Coherent and convex monetary risk measures for unbounded càdlàg processes

Assume that the random future evolution of values is modelled in continuous time. Then, a risk measure can be viewed as a functional on a space of continuous-time stochastic processes. In this paper we study coherent and convex monetary risk measures on the space of all càdlàg processes that are adapted to a given filtration. We show that if such risk measures are required to be real-valued, then they can only depend on a stochastic process in a way that is uninteresting for many applications. Therefore, we allow them to take values in ( −∞, ∞]. The economic interpretation of a value of ∞ is that the corresponding financial position is so risky that no additional amount of money can make it acceptable. The main result of the paper gives different characterizations of coherent or convex monetary risk measures on the space of all bounded adapted càdlàg processes that can be extended to coherent or convex monetary risk measures on the space of all adapted càdlàg processes. As examples we discuss a new approach to measure the risk of an insurance company and a coherent risk measure for unbounded càdlàg processes induced by a so called m-stable set.Due to errors during the typesetting process, this article was published incorrectly in Finance Stoch 9(3):369–387 (2005). The address of the first author was printed incorrectly, and in the whole paper the angular brackets were misprinted as [ ]. The complete corrected article is given here. The online version of the original paper can be found at: http://dx.doi.org/10.1007/s00780-004-0150-7     

A super-replication theorem in Kabanov’s model of transaction costs

We prove a general version of the super-replication theorem, which applies to Kabanov’s model of foreign exchange markets under proportional transaction costs. The market is described by a matrix-valued càdlàg bid-ask process evolving in continuous time. We propose a new definition of admissible portfolio processes as predictable (not necessarily right- or left- continuous) processes of finite variation related to the bid-ask process by economically meaningful relations. Under the assumption of existence of a strictly consistent price system (SCPS), we prove a closedness property for the set of attainable vector-valued contingent claims. We then obtain the super-replication theorem as a consequence of that property, thus generalizing to possibly discontinuous bid-ask processes analogous results obtained by Kabanov (Financ. Stoch. 3, 237–248, 1999), Kabanov and Last (Math. Financ. 12, 63–70, 2002) and Kabanov and Stricker (Advances in Finance and Stochastics: Essays in Honour of Dieter Sondermann, pp 125–136, 2002). Rásonyi’s counter-example (Lecture Notes in Mathematics 1832, 394–398, 2003) served as an important motivation for our approach.     

A note on the condition of no unbounded profit with bounded risk

As a corollary to Delbaen and Schachermayer’s fundamental theorem of asset pricing (Delbaen in Math. Ann. 300:463–520, 1994; Stoch. Stoch. Rep. 53:213–226, 1995; Math. Ann. 312:215–250, 1998), we prove, in a general finite-dimensional semimartingale setting, that the no unbounded profit with bounded risk (NUPBR) condition is equivalent to the existence of a strict sigma-martingale density. This generalizes the continuous-path result of Choulli and Stricker (Séminaire de Probabilités XXX, pp. 12–23, 1996) to the càdlàg case and extends the recent one-dimensional result of Kardaras (Finance and Stochastics 16:651–667, 2012) to the multidimensional case. It also refines partially the second main result of Karatzas and Kardaras (Finance Stoch. 11:447–493, 2007) concerning the existence of an equivalent supermartingale deflator. The proof uses the technique of numéraire change.     

On the existence of shadow prices

For utility maximization problems under proportional transaction costs, it has been observed that the original market with transaction costs can sometimes be replaced by a frictionless shadow market that yields the same optimal strategy and utility. However, the question of whether or not this indeed holds in generality has remained elusive so far. In this paper, we present a counterexample which shows that shadow prices may fail to exist. On the other hand, we prove that short selling constraints are a sufficient condition to warrant their existence, even in very general multi-currency market models with possibly discontinuous bid–ask spreads.     

The fundamental theorem of asset pricing for continuous processes under small transaction costs

A version of the fundamental theorem of asset pricing is proved for continuous asset prices with small proportional transaction costs. Equivalence is established between: (a) the absence of arbitrage with general strategies for arbitrarily small transaction costs ${\varepsilon > 0}${\varepsilon > 0}, (b) the absence of free lunches with bounded risk for arbitrarily small transaction costs ${\varepsilon > 0}${\varepsilon > 0}, and (c) the existence of e{\varepsilon}-consistent price systems—the analogue of martingale measures under transaction costs—for arbitrarily small ${\varepsilon > 0}${\varepsilon > 0}. The proof proceeds through an explicit construction, as opposed to the usual separation arguments. The paper concludes comparing numéraire-free and numéraire-based notions of admissibility, and the corresponding martingale and local martingale properties for consistent price systems.     

Risk arbitrage and hedging to acceptability under transaction costs

The classical discrete-time model of proportional transaction costs relies on the assumption that a feasible portfolio process has solvent increments at each step. We extend this setting in two directions, allowing convex transaction costs and assuming that increments of the portfolio process belong to the sum of a solvency set and a family of multivariate acceptable positions, e.g. with respect to a dynamic risk measure. We describe the sets of superhedging prices, formulate several no (risk) arbitrage conditions and explore connections between them. In the special case when multivariate positions are converted into a single fixed asset, our framework turns into the no-good-deals setting. However, in general, the possibilities of assessing the risk with respect to any asset or a basket of assets lead to a decrease of superhedging prices and the no-arbitrage conditions become stronger. The mathematical techniques rely on results for unbounded and possibly non-closed random sets in Euclidean space.

     

The scaling limit of superreplication prices with small transaction costs in the multivariate case

Kusuoka (Ann. Appl. Probab. 5:198–221, 1995) showed how to obtain non-trivial scaling limits of superreplication prices in discrete-time models of a single risky asset which is traded at properly scaled proportional transaction costs. This article extends the result to a multivariate setup where the investor can trade in several risky assets. The \(G\)-expectation describing the limiting price involves models with a volatility range around the frictionless scaling limit that depends not only on the transaction costs coefficients, but also on the chosen complete discrete-time reference model.     

Rational expectations equilibrium with transaction costs in financial markets

We obtain a closed-form solution to a rational expectations equilibrium model with transaction costs in the framework of Grossman and Stiglitz [1980. American Economic Review 70, 543–566]. Individual private information incorporated into prices is reduced due to suppressed trading activities by transaction costs. The fraction of informed traders in equilibrium increases (decreases) with transaction costs when the costs are low (high). The informativeness of prices decreases with transaction costs.     

Dynamic Volatility Trading Strategies in the Currency Option Market

The conditional volatility of foreign exchange rates can be predicted using GARCH models or implied volatility extracted from currency options. This paper investigates whether these predictions are economically meaningful in trading strategies that are designed only to trade volatility risk. First, this article provides new evidence on the issue of information content of implied volatility and GARCH volatility in forecasting future variance. In an artificial world without transaction costs both delta-neutral and straddle trading stratgies lead to significant positive profits, regardless of which volatility prediction method is used. Specifically, the agent using the Implied Stochastic Volatility Regression method (ISVR) earns larger profits than the agent using the GARCH method. Second, it suggests that the currency options market is informationally efficient. After accounting for transaction costs, which are assumed to equal one percent of option prices, observed profits are not significantly differentfrom zero in most trading strategies. Finally, these strategies offered returns have higher Sharpe ratio and lower correlation with several major asset classes. Consequently, hedge funds and institutional investors who are seeking alternative “marketneutral” investment methods can use volatility trading to improvethe risk-return profile of their portfolio through diversification. This revised version was published online in November 2006 with corrections to the Cover Date.     

Market liquidity,closeout procedures and initial margin for CCPs*

ABSTRACT

Closeout procedures enable central counterparties (CCPs) to respond to events that challenge the continuity of their normal operations, most frequently triggered by the default of one or more clearing members. The procedures typically entail three main phases: splitting, hedging, and liquidation. Together, these ensure the regularity of the settlement process through the prudent and orderly liquidation of the defaulters’ portfolios. Traditional approaches to CCPs’ margin requirements typically assume a simple closeout profile, not accounting for the ‘real life’ constraints embedded in the management of a default. The paper proposes an approach to assess how distinct closeout strategies may expose a CCP to different sets of risks and costs taking into account real-life frictions. The proposed approach enables the evaluation of a full spectrum of hedging strategies and the assessment of the trade-offs between the risk-reducing benefits of hedging and the transaction costs associated with it. Using an unexplored set of transactional level data, the proposed framework is evaluated assuming the hypothetical default of a real CCP clearing member. We consider the worst-case loss of a large interest rate swap portfolio observed over the past 10 years (i.e. 2005–2015) and show that an efficient hedging strategy which minimises risk may not be optimal when transaction costs are taken into account. The empirical analysis suggests that transaction costs are a significant factor and should be accounted for when designing a hedging strategy. Specifically, it is shown that the risk-reducing benefits arising from more tailored hedging strategies may introduce higher transaction costs, and therefore may change the effectiveness of the strategies.     

Liquidity risk,price impacts and the replication problem

We extend a linear version of the liquidity risk model of Çetin et al. (Finance Stoch. 8:311–341, 2004) to allow for price impacts. We show that the impact of a market order on prices depends on the size of the transaction and the level of liquidity. We obtain a simple characterization of self-financing trading strategies and a sufficient condition for no arbitrage. We consider a stochastic volatility model in which the volatility is partly correlated with the liquidity process and show that, with the use of variance swaps, contingent claims whose payoffs depend on the value of the asset can be approximately replicated in this setting. The replicating costs of such payoffs are obtained from the solutions of BSDEs with quadratic growth, and analytical properties of these solutions are investigated.     

The no-arbitrage condition and financial markets with transaction costs and heterogeneous information: The bid-ask spread

This paper investigates the implications of the no-arbitrage (NA) condition in markets with transaction costs and heterogeneous information. Dermody and Prisman (1993) showed that, in financial markets with increasing marginal transaction costs, the NA condition is equivalent to the existence of a valuation operator. They explore the exact dependence of this operator on the structure of transaction costs. They show that equilibrium prices in the “corresponding” perfect markets plus a certain factor determine the valuation operator in markets with increasing marginal transaction costs. This paper emphasizes that their result is applicable to financial markets with decreasing marginal transaction costs. Furthermore, this paper shows that, in financial markets with transaction costs and heterogeneous information, the NA condition imposes a constraint on the bid-ask spread.     

The valuation of American call options and the expected ex-dividend stock price decline

This study focuses on the ex-dividend stock price decline implicit within the valuation of American call options on dividend-paying stocks. The Roll (1977) American call option pricing formula and the observed structure of CBOE call option transaction prices are used to infer the expected ex-dividend stock price decline as a proportion of the amount of the dividend. The relative decline is shown to be not meaningfully different from one, confirming some recent evidence from studies which examined stock prices in the days surrounding ex-dividend.     

Intraday pairs trading strategies on high frequency data: the case of oil companies

This paper introduces novel ‘doubly mean-reverting’ processes based on conditional modelling of model spreads between pairs of stocks. Intraday trading strategies using high frequency data are proposed based on the model. This model framework and the strategies are designed to capture ‘local’ market inefficiencies that are elusive for traditional pairs trading strategies with daily data. Results from real data back-testing for two periods show remarkable returns, even accounting for transaction costs, with annualized Sharpe ratios of 3.9 and 7.2 over the periods June 2013–April 2015 and 2008, respectively. By choosing the particular sector of oil companies, we also confirm the observation that the commodity price is the main driver of the share prices of commodity-producing companies at times of spikes in the related commodity market.     

Endogenous informed trading in the presence of trading costs: Theory and evidence

The basic premise of the model we propose is that market frictions (trading costs) force traders with market-wide information to strategically choose which securities to trade in. We study the effect of recognizing trading costs on the choices of informed traders and the resulting statistical properties of security prices. Specifically, we show that (1) stocks with intermediate β's have the least informative prices, even though they are traded by the greatest number of informed traders; (2) for high β securities, the contemporaneous correlation of prices is close to the correlation in fundamental values; (3) a security with a higher β, higher volume of liquidity trading and lower idiosyncratic variance is more likely to lead another security. With market capitalization as a proxy for the level of liquidity trading, these specific predictions of the model on the lead–lag relationship are also shown to be strongly supported by the data.     

Transaction cost optimization for online portfolio selection

To improve existing online portfolio selection strategies in the case of non-zero transaction costs, we propose a novel framework named Transaction Cost Optimization (TCO). The TCO framework incorporates the L1 norm of the difference between two consecutive allocations together with the principles of maximizing expected log return. We further solve the formulation via convex optimization, and obtain two closed-form portfolio update formulas, which follow the same principle as Proportional Portfolio Rebalancing (PPR) in industry. We empirically evaluate the proposed framework using four commonly used data-sets. Although these data-sets do not consider delisted firms and are thus subject to survival bias, empirical evaluations show that the proposed TCO framework may effectively handle reasonable transaction costs and improve existing strategies in the case of non-zero transaction costs.     

HIGH‐FREQUENCY EXCHANGE‐RATE PREDICTION WITH AN ARTIFICIAL NEURAL NETWORK

This paper examines how market microstructure variables can be used to forecast foreign exchange (FX) rates at frequencies of one to several minutes. We use a unique FX dataset of global inter‐dealer electronic transactions and applied the artificial neural network (ANN) as the predicting model. The immediately preceding bid and ask prices are significant factors in these predictions, which is in keeping with market microstructure theory. These microstructure factors have not been tested in an ANN model before. High‐frequency trading strategies based on the ANN model are shown to be profitable even when transaction costs are included. Copyright © 2012 John Wiley & Sons, Ltd.     

Asymptotic arbitrage with small transaction costs

We give characterizations of asymptotic arbitrage of the first and second kind and of strong asymptotic arbitrage for a sequence of financial markets with small proportional transaction costs λ _n on market n, in terms of contiguity properties of sequences of equivalent probability measures induced by λ _n -consistent price systems. These results are analogous to the frictionless case; compare (Kabanov and Kramkov in Finance Stoch. 2:143–172, 1998; Klein and Schachermayer in Theory Probab. Appl. 41:927–934, 1996). Our setting is simple, each market n contains two assets. The proofs use quantitative versions of the Halmos–Savage theorem (see Klein and Schachermayer in Ann. Probab. 24:867–881, 1996) and a monotone convergence result for nonnegative local martingales. Moreover, we study examples of models which admit a strong asymptotic arbitrage without transaction costs, but with transaction costs λ _n >0 on market n; there does not exist any form of asymptotic arbitrage. In one case, (λ _n ) can even converge to 0, but not too fast.     

Inferring the Components of the Bid-Ask Spread: Theory and Empirical Tests

The relation between the square of the quoted bid-ask spread and two serial covariances—the serial covariance of transaction returns and the serial covariance of quoted returns—is modeled as a function of the probability of a price reversal, π, and the magnitude of a price change, ?, where ? is stated as a fraction of the quoted spread. Different models of the spread are contrasted in terms of the parameters, π and ?. Using data on the transaction prices and price quotations for NASDAQ/NMS stocks, π and ? are estimated and the relative importance of the components of the quoted spread—adverse information costs, order processing costs, and inventory holding costs—is determined.