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.     

Fundamental strength strategy: The role of investor sentiment versus limits to arbitrage

This paper evaluates the return predictability of fundamental strength in a two-dimensional framework that considers both investor sentiment and limits to arbitrage simultaneously. Sentiment and limits to arbitrage have independent and overlapping explanatory power on the return predictability of fundamental strength. The return predictability of fundamental strength is more pronounced among stocks with high arbitrage costs following high sentiment. Among stocks with low arbitrage costs, the fundamental strength strategy is profitable only following high sentiment. However, among stocks with high arbitrage costs, the same strategy can earn economically and statistically significant profits even following low sentiment. Consistent with Miller (1977), we emphasize the interaction of sentiment and limits to arbitrage on stock valuation.     

Fundamental theorems of asset pricing for piecewise semimartingales of stochastic dimension

This paper has two purposes. The first is to extend the notions of an n-dimensional semimartingale and its stochastic integral to a piecewise semimartingale of stochastic dimension. The properties of the former carry over largely intact to the latter, avoiding some of the pitfalls of infinite-dimensional stochastic integration. The second purpose is to extend two fundamental theorems of asset pricing (FTAPs): the equivalence of no free lunch with vanishing risk to the existence of an equivalent sigma-martingale measure for the price process, and the equivalence of no arbitrage of the first kind to the existence of an equivalent local martingale deflator for the set of nonnegative wealth processes.     

Asymptotic arbitrage and numéraire portfolios in large financial markets

This paper deals with the notion of a large financial market and the concepts of asymptotic arbitrage and strong asymptotic arbitrage (both of the first kind) introduced in Probab. Theory Appl. 39, 222–229 (1994) and in Finance Stoch. 2, 143–172 (1998). We show that the arbitrage properties of a large market are completely determined by the asymptotic behavior of the sequence of the numéraire portfolios related to small markets. The obtained criteria can be expressed in terms of contiguity, entire separation, and Hellinger integrals, provided that these notions are extended to sub-probability measures. As examples, we consider market models on finite probability spaces, semimartingale models, and diffusion models. We also examine a discrete-time infinite horizon market model with one log-normal stock. This work was supported by Southern Federal University, grant No. 26 “Mathematical Finance” and by RFBR, grant 07-01-00520.     

Asymptotic arbitrage in large financial markets

A large financial market is described by a sequence of standard general models of continuous trading. It turns out that the absence of asymptotic arbitrage of the first kind is equivalent to the contiguity of sequence of objective probabilities with respect to the sequence of upper envelopes of equivalent martingale measures, while absence of asymptotic arbitrage of the second kind is equivalent to the contiguity of the sequence of lower envelopes of equivalent martingale measures with respect to the sequence of objective probabilities. We express criteria of contiguity in terms of the Hellinger processes. As examples, we study a large market with asset prices given by linear stochastic equations which may have random volatilities, the Ross Arbitrage Pricing Model, and a discrete-time model with two assets and infinite horizon. The suggested theory can be considered as a natural extension of Arbirage Pricing Theory covering the continuous as well as the discrete time case.     

No arbitrage of the first kind and local martingale numéraires

A supermartingale deflator (resp. local martingale deflator) multiplicatively transforms nonnegative wealth processes into supermartingales (resp. local martingales). A supermartingale numéraire (resp. local martingale numéraire) is a wealth process whose reciprocal is a supermartingale deflator (resp. local martingale deflator). It has been established in previous works that absence of arbitrage of the first kind (\(\mbox{NA}_{1}\)) is equivalent to the existence of the (unique) supermartingale numéraire, and further equivalent to the existence of a strictly positive local martingale deflator; however, under \(\mbox{NA}_{1}\), a local martingale numéraire may fail to exist. In this work, we establish that under \(\mbox{NA}_{1}\), a supermartingale numéraire under the original probability \(P\) becomes a local martingale numéraire for equivalent probabilities arbitrarily close to \(P\) in the total variation distance.     

Implied Foreign Exchange Risk Premia

This paper uses implied volatilities from foreign exchange option prices and the results of no‐arbitrage theory to estimate foreign exchange risk premia. In particular, under the assumption of no‐arbitrage, the foreign exchange risk premium is driven by the difference between investors’ market prices of risk in the two currencies. In an international economy with three currencies, sterling, US dollar and Deutschemark, we can use the information on implied volatilities of the three cross rates to derive estimates of implied or ex ante market prices of risk and of foreign exchange risk premia. The foreign exchange risk premia estimates are then compared to survey‐based risk premia.     

Efficient monetary allocations and the illiquidity of bonds

We construct a monetary economy with heterogeneity in discounting and consumption risk. Agents can insure against this risk with money and nominal government bonds, but all trades must be monetary. We demonstrate that a deflationary policy à la Friedman cannot sustain the constrained-efficient allocation as no-arbitrage imposes too stringent a bound on the return money can pay. The constrained-efficient allocation can be sustained when bonds have positive yields and, under certain conditions, only if they are illiquid. Illiquidity, meaning that bonds cannot be transformed into consumption as easily as cash, is necessary to eliminate arbitrage opportunities due to disparities in shadow interest rates.     

Factorization of European and American option prices under complete and incomplete markets

In a standard option-pricing model, with continuous-trading and diffusion processes, this paper shows that the price of one European-style option can be factorized into two intuitive components: One robust, X₀, which is priced by arbitrage, and a second, Π₀, which depends on a risk orthogonal to the traded securities. This result implies the following: (1) In an incomplete market, these parts represent the price of a hedging portfolio, which is unique, and a premium, which depends only on the risk premiums associated with the residual risk, respectively. (2) In a complete market, it allows factoring the contribution of the different sources of risk to the final option price. For example, in a stochastic volatility model, we can quantify the impact on the option price of volatility risk relative to market risk, Π₀ and X₀, respectively. Hence, certain misspricings in option markets can be directly related to the premium, Π₀. (3) Moreover, these results extend to American securities, which have a third component – an additional early-exercise premium.     

Empirical investigation of stock index futures market efficiency: the case of the Athens Derivatives Exchange

Pricing and trading practices in the Athens Derivatives Exchange, a newly established derivatives market, result in significant futures arbitrage profit opportunities for low-cost traders. We find that a large part of the mispricing is due to transaction costs, but additional factors, such as anticipated volatility and time to maturity, also contribute. Ex ante tests reveal significant arbitrage opportunities that could have been exploited up to 30 min after they had been identified. All different tests employed indicate that the derivatives market was inefficient during its early trading history because arbitrage opportunities persisted even after other market impact costs were taken into consideration.     

Arbitrage boundaries,treasury bills,and covered interest parity

This paper uses the covered interest parity relationship to define upper and lower arbitrage boundaries on domestic Treasury bill (t-bill) rates. Adapting the methodology for estimating frontier functions, empirical tests are specified to determine whether the arbitrage boundaries provide a binding constraint on the behavior of the domestic t-bill rate. Empirical evidence is presented for the covered Canadian t-bill rate which indicates that at various times the US t-bill rate has been a binding lower boundary and the Euro–US ollar deposit rate has provided a binding upper boundary.     

Evidence on the arbitrage efficiency of SPI index futures and options markets

This paper is concerned with arbitrage opportunities in the futures and futures option contracts traded on the Sydney Futures Exchange (SFE) within a put-call-futures-parity (PCFP) framework. Tick-by-tick transaction price data are employed so that the futures contracts, the call futures options and the put futures options can be matched within a one-minute interval. This paper also takes into account the realistic transaction costs that an arbitrager has to incur, including the implicit bid-ask spread. A thorough ex post analysis is first carried out. The results reveal a significant number of violations of the PCFP in the sample. Ex ante tests are then conducted whereby ex post profitable arbitrage strategies, signified by the matched trios of futures, put and call contracts, are executed with lags up to 3 min. The ex ante results are similar to the ex post results. However, further analysis reveals that the exploitability of the identified arbitrage opportunities is very limited due to the small trading volumes of the futures and options contracts. Thus, we conclude that there is no strong evidence against the arbitrage efficiency between the SPI index futures and options markets in Australia.      

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. Rásonyi 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 Rásonyi. In the last section, we deduce the Rásonyi theorem from our general result by using specific features of discrete-time models.     

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.     

Arbitrage risk and market efficiency

This article explores arbitràge risk and models a testable hypothesis for studies in the treasury bill futures market efficiency. The modern mean-variance theory applied to a hedged arbitrage portfolio is used for the analysis. For a given expected arbitrage profit, we derive minimum variance arbitrage (MVA) conditions. A minimum variance arbitrage line (MVAL) is then derived to show the risk-return tradeoff for arbitrage. Market efficiency conditions are discussed by taking into account arbitrage risk along with bid-ask spreads. The analysis in this study helps explain the puzzle of inefficiencies in the T-bill futures market. Because refinancing and variation margin (due to marking-to-market) are required for arbitrage using futures trading in general, our ex ante arbitrage model using the case of T-bill futures can be applied to other futures markets.     

Tax arbitrage in government bonds: A suggested methodology with policy implications

This paper develops a geometric methodology with which to analyze the no-arbitrage condition, with special reference to tax arbitrage in government bonds. Using this methodology, it is shown that a country's bond-issuing authority might be able to painlessly avoid market equilibria which is likely to induce tax arbitrage activities. The simple bond-issuing policy which will achieve this goal is identified, and its limitations are discussed. An examination of the Canadian and Israeli bond markets shows that adopting the prescribed bond-issuing policy does not meaningfully impinge on the bond-issuing authority's ability to sell bonds.     

International tax arbitrage,financial parity conditions and preferential capital gains taxation

Using a finite-horizon general equilibrium model with uncertainty and money, we characterize situations where tax arbitrage opportunities may arise for international portfolio investors in an economy with heterogeneous capital income taxation when interest income and capital gains/losses are taxed differentially for some agents. We derive tax-modified uncovered interest parity conditions, Fisher conditions and forward prices similar to the no-tax ones, but augmented by tax-induced ‘risk-premium’ terms; covered interest parity and Fisher conditions remain unaffected by the introduction of capital income taxes as we bound tax-based arbitrage without restricting arbitrage per se.     

Term Structure Models: A Perspective from the Long Rate

Term structure models based on dynamic asset-pricing theory are discussed by taking a perspective from the long rate. This paper partially answers two questions about the asymptotic behavior of yields on default-free zero-coupon bonds: in frictionless markets having no arbitrage, what should the behavior be; and, in known term structure models, what can the behavior be.

In frictionless markets having no arbitrage, yields of all maturities should be positive and uniformly bounded from above. The yield curve should level out as term to maturity increases. Slopes with large absolute values occur only in the early maturities. In a continuous-time framework, the longer the maturity of the yield is, the less volatile it will be. The long rate should be a nondecreasing process. Furthermore, the long rate in continuous-time factor models with nonsingular volatility matrices should be a nondecreasing deterministic function.

In the Black, Derman, and Toy model and factor models with the short rate having the mean reversion property, yields of all maturities are uniformly bounded from above. The long rate in the Duffie and Kan model with the mean reversion property is a constant. The long rate in the Heath, Jarrow, and Morton model can be infinite or a nondecreasing process. Examples with the long rate increasing are given in this paper. A model with the long rate and short rate as two state variables is then obtained.     

No arbitrage conditions for simple trading strategies

Strict local martingales may admit arbitrage opportunities with respect to the class of simple trading strategies. (Since there is no possibility of using doubling strategies in this framework, the losses are not assumed to be bounded from below.) We show that for a class of non-negative strict local martingales, the strong Markov property implies the no arbitrage property with respect to the class of simple trading strategies. This result can be seen as a generalization of a similar result on three dimensional Bessel process in Delbaen and Schachermayer (Math Finance 4:343–348, 1994). We also provide no arbitrage conditions for stochastic processes within the class of simple trading strategies with shortsale restriction.     

Forecasting daily variability of the S&P 100 stock index using historical,realised and implied volatility measurements

The increasing availability of financial market data at intraday frequencies has not only led to the development of improved volatility measurements but has also inspired research into their potential value as an information source for volatility forecasting. In this paper, we explore the forecasting value of historical volatility (extracted from daily return series), of implied volatility (extracted from option pricing data) and of realised volatility (computed as the sum of squared high frequency returns within a day). First, we consider unobserved components (UC-RV) and long memory models for realised volatility which is regarded as an accurate estimator of volatility. The predictive abilities of realised volatility models are compared with those of stochastic volatility (SV) models and generalised autoregressive conditional heteroskedasticity (GARCH) models for daily return series. These historical volatility models are extended to include realised and implied volatility measures as explanatory variables for volatility. The main focus is on forecasting the daily variability of the Standard & Poor's 100 (S&P 100) stock index series for which trading data (tick by tick) of almost 7 years is analysed. The forecast assessment is based on the hypothesis of whether a forecast model is outperformed by alternative models. In particular, we will use superior predictive ability tests to investigate the relative forecast performances of some models. Since volatilities are not observed, realised volatility is taken as a proxy for actual volatility and is used for computing the forecast error. A stationary bootstrap procedure is required for computing the test statistic and its p-value. The empirical results show convincingly that realised volatility models produce far more accurate volatility forecasts compared to models based on daily returns. Long memory models seem to provide the most accurate forecasts.