by
Lorenzo F. Naranjo
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Department of Finance
New York University
May 2009
Abstract
The first essay examines theoretically and empirically how borrowing and short-selling costs affect the pricing of derivatives contracts. In the presence of such costs, an agent trading a derivatives contract is unable to perfectly hedge the derivative if she is exposed to exogenous demand shocks. I derive an equilibrium model of the futures market where demand imbalances from traders affect the price of derivatives contracts. In the model, the price of a derivative depends on the risk- free rate and a latent demand factor. I estimate the model using S&P 500 index futures and the Kalman filter, and find that the latent demand factor is priced. I also find that the latent demand factor is closely related to proxies for demand pressure in the futures market, such as large speculator positions as well as investor sentiment. The demand factor is positive (negative) when buying (selling) pressure is high and is difficult to borrow (to short-sell the underlying). I also find that demand imbalances are correlated across different indexes, for both futures and options.
The second essay studies the stochastic behavior of implied interest rates from derivatives contracts. I start from the observation that there are many proxies for the short-term interest rate that are used in asset pricing. Yet, they behave differently, especially in periods of economic stress. Derivatives markets offer a unique laboratory to extract a short-term borrowing and lending rate available to all investors that is relatively free from liquidity and credit effects. Interestingly, implied interest rates do not resemble benchmark interest rates such as the three- month T-bill rate or LIBOR, but instead are much more volatile. I argue that the volatility in the implied short-term rate in futures and option markets is due to frictions arising from borrowing and short-selling costs. By using the techniques developed in the first essay of this dissertation, I propose a methodology to filter the “true” risk free rate from noisy implied interest rates. The risk-free rate that results from this estimation has time-series properties similar to Treasury and LIBOR rates, and anticipates interest rate changes. The spread between the risk-free and the T-bill rate correlates with liquidity proxies of the Treasury market, while the spread between LIBOR and the risk-free rate is related to economic distress.
The third essay examines the effect of improvements in relative rankings among stocks. The fact that the momentum effect survived the widely cited paper by Jegadeesh and Titman (1993) is itself evidence in favor of a behavioral explanation of this hypothesis. The experimental design of Jegadeesh and Titman (2001) excludes the possibility that the results are some artifact of thin trading or small stocks. However, to validate the behavioral hypothesis we must consider other possible implications. If the momentum effect is indeed under-reaction and delayed overreaction to news events as suggested in Jegadeesh and Titman (2001), then we should anticipate an even greater effect when there are improvements in relative rankings. We refer to this as the acceleration hypothesis and find that it is a significant and distinct factor that has interesting implications for the cross section of security returns.
1.1 Introduction
In this paper I study theoretically and empirically the effect of borrowing and short-selling costs in the pricing of derivatives contracts. In the presence of such costs, an arbitrageur that provides liquidity in a particular derivatives market will be unable to perfectly hedge the derivative if she is exposed to exogenous demand shocks. Thus, demand imbalances for the derivative affect prices, in a similar manner as in the model originally studied by Garleanu, Pedersen, and Poteshman (2007) for options, and Vayanos and Vila (2007) for fixed-income markets.
In perfect markets, it is well-known that the cost of carrying forward a hedged derivatives position should be equal to the risk-free rate minus the dividend yield of the underlying asset. Thus, in frictionless markets it is possible to use derivatives contracts to borrow or lend funds at the risk-free rate. Borrowing funds is accomplished by selling the derivative and buying the risky asset, whereas lending funds is accomplished by buying the derivative and short-selling the risky asset.
In general, though, there are costs faced by agents when they want to borrow funds or short-sell the risky asset. As a result, if a group of agents borrow funds using the derivatives market, the equilibrium derivative’s price should reflect the costs associated with borrowing to preclude arbitrage opportunities. Agents borrowing through the derivatives market generate demand pressure for the derivative, affecting its price because of the borrowing costs. A similar effect follows if a group of agents generate demand pressure in the derivatives market to short-sell the risky asset. Thus, in the presence of borrowing and short-selling costs the equilibrium price of a derivatives contract depends on its demand imbalance.
Even though the intuition can be applied to price any derivative contract, in this paper I specialize the analysis to the pricing of futures contracts. There are several reasons for doing this. First, the pricing formulas can be derived in closed-form, which is convenient for empirical tests of the model. Second, futures contracts are very simple instruments with linear payoffs, which makes their valuation more robust to model assumptions. Third, futures contracts have been trading for a long time, which provides with a long time series of observations to use in empirical tests. Fourth, futures contracts are very liquid instruments, which makes their prices less susceptible to deviate from fair value for liquidity reasons.
The model setup is described in Section 1.2. There are two types of agents: arbitrageurs and traders. Arbitrageurs are rational agents that maximize their expected utility and operate in a competitive market in which there are no-arbitrage opportunities. Traders can be speculators or hedgers, although I do not distinguish between them and assume that all traders have an exogenous demand for the derivative. Arbitrageurs take the opposite position in the derivatives market and hedge their position using the risky asset and risk-free bonds.
In the model arbitrageurs pay borrowing and short-selling costs. One way to motivate this is to assume that arbitrageurs trade with a bank that provides brokering services. In order to derive a closed-form solution for the model, I make the simplifying assumption that borrowing costs increase linearly with demand. The assumption is consistent with the fact that borrowing rates may differ from lending rates, and that it is more expensive to borrow larger amounts of the riskless or risky asset. If traders take a long position in futures contracts, arbitrageurs will take a short position in the futures market. In order to hedge the additional risk carried by the futures, arbitrageurs will simultaneously take a long position in the risky asset. If this position is sufficiently large, they will have to borrow and pay borrowing costs. Equivalently, if traders short the futures, arbitrageurs will have to take the opposite position in the futures market and hedge it by selling the risky asset. If the shorting demand by traders is sufficiently large, arbitrageurs will have to sell short the risky asset, incurring in short-selling costs.
In Section 1.3 I derive an equilibrium model of the futures market where demand imbalances from traders affect the price of futures contracts. The mechanism is as follows. If traders’ demand is positive, then arbitrageurs short the derivative, buy the risky asset and borrow. In equilibrium, arbitrageurs borrow when traders’ demand is positive, which is precisely when borrowing costs are high. In a competitive market, arbitrageurs set the price of the derivative such that they are indifferent between taking the opposite side of the trade or doing nothing. This equilibrium price will be higher than the price obtained in an otherwise equivalent frictionless economy. A similar logic applies if traders want to short the derivative. In that case, arbitrageurs buy the derivative, short the underlying asset and pay short-selling costs. In equilibrium, the price of the derivative should be set lower than in an otherwise equivalent frictionless economy in order to motivate arbitrageurs to take the long position.
In this economy the futures price in the presence of borrowing and short-selling costs depends on the risk-free rate and a latent demand factor. Essentially, the model predicts that the difference between the usual cost-of-carry formula and observed prices is given by the latent demand factor. Thus, derivatives “mispricings” could just be proxying for demand pressure and borrowing and short-selling costs. Moreover, the futures price is equal to the expected spot price under a “local” equivalent risk-neutral measure, implying that arbitrageurs receive a time-varying risk premium for providing liquidity to traders.
The theory has several testable implications. First, the model predicts the existence of a risk premium for the demand factor. Intuitively, if borrowing and short-selling costs are high when demand is high, and vice-versa, then the arbitrageur is exposed to an additional source of risk when hedging her portfolio and commands a premium. Second, the latent demand factor should be related to proxies for demand pressure in the futures market. In other words, proxies for demand pressure in the futures market should be priced factors in the term-structure of futures prices. Third, the demand factor in equilibrium should be related to the borrowing and short-selling costs of the marginal trader. Thus, the model predicts that the latent demand factor is positive when buying pressure is high and it is difficult to borrow irrespective of short-selling costs. Similarly, the latent demand factor is negative when selling pressure is high and it is difficult to short-sell the underlying asset regardless of borrowing costs. Fourth, if demand pressure for derivatives is driven by a common factor, deviations from fair-value should be correlated across markets and derivative instruments.
I estimate the model parameters and state variables using S&P 500 index futures and the Kalman filter in Section 1.4. This is possible because the model delivers a closed-form solution for the valuation of futures contracts and the state- variables follow a system of Gaussian processes. I estimate sixteen parameters by maximizing the likelihood function of price innovations.
The estimation reveals that the demand factor that enters the pricing formula is significantly priced. Thus, engaging in index futures arbitrage is risky and should be compensated. This provides a rationale for the existence of a large industry dedicated to arbitrage deviations from fair-value in derivatives markets. Any agent who has a competitive advantage in borrowing cheaper than others enjoys an economic surplus.
I also verify that the latent demand factor is related to proxies for demand pressure in the futures market, such as large speculators positions in S&P 500 futures and market sentiment for large institutional investors (Han, 2008). If on average the pressure is generated by large speculators, it follows that hedgers are actually paid a premium for the services they provide.
Moreover, I find that the latent demand factor is positive when buying pressure is high and it is difficult to borrow independently of short-selling costs. The opposite effect is also true. The latent demand factor is negative when selling pressure is high and it is difficult to short-sell the risky asset, independently of borrowing costs. This implies that borrowing costs affect the pricing of derivatives only when there is demand pressure to buy the derivative but not to sell it. Similarly, short-selling costs are relevant only when when there is demand pressure to short the derivative but not to buy it.
Finally, I look at whether latent demand imbalances co-move across instruments and markets. Since the demand factor is essentially capturing deviations from fair value in a market without frictions, it is possible to obtain estimates of this factor for other indexes and also put-call parity relations on stock index options. I find that mispricings in futures and put-call parity relations in three indexes (S&P 500, DJIA and Nasdaq 100) are significantly correlated with each other and also with the sentiment proxy, suggesting co-movement in demand imbalances across instruments and markets.
This paper is closely related to two papers that show how demand imbalances for a derivative affect its price. Garleanu et al. (2007) show how demand for options affect options prices, while Vayanos and Vila (2007) analyze a similar effect for fixed-income markets. The main difference with these two papers is that I analyze both theoretically and empirically the effect of borrowing and short-selling costs in the pricing of derivatives.
Also, this study contributes to the literature that studies index futures arbitrage. The main focus of this literature has been to understand absolute mispricings, whereas this paper contributes understanding the sign and magnitude of mispricings.
Finally, this paper fits into a broader literature that looks at how market frictions prevent arbitrageurs in some circumstances from profiting of arbitrage opportunities. Effectively, I show that borrowing costs open a channel through which demand imbalances in the derivatives market can affect no-arbitrage prices, even when the payoff is linear in the underlying asset.
Chapter 2
Implied Interest Rates in a Market with Frictions
2.1 Introduction
Measuring and understanding the risk-free rate is a fundamental question in financial economics. The risk-free rate is a required input in many theories that are extensively used by academics and practitioners in finance, like the CAPM, the APT and the no-arbitrage valuation of derivatives products. The valuation of real and financial assets depends crucially on using the correct risk-free rate. In complete markets, the risk-free rate is uniquely determined as the conditional expectation of the inverse of the stochastic discount factor. In incomplete markets, however, the risk-free rate might not trade at all. If that is the case, there might be several ways of defining what we understand for the risk-free rate (see e.g. Cochrane, 2005, Section 6.5). In this paper, I define the risk-free rate for a particular maturity to be the yield of a zero-coupon bond that is free of liquidity and credit risk effects.
In Section 2.2 I start analyzing the two main proxies for the short-term interest rate that are commonly used in asset pricing. On the one hand, it is common for empirical researchers in finance to use the yield on three-month T-bills as a proxy for the risk-free rate. The intuition for this practice is simple: T-bills are backed by the full faith and credit of the U.S. government. As such, they are the safest investment available for an investor whose consumption is denominated in U.S. dollars. One problem with using T-bill yields as a proxy of the risk-free rate is that only the U.S. government can borrow at this rate. Also, Treasury rates can exhibit periods of flight-to-liquidity during which investors are willing to pay more for the benefit of holding a liquid security (Longstaff, 2004). On the other hand, it is common for practitioners to use LIBOR rates as a proxy for the risk-free rate when valuing derivatives contracts. The intuition for this practice is also simple: practitioners regard LIBOR as their opportunity cost of capital. However, LIBOR represents the interest rate charged on an uncollateralized loan between banks and hence is subject to credit risk.
As a result, these interest rates behave differently, especially in periods of economic stress. As an example, Figure 2.1 shows how Treasury and LIBOR rates have being drifting apart during the 2007-08 credit and liquidity crisis. The so- called TED spread, defined as the difference between LIBOR and Treasury rates, was at an all time high (461 bp) during October 2008.
Fortunately, there are other markets that investors can use to lend or borrow from which it is possible to infer short-term interest rates. For example, an investor can borrow by entering into a long position in a forward contract and selling the underlying asset. Similarly, an investor can lend by shorting a forward and buying the underlying asset. If this transaction is performed through an organized exchange, standard features such as margin requirements and the existence of a clearing corporation significantly reduce the credit risk of the transaction. Also, since derivatives contracts are in zero net-supply, flight-to-liquidity problems are mitigated. Thus, derivatives markets offer a unique laboratory to extract a short- term borrowing and lending rate available to all investors that is free from liquidity and credit risk effects.
In perfect markets, it is well-known that the cost of carrying a forward position should be equal to the risk-free rate minus the underlying asset’s dividend yield. Thus, in frictionless markets2 it is possible to use forward and futures contracts as substitutes for risk-free bonds to derive risk-free rate estimates.
In Section 2.4 I study the implied risk-free rate obtained from futures contracts and put-call parity relations written on major indexes: S&P 500, Nasdaq 100 and Dow Jones Industrial Average (DJIA). The data used in this section is described in Section 2.3. I find that on average implied interest rates from both futures and options lie between Treasury and LIBOR rates. From January 1998 to December 2007, three-month rates implied from futures prices are on average 48 bp above Treasury and 5 bp below LIBOR, whereas implied interest rates from options are on average 50 bp above Treasury and 3 bp below LIBOR. Thus, implied interest rates are very similar regardless of whether they are extracted from futures or options, and are on average much closer to borrowing (LIBOR) rather than lending (Treasury) rates. This result is consistent with the common industry practice of using LIBOR rates as a proxy for the risk-free rate when valuing derivatives contracts, and also with previous findings in the literature (Brenner and Galai, 1986).
Interestingly, the time-series of implied interest rates do not resemble that of benchmark interest rates such as the three-month T-bill rate or LIBOR, but instead is much more volatile. As discussed in Chapter 1, the phenomenon is expected if we account for market frictions. In the presence of borrowing and short-selling costs, an arbitrageur that provides liquidity in a particular derivatives market will be unable to hedge the derivative perfectly if she is exposed to exogenous demand shocks. In this case demand imbalances for the derivative will affect prices, making implied interest rates to be correlated with demand, increasing their volatility and affecting their level.
However, it is possible to use the methodology outlined in Chapter 1 to estimate the risk-free rate implied in derivatives prices. Using the Kalman filter, it is possible to infer the time-series of the risk-free rate from the price of a derivative contract. The main identifying assumption used in the estimation is that the risk-free rate is much more persistent than the demand factor driving the “mispricing” of the derivative.
I restrict the estimation of the risk-free rate to S&P 500 futures contracts because they provide with the longest time-series of observations, and they are one of the most liquid derivatives contracts available. In Section 2.5 I show that the risk-free rate that results from this estimation has similar time-series properties as Treasury and LIBOR rates.
The estimated spot rate is on average 15 bp above the fed funds rate, and short-term rates are on average lower than implied rates computed in Section 2.4. Most interestingly, the spot rate seems to be forecasting the Federal funds rate, suggesting that futures markets anticipate changes in short-term interest rates. Also, the spread between LIBOR and the risk-free rate is high in periods of market stress, as proxied by the VIX and the credit spread between AAA and BAA bonds. The spread between the risk-free rate and three-month T-bill rates is high in periods of market illiquidity. These results suggest that the estimated risk-free rate is less affected by liquidity and credit risk than Treasury rates and LIBOR, respectively.
While there is a huge literature on dynamic term structure modeling3 using Treasury bonds and LIBOR rates, the use of derivatives for extracting information about the risk-free rate has been mostly ignored by financial economists. A notable exception is Brenner and Galai (1986), who are probably the first to estimate implied risk-free rates from put-call parity relations on stock option prices. In related work, Brenner et al. (1990) also look at implied risk-free rates using Nikkei index futures data. Liu, Longstaff, and Mandell (2006) obtain implied risk-free rates from plain-vanilla swap contracts, but they use the three-month General Collateral (GC) repo rate as a proxy for the three-month risk-free rate. I make no initial assumptions about what the risk-free rate should be. Feldhütter and Lando (2008) also use swap data to estimate the risk-free rate. However, I use a much larger time-series and I cross-validate my implied rate with different assets. I also account for the fact that demand pressure can affect derivatives prices, distorting implied risk-free rate estimates in significant ways.
Chapter 3
Momentum and the Acceleration Hypothesis (joint with Stephen Brown )
Introduction
A seminal paper by Jegadeesh and Titman (1993) showing that there is strong evidence of serial dependence in the return rankings of stocks has been highly influential in subsequent research for both by academics and practitioners. To date there have been 305 citations of this research in refereed journal publications, and altogether 1188 citations including citations in unpublished working papers that have been electronically circulated1. The results of this research are widely accepted among practitioners and have been used to develop trading strategies both in the United States and abroad. The analysis has been extended in a number of ways to examine whether it applies in other countries (for example Rouwenhorst, 1998) and to other financial statistics (for example, earnings reports). Despite the influence of this research and the importance of its application, the momentum phenomenon remains mysterious. What is the source of this momentum and is it likely to persist in the context of numerous trading strategies designed to exploit it?
One curious finding difficult to reconcile with rational market behavior is the result that the strength of the momentum effect is if anything greater after the effect was described and published in 1993 (Jegadeesh and Titman, 2001). While the results of the 1993 paper appear to diminish and even disappear when the 1993 study is extended to 2004 and to NASDAQ listed stocks (Table 3.1), Jegadeesh and Titman (2001) exclude microcap stocks from the analysis. The results are if anything much stronger once we exclude from the analysis stocks trading under $5.00 (Table 3.2). This result is important, not only as a post sample test of the momentum hypothesis, but also because it excludes the possibility that the momentum effect is an artifact of thin trading and/or small stocks in the sample.
These results leave standing the behavioral hypothesis of Jegadeesh and Titman (2001). These results alone however cannot exclude other possible explanations for the observed momentum effect. One approach is to examine other possible implications of the behavioral hypothesis. The purpose of this paper is to examine one such implication. If the momentum effect arises from delayed overreactions that are eventually reversed, then we should anticipate that improvements in the relative ranking of stocks have an even more extreme effect. We identify this additional implication as the acceleration hypothesis and find that it is a significant and distinct factor that has implications that differ for small and large traded equities. This would appear to be confirmation of the behavioral hypothesis.
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