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.     

A Rude Awakening: Internet Shakeout in 2000

This study explores various value-drivers of business-to-consumer (B2C) Internet companies' share prices both before and after the market correction in the spring of 2000. Although many market observers had predicted that the shakeout would eventually occur (e.g., Perkins and Perkins 1999), the ultimate and previously unanswered challenge lay identifying which stocks would fall and which ones would survive the shakeout. We develop an empirical valuation model and provide evidence that the Internet stocks that this model suggests were relatively over-valued prior to the Internet stock market correction experienced relatively larger drops in their price-to-sales ratios when the shakeout occurred. This result is robust to the inclusion of competing explanatory variables suggested by the economics literature related to industry rationalizations.We examine the ability of a valuation model comprised of both financial (accounting) variables and nonfinancial web traffic metrics to explain Internet companies' market values during each of 1999 and 2000. Our findings suggest that the reach and stickiness web traffic performance measures are value-relevant to the share prices of Internet companies in each of 1999 and 2000. Our findings of significance for the year 2000 contradict the recent claims of some analysts that web traffic measures are no longer important. We also explore the valuation role of our proxy for B2C companies' current rate of cash burn and find that this proxy is a significant value-driver in each of 1999 and 2000, but with differential valuation implications for each period. Our results suggest that the market was favorably disposed towards Internet companies' aggressive cash expenditures in 1999, but appeared to adopt a more critical view of Internet companies' cash burn rates in 2000. Our results further suggest that investors adopted a more skeptical attitude towards expenditures on intangible investments as the Internet sector began to mature. We find that investors appear to implicitly capitalize product development (R&D) and advertising expenses (customer acquisition costs) during the earlier period when the market was more optimistic about the prospects of B2C companies. However, only product development costs are implicitly capitalized into value, on average, subsequent to the shakeout in the spring of 2000. Finally, we provide statistical evidence to support the conjecture that different parameter vectors characterize the estimated market valuation models for each of 1999 and 2000. Overall, our study provides a preliminary view of the shakeout and maturation of one of the most important New Economy industries to emerge to date–the Internet.     

Universal bounds for asset prices in heterogeneous economies

We establish universal bounds for asset prices in heterogeneous complete market economies with scale invariant preferences. Namely, for each agent in the economy we consider an artificial homogeneous economy populated solely by this agent, and calculate the “homogeneous” price of an asset in each of these economies. Dumas (Rev. Financ. Stud. 2, 157–188, [1989]) conjectured that the risk free rate in a heterogeneous economy must lie in the interval determined by the minimal and maximal of the “homogeneous” risk free rates. We show that the answer depends on the risk aversions of the agents in the economy: the upper bound holds when all risk aversions are smaller than one, and the lower bound holds when all risk aversions are larger than one. The bounds almost never hold simultaneously. Furthermore, we prove these bounds for arbitrary assets.      

Arbitrage Opportunities in Diverse Markets via a Non-equivalent Measure Change

We study arbitrage opportunities in diverse markets as introduced by Fernholz (J Math Econ 31:393–417, 1999). By a change of measure technique we are able to generate a variety of diverse markets. The construction is based on an absolutely continuous but non-equivalent measure change which implies the existence of instantaneous arbitrage opportunities in diverse markets. For this technique to work, we single out a crucial non-degeneracy condition. Moreover, we discuss the dynamics of the price process under the new measure as well as further applications.Both authors gratefully acknowledge financial support by the National Centre of Competence in Research “Financial Valuation and Risk Management” (NCCR FINRISK). The NCCR FINRISK is a research program supported by the Swiss National Science Foundation. The first author also thanks Credit Suisse Group for financial support.     

Options Arbitrage in Imperfect Markets

Option valuation models are based on an arbitrage strategy—hedging the option against the underlying asset and rebalancing continuously until expiration—that is only possible in a frictionless market. This paper simulates the impact of market imperfections and other problems with the “standard” arbitrage trade, including uncertain volatility, transactions costs, indivisibilities, and rebalancing only at discrete intervals. We find that, in an actual market such as that for stock index options, the standard arbitrage is exposed to such large risk and transactions costs that it can only establish very wide bounds on equilibrium options prices. This has important implications for price determination in options markets, as well as for testing of valuation models.     

Recent Developments in Credit Markets

We summarize recent developments in the credit derivative markets. We show the role of dependence between individual debtors in portfolio derivatives in a study of implied correlation. The risk of changing dependence structures between stock and bond markets becomes evident in an example of capital structure arbitrage. How credit derivatives can introduce new risks is illustrated by the example of “overlay” in basket derivatives.     

Beyond cash-additive risk measures: when changing the numéraire fails

We discuss risk measures representing the minimum amount of capital a financial institution needs to raise and invest in a pre-specified eligible asset to ensure it is adequately capitalized. Most of the literature has focused on cash-additive risk measures, for which the eligible asset is a risk-free bond, on the grounds that the general case can be reduced to the cash-additive case by a change of numéraire. However, discounting does not work in all financially relevant situations, especially when the eligible asset is a defaultable bond. In this paper, we fill this gap by allowing general eligible assets. We provide a variety of finiteness and continuity results for the corresponding risk measures and apply them to risk measures based on value-at-risk and tail value-at-risk on L ^p spaces, as well as to shortfall risk measures on Orlicz spaces. We pay special attention to the property of cash subadditivity, which has been recently proposed as an alternative to cash additivity to deal with defaultable bonds. For important examples, we provide characterizations of cash subadditivity and show that when the eligible asset is a defaultable bond, cash subadditivity is the exception rather than the rule. Finally, we consider the situation where the eligible asset is not liquidly traded and the pricing rule is no longer linear. We establish when the resulting risk measures are quasiconvex and show that cash subadditivity is only compatible with continuous pricing rules.     

On fairness of systemic risk measures

Finance and Stochastics - In our previous paper “A unified approach to systemic risk measures via acceptance sets” (Mathematical Finance, 2018), we have introduced a general class of...     

The Dynamics of the S&P 500 Implied Volatility Surface

This empirical study is motivated by the literature on “smile-consistent” arbitrage pricing with stochastic volatility. We investigate the number and shape of shocks that move implied volatility smiles and surfaces by applying Principal Components Analysis. Two components are identified under a variety of criteria. Subsequently, we develop a “Procrustes” type rotation in order to interpret the retained components. The results have implications for both option pricing and hedging and for the economics of option pricing. This revised version was published online in June 2006 with corrections to the Cover Date.     

The makings of an information leader: the intraday price discovery process for individual stocks in the DJIA

The main purpose of this paper is to construct an intraday arbitrage price series for each stock in the DJIA using information in the Diamond Trust Fund ETF. We then compute the information shares (Hasbrouck in J Finan 50(4):1175–1199, 1995) for the actual versus the arbitrage prices for each stock. While previous literature documents that ETFs lead stock indices in information origination, we find that some firms are “information leaders” in that the information share that comes from the stock price is larger than that which comes from the ETF-related arbitrage price. Further analysis is conducted to uncover the firm-specific factors that are related to a stock’s role in information generation.     

Optimal investment and contingent claim valuation in illiquid markets

This paper extends basic results on arbitrage bounds and attainable claims to illiquid markets and general swap contracts where both claims and premiums may have multiple payout dates. Explicit consideration of swap contracts is essential in illiquid markets where the valuation of swaps cannot be reduced to the valuation of cumulative claims at maturity. We establish the existence of optimal trading strategies and the lower semicontinuity of the optimal value of optimal investment under conditions that extend the no-arbitrage condition in the classical linear market model. All results are derived with the “direct method” without resorting to duality arguments.     

On the range of options prices

In this paper we consider the valuation of an option with time to expiration and pay-off function which is a convex function (as is a European call option), and constant interest rate , in the case where the underlying model for stock prices is a purely discontinuous process (hence typically the model is incomplete). The main result is that, for “most” such models, the range of the values of the option, using all possible equivalent martingale measures for the valuation, is the interval , this interval being the biggest interval in which the values must lie, whatever model is used.     

Valuation before and after tax in the discrete time, finite state no arbitrage model

We establish necessary and sufficient conditions for a linear taxation system to be neutral—within the multi-period discrete time “no arbitrage” model—in the sense that valuation is invariant to the exact sequence of tax rates, realization dates as well as immune to timing options attempting to twist the time profile of taxable income through wash sale transactions.

“In the study of investments, taxes are largely a source of embarrassment to financial economists.” (Introduction to Dybvig and Ross 1986) “Accordingly, my approach in this chapter is to examine the restrictions on the income measurement rules applicable to financial instruments implied by the requirement that the rules be linear. . . . Linearity is a desideratum of a tidy tax system.” (Bradford 2000, p. 373–374)