environments.

Research Papers on Martingale Forecasts:

– Discussion of research papers on Martingale forecasts, particularly the work of Nassim Taleb and Dhruv.

– Emphasis on the importance of skin in the game and consistent updating schemes in forecasts.

– Identification of forecasts without skin in the game as unreliable.

– Acknowledgment of the growth in literature on Martingale forecasts and appreciation for the work using the quantitative finance approach.

Notes by: MythicNeutron