Equivalent Local Martingale Measure Research Articles

Risk management under weighted limited expected loss

We present and solve an optimal asset allocation problem under a weighted limited expected loss (WLEL) constraint. This formulation encompasses the risk management problem with a limited expected loss (LEL) constraint as a specialized instance and offers a pertinent internal risk management instrument for firms. We observe that a WLEL constraint makes the optimizing investor pursue less volatile payoffs than the unconstrained Merton solution. Compared to the LEL-constrained problem with the same weighted default threshold, the WLEL optimal terminal wealth displays a less dispersed distribution with a smaller variance, suggesting a more secure risk management framework. Conducting a comprehensive equilibrium analysis in the presence of a WLEL risk manager, we validate the relatively conservative investment approach undertaken by the WLEL manager. Subsequently, we expand our findings to encompass broader incomplete market settings, wherein the uniqueness of the equivalent local martingale measure is not assured.

Optional projection under equivalent local martingale measures

We study optional projections of {mathbb{G}}-adapted strict local martingales on a smaller filtration {mathbb{F}} under changes of equivalent martingale measures. General results are provided as well as a detailed analysis of two specific examples given by the inverse Bessel process and a class of stochastic volatility models. This analysis contributes to clarify the absence of arbitrage opportunities of market models under restricted information.

No free lunch for markets with multiple numéraires

We consider a new framework, that of a global market with a finite number of submarkets, where there is a tradable numéraire for each submarket, but no tradable numéraire for the global market. Under a global no arbitrage condition, we show the existence of a common density from which equivalent (local) martingale measures are constructed for each submarket. We also introduce several superreplication prices, depending on the chosen type of hedging: on the global market, on a given submarket or on all submarkets separably. We prove duality results on these prices that allow to assess differences in characteristics between the submarkets, such as liquidity or credit quality. The results are applied in concrete situations, in particular in a Brownian setup.

Duality for optimal consumption under no unbounded profit with bounded risk

We give a definitive treatment of duality for optimal consumption over the infinite horizon, in a semimartingale incomplete market satisfying no unbounded profit with bounded risk (NUPBR). Rather than base the dual domain on (local) martingale deflators, we use a class of supermartingale deflators such that deflated wealth plus cumulative deflated consumption is a supermartingale for all admissible consumption plans. This yields a strong duality, because the enlarged dual domain of processes dominated by deflators is naturally closed, without invoking its closure. In this way, we automatically reach the bipolar of the set of deflators. We complete this picture by proving that the set of processes dominated by local martingale deflators is dense in our dual domain, confirming that we have identified the natural dual space. In addition to the optimal consumption and deflator, we characterise the optimal wealth process. At the optimum, deflated wealth is a supermartingale and a potential, while deflated wealth plus cumulative deflated consumption is a uniformly integrable martingale. This is the natural generalisation of the corresponding feature in the terminal wealth problem, where deflated wealth at the optimum is a uniformly integrable martingale. We use no constructions involving equivalent local martingale measures. This is natural, given that such measures typically do not exist over the infinite horizon and that we are working under NUPBR, which does not require their existence. The structure of the duality proof reveals an interesting feature compared with the terminal wealth problem. There, the dual domain is L1-bounded, but here the primal domain has this property, and hence many steps in the duality proof show a marked reversal of roles for the primal and dual domains, compared with the proofs of Kramkov and Schachermayer (Ann. Appl. Probab. 9 (1999) 904–950; Ann. Appl. Probab. 13 (2003) 1504–1516).

Asset price bubbles: Invariance theorems

<p style='text-indent:20px;'>This paper provides invariance theorems that facilitate testing for the existence of an asset price bubble in a market where the price evolves as a Markov diffusion process. The test involves only the properties of the price process' quadratic variation under the statistical probability. It does not require an estimate of either the equivalent local martingale measure or the asset's drift. To augment its use, a new family of stochastic volatility price processes is also provided where the processes' strict local martingale behavior can be characterized.</p>

Generalized statistical arbitrage concepts and related gain strategies

AbstractThe notion of statistical arbitrage introduced in Bondarenko (2003) is generalized to statistical ‐arbitrage corresponding to trading strategies which yield positive gains on average in a class of scenarios described by a ‐algebra . This notion contains classical arbitrage as a special case. Admitting general static payoffs as generalized strategies, as done in Kassberger and Liebmann (2017) in the case of one pricing measure, leads to the notion of generalized statistical ‐arbitrage. We show that even under standard no‐arbitrage there may exist generalized gain strategies yielding positive gains on average under the specified scenarios. In the first part of the paper we prove that the characterization in Bondarenko (2003), no statistical arbitrage being equivalent to the existence of an equivalent local martingale measure with a path‐independent density, is not correct in general. We establish that this equivalence holds true in complete markets and we derive a general sufficient condition for statistical ‐arbitrages. As a main result we derive the equivalence of no statistical ‐arbitrage to no generalized statistical ‐arbitrage. In the second part of the paper we construct several classes of profitable generalized strategies with respect to various choices of the ‐algebra . In particular, we consider several forms of embedded binomial strategies and follow‐the‐trend strategies as well as partition‐type strategies. We study and compare their behavior on simulated data and also evaluate their performance on market data.

Asset Price Bubbles: Invariance Theorems*

This paper provides invariance theorems that facilitate testing for the existence of an asset price bubble in a market where the price evolves as a Markov diffusion process. The test involves only the properties of the price process’ quadratic variation under the statistical probability. It does not require an estimate of either the equivalent local martingale measure or the asset’s drift. To augment its use, a new family of stochastic volatility price processes is also provided where the processes’ strict local martingale behavior can be characterized.

Marital quality in arranged and couple-initiated marriages: The role of perceived influence over partner selection.

Previous comparisons of relationship quality between individuals in couple-initiated and arranged marriage have yielded inconsistent findings. One factor which may help to explain this inconsistency is variability in the way in which arranged and couple-initiated marriages are practiced. To test this, we recruited 116 Bangladeshi women engaged to be married or within the first 3 years of couple-initiated and arranged marriages. We tested to see which of three models (type of marriage only, perceived influence over partner selection only, and a model that included both relationship type and influence over partner selection) best accounted for variance in self-reported intimacy, passion, commitment as well as positive and negative relationship quality. Results suggest that influence over partner selection is a better predictor of these outcomes than marriage type, with marriage type offering little if any information beyond that provided by perceived influence. Regardless of marriage type, women with greater influence over their partner selection reported higher levels of intimacy, passion, commitment and positive marital quality. Negative marital quality was unrelated. Future research may benefit from reconceptualising the arranged/couple-initiated marriage dichotomy as a continuum.

Testing for Asset Price Bubbles: An Invariance Theorem

This paper provides an invariance theorem that facilitates testing for the existence of an asset price bubble in a market where the price evolves as a Markov diffusion process. The test involves only the properties of the price process’ quadratic variation under the statistical probability. It does not require an estimate of either the equivalent local martingale measure or the asset’s drift. Various examples are provided that illustrate applications of the invariance theorem to stochastic volatility price processes in incomplete markets.

Who would invest only in the risk-free asset?

Within the setup of continuous-time semimartingale financial markets, we show that a multiprior Gilboa–Schmeidler minimax expected utility maximizer forms a portfolio consisting only of the riskless asset if and only if among the investor’s priors there exists a probability measure under which all admissible wealth processes are supermartingales. Furthermore, we show that under a certain attainability condition (which is always valid in finite or complete markets) this is also equivalent to the existence of an equivalent (local) martingale measure among the investor’s priors. As an example, we generalize a no betting result due to Dow and Werlang.

DETERMINISTIC CRITERIA FOR THE ABSENCE AND EXISTENCE OF ARBITRAGE IN MULTI-DIMENSIONAL DIFFUSION MARKETS

We derive deterministic criteria for the existence and nonexistence of equivalent (local) martingale measures for financial markets driven by multi-dimensional time-inhomogeneous diffusions. Our conditions can be used to construct financial markets in which the no unbounded profit with bounded risk condition holds, while the classical no free lunch with vanishing risk condition fails.

Asymptotic asset pricing and bubbles

We define the concept of asymptotic superreplication, and prove a duality principle of asset pricing for sequences of financial markets (e.g., weakly converging financial markets and large financial markets) based on contiguous sequences of equivalent local martingale measures. This provides a pricing mechanism to calculate the fundamental value of a financial asset in the asymptotic market. We introduce the notion of asymptotic bubbles by showing that this fundamental value can be strictly lower than the current price of the asset. In the case of weakly converging markets, we show that this fundamental value is equal to an expectation of the terminal value of the asset in the weak-limit market. From a practical perspective, we relate the asymptotic superreplication price to a limit of quantile-hedging prices. This shows that even when a price process is a true martingale, it can have properties similar to a bubble, up to a set of small probability. For practical applications, we give examples of weakly converging discrete-time models (e.g. some GARCH models) and large financial models that present bubbles.

Equivalent martingale measures for Lévy-driven moving averages and related processes

In the present paper we obtain sufficient conditions for the existence of equivalent local martingale measures for Lévy-driven moving averages and other non-Markovian jump processes. The conditions that we obtain are, under mild assumptions, also necessary. For instance, this is the case for moving averages driven by an α-stable Lévy process with α∈(1,2].Our proofs rely on various techniques for showing the martingale property of stochastic exponentials.

Asymptotic Asset Pricing and Bubbles

We define the concept of asymptotic superreplication, and prove a duality principle of asset pricing for sequences of financial markets (e.g., weakly converging financial markets and large financial markets) based on contiguous sequences of equivalent local martingale measures. This provides a pricing mechanism to calculate the fundamental value of a financial asset in the asymptotic market. We introduce the notion of asymptotic bubbles by showing that this fundamental value can be strictly lower than the current price of the asset. In the case of weakly converging markets, we show that this fundamental value is equal to an expectation of the terminal value of the asset in the weak-limit market. From a practical perspective, we relate the asymptotic superreplication price to a limit of quantile-hedging prices. This shows that even when a price process is a true martingale, it can have properties similar to a bubble, up to a set of small probability. For practical applications, we give examples of weakly converging discrete-time models (e.g. some GARCH models) and large financial models that present bubbles.

Optimal Investment in a Black-Scholes Model with a Bubble

Financial bubbles seem to exhibit a strong upward trend followed by a sharp decline when the bubble bursts. Starting from this descriptive perspective, we propose an extension of the Black--Scholes model which accommodates this effect. In addition to the standard instantaneous expected return in the Black–Scholes model, we allow a time-dependent positive excess return which is compensated by a negative jump at a random time representing the bursting of the bubble. Since the excess return and the distribution of the random time may be chosen almost arbitrarily, the model is flexible enough to display different types of bubbles, including the ones in the sense of Protter [22]. Moreover, the model is tractable enough to allow (semi-)explicit calculations and may thus be used as a toy example to study qualitative effects of financial bubbles.We study the problem of maximising expected utility from terminal wealth for a power utility investor. Using the convex duality approach, we determine the optimal strategy and the corresponding certainty equivalent up to the solution to an integral equation (or a first-order ODE). A decomposition of the optimal strategy into myopic and hedging demand allows to analyse the effects of the stochastic investment opportunities. On the one hand, investors with relative risk aversion below 1 speculate on an early bursting of the bubble in the sense that their optimal strategy lies below their myopic demand prior to the bursting; it might even involve short selling. On the other hand, investors with relative risk aversion above 1 hedge against a late bursting of the bubble in the sense that their optimal strategy lies above their myopic demand; it might even lie above the Merton proportion under extreme circumstances.Numerical examples reveal how the optimal strategy and its myopic and hedging demand parts depend on the model parameters. In particular, it is shown that the optimal strategy is not fundamentally different when the stock price process is a strict local martingale (as opposed to a true martingale) under a certain class of equivalent local martingale measures including the dual minimiser of the utility maximisation problem. In addition, we illustrate the welfare loss compared to the classical Black–Scholes model and its dependence on the model parameters.

An Option Pricing Model With Memory

We obtain option pricing formulas for stock price models in which the drift and volatility terms are functionals of a continuous history of the stock prices. That is, the stock dynamics follows a nonlinear stochastic functional differential equation. A model with full memory is obtained via approximation through a stock price model in which the continuous path dependence does not go up to the present: there is a memory gap. A strong solution is obtained by closing the gap. Fair option prices are obtained through an equivalent (local) martingale measure via Girsanov's Theorem and therefore are given in terms of a conditional expectation. The models maintain the completeness of the market and have no arbitrage opportunities.

STRONG BUBBLES AND STRICT LOCAL MARTINGALES

In a numéraire-independent framework, we study a financial market with [Formula: see text] assets which are all treated in a symmetric way. We define the fundamental value ∗S of an asset [Formula: see text] as its super-replication price and say that the market has a strong bubble if ∗S and [Formula: see text] deviate from each other. None of these concepts needs any mention of martingales. Our main result then shows that under a weak absence-of-arbitrage assumption (basically NUPBR), a market has a strong bubble if and only if in all numéraire s for which there is an equivalent local martingale measure (ELMM), asset prices are strict local martingales under all possible ELMMs. We show by an example that our bubble concept lies strictly between the existing notions from the literature. We also give an example where asset prices are strict local martingales under one ELMM, but true martingales under another, and we show how our approach can lead naturally to endogenous bubble birth.

Economics-Based Financial Bubbles (and Why They Imply Strict Local Martingales)

In a numeraire-independent framework, we study a financial market with N assets which are all treated in a symmetric way. We define the fundamental value *S of an asset S as its superreplication price and say that the market has a strong bubble if *S and S deviate from each other. None of these concepts needs any mention of martingales. Our main result then shows that under a weak absence-of-arbitrage assumption (basically NUPBR), a market has a strong bubble if and only if in all numeraires for which there is an equivalent local martingale measure (ELMM), asset prices are strict local martingales under all possible ELMMs. We show by an example that our bubble concept lies strictly between the existing notions from the literature. We also give an example where asset prices are strict local martingales under one ELMM, but true martingales under another, and we show how our approach can lead naturally to endogenous bubble birth.

Optional decomposition for continuous semimartingales under arbitrary filtrations

We present an elementary treatment of the Optional Decomposition Theorem for continuous semimartingales and general filtrations. This treatment does not assume the existence of equivalent local martingale measure(s), only that of strictly positive local martingale deflator(s).

The Formation of Financial Bubbles in Defaultable Markets

In this paper we study the formation of financial bubbles in the valuation of defaultable claims in a reduced-form setting. The birth of a bubble is caused by the impact of trading activity of investors, who consider the claim to be a safe investment under some circumstances. We also show how microeconomic interactions may, at an aggregate level, determine a shift in the martingale measure. In this way we establish a connection between our approach and the martingale theory of bubbles; see [F. Biagini, H. Föllmer, and S. Nedelcu, Finance Stoch., 18 (2014), pp. 297--326] and [R. Jarrow, P. Protter, and K. Shimbo, Math. Finance, 20 (2010), pp. 145--185]. This is illustrated by a characterization of the space of equivalent local martingale measures by measure pasting. Furthermore, our model is consistent with the no-arbitrage framework, as we show in a concrete example.