CRIF Working Paper series

Liquidity and Contagion in Financial Markets

David Backus et al. — Wed, 30 Apr 2008 12:43:56 PDT

This paper presents a model on contagion in financial markets. We use a bank run framwork as a mechanism to initiate a crisis and argues that liquidity crunch and imperfect information are the key culprits for a crisis to be contagious. The model proposes that a crisis is more likely to be contagious when (1) banks have similar cost-efficiency structures (clustering) and (2) a large fraction of the investment is in the illiquid sector (illiquidity). The latter is an endogenous decision made by the banks. It increases with (1) the prospect of the risky asset (risk-return trade-off) and (2) the fraction of patient consumers (liquidity demand).

Accounting for Biases in Black-Scholes

David Backus et al. — Wed, 30 Apr 2008 12:06:05 PDT

Prices of currency options commonly deffer from the Black-Scholes formula along two dimensions: implied volatilities vary by strike price (volatility smiles) and maturity (implied volatility of at-the-money options increases, on average, with maturity). We account for both using Gram-Charlier expansions to approximate the conditional distribution of the logarithm of the price of the underlying security. In this setting, volatility is approximately a quadratic function of moneyness, a result we use to infer skewness and kurtosis from volatility smiles. Evidence suggests that both kurtosis in currency prices and biases in Black-Scholes option prices decline with maturity.

A Dynamic Equilibrium Model of Real Exchange Rates with General Transaction Costs

GAUTAM GOSWAMI et al. — Wed, 30 Apr 2008 11:13:12 PDT

We study the behavior of real exchange rates in a two-country dynamic equilibrium model. In this model, consumers can only consume domestic goods but can invest costlessly in capital stocks of both countries. Nevertheless, transporting goods between the two countries is costly and, hence, the rebalancing of the capital stock can only happen finitely often. We propose a realistic cost structure for goods transportation, wherein the total cost increases with the amount of shipment but the unit cost decreases with it due to economies of scale. Given such a cost structure, the optimal decisions on when and how much to transfer need to be determined jointly. The dual decision depends upon the magnitude of economies of scale, the production technology specifications, and the consumer preferences. The model can reconcile the observed large short-term volatility of the real exchange rate with its slow convergence to parity. Further, the drift and diffusion of the real exchange rate are not uniquely determined by the real exchange rate level. The dynamics of the real exchange rate can only be determined by a joint analysis of the real exchange rate and the underlying economic fundamentals such as the capital stock imbalance between the two countries.

Jumps and Dynamic Asset Allocation

Liuren Wu — Wed, 30 Apr 2008 08:47:18 PDT

This paper provides a general framework for analyzing optimal dynamic asset allocation problems in economies with infrequent events and where the investment opportunities are stochastic and predictable. Analytical solutions are obtained, with which I do a thorough comparative study of the impacts of jumps on the dynamic decision. I also calibrate the model to the U.S. equity market and assess the quantitative impacts of jumps under a dynamic environment. I find that jump risk not only makes the investor's allocation more conservative overall but also makes her dynamic portfolio rebalancing less dramatic over time.

Term Structure of Interest Rates, Yield Curve Residuals, and the Consistent Pricing of Interest Rates and Interest Rate Derivatives

MASSOUD HEIDARI et al. — Tue, 29 Apr 2008 12:47:22 PDT

Dynamic term structure models (DTSMs) price interest rate derivatives based on the modelimplied fair values of the yield curve, ignoring any pricing residuals on the yield curve that are either from model approximations or market imperfections. In contrast, option pricing in practice often takes the market observed yield curve as given and focuses exclusively on the specification of the volatility structure of forward rates. Thus, if any errors exist on the observed yield curve, they will be carried over permanently. This paper proposes a new framework that consistently prices both interest rates and interest rate derivatives. In particular, under such a framework, instead of making a priori assumptions, we allow the data on interest rates and interest rate derivatives to dictate the dynamics of the yield curve residuals, as well as their impact on the pricing of interest rate derivatives. Specifically, we propose an m+n model structure. The first m factors capture the systematic movement of the yield curve and hence are referred to as the yield curve factors. The latter n factors are derived from the residuals on the yield curve and are labeled as the residual factors. We estimate a simple 3+3 Gaussian affine example using eight years of data on U.S. dollar LIBOR/swap rates and interest rate caps. The model performs well in pricing both interest rates and interest rate derivatives. Furthermore, we find that small residuals on the yield curve can have large impacts on the pricing of interest rate caps. Under the estimated model, the three Gaussian yield curve factors explain over 99.5 percent of the variation on the yield curve, but only account for less than 25 percent of the variation in the cap implied volatility. Incorporating the three residual factors improves the explained variance in cap implied volatility to over 95 percent. We investigate the reasons behind the “amplification” of yield curve residuals in pricing interest rate derivatives and find that the yield curve residuals are a recurring phenomenon, not a one-time event. Hence, the dynamics of the residuals influence option prices even if the current residual level is zero. We also find that the residuals concentrate on the two ends of the yield curve and are more transient than the original interest rate series, both of which, we argue, contribute to the amplification effect.

Time-Varying Arrival Rates of Informed and Uninformed Trades

DAVID EASLEY et al. — Tue, 29 Apr 2008 12:47:20 PDT

In this paper we extend the model of Easley and O’Hara (1992) to allow the arrival rates of informed and uninformed trades to be time-varying and forecastable. We specify a generalized autoregressive bivariate process for the arrival rates of informed and uninformed trades and estimate the model on 16 actively traded stocks on the New York Stock Exchange over 15 years of transaction data. Our results show that uninformed trades are highly persistent. Uninformed order arrivals clump together, with high uninformed volume days likely to follow high uninformed volume days, and conversely. This behavior is consistent with the passive characterization of the uninformed found in the literature. But we do find an important difference in how the uninformed behave; they avoid trading when the informed are forecasted to be present. Informed trades also exhibit complex patterns, but these patterns are not consistent with the strategic behavior posited in the literature. The informed do not appear to hide in order flow, but instead they trade persistently. We also investigate the correlation between the arrival rates of trades and trade composition on market volatility, liquidity and depth. We find that although volatility increases with the forecasted arrival rates of total trades, it is relatively independent of the forecasted composition of the trade. We use the opening bid-ask spread as a measure of market liquidity. We find that as the number of trades increases over time, the relative proportion of informed trades decreases and hence, spreads become narrower and the market becomes more liquid. Finally, we compute the price impact curve of consecutive buy orders and report the half life of the price impact as a measure of market depth. We find a positive correlation between the half life and total trades indicating that the market is deeper in presence of more trades.

Are Interest Rate Derivatives Spanned by the Term Structure of Interest Rates?

Massoud Heidari et al. — Tue, 29 Apr 2008 12:25:36 PDT

We investigate whether the same finite dimensional dynamic system spans both interest rates (the yield curve) and interest rate options (the implied volatility surface). We find that the options market exhibits factors independent of the underlying yield curve. While three common factors are adequate to capture the systematic movement of the yield curve, we need three additional factors to capture the movement of the implied volatility surface.

Asset Pricing Under The Quadratic Class

Markus Leippold et al. — Tue, 29 Apr 2008 12:22:30 PDT

We identify and characterize a class of term structure models where bond yields are quadratic functions of the state vector. We label this class the quadratic class and aim to lay a solid theoretical foundation for its future empirical application. We consider asset pricing in general and derivative pricing in particular under the quadratic class. We provide two general transform methods in pricing a wide variety of fixed income derivatives in closed or semi-closed form. We further illustrate how the quadratic model and the transform methods can be applied to more general settings.

Design and Estimation of Quadratic Term Structure Models

Markus Leippold et al. — Tue, 29 Apr 2008 12:22:26 PDT

We consider the design and estimation of quadratic term structure models. We start with a list of stylized facts on interest rates and interest rate derivatives, classified into three layers: (1) general statistical properties, (2) forecasting relations, and (3) conditional dynamics. We then investigate the implications of each layer of property on model design and strive to establish a mapping between evidence and model structures. We calibrate a two-factor model that approximates these three layers of properties well, and illustrate how the model can be applied to pricing interest rate derivatives.

Static Hedging of Standard Options

PETER CARR et al. — Tue, 29 Apr 2008 12:13:41 PDT

We consider the hedging of derivative securities when the price movement of the underlying asset can exhibit random jumps. Under a one factor Markovian setting, we derive a spanning relation between a long term option and a continuum of short term options. We then apply this spanning relation to the static hedging of long term options with a finite choice of short term, more liquid options based on a quadrature rule. We use Monte Carlo simulation to determine the hedging error introduced by the quadrature approximation and compare this hedging error to the hedging error from a delta hedging strategy based on daily rebalancing in the underlying futures. The simulation results indicate that the two types of strategies have comparable hedging effectiveness in the classic Black-Scholes environment, but that our static hedging strategy strongly outperforms the dynamic delta-hedging strategy when the underlying asset price movement is governed by Merton (1976)’s jump diffusion model. Further simulation exercises indicate that these results are robust to model misspecification, so long as one performs ad hoc adjustments based on the observed implied volatility. We also compare the hedging effectiveness of the two types of strategies using more than six years of data on S&P 500 index options. We find that a static hedge using just five call options outperforms daily rebalancing on the delta hedging with the underlying futures. The consistency of this result with our jump model simulations lends empirical support for the existence of jumps of random size in the movement of the S&P 500 index. We also find that our static strategy performs best when the maturity of the options in the hedging portfolio is close to the maturity of the target option being hedged. As the maturity gap increases, the hedging performance deteriorates moderately, indicating the likely existence of additional random factors such as stochastic volatility.

Predictable Changes in Yields and Forward Rates

David Backus et al. — Tue, 29 Apr 2008 12:13:33 PDT

We make two contributions to the study of interest rates. The first is to characterize their dynamics in a new way. We estimate forecasting relations based on one-period changes in forward rates, which are more easily compared than earlier work on yields to the stationary theory of bond pricing. The second is to approximate these dynamics and other salient features of interest rates with an affine model. We show that models with "negative" factors come closer to accounting for the properties of interest rates, including their dynamics, than multifactor Cox-Ingersoll-Ross models.

What Type of Process Underlies Options? A Simple Robust Test

PETER CARR et al. — Tue, 29 Apr 2008 11:54:10 PDT

We develop a simple robust test for the presence of continuous and discontinuous (jump) components in the price of an asset underlying an option. Our test examines the prices of at-the-money and out-of-the-money options as the option maturity approaches zero. We show that these prices converge to zero at speeds which depend upon whether the sample path of the underlying asset price process is purely continuous, purely discontinuous, or a mixture of both. By applying the test to S&P 500 index options data, we conclude that the sample path behavior of this index contains both a continuous component and a jump component. In particular, we find that while the presence of the jump component varies strongly over time, the presence of the continuous component is constantly felt. We investigate the implications of the evidence for parametric model specifications.

The Finite Moment Log Stable Process and Option Pricing

PETER CARR et al. — Tue, 29 Apr 2008 11:00:25 PDT

We document a surprising pattern in market prices of S&P 500 index options. When implied volatilities are graphed against a standard measure of moneyness, the implied volatility smirk does not flatten out as maturity increases up to the observable horizon of two years. This behavior contrasts sharply with the implications of many pricing models and with the asymptotic behavior implied by the central limit theorem (CLT). We develop a parsimonious model which deliberately violates the CLT assumptions and thus captures the observed behavior of the volatility smirk over the maturity horizon. Calibration exercises demonstrate its superior performance against several widely used alternatives.

Time-Changed L´evy Processes and Option Pricing

Pete Carr et al. — Tue, 29 Apr 2008 10:55:29 PDT

As is well known, the classic Black-Scholes option pricing model assumes that returns follow Brownian motion. It is widely recognized that return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to non-normal return innovations. Second, return volatilities vary stochastically over time. Third, returns and their volatilities are correlated, often negatively for equities. We propose that time-changed L´evy processes be used to simultaneously address these three facets of the underlying asset return process. We show that our framework encompasses almost all of the models proposed in the option pricing literature. Despite the generality of our approach, we show that it is straightforward to select and test a particular option pricing model through the use of characteristic function technology.

Currency Union and Real Exchange Rate Behavior

James R. Lothian et al. — Tue, 29 Apr 2008 10:52:50 PDT

In this paper we study the behavior of the real exchange rate of three North American currencies vis-a-vis the U.S. dollar: the Canadian dollar the Mexican peso, and the Panamanian Balboa. Our principal object is to design an experiment in which meaningful comparisons of behavior across regimes would be possible. In the main we were unable to find any. The allegation of problems created due to aggregating data across regimes therefore receives no support at all in these data. A second criterion for choosing the countries in our sample was differences in level of economic development. The object here was to provide ample leeway for real variables to operate. For Mexico such factors do not appear to matter. For Panama they might be of some importance, but a modified form of PPP nevertheless continues to perform well.

Equity Returns and Inflation: The Puzzlingly Long Lags

James R. Lothian et al. — Tue, 29 Apr 2008 10:49:54 PDT

This paper examines data for stock prices and price levels of 14 developed countries during the post-WWII era and compares their behavior in that sample with behavior over the past two centuries in the UK and the US. Contrary to much of the literature of the past several decades, we find that nominal equity prices do, in fact, keep pace with movements in the overall price level. Our results suggest, however, that this is only the case over long periods. The puzzle therefore is not that equities fail the test as inflation hedges, as had been quite widely believed, but that they take so long to pass.

Real Exchange-Rate Behaviour under Fixed and Floating Exchange Rate Regimes

James R. Lothian et al. — Tue, 29 Apr 2008 10:47:15 PDT

In this paper we examine the stability of the real exchange rate and the macroeconomic effects of alternative exchange-rate regimes, including currency union, on real exchange-rate behaviour. We focus on the Irish punt in order to exploit its diversity of experience over different nominal exchange rate regimes. We make both temporal and cross-country comparisons of real-exchange-rate stability for the Irish punt with sterling, the US dollar and the German mark. We reach two conclusions on the basis of our results. The first is that for Ireland, as for most other countries, purchasing power parity provides a reasonably good description of actual exchange rate behaviour over the long run. Our second principal conclusion concerns regime effects. Currency union appears to matter. The real exchange rates we analyse are unambiguously less variable under curr ency union than under alternative exchange-rate systems. Otherwise, however, we find no clear-cut differences in behaviour across regimes.

International Money and Common Currencies in Historical Perspective

Gerald P. Dwyer Jr. et al. — Tue, 29 Apr 2008 08:47:16 PDT

We review the history of international monies and the theory related to their adoption and use. There are four key characteristics of these currencies: high unitary value; relatively low inflation rates for long periods; issuance by major economic and trading powers; and spontaneous, as opposed to planned, adoption internationally. The economic theory of the demand for money provides support for the importance of these characteristics. The value of a unit is arbitrary for a fiat money, but the other characteristics are likely to be important for determining any fiat money that will be the international money in the future. If the euro continues to exist for the next half century or so and has a relatively stable value, we conclude that the euro is likely to be serious competition for the dollar as the international money.

Has International Financial Integration Increased?

Lawrence G. Goldberg et al. — Tue, 29 Apr 2008 08:37:12 PDT

This paper compares the behavior of real interest rate differentials across the major countries under the Bretton Woods regime and the regime of floating exchanges that replaced it. The primary object is to investigate both the extent of market integration and its changes over time. For all fifteen possible country pairs real interest differentials are mean reverting, and in two-thirds of these cases indistinguishable from zero statistically. For all country pairs on average and for most such pairs individually, moreover, the estimated differentials are not appreciably different in absolute value than the differentials that we estimate for various money-market rates within the United States. Additional evidence points to a narrowing of differentials under floating rates over time and an increase in speeds of convergence. “The rate of interest plays a central role in two great branches of economic science, – the theory of prices, and the theory of distribution. The role of the rate of interest in the theory of prices applies to the deter mination of the prices of wealth, proper ty, and services.” Irving Fisher (1907, p. 225). In an integrated world economy, real rates of interest on physical assets will tend to converge. So too will real rates of interest on financial assets like bonds, which are much more dir ectly observable. Real interest rate equalization is, therefore, the broadest, and arguably most theoretically appealing, of the various measures of financial integration. In this paper, we examine the behavior of cross-countr y real interest rate differentials for the United States and five other major industrial countries vis-à-vis one another during the last decade and a half of the Bretton Woods period and under the current regime of floating rates that replaced it. We investigate both the extent of financial market integration per se and whether and how it may have changed through time. We focus on three issues specifically: whether real interest rate differentials, if not literally zero, are at least small in absolute value and hence consistent with financial integration in the presence of crosscountry differences in risk; whether they are mean reverting, and hence indicative of long-run equilibrium; and whether and how their behavior has differed across exchange-r ate regimes. As a first step in this investigation, we examine separately the time-series behavior of the individual countries' real interest rates and their nominal rate and inflation components. We find after allowing for a structur al break in 1980 that real inter est rates in the six countries are stationary. We find further that cross-country differentials are invariant to the change in regime. Fluctuations in differentials occur periodically over the sample period, but while somewhat persistent, in the end prove transitory. For all fifteen possible country pairs real-interest- rate differentials are mean reverting, and in two-thirds of these cases indistinguishable from zero statistically. Additional evidence points to a narrowing of these cross-country differentials through time and an increase in speeds of convergence. Has international financial integration, therefore, increased? Some of this evidence suggests that it has; almost none suggests the opposite. Viewed from an absolute standpoint, moreover, the degree of integration appears to be not drastically different from what one finds comparing the behaviors of spreads between the nominal rates yielded by different domestic financial instruments. If the markets for these instruments can be considered well integrated, as they commonly are believed to be, then the implication is obvious -- so also international markets.

The Internationalization of Money and Finance and the Globalization of Financial Markets

James R. Lothian — Tue, 29 Apr 2008 08:11:19 PDT

In this paper I combine long multi-country time series data for interest rates and stock returns with the institutional evidence for much earlier centuries amassed by economic historians to study the question of financial globalization and how it has altered since the late classical era. At their longest, for Dutch and English short-term interest rates, the quantitative data that I use extend back slightly more than three centuries. The institutional history provides information on an additional millennium’s worth of experience. The conclusion that I reach is that the internationalization of money and finance and the globalization of financial markets are not new phenomena. They are part of an evolutionary process that began much earlier and that has continued, albeit with periodic interruptions and reversals, for many centuries. What we see today is simply the latest and most advanced manifestation of this process.