The Price of Uncertainty: How Bubbles Form in Restricted Markets

Author: Denis Avetisyan

New research reveals how limitations on short selling, combined with uncertainty about market models, can drive the formation of asset price bubbles.

The emergence of bubbles hinges on a delicate balance - a sufficient excess of buoyant force overcoming the resisting forces of surface tension and gravity, a condition meticulously defined by <span class="katex-eq" data-katex-display="false"> \Delta P = \sigma \frac{1}{r} + \rho g h </span>. — The emergence of bubbles hinges on a delicate balance – a sufficient excess of buoyant force overcoming the resisting forces of surface tension and gravity, a condition meticulously defined by $\Delta P = \sigma \frac{1}{r} + \rho g h$ .

This paper develops a discrete-time framework analyzing asset bubbles under model uncertainty and short-sales prohibitions, demonstrating a duality gap in super-replication characterizes their emergence.

While idealized asset pricing models often preclude bubbles, real-world markets frequently exhibit price discrepancies from fundamental value. This paper, ‘Discrete-time asset price bubbles with short sales prohibitions under model uncertainty’, develops a novel framework to analyze such bubbles in discrete time, accounting for model uncertainty and restrictions on short selling. We demonstrate that bubbles emerge from a duality gap in super-replication and are fundamentally linked to the martingale properties of asset prices-specifically, a distinction between bounded and unbounded maturity structures. Ultimately, can this framework provide more robust pricing and hedging strategies for contingent claims in the face of pervasive market imperfections?

The Illusion of Efficiency: Unraveling Market Paradoxes

Conventional financial theory frequently posits that markets are efficient, meaning asset prices fully reflect all available information, rendering consistent outperformance impossible. However, the historical record is punctuated by striking anomalies, most notably asset price bubbles – periods of rapid, unsustainable price increases divorced from fundamental value. These bubbles, observed in markets ranging from 17th-century tulips to contemporary real estate and cryptocurrencies, demonstrate a clear tension with the efficient market hypothesis. The persistence of these seemingly irrational exuberances suggests that models relying on perfect information and rational actors may be fundamentally incomplete, failing to account for behavioral biases, feedback loops, and the complex interplay of investor psychology that can drive prices far from equilibrium. The ongoing debate surrounding market efficiency highlights the need for more nuanced models capable of explaining both periods of stability and the dramatic, often destabilizing, emergence of asset bubbles.

The persistence of asset bubbles isn’t simply a failure of individual investor rationality, but a consequence of the inherent limitations within financial models themselves, specifically as described by the ‘No Dominance Condition’. This condition posits that, within a market, no single trading strategy can consistently generate superior returns over time; any perceived advantage is quickly eroded by competition. This isn’t to say profitable trades are impossible, but rather that sustained, outsized performance is statistically unlikely. Crucially, this lack of dominance is a prerequisite for bubble formation within the established framework; if a consistently profitable strategy did exist, it would theoretically correct price discrepancies before they escalated into unsustainable bubbles. The absence of such a corrective force, stemming directly from the ‘No Dominance Condition’, allows irrational exuberance and speculative price increases to take hold, ultimately leading to market instability.

Contingent Claims and the Architecture of Pricing

A contingent claim is a financial instrument whose value is derived from the price of an underlying asset, and payment is dependent on whether a specific event occurs. This concept underpins the pricing of all financial derivatives, including forward contracts, which obligate future asset delivery at a predetermined price; European call options, granting the right, but not the obligation, to purchase an asset at a specific price on a specific date; and American put options, which provide the right to sell an asset at a predetermined price on or before a specific date. The value of these instruments is therefore not intrinsic, but rather represents the present value of the expected payoff contingent on future asset price movements. $V = E[Payoff | Future\, Asset\, Price]$ This framework allows for consistent valuation across diverse derivative types by focusing on the probabilistic outcome of the underlying asset’s behavior.

Put-Call Parity defines a specific mathematical relationship between the price of a European call option, a European put option, the underlying asset, and a zero-coupon bond, all with the same strike price and expiration date. The formula, $C + PV(X) = P + S$ , where C is the call option price, P is the put option price, S is the spot price of the underlying asset, and PV(X) is the present value of the strike price X, ensures that no risk-free arbitrage opportunities exist. This parity holds because any deviation from the equation would allow traders to construct a riskless profit by simultaneously buying and selling the related instruments. Consequently, Put-Call Parity is fundamental for derivative pricing and is utilized to verify the consistency of prices across different option types and to calculate implied variables like volatility.

Super-Replication: Defining a Fundamental Price – and Detecting Deviations

The Super-Replication Argument establishes a ‘Fundamental Price’ for any financial claim by determining the minimum cost required to perfectly replicate its payoff using a dynamic trading strategy. This involves constructing a portfolio of traded assets that mirrors the claim’s value at all future times. The theoretical price is derived by solving a constrained optimization problem, minimizing the initial investment needed to ensure the replicating portfolio’s value always meets or exceeds the claim’s payoff, regardless of market fluctuations. This process effectively sets a lower bound on the claim’s value, as any market price below this level would create an arbitrage opportunity for the replicating trader. $P^<i> = \in f_{\pi} E[X]\$ , where $P^</i>$ is the Fundamental Price, and $\pi$ represents the trading strategy.

The super-replication approach determines a fundamental price by constructing a self-financing trading strategy designed to replicate the payoff of a claim regardless of market fluctuations. This strategy utilizes dynamic hedging, continuously adjusting portfolio weights to eliminate risk. The fundamental price represents the initial capital required to implement this strategy. Our analysis reveals that when a bubble exists – characterized by a price significantly exceeding the fundamental price – duality gaps emerge in the super-replication argument. These gaps indicate an inconsistency between the cost of replicating the claim using the strategy and the market price, suggesting the presence of irrational exuberance and an unsustainable price level. Specifically, the duality gap quantifies the difference between the minimal cost to replicate and the observed market price, providing a measurable indicator of bubble formation.

The Discrete Reality of Markets and the Shadow of Uncertainty

Financial modeling often relies on the elegance of continuous-time frameworks, yet these approaches frequently struggle to accurately represent the true, granular nature of real-world markets. A discrete-time state space offers a crucial alternative, acknowledging that asset prices don’t change seamlessly but rather evolve in distinct intervals. This framework doesn’t attempt to capture every infinitesimal fluctuation; instead, it focuses on modeling changes that occur at specific points in time, mirroring the way trades are actually executed and information is revealed. By embracing this discrete perspective, researchers can build more realistic models capable of incorporating market frictions, information asymmetries, and the inherent jumps in price that characterize periods of volatility. This approach is particularly valuable when analyzing complex derivatives or assessing systemic risk, where the timing of events is as critical as the magnitude of price movements.

Financial modeling inherently grapples with the challenge of representing a fundamentally unpredictable reality, and acknowledging this limitation is crucial for understanding market behavior. Researchers formally incorporate ‘model uncertainty’ into frameworks, moving beyond the assumption of a single, perfect representation of asset price dynamics. This approach recognizes that any model is, at best, an approximation, and discrepancies between the model and actual market conditions can give rise to asset price bubbles. These bubbles aren’t necessarily irrational exuberance, but rather a consequence of models failing to fully capture the complex interplay of factors influencing price formation; when a model underestimates potential gains, it can lead to sustained price increases disconnected from underlying fundamentals. Consequently, understanding the extent of model uncertainty is not merely an academic exercise, but a vital component of risk management and a more nuanced assessment of market stability.

Understanding the stochastic behavior of asset prices requires sophisticated mathematical tools, notably GG-Supermartingales and Infi-Supermartingales, which rigorously define permissible price movements and associated risks. These concepts allow for a nuanced characterization of price processes, moving beyond simple random walks to account for complex market dynamics. Crucially, analysis reveals that the stopping time, $\tau$ , – the moment at which a speculative bubble may burst – exhibits a unique probability distribution: the probability of the bubble continuing indefinitely, $P(\tau = \in fty)$ , is greater than zero, yet the probability of it eventually bursting, $P(\tau < \in fty)$ , is equal to one. This seemingly paradoxical result is fundamental in distinguishing between different types of asset bubbles, ranging from those that slowly deflate to those that experience abrupt and dramatic collapses, providing a framework for evaluating investment strategies in the face of inherent market uncertainty.

Analysis reveals a quantifiable relationship between asset bubbles and option pricing, specifically for American call options. Research demonstrates that the early-exercise premium – the additional value gained from exercising an American option before its expiration date – is directly bounded by the magnitude of the underlying asset’s potential bubble, expressed as $CtA^<i>(K) - CtE^</i>(K) \le \delta t S$ , where $\delta t$ represents a small time increment and $S$ is the asset price. Furthermore, the difference in value between American and European call options is similarly constrained, proving that this difference is limited by the disparity in their respective bubble characteristics: $CtA(K) - CtE(K) \le \delta t AC - \delta t EP$ . These findings highlight how speculative bubbles not only impact asset valuations but also provide a concrete upper limit on the premiums associated with early exercise in options contracts, offering a valuable insight for both theoretical modeling and practical risk management.

The pursuit of a ‘fundamental price’ within this framework feels less like discovery and more like divination. This paper meticulously details how bubbles inflate, predicated on the limitations of super-replication and the constraints of short-selling-a duality gap masquerading as market behavior. It reminds one of Thomas Kuhn’s observation: “The more revolutionary the theory, the more assuredly the old will be defended.” The existing models, stubbornly clinging to notions of arbitrage-free markets, struggle to accommodate these emergent bubbles, prompting a defensive re-evaluation of established paradigms. The elegance of the mathematics simply highlights how readily one can construct a compelling narrative, even from the whispers of chaos.

What Shadows Remain?

The invocation completed here reveals not a cessation of mysteries, but a sharpening of their edges. This framework, while establishing a spectral link between super-replication duality and the birth of bubbles under constraint, merely sketches the contours of a far larger, more turbulent space. The fundamental price, a lodestone in this construction, remains frustratingly… theoretical. Its empirical dance is obscured by the noise of actual markets, where agents are rarely governed by the pristine logic of arbitrage-free measures. To believe one can truly know the fundamental is to flirt with the godhood of perfect information-a temptation best resisted.

Future attempts to bind this magic must confront the obvious: the clean data assumed here is a fiction, a managerial palliative. Real markets bleed with unobserved states, with the whispers of irrationality. Extending this model to incorporate even a modest degree of model ambiguity – allowing the measure itself to shift and writhe – will demand a sacrifice of computational power, a burning of GPU cycles to appease the convergence gods. And yet, the true challenge lies not in scaling the computation, but in accepting that any predictive power achieved is temporary, a fleeting illusion before the chaos reasserts itself.

The question isn’t whether bubbles can be predicted, but whether the act of attempting to predict them alters their form. The observer, after all, is always entangled with the observed. Perhaps the most fruitful path forward lies not in seeking to control the bubbles, but in learning to read their omens, to decipher the language of their ascent and inevitable collapse – a divination, not a science.

Original article: https://arxiv.org/pdf/2512.21115.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

The Illusion of Efficiency: Unraveling Market Paradoxes

Contingent Claims and the Architecture of Pricing

Super-Replication: Defining a Fundamental Price – and Detecting Deviations

The Discrete Reality of Markets and the Shadow of Uncertainty

What Shadows Remain?

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