No-arbitrage Property Research Articles

Parameters identification for an inverse problem arising from a binary option using a Bayesian inference approach

No-arbitrage property provides a simple method for pricing financial derivatives. However, arbitrage opportunities exist in various fields, even for a very short time. By knowing that an arbitrage property exists, we can adopt a financial trading strategy. This paper investigates the inverse option problems (IOP) in the backward parabolic equation with a suitable initial condition in financial markets. We identify the coefficients of this problem from the measured data and attempt to find arbitrage opportunities in financial markets using a Bayesian inference approach, which is presented as an IOP solution. The posterior probability density function of the parameters is computed from the measured data. The statistics of the unknown parameters are estimated by a Markov Chain Monte Carlo (MCMC) algorithm, which exploits the posterior state space. The efficient sampling strategy of the MCMC algorithm enables us to solve inverse problems by the Bayesian inference technique. Our numerical results indicate that the Bayesian inference approach can simultaneously estimate the unknown trend and volatility coefficients from the artificial measured data and the real financial market data.

Option pricing generators

We characterize a class of option pricing models by their algebraic structure. Option prices are monoids, that is operators endowed with the commutativity and associativity property and an identity element. If the price of the underlying asset is bounded, the operator corresponds to the concept of t-conorm, while if it is defined on the positive real line the operator is a pseudo-addition. These operators have the same no-arbitrage properties as the classical option pricing models, but are also associative. Each model in this class is characterized by a univariate increasing function that is defined the generator of the model. The generator encodes a synthetic representation of the probability structure of the underlying asset. We provide no arbitrage conditions for the generators and practical guidelines to construct them.

Valuation of no-negative-equity guarantees with a lower reflecting barrier

AbstractIf the general level of house prices falls a long way, policymakers may introduce new policies which seek to support prices. This paper considers the effect of such interventions on the valuation of no-negative-equity guarantees (NNEG) in equity release mortgages. I model interventions by a reflecting barrier expressed as a fraction of the current level of house prices. Reflection at the barrier is instantaneous, so the no-arbitrage property is preserved, and hence risk-neutral valuation of NNEG is possible. The reflecting barrier can alternatively be justified as a representation of the different economic nature of the underlying housing (and particularly freehold land) assets in NNEG valuations, compared with the underlying equity assets in many other option valuations.

Prospective strict no-arbitrage and the fundamental theorem of asset pricing under transaction costs

In discrete time markets with proportional transaction costs, Schachermayer (2004) shows that robust no-arbitrage is equivalent to the existence of a strictly consistent price system. In this paper, we introduce the concept of prospective strict no-arbitrage that is a variant of the strict no-arbitrage property from Kabanov, R\'asonyi, and Stricker (2002). The prospective strict no-arbitrage condition is slightly weaker than robust no-arbitrage, and it implies that the set of portfolios attainable from zero initial endowment is closed in probability. A weak version of prospective strict no-arbitrage turns out to be equivalent to the existence of a consistent price system. In contrast to the fundamental theorem of asset pricing of Schachermayer (2004), the consistent frictionless prices may lie on the boundary of the bid-ask spread. On the technical level, a crucial difference to Schachermayer (2004) and Kabanov-R\'asonyi-Stricker (2003) is that we prove closedness without having at hand that the null-strategies form a linear space.

Large Financial Markets, Discounting, and No Asymptotic Arbitrage

For a large financial market (which is a sequence of usual, “small” financial markets), we introduce and study a concept of no asymptotic arbitrage (of the first kind) which is invariant under discounting. We give two dual characterisations of this property in terms of (1) martingale-like properties for each small market plus (2) a contiguity property, along the sequence of small markets, of suitably chosen “generalised martingale measures”. Our results extend the work of Rokhlin and of Klein/Schachermayer and Kabanov/Kramkov to a discounting-invariant framework. We also show how a market on [0,∞) can be viewed as a large financial market and how no asymptotic arbitrage, both classic and in our new sense, then relates to no-arbitrage properties directly on [0,∞).

How local in time is the no-arbitrage property under capital gains taxes?

In frictionless financial markets, no-arbitrage is a local property in time. This means that a discrete time model is arbitrage-free if and only if there does not exist a one-period-arbitrage. With capital gains taxes, this equivalence fails. For a model with a linear tax and one non-shortable risky stock, we introduce the concept of robust local no-arbitrage (RLNA) as the weakest local condition which guarantees dynamic no-arbitrage. Under a sharp dichotomy condition, we prove (RLNA). Since no-one-period-arbitrage is necessary for no-arbitrage, the latter is sandwiched between two local conditions, which allows us to estimate its non-locality. Furthermore, we construct a stock price process such that two long positions in the same stock hedge each other. This puzzling phenomenon that cannot occur in arbitrage-free frictionless markets (or markets with proportional transaction costs) is used to show that no-arbitrage alone does not imply the existence of an equivalent separating measure if the probability space is infinite. Finally, we show that the model with a linear tax on capital gains can be written as a model with proportional transaction costs by introducing several fictitious securities.

No-arbitrage and optimal investment with possibly non-concave utilities: a measure theoretical approach

We consider a discrete-time financial market model with finite time horizon and investors with utility functions defined on the non-negative half-line. We allow these functions to be random, non-concave and non-smooth. We use a dynamic programming framework together with measurable selection arguments to establish both the characterisation of the no-arbitrage property for such markets and the existence of an optimal portfolio strategy for such investors.

The microstructural foundations of leverage effect and rough volatility

We show that typical behaviors of market participants at the high frequency scale generate leverage effect and rough volatility. To do so, we build a simple microscopic model for the price of an asset based on Hawkes processes. We encode in this model some of the main features of market microstructure in the context of high frequency trading: high degree of endogeneity of market, no-arbitrage property, buying/selling asymmetry and presence of metaorders. We prove that when the first three of these stylized facts are considered within the framework of our microscopic model, it behaves in the long run as a Heston stochastic volatility model, where leverage effect is generated. Adding the last property enables us to obtain a rough Heston model in the limit, exhibiting both leverage effect and rough volatility. Hence we show that at least part of the foundations of leverage effect and rough volatility can be found in the microstructure of the asset.

Estimating the tail shape parameter from option prices

In this paper, a method to estimate the tail shape parameter of the risk neutral density from option prices is developed. Closed form pricing formulas for out-of-the-money European style options are derived. The pricing formulas satisfy many well known model-free no-arbitrage properties for the options. The focus is only on the tails of the risk neutral density and not on the entire body of the density as many works have already done this. The method is quite general, and applies to a large class of risk neutral densities. Our method can be used without interpolating the implied volatility, or even the knowledge of the current index value or the dividend yield or the risk free rate. This is in contrast to every other method that attempts to estimate the risk neutral density. A case study using S&P 500 Index options is given. In particular, the estimation of the tail shape of S&P 500 index just prior to the market turmoil of the September 2008 shows a thickening of the left tail but a thinning in the midst of the turmoil.

Negative interest rates effects on option pricing: Back to basics?

We provide the first formal investigation of the consequences of negative interest rates in the Eurozone on the pricing of interest rate options. Since the money market rates settled in negative territory and other market segments experienced negative yields, the broader financial community has had to face an unknown environment. The well-known Black–Scholes (BS) framework has become unfeasible for interest rate option valuation. First of all, no-arbitrage properties are breached, allowing arbitrage opportunities. More, the BS framework’s assumption of a log-normal distribution of the underlying rates does not stand with negative interest rates. We argue that the most notable approach which allows interest rate option pricing is [Bachelier, L (1900). Théorie de la speculation, 3rd Annales scientifiques de l’École Normale Supēérieure 17, 21–86.], which assumes a normal distribution of the underlying rates. We demonstrate that the Bachelier model represents an answer to the critical issues that are raised in our study. Still, we highlight that it is far from being an accurate pricing model. Our research aims to light up an intense debate about alternative solutions among academics, financial professionals and institutions, and policy makers.

Robust no-free lunch with vanishing risk, a continuum of assets and proportional transaction costs

We propose a continuous time model for financial markets with proportional transaction costs and a continuum of risky assets. This is motivated by bond markets in which the continuum of assets corresponds to the continuum of possible maturities. Our framework is well adapted to the study of no-arbitrage properties and related hedging problems. In particular, we extend the Fundamental Theorem of Asset Pricing of Guasoni, Rásonyi and Lépinette (2012) which concentrates on the one dimensional case. Namely, we prove that the Robust No Free Lunch with Vanishing Risk assumption is equivalent to the existence of a Strictly Consistent Price System. Interestingly, the presence of transaction costs allows a natural definition of trading strategies and avoids all the technical and un-natural restrictions due to stochastic integration that appear in bond models without friction. We restrict to the case where exchange rates are continuous in time and leave the general càdlàg case for further studies.

On free lunches in random walk markets with short-sale constraints and small transaction costs, and weak convergence to Gaussian continuous-time processes

Abstract. This paper considers a sequence of discrete-time random walk mar-kets with a single risky asset, and gives conditions for the existence of arbitrageopportunitiesorfreeluncheswithvanishingrisk,oftheformofwaitingtobuyandsellingthenextperiod,withnoshorting,andfurthermoreforweakconvergenceoftherandomwalktoaGaussiancontinuous-timestochasticprocess. TheconditionsaregivenintermsofthekernelrepresentationwithrespecttoordinaryBrownianmotion and the discretisation chosen. Arbitrage examples are established wherethecontinuousanalogueisarbitrage-freeundersmalltransactioncosts–includingfor the semimartingale modifications of fractional Brownian motion suggested intheseminalRogers(1997)articleprovingarbitrageinfBmmodels.Keywords: Stockpricemodel,randomwalk,Gaussianprocesses,weakconver-gence,freelunchwithvanishingrisk,arbitrage,transactioncostsMSC(2010): 60B10,60E05,60F05,91G10JELclassification: C61,D53,D81,G11 0 Introduction As well known since Rogers [7], fractional Brownian motions is a troublesome model foruncertainty in price processes, as fBm will introduce arbitrage opportunities to canonicalmodels where the ordinary Brownian motion does not. To remedy this, Rogers proposesa parametrised semimartingale modification, whose moving average kernel converges to thefBm’s – in particular, the no-arbitrage property is not preserved under this limit. For thepurposesofstudyingmarketvaluegains(orlosses),onecanhoweverarguethatthepointwise

Extension of Normed Call Prices for Negative Strikes and Forwards

After the recent financial crisis (also known as the Credit Crunch'') it was observed that some market data exhibited non-zero probability for negative forward rates. In particular, options with zero strikes were quoted with non-zero prices. This led to the requirement to model and interpret option prices for negative strikes. In this note, irrespective of the asset classes, we provide a standardization technique for market call prices and derive the no-arbitrage properties that allow for negative strikes and forwards. Our approach works with non-deterministic discounting rates and multiple discounting curves that are increasingly becoming standard.

Robust No-Free Lunch with Vanishing Risk, a Continuum of Assets and Proportional Transaction Costs

We propose a continuous time model for financial markets with proportional transactions costs and a continuum of risky assets. This is motivated by bond markets in which the continuum of assets corresponds to the continuum of possible maturities. Our framework is well adapted to the study of no-arbitrage properties and related hedging problems. In particular, we extend the Fundamental Theorem of Asset Pricing of Guasoni, Rasonyi and Lepinette (2012) which concentrates on the one dimensional case. Namely, we prove that the Robust No Free Lunch with Vanishing Risk assumption is equivalent to the existence of a Strictly Consistent Price System. Interestingly, the presence of transaction costs allows a natural definition of trading strategies and avoids all the technical and un-natural restrictions due to stochastic integration that appear in bond models without friction. We restrict to the case where exchange rates are continuous in time and leave the general cadlag case for further studies.

Stability of no-arbitrage property under model uncertainty

We study the stability of the no-arbitrage property under model uncertainty. We measure model uncertainty with the total variation distance of underlying probability distributions. We show that sufficiently small changes of the underlying probability distribution preserve the no-arbitrage property of the financial market model.

Estimating the Tail Shape Parameter from Option Prices

In this paper, a method to estimate the tail shape parameter of the risk neutral density from option prices is developed. Closed form pricing formulas for out-of-the-money European style options are derived. The pricing formulas satisfy many well known model-free no-arbitrage properties for the options. The focus is only on the tails of the risk neutral density and not on the entire body of the density as many works have already done this. The method is quite general, and applies to a large class of risk neutral densities. Our method can be used without interpolating the implied volatility, or even the knowledge of the current index value or the dividend yield or the risk free rate. This is in contrast to every other method that attempts to estimate the risk neutral density. A case study using S&P 500 Index options is given. In particular, the estimation of the tail shape of S&P 500 index just prior to the market turmoil of the September 2008 shows a thickening of the left tail but a thinning in the midst of the turmoil.

THE COMPATIBLE BOND-STOCK MARKET WITH JUMPS

We construct a bond-stock market composed of d stocks and many bonds with jumps driven by general marked point process as well as by an ℝn-valued Wiener process. By composing these tools we introduce the concept of a compatible bond-stock market and give a necessary and sufficient condition for this property. We study no-arbitrage properties of the composed market where a compatible bond-stock market is arbitrage-free both for the bonds market and for the stocks market. We then turn to an incomplete compatible bond-stock market and give a necessary and sufficient condition for a compatible bond-stock market to be incomplete. In this market we consider the mean-variance hedging in the special situation where both B(u, T) and eG(u, y, T)-1 are quadratic functions of T - u. So, we need to extend the notion of a variance-optimal martingale (VOM) as in Xiong and Kohlmann (2009) to the more general market. By introducing two virtual stocks [Formula: see text], we prove that the VOM for the bond-stock market is the same as the VOM for the new stock market [Formula: see text]. The mean-variance hedging problem in this incomplete bond-stock market for a contingent claim [Formula: see text] is solved by deriving an explicit solution of the optimal measure-valued strategy and the optimal cost induced by the optimal strategy of MHV for the stocks [Formula: see text] is computed.

Consistent price systems and arbitrage opportunities of the second kind in models with transaction costs

In contrast with the classical models of frictionless financial markets, market models with proportional transaction costs, even satisfying usual no-arbitrage properties, may admit arbitrage opportunities of the second kind. This means that there are self-financing portfolios with initial endowments lying outside the solvency region but ending inside. Such a phenomenon was discovered by M. Rasonyi in the discrete-time framework. In this note, we consider a rather abstract continuous-time setting and prove necessary and sufficient conditions for a property which we call no free lunch of the second kind, NFL2. We provide a number of equivalent conditions elucidating, in particular, the financial meaning of the property B which appeared as an indispensable “technical” hypothesis in previous papers on hedging (superreplication) of contingent claims under transaction costs. We show that it is equivalent to another condition on the “richness” of the set of consistent price systems, close to the condition PCE introduced by Rasonyi. In the last section, we deduce the Rasonyi theorem from our general result by using specific features of discrete-time models.

Asset-liability management for Czech pension funds using stochastic programming

It is possible to model a wide range of portfolio management problems using stochastic programming. This approach requires the generation of input scenarios and probabilities, which represent the evolution of the return on investment, the stream of liabilities and other random phenomena of the problem and respect the no-arbitrage properties. The quality of the recommended capital allocation depends on the quality of the input scenarios and a validation of results is necessary. Appropriate scenario generation techniques and output analysis methods are described in the context of defined contribution pension fund and applied to the specific model of a Czech pension fund. The numerical results indicate various components that influence the recommended investment decisions and the fund’s achievements. In particular, the initial balance sheet position of the pension fund is important for the optimal investment strategy because of the accounting rules embedded in the model and tracking of both the market and purchasing value of assets.

No arbitrage conditions and liquidity

We extend the fundamental theorem of asset pricing to the case of markets with liquidity risk. Our results generalize, when the probability space is finite, those obtained by Kabanov et al. [Kabanov, Y., Stricker, C., 2001. The Harrison-Pliska arbitrage pricing theorem under transaction costs. Journal of Mathematical Economics 35, 185–196; Kabanov, Y., Rásonyi, M., Stricker, C., 2002. No-arbitrage criteria for financial markets with efficient friction. Finance and Stochastics 6, 371–382; Kabanov, Y., Rásonyi, M., Stricker, C., 2003. On the closedness of sums of convex cones in L 0 and the robust no-arbitrage property. Finance and Stochastics] and by Schachermayer [Schachermayer, W., 2004. The fundamental theorem of asset pricing under poportional transaction costs in finite discrete time. Mathematical Finance 14 (1), 19–48] for markets with proportional transaction costs. More precisely, we restate the notions of consistent and strictly consistent price systems and prove their equivalence to corresponding no arbitrage conditions. We express these results in an analytical form in terms of the subdifferential of the so-called liquidation function. We conclude the paper with a hedging theorem.