Arbitrage opportunities and their implications to derivative hedging

We explore the role that random arbitrage opportunities play in hedging financial derivatives. We extend the asymptotic pricing theory presented by Fedotov and Panayides [Stochastic arbitrage return and its implication for option pricing, Physica A 345 (2005) 207–217] for the case of hedging a derivative when arbitrage opportunities are present in the market. We restrict ourselves to finding hedging confidence intervals that can be adapted to the amount of arbitrage risk an investor will permit to be exposed to. The resulting hedging bands are independent of the detailed statistical characteristics of the arbitrage opportunities.     

Stochastic relaxational dynamics applied to finance: Towards non-equilibrium option pricing theory

Non-equilibrium phenomena occur not only in the physical world, but also in finance. In this work, stochastic relaxational dynamics (together with path integrals) is applied to option pricing theory. Equilibrium in financial markets is defined as the absence of arbitrage, i.e. profits “for nothing”. A recently proposed model (by Ilinski et al.) considers fluctuations around this equilibrium state by introducing a relaxational dynamics with random noise for intermediate deviations called “virtual” arbitrage returns. In this work, the model is incorporated within a martingale pricing method for derivatives on securities (e.g. stocks) in incomplete markets using a mapping to option pricing theory with stochastic interest rates. The arbitrage return is considered as a component of a fictitious short-term interest rate in a virtual world. The influence of intermediate arbitrage returns on the price of derivatives in the real world can be recovered by performing an average over the (non-observable) arbitrage return at the time of pricing. Using a famous result by Merton and with some help from the path integral method, exact pricing formulas for European call and put options under the influence of virtual arbitrage returns (or intermediate deviations from economic equilibrium) are derived where only the final integration over initial arbitrage returns needs to be performed numerically. This result, which has not been given previously and is at variance with results stated by Ilinski et al., is complemented by a discussion of the hedging strategy associated to a derivative, which replicates the final payoff but turns out to be not self-financing in the real world, but self-financing when summed over the derivative's remaining life time. Numerical examples are given which underline the fact that an additional positive risk premium (with respect to the Black-Scholes values) is found reflecting extra hedging costs due to intermediate deviations from economic equilibrium. Received 16 June 1999 and Received in final form 26 September 1999     

A quantum model of option pricing: When Black-Scholes meets Schrödinger and its semi-classical limit

The Black-Scholes equation can be interpreted from the point of view of quantum mechanics, as the imaginary time Schrödinger equation of a free particle. When deviations of this state of equilibrium are considered, as a product of some market imperfection, such as: Transaction cost, asymmetric information issues, short-term volatility, extreme discontinuities, or serial correlations; the classical non-arbitrage assumption of the Black-Scholes model is violated, implying a non-risk-free portfolio. From Haven (2002) [1] we know that an arbitrage environment is a necessary condition to embedding the Black-Scholes option pricing model in a more general quantum physics setting. The aim of this paper is to propose a new Black-Scholes-Schrödinger model based on the endogenous arbitrage option pricing formulation introduced by Contreras et al. (2010) [2]. Hence, we derive a more general quantum model of option pricing, that incorporates arbitrage as an external time dependent force, which has an associated potential related to the random dynamic of the underlying asset price. This new resultant model can be interpreted as a Schrödinger equation in imaginary time for a particle of mass 1/σ² with a wave function in an external field force generated by the arbitrage potential. As pointed out above, this new model can be seen as a more general formulation, where the perfect market equilibrium state postulated by the Black-Scholes model represent a particular case. Finally, since the Schrödinger equation is in place, we can apply semiclassical methods, of common use in theoretical physics, to find an approximate analytical solution of the Black-Scholes equation in the presence of market imperfections, as it is the case of an arbitrage bubble. Here, as a numerical illustration of the potential of this Schrödinger equation analogy, the semiclassical approximation is performed for different arbitrage bubble forms (step, linear and parabolic) and compare with the exact solution of our general quantum model of option pricing.     

Quantum-Like Tunnelling and Levels of Arbitrage

We apply methods of wave mechanics to financial modelling. We proceed by assigning a financial interpretation to wave numbers. This paper makes a plea for the use of the concept of ‘tunnelling’ (in the mathematical formalism of quantum mechanics) in the modelling of financial arbitrage. Financial arbitrage is a delicate concept to model in social science (i.e. in this case economics and finance) as its presence affects the precision of benchmark financial asset prices. In this paper, we attempt to show how ‘tunnelling’ can be used to positive effect in the modelling of arbitrage in a financial asset pricing context.     

Dynamic option pricing with endogenous stochastic arbitrage

Only few efforts have been made in order to relax one of the key assumptions of the Black-Scholes model: the no-arbitrage assumption. This is despite the fact that arbitrage processes usually exist in the real world, even though they tend to be short-lived. The purpose of this paper is to develop an option pricing model with endogenous stochastic arbitrage, capable of modelling in a general fashion any future and underlying asset that deviate itself from its market equilibrium. Thus, this investigation calibrates empirically the arbitrage on the futures on the S&P 500 index using transaction data from September 1997 to June 2009, from here a specific type of arbitrage called “arbitrage bubble”, based on a t-step function, is identified and hence used in our model. The theoretical results obtained for Binary and European call options, for this kind of arbitrage, show that an investment strategy that takes advantage of the identified arbitrage possibility can be defined, whenever it is possible to anticipate in relative terms the amplitude and timespan of the process. Finally, the new trajectory of the stock price is analytically estimated for a specific case of arbitrage and some numerical illustrations are developed. We find that the consequences of a finite and small endogenous arbitrage not only change the trajectory of the asset price during the period when it started, but also after the arbitrage bubble has already gone. In this context, our model will allow us to calibrate the B-S model to that new trajectory even when the arbitrage already started.     

Triangular arbitrage as an interaction among foreign exchange rates

We first show that there are in fact triangular arbitrage opportunities in the spot foreign exchange markets, analyzing the time dependence of the yen–dollar rate, the dollar–euro rate and the yen–euro rate. Next, we propose a model of foreign exchange rates with an interaction. The model includes effects of triangular arbitrage transactions as an interaction among three rates. The model explains the actual data of the multiple foreign exchange rates well.     

Beyond Black-Scholes: semimartingales and Lévy processes for option pricing

We consider the problem of option pricing when the underlying asset follows a general semimartingale process. After reviewing the foundations of arbitrage pricing theory for semimartingales and the link with Lévy processes, we introduce a general method to price options in this framework based on Fourier and Wavelet analysis. Received 4 September 2000     

Long-memory volatility in derivative hedging

The aim of this work is to take into account the effects of long memory in volatility on derivative hedging. This idea is an extension of the work by Fedotov and Tan [Stochastic long memory process in option pricing, Int. J. Theor. Appl. Finance 8 (2005) 381–392] where they incorporate long-memory stochastic volatility in option pricing and derive pricing bands for option values. The starting point is the stochastic Black–Scholes hedging strategy which involves volatility with a long-range dependence. The stochastic hedging strategy is the sum of its deterministic term that is classical Black–Scholes hedging strategy with a constant volatility and a random deviation term which describes the risk arising from the random volatility. Using the fact that stock price and volatility fluctuate on different time scales, we derive an asymptotic equation for this deviation in terms of the Green's function and the fractional Brownian motion. The solution to this equation allows us to find hedging confidence intervals.     

Analytic solutions for optimal statistical arbitrage trading

In this paper we derive analytic formulae for statistical arbitrage trading where the security price follows an Ornstein-Uhlenbeck process. By framing the problem in terms of the first-passage time of the process, we derive expressions for the mean and variance of the trade length and the return. We examine the problem of choosing an optimal strategy under two different objective functions: the expected return, and the Sharpe ratio. An exact analytic solution is obtained for the case of maximising the expected return.     

Semi-Markov regime switching interest rate models and minimal entropy measure

In this paper, we present a discrete time regime switching binomial-like model of the term structure where the regime switches are governed by a discrete time semi-Markov process. We model the evolution of the prices of zero-coupon when given an initial term structure as in the model by Ho and Lee that we aim to extend. We discuss and derive conditions for the model to be arbitrage free and relate this to the notion of martingale measure. We explicitly show that due to the extra source of uncertainty coming from the underlying semi-Markov process, there are an infinite number of equivalent martingale measures. The notion of path independence is also studied in some detail, especially in the presence of regime switches. We deal with the market incompleteness by giving an explicit characterization of the minimal entropy martingale measure. We give an application to the pricing of a European bond option both in a Markov and semi-Markov framework. Finally, we draw some conclusions.     

“String” formulation of the dynamics of the forward interest rate curve

We propose a formulation of the term structure of interest rates in which the forward curve is seen as the deformation of a string. We derive the general condition that the partial differential equations governing the motion of such string must obey in order to account for the condition of absence of arbitrage opportunities. This condition takes a form similar to a fluctuation-dissipation theorem, albeit on the same quantity (the forward rate), linking the bias to the covariance of variation fluctuations. We provide the general structure of the models that obey this constraint in the framework of stochastic partial (possibly non-linear) differential equations. We derive the general solution for the pricing and hedging of interest rate derivatives within this framework, albeit for the linear case (we also provide in the appendix a simple and intuitive derivation of the standard European option problem). We also show how the “string” formulation simplifies into a standard N-factor model under a Galerkin approximation. Received: 30 January 1998 / Revised: 12 February 1998 / Accepted: 16 February 1998     

Key Roles of Crypto-Exchanges in Generating Arbitrage Opportunities

The evolving crypto-currency market is seen as dynamic, segmented, and inefficient, coupled with a lack of regulatory oversight, which together becomes conducive to observing the arbitrage. In this context, a crypto-network is designed using bid/ask data among 20 crypto-exchanges over a 2-year period. The graph theory technique is employed to describe the network and, more importantly, to determine the key roles of crypto-exchanges in generating arbitrage opportunities by estimating relevant network centrality measures. Based on the proposed arbitrage ratio, Gatecoin, Coinfloor, and Bitsane are estimated as the best exchanges to initiate arbitrage, while EXMO and DSX are the best places to close it. Furthermore, by means of canonical correlation analysis, we revealed that higher volatility and the decreasing price of dominating crypto-currencies and CRIX index signal bring about a more likely arbitrage appearance in the market. The findings of research include pre-tax and after-tax arbitrage opportunities.     

A Mean–variance analysis of arbitrage portfolios

Based on the careful analysis of the definition of arbitrage portfolio and its return, the author presents a mean–variance analysis of the return of arbitrage portfolios, which implies that Korkie and Turtle's results ( B. Korkie, H.J. Turtle, A mean–variance analysis of self-financing portfolios, Manage. Sci. 48 (2002) 427–443) are misleading. A practical example is given to show the difference between the arbitrage portfolio frontier and the usual portfolio frontier.     

Constrained information minority game: How was the night at El Farol?

We introduce a variation of the El Farol Game in which the only players who surely know the outcome of the last turn of the game are those who actually attended the bar. Other players may receive this information with reduced probability. This information can be transmitted by another player who actually attended the bar in the last turn of the game or from the media. We show that since this game is not organized around the socially optimal point, arbitrage opportunities may arise. Therefore, we study how these opportunities can be exploited by an agent. An interesting application of this model is the market of goods being auctioned, such as cars being repossessed. The results obtained here seem to closely reflect the dynamics of this market in Brazil.     

Deterministic factors of stock networks based on cross-correlation in financial market

The stock market has been known to form homogeneous stock groups with a higher correlation among different stocks according to common economic factors that influence individual stocks. We investigate the role of common economic factors in the market in the formation of stock networks, using the arbitrage pricing model reflecting essential properties of common economic factors. We find that the degree of consistency between real and model stock networks increases as additional common economic factors are incorporated into our model. Furthermore, we find that individual stocks with a large number of links to other stocks in a network are more highly correlated with common economic factors than those with a small number of links. This suggests that common economic factors in the stock market can be understood in terms of deterministic factors.     

Toward a theory of marginally efficient markets

Empirical evidence suggests that even the most competitive markets are not strictly efficient. Price histories can be used to predict near future returns with a probability better than random chance. Many markets can be considered as favorable games, in the sense that there is a small probabilistic edge that smart speculators can exploit. We propose to identify this probability using conditional entropy concept. A perfect random walk has this entropy maximized, and departure from the maximal value represents a price history's predictability. We propose that market participants should be divided into two categories: producers and speculators. The former provides the negative entropy into the price, upon which the latter feed. We show that the residual negative entropy can never be arbitraged away: infinite arbitrage capital is needed to make the price a perfect random walk.     

Pólya number and first return of bursty random walk: Rigorous solutions

The recurrence properties of random walks can be characterized by Pólya number, i.e., the probability that the walker has returned to the origin at least once. In this paper, we investigate Pólya number and first return for bursty random walk on a line, in which the walk has different step size and moving probabilities. Using the concept of the Catalan number, we obtain exact results for first return probability, the average first return time and Pólya number for the first time. We show that Pólya number displays two different functional behavior when the walk deviates from the recurrent point. By utilizing the Lagrange inversion formula, we interpret our findings by transferring Pólya number to the closed-form solutions of an inverse function. We also calculate Pólya number using another approach, which corroborates our results and conclusions. Finally, we consider the recurrence properties and Pólya number of two variations of the bursty random walk model.     

Walks on Apollonian networks

We carry out comparative studies of random walks on deterministic Apollonian networks (DANs) and random Apollonian networks (RANs). We perform computer simulations for the mean first-passage time, the average return time, the mean-square displacement, and the network coverage for the unrestricted random walk. The diffusions both on DANs and RANs are proved to be sublinear. The effects of the network structure on the dynamics and the search efficiencies of walks with various strategies are also discussed. Contrary to intuition, it is shown that the self-avoiding random walk, which has been verified as an optimal local search strategy in networks, is not the best strategy for the DANs in the large size limit.