Existence Of Equivalent Martingale Measure Research Articles

Detection of arbitrage opportunities in multi-asset derivatives markets

Abstract We are interested in the existence of equivalent martingale measures and the detection of arbitrage opportunities in markets where several multi-asset derivatives are traded simultaneously. More specifically, we consider a financial market with multiple traded assets whose marginal risk-neutral distributions are known, and assume that several derivatives written on these assets are traded simultaneously. In this setting, there is a bijection between the existence of an equivalent martingale measure and the existence of a copula that couples these marginals. Using this bijection and recent results on improved Fréchet–Hoeffding bounds in the presence of additional information on functionals of a copula by [18], we can extend the results of [33] on the detection of arbitrage opportunities to the general multi-dimensional case. More specifically, we derive sufficient conditions for the absence of arbitrage and formulate an optimization problem for the detection of a possible arbitrage opportunity. This problem can be solved efficiently using numerical optimization routines. The most interesting practical outcome is the following: we can construct a financial market where each multi-asset derivative is traded within its own no-arbitrage interval, and yet when considered together an arbitrage opportunity may arise.

Wealth optimization models on jump-diffusion model

In this paper, we study the optimization problems of minimization of shortfall risk under the jump diffusion model. Jump-diffusion asset price model is driven by nonexplosive counting process that is more general than Poisson process. We take the expected discount utility of shortfall risk as our objective function. Applying Legendre transform and stochastic analysis methods,the existence of equivalent martingale measure and the optimal portfolio is proved. Furthermore, the value function, the optimal portfolio, and the optimal terminal wealth are also proposed.

Pricing the European call option in the model with stochastic volatility driven by Ornstein–Uhlenbeck process. Exact formulas

We consider the Black--Scholes model of financial market modified to capture the stochastic nature of volatility observed at real financial markets. For volatility driven by the Ornstein--Uhlenbeck process, we establish the existence of equivalent martingale measure in the market model. The option is priced with respect to the minimal martingale measure for the case of uncorrelated processes of volatility and asset price, and an analytic expression for the price of European call option is derived. We use the inverse Fourier transform of a characteristic function and the Gaussian property of the Ornstein--Uhlenbeck process.

On the fundamental theorem of asset pricing

It is well known that existence of equivalent martingale measure (EMM) is essentially equivalent to absence of arbitrage. In this paper, we give an overview of this connection and also include material on this theme from an unpublished article written with Professor Kallianpur. This is included in the book by Kallianpur (7) on Option pricing. In that article, we had introduced a concept of No Approximate Arbitrage with Controlled Risk - NAACR which turns out to be equivalent to the existence of equivalent martingale measure. This seems to be the only result characterising EMM in terms of absence of (suitable notion of) arbitrage in the class simple strategies. Moreover, the proof of this assertion is purely functional analytic, without invoking semimartingales and stochastic integration.

Exponential stock models driven by tempered stable processes

We investigate exponential stock models driven by tempered stable processes, which constitute a rich family of purely discontinuous Lévy processes. With a view of option pricing, we provide a systematic analysis of the existence of equivalent martingale measures, under which the model remains analytically tractable. This includes the existence of Esscher martingale measures and martingale measures having minimal distance to the physical probability measure. Moreover, we provide pricing formulae for European call options and perform a case study.

Martingales and arbitrage: a new look

This paper addresses the equivalence between the absence of arbitrage and the existence of equivalent martingale measures. The equivalence will be established under quite weak assumptions since there are no conditions on the set of trading dates (it may be finite or countable, with bounded or unbounded horizon, etc.) or on the trajectories of the price process (for instance, they do not have to be right-continuous).

A note on the CIR process and the existence of equivalent martingale measures

This note shows that in a model where historical stock price follows a Cox–Ingersoll–Ross process, an equivalent martingale measure does not exist except when k θ = 0 .

Risk-neutral valuation with infinitely many trading dates

The first Fundamental Theorem of Asset Pricing establishes the equivalence between the absence of arbitrage in financial markets and the existence of Equivalent Martingale Measures, if appropriate conditions hold. Since the theorem may fail when dealing with infinitely many trading dates, this paper draws on the A.A. Lyapunov Theorem in order to retrieve the equivalence for complete markets such that the Sharpe Ratio is adequately bounded.

Universal state prices and asymmetric information

This note gives a simple, robust, and arbitrage-free example of the absence of a fixed vector of state prices that applies to asymmetrically informed agents. In the same sense, the example is such that there is no universal equivalent martingale measure. The example has a finite number of agents, states, and periods. In this example, moreover, each of the asymmetrically informed agents has complete market. The example is consistent with a full rational expectations general equilibrium with learning from prices. Duffie and Huang (1986) showed conditions implying the existence of a universal equivalent martingale measure. These conditions include the assumption that one of the agents has more information than each of the others. When there is no agent whose information dominates in this sense, there is no particular reason to believe in the existence of an equivalent martingale measure (EMM) that applies to all. Our example of non-existence depends on a special information structure, as it must since prices are generically fully revealing with a finite number of states. Given this special information structure, the example is robust in that no perturbation of the payoffs of the securities allows for existence of universal state prices.

No Arbitrage in Discrete Time Under Portfolio Constraints

In frictionless securities markets, the characterization of the no‐arbitrage condition by the existence of equivalent martingale measures in discrete time is known as the fundamental theorem of asset pricing. In the presence of convex constraints on the trading strategies, we extend this theorem under a closedness condition and a nondegeneracy assumption. We then provide connections with the superreplication problem solved in Föllmer and Kramkov (1997).

The fundamental theorem of asset pricing with cone constraints

In frictionless securities markets, the characterization of the no arbitrage condition by the existence of equivalent martingale measures in discrete time is known as the fundamental Theorem of asset pricing. In the presence of cone constraints on the trading strategies, we extend the fundamental theorem of asset pricing under a nondegeneracy assumption. We also prove a one-period version of this theorem when there are transaction costs.

Complications with stochastic volatility models

We show a class of stock price models with stochastic volatility for which the most natural candidates for martingale measures are only strictly local martingale measures, contrary to what is usually assumed in the finance literature. We also show the existence of equivalent martingale measures, and provide one explicit example.

Complications with stochastic volatility models

We show a class of stock price models with stochastic volatility for which the most natural candidates for martingale measures are only strictly local martingale measures, contrary to what is usually assumed in the finance literature. We also show the existence of equivalent martingale measures, and provide one explicit example.

Existence of Equivalent Martingale Measures in Finite Dimensional Securities Markets

Given the information structure and the number of securities, wecharacterize the set of all price–dividend systemsfor which there exist a dynamic portfolio with strictly positive value through time, and a strictly positive probabilityQsuch that, expressing prices and dividends in units of such portfolio, every gain process is aQ-martingale. We also weaken our characterization by showing that the existence of a trading strategy with strictly positive terminal payoffs only isgenerically sufficientfor the existence of an equivalent martingale measure. Finally, an example shows that our results robustely extend the current literature on the topic.Journal of Economic LiteratureClassification Numbers: D52, G10, G12

A quantitative and a dual version of the Halmos-Savage theorem with applications to mathematical finance

The celebrated theorem of Halmos and Savage implies that if M is a set of $\mathbb{P}$-absolutely continuous probability measures Q on $(\Omega, \mathscr{F}, \mathbb{P})$ such that each $A \in \mathscr{F}, \mathbb{P}(A) > 0$ is charged by some $Q\in M$, that is, $Q(A) > 0$ (where the choice of Q depends on the set A), then -- provided M is closed under countable convex combinations -- we can find $Q_0 \in M$ with full support; that is, $\mathbb{P}(A) > 0$ implies $Q_0(A) > 0 $. We show a quantitative version: if we assume that, for $\varepsilon > 0$ and $\delta > 0$ fixed, $\mathbb{P}(A)> \varepsilon$ implies that there is $Q \in M$ and $Q(A) > \delta$, then there is $Q_0 \in M$ such that $\mathbb{P}(A) >4 \varepsilon$ implies $Q_0(A)>\varepsilon^2 \delta/2$. This version of the Halmos-Savage theorem also allows a "dualization" which we also prove in a quantitative and qualitative version. We give applications to asymptoic problems arising in mathematical finance and pertaining to the relation of the concept of "no arbitrage" and the existence of equivalent martingale measures for a sequence of stochastic processes.

A Necessary and Sufficient Condition for Absence of Arbitrage with Tame Portfolios

We characterize absence of arbitrage with tame portfolios in the case of invertible volatility matrix. As a corollary we get that, under a certain condition, absence of arbitrage with tame portfolios is characterized by the existence of the so-called equivalent martingale measure. Without that condition, the existence of equivalent martingale measure is equivalent to absence of approximate arbitrage. The proofs are probabilistic and are based on a construction of two specific arbitrages. Some examples are provided.

A Remark on Arbitrage and Martingale Measure

In the option pricing theory, as was pointed out in Harrison-Kreps [6], the most important fact is that the absence of arbitrage follows from the existence of equivalent martingale measure for the price process of securities. There are several attempts to show the converse statement that the absence of arbitrage implies the existence of an equivalent martingale measure. In the discrete time case, the proof for the most general case has been given by Dalang-Morton-Willinger [2] and Schachermayer [9]. On the continuous time case Strieker [11] gave beautiful results, and they were extended by Delbaen [3], Schachermayer [10] and Delbaen-Schachermayer [4], [5]. In particular, the Mackey topology is cleverly used in Delbaen [3], and his result is quite satisfactory in the case where price processes are path-wise continuous. In this paper, we give some remarks on Orlicz spaces, Mackey topologies, and results by Ansel-Strieker [1] and Delbaen [3]. Then we give a certain necessary and sufficient conditions for the existence of an equivalent martingale measure.