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Abstract:
This paper examines the different factors that have contributed to the subprime mortgage credit crisis: the search for yield enhancement, agency problems, lax underwriting standards, failure by the rating agencies to identify a changing environment, poor risk management by financial institutions, lack of transparency, the limitation of extant valuation models and the failure of regulators to understand the implications of the changing environment for the financial system. The paper addresses the different issues and offers suggestions on how to move forward.

Subprime mortgages, SIVs, monolines, transparency, valuation

Abstract:
This paper introduces the concept of statistical arbitrage, a long horizon trading opportunity that generates a riskless profit and is designed to exploit persistent anomalies. Statistical arbitrage circumvents the "joint hypothesis" dilemma of traditional market efficiency tests because its definition is independent of any equilibrium model and its existence is incompatible with market efficiency. We provide a methodology to test for statistical arbitrage and then empirically investigate whether momentum and value trading strategies constitute statistical arbitrage opportunities. Despite controlling for transaction costs and the influence of small stocks, we find evidence that these strategies generate statistical arbitrage. Furthermore, their profitability does not appear to decline over time.

market efficiency, statistical arbitrage, arbitrage, momentum, value

Abstract:
When I was I thinking about what to discuss in this address, my mind wandered across many topics. I first thought about discussing martingale probability measures and the topologies of asset pricing. I quickly discarded this topic for obvious reasons(and one not so obvious). The not so obvious one is that I was asked to avoid using any equations. I next thought about preaching on the evils of 'value at risk,' but learned that Steve Ross had talked about this in the past. One doesn't want to imitate a master. So, I rejected that topic. By then I was desperate. To illustrate how much so, I even thought about a banking topic=FE asset/liability management. I rejected this as well because after lunch, there is always the danger of putting everyone to sleep.

Then, an inspiration hit. I remembered when I was a graduate student in the late 70s at MIT, how much I enjoyed listening to Bob Merton and Fischer Black talk about the historical development of the Black-Scholes formula. Who did what, and when? What were the stumbling blocks in the development, in particular, solving the Black-Scholes partial differential equation. To this day, I still like recounting those stories to my students.

So, I thought I would share my insights and recollections on the development process of the Heath-Jarrow-Morton term structure model with you. In the process, I will take advantage of the historic progression to discuss the salient issues regarding its derivation and implementation. In addition, I will also discuss its extensions to foreign currency and credit derivatives. Finally, I will share my predictions with you on the future of research in this area, and on the future of the field of financial engineering itself.

Abstract:
This paper investigates the forecasting accuracy of bankruptcy hazard rate models for U.S. companies over the time period 1962 - 1999 using both yearly and monthly observation intervals. The contribution of this paper is multiple-fold. One, using an expanded bankruptcy database we validate the superior forecasting performance of Shumway's (2001) model as opposed to Altman (1968) and Zmijewski (1984). Two, we demonstrate the importance of including industry effects in hazard rate estimation. Industry groupings are shown to significantly affect both the intercept and slope coefficients in the forecasting equations. Three, we extend the hazard rate model to apply to financial firms and monthly observation intervals. Due to data limitations, most of the existing literature employs only yearly observations. We show that bankruptcy prediction is markedly improved using monthly observation intervals. Fourth, consistent with the notion of market efficiency with respect to publicly available information, we demonstrate that accounting variables add little predictive power when market variables are already included in the bankruptcy model.

Bankruptcy Prediction, Hazard Models, Industry Effects, Reduced Form Credit Risk Models

Abstract:
Recent advances in the theory of credit risk allow the use of standard term structure machinery for default risk modeling and estimation. The empirical literature in this area often interprets the drift adjustments of the default intensity's diffusion state variables as the only default risk premium. We show that this interpretation implies a restriction on the form of possible default risk premia, which can be justified through exact and approximate notions of "diversifiable default risk." The equivalence between the empirical and martingale default intensities that follows from diversifiable default risk greatly facilitates the pricing and management of credit risk. We emphasize that this is not an equivalence in distribution, and illustrate its importance using credit spread dynamics estimated in Duffee (1999). We also argue that the assumption of diversifiability is implicitly used in certain existing models of mortgage-backed securities.

Abstract:
Recent advances in the theory of credit risk allow the use of standard term structure machinery for default risk modeling and estimation. The empirical literature in this area often interprets the drift adjustments of the default intensity's diffusion state variables as the only default risk premium. We show that this interpretation implies a restriction on the form of possible default risk premia, which can be justified through exact and approximate notions of ``diversifiable default risk.'' The equivalence between the empirical and martingale default intensities that follows from diversifiable default risk greatly facilitates the pricing and management of credit risk. We emphasize that this is not an equivalence in distribution, and illustrate its importance using credit spread dynamics estimated in Duffee (1999). We also argue that the assumption of diversifiability is implicitly used in certain existing models of mortgage-backed securities.

Abstract:
Motivated by recent financial crises in East Asia and the U.S. where the downfall of a small number of firms had an economy-wide impact, this paper generalizes existing reduced-form models to include default intensities dependent on the default of a counterparty. In this model, firms have correlated defaults due not only to an exposure to common risk factors, but also to firm-specific risks that are termed ``counterparty risks.'' Numerical examples illustrate the effect of counterparty risk on the pricing of defaultable bonds and credit derivatives such as default swaps.

Abstract:
Statistical arbitrage enables tests of market efficiency which circumvent the joint-hypotheses dilemma. This paper makes several contributions to the statistical arbitrage framework. First, we enlarge the set of statistical arbitrage opportunities in Hogan, Jarrow, Teo, and Warachka (2004) to avoid penalizing incremental trading profits with positive deviations from their expected value. Second, we provide a statistical methodology to remedy the lack of consistency and statistical power in their Bonferroni approach. In addition, this procedure allows for autocorrelation and non-normality in trading profits. Third, we apply our tests to a wide range of trading strategies based on stock momentum, stock value, stock liquidity, and industry momentum. Over 50% of these strategies are found to violate market efficiency. We also identify dominant trading strategies which converge to arbitrage most rapidly.

Market Efficiency, Financial Anomalies

Abstract:
This paper characterizes the liquidity discount, the difference between the market value of a large trader's position and its value when liquidated. This discount occurs whenever traders face downward sloping demand curves for shares and execution lags in selling shares. This characterization enables one to modify the standard value at risk (VAR) computation to include liquidity risk.

Liquidity Risk, Large Trader, Value at Risk

Abstract:
This paper studies asset price bubbles in a continuous time model using the local martingale framework. Providing careful definitions of the asset's market and fundamental price, we characterize all possible price bubbles in an incomplete market satisfying the "no free lunch with vanishing risk (NFLVR)" and "no dominance" assumptions. We show that the two leading models for bubbles as either charges or as strict local martingales, respectively, are equivalent. We propose a new theory for bubble birth which involves a nontrivial modification of the classical martingale pricing framework. This modification involves the market exhibiting different local martingale measures across time - a possibility not previously explored within the classical theory. Finally, we investigate the pricing of derivative securities in the presence of asset price bubbles, and we show that: (i) European put options can have no bubbles, (ii) European call options and discounted forward prices have bubbles whose magnitudes are related to the asset's price bubble, (iii) with no dividends, American call options may be exercised early, (iv) European put-call parity in market prices must always hold, regardless of bubbles, and (v) futures price bubbles can exist and they are independent of the underlying asset's price bubble. Many of these results stand in contrast to those of the classical theory. We propose, but do not implement, some new tests for the existence of asset price bubbles using derivative securities.

Abstract:
We develop a reduced-form approach for valuing callable corporate bonds by characterizing the call probability via an intensity process. Asymmetric information and market frictions justify the existence of a call-arrival intensity from the market's perspective. Our approach extends the reduced-form model of Duffie and Singleton (1999) for defaultable bonds to callable bonds and can capture some important differences between call and default decisions.We also provide one of the first comprehensive empirical analyses of callable bonds using both our approach and the traditional approach of valuing callable bonds as American options. Empirical results show that the reduced-form model fits callable bond price data well and outperforms the traditional approach in both in-sample and out-of-sample applications.

Callable bond, reduced-form

Abstract:
Although relatively obscure, the market for distressed real estate tax liens exists in over 30 U.S. states, with a market size estimated to be around 20 billion dollars. While this niche asset class is relatively unknown to academics, internet advertising hypes tax liens to the populace as providing extraordinary returns. Not yet studied scientifically, this market provides a fertile and untouched arena for the application of asset pricing theory. Using insights from several areas of asset pricing, we formulate and test a pricing model for tax liens. The empirical evidence supports the general validity of the pricing model and suggests avenues for extensions and further research. In addition, we show that the tax lien markets are unfair towards property owners and provide extraordinary returns to investors, lending some credibility to the industry claims.

Tax Liens, Tax Certificates, Asset Pricing, Investment

Abstract:
We develop a reduced-form approach for valuing callable corporate bonds by characterizing the call probability via an intensity process. Asymmetric information and market frictions justify the existence of a call-arrival intensity from the market's perspective. Our approach both extends the reduced-form model of Duffie and Singleton (1999) for defaultable bonds to callable bonds and captures some important differences between call and default decisions. We also provide one of the first comprehensive empirical analyses of callable bonds using both our model and the more traditional American option approach for valuing callable bonds. Our empirical results show that the reduced-form model fits callable bond prices well and that it outperforms the traditional approach both in- and out-of-sample.

Abstract:
This paper proposes an aggregate deposit insurance premium design that is risk-based in the sense that the premium structure ensures the deposit insurance system has a target of survival over the longer term. Such a premium system naturally exceeds the actuarily fair value and leads to a growth in the insurance fund over time. The proposed system builds in a swap in premia that reduces premia when fund size exceeds a threshold. In addition, we build in a swap contract that trades premia in good times for relief in bad times.

Deposit insurance, Procyclicality, Default probabilty, Loss distributions

Abstract:
This book is a collection of original papers by Robert Jarrow that contributed to significant advances in financial economics. Divided into three parts, Part I concerns option pricing theory and its foundations. The papers here deal with the famous Black-Scholes-Merton model, characterizations of the American put option, and the first applications of arbitrage pricing theory to market manipulation and liquidity risk.

Part II relates to pricing derivatives under stochastic interest rates. Included is the paper introducing the famous Heath-Jarrow-Morton (HJM) model, together with papers on topics like the characterization of the difference between forward and futures prices, the forward price martingale measure, and applications of the HJM model to foreign currencies and commodities.

Part III deals with the pricing of financial derivatives considering both stochastic interest rates and the likelihood of default. Papers cover the reduced form credit risk model, in particular the original Jarrow and Turnbull model, the Markov model for credit rating transitions, counterparty risk, and diversifiable default risk.

Derivatives, Options, Hedging, HJM, Black–Scholes, Forwards, Futures, Martingale Measure, Calls, Puts, Market Manipulation, Margin Requirements

Abstract:
This note defines the premium of a put option on the firm as a measure of insolvency risk. The put premium is not a coherent risk measure as defined by Artzner et al. (1999). It satisfies all the axioms for a coherent risk measure except one, the translation invariance axiom. However, it satisfies a weakened version of the translation invariance axiom that we label translation monotonicity. The put premium risk measure generates an acceptance set that satisfies the regularity Axioms 2.1,2.4 of Artzner et al. (1999). In fact, this is a general result for any risk measure satisfying the same risk measure axioms as the put premium. Finally, the coherent risk measure generated by the put premium's acceptance set is the minimal capital required to protect the firm against insolvency uniformly across all states of nature.

Abstract:
This article presents the theory of option pricing with random volatilities in complete markets. As such, it makes two contributions. First, the newly developed martingale measure technique is used to synthesize results dating from Merton (1973) through Eisenberg, (1985, 1987). This synthesis illustrates how Merton's formula, the CEV formula, and the Black-Scholes formula are special cases of the random volatility model derived herein. The impossibility of obtaining a self-financing trading strategy to duplicate an option in incomplete markets is demonstrated. This omission is important because option pricing models are often used for risk management, which requires the construction of synthetic options. Second, we derive a new formula, which is easy to interpret and easy to program, for pricing options given a random volatility. This formula (for a European call option) is seen to be a weighted average of Black-Scholes values, and is consistent with recent empirical studies finding evidence of mean-reversion in volatilities.

option pricing, synthetic options, martingale measure, European call option

Abstract:
We analyze the effect various delivery options embedded in commodity futures contracts have on the futures price. The two embedded options considered are the timing and location options. We show that early delivery is always optimal when only a timing option is present, but not so with joint options. The estimates of the combined options are much smaller than the comparable estimates for the timing option alone. The average value of the joint option is about 5% of the average basis on the first day of the maturity month. This suggests that joint options can increase deliverable supplies while potentially having only a small effect on basis behavior.

Commodity futures, delivery option

Abstract:
This paper provides an empirical implementation of a reduced form credit risk model that incorporates both liquidity risk and correlated defaults. Liquidity risk is modeled as a convenience yield and default correlation is modeled via an intensity process that depends on market factors. Various different liquidity risk and intensity process models are investigated. Firstly, the evidence supports a non-zero liquidity premium that is firm specific, reflecting idiosyncratic and not systematic risk. Secondly, the credit risk model with correlated defaults fits the data quite well with an average R wedge 2 of .87 and a pricing error of only 1.1 percent.

Credit Risk

Abstract:
This paper uses an HJM model to price TIPS and related derivative securities. First, using the market prices of TIPS and ordinary U.S. Treasury securities, both the real and nominal zero-coupon bond price curves are obtained using standard coupon-bond price stripping procedures. Next, a three-factor arbitrage-free term structure model is fit to the time series evolutions of the CPI-U and the real and nominal zero-coupon bond price curves. Then, using these estimated term structure parameters, the validity of the HJM model for pricing TIPS is confirmed via its hedging performance. Lastly, the usefulness of the pricing model is illustrated by valuing call options on the inflation index.

Abstract:
This paper compares structural versus reduced form credit risk models from an information based perspective. We show that the difference between these two model types can be characterized in terms of the information assumed known by the modeler. Structural models assume that the modeler has the same information set as the firm's manager - complete knowledge of all the firm's assets and liabilities. In most situations, this knowledge leads to predictable default time. In contracts, reduced form models assume that the modeler has the same information set as the market - incomplete knowledge of the firm's condition. In most cases, the imperfect knowledge leads to an inaccessible default time. As such, we argue that the key distinction between structural and reduced form models is not whether the default is predictable or inaccessible, but whether the information set is observed by the market or not. Consequently, for pricing and hedging, reduced form models are the preferred methodology.

Credit risk, structural models, reduced form models, hazard rate models, risky debt, credit derivatives

Abstract:
This paper uses a reduced form credit risk model to estimate default probabilities implicit in equity prices. For a cross-section of firms, a time-series regression of monthly equity returns is estimated. We show that it is feasible to infer the firm's probability of default implicit in equity returns. However, the existence of price bubbles and the difficulty in modeling equity price risk premium confound the estimation of these default probabilities, generating potentially biased estimates with large standard errors. Comparing these default intensities with those obtained from historical data or implicitly from debt prices confirms this result.

Credit risk

Abstract:
We provide an approach to the market valuation of deposit insurance that is based on reduced-form methods for the pricing of fixed-income securities under default risk. By reference to bank debt prices as well as qualitative-response models of the probability of bank failure, we suggest how a risk-neutral valuation model for deposit insurance can be applied both to the calculation of fair-market deposit insurance premia and to the valuation of long-term claims against the insurer.

Deposit insurance pricing, risk-neutral default probability, bank failure

Abstract:
This article presents a new methodology for estimating recovery rates and the (pseudo) default probabilities implicit in both debt and equity prices. In this methodology, recovery rates and default probabilities are correlated and depend on the state of the macroeconomy. This approach makes two contributions: First, the methodology explicitly incorporates equity prices in the estimation procedure. This inclusion allows the separate identification of recovery rates and default probabilities and the use of an expanded and relevant data set. Equity prices may contain a bubble component - which is essential in light of recent experience with Internet stocks. Second, the methodology explicitly incorporates a liquidity premium in the estimation procedure - which is also essential in light of the large observed variability in the yield spread between risky debt and U.S. Treasury securities and the illiquidities present in risky-debt markets.

Abstract:
This paper presents a new definition of market completeness that is independent of the notions of no arbitrage and equivalent martingale measures. Our definition has many advantages, all shown herein. First, it preserves the Second Fundamental Theorem of Asset Pricing, even in complex economies. Second, under our definition, the market can be complete yet arbitrage opportunities exist. This is important in practice, and stands in contrast to the traditional definitions. Third, under the assumption of no arbitrage and when used in the standard models, our definition is equivalent to the traditional one.

Abstract:
This paper provides a new methodology for pricing and hedging derivative securities involving credit risk. Two types of credit risks are considered. The first is where the asset underlying the derivative security may default. The second is where the writer of the derivative security may default. We apply the foreign currency analogy of Jarrow and Turnbull (1991) to decompose the dollar payoff from a risky security into a certain payoff and a "spot exchange rate". Arbitrage free valuation techniques are then employed. This methodology can be applied to corporate debt and OTC derivatives, such as swaps and caps.

Abstract:
This paper studies a new theory for pricing options in a large trader economy. This theory necessitates studying the impact that derivative security markets have on market manipulation. In an economy with a stock, money market account, and a derivative security, it is shown, by example, that the introduction of the derivative security generates market manipulation trading strategies that would otherwise not exist. A sufficient condition is provided on the price process such that no additional market manipulation trading strategies are introduced by a derivative security. Options are priced under this condition, where it is shown that the standard binomial option model still applies but with random volatilities.

Abstract:
The Black-Scholes formula is the "industry standard" for pricing options on a variety of instruments. This paper shows that even when markets are incomplete, the Black- Scholes option pricing formula can arise in an equilibrium merely from self-fulfilling beliefs that it is the correct pricing formula. In such an equilibrium, the presence of traders speculating on the stock's underlying price movements can cause implied volatilities to deviate from future volatilities. This result provides a possible explanation for the recent empirical evidence supporting an "overreactions" effect in implied volatilities.

Abstract:
A Monte Carlo based Bayesian estimator of contingent claim models formally incorporating model error and parameter uncertainty is introduced. Estimate and prediction on the basis of a model extended by a regression of (functions of) variables and parameters or a non parametric expansion of the model is also proposed. For these models, we show how to make draws from the exact posterior distributions of the parameters or any function of interest such as hedge ratios or option values. It is crucial to obtain exact posterior or predictive densities in a Bayesian setup because (1) typical applications may involve small sample situations, (2) in an updating setup one will want to incorporate specific prior information, and (3) parameters and on linear functions thereof will be shown to be non-normal thus rendering standard asymptotic inference ineffective. The Bayesian Monte Carlo estimators derived allow the simulation of the exact posterior densities of parameters and deterministic functions of parameters, e.g., Delta, Gamma, Vega, Theta, model value, and the exact predictive densities of the contingent claim prices. We provide both within sample (residual analysis), and out of sample (predictive) specification tests which can be used in a dynamic trading system. Finally, the model is extended by allowing for the (small) probability of observation with a larger model error. This produces the posterior probability of each observation being an outlier, and may help distinguish market from model error.

Abstract:
This paper extends the known results on the equivalence between market completeness and the uniqueness of martingale measures for finite asset economies, to the infinite asset case. Our arguments employ results from the theory of linear operators between locally convex topological vector spaces. This theory of linear operators provides an operational approach to the issue of completeness and uniqueness, that is also more closely connected with the mainstream of empirical asset pricing, than was hitherto available.

Abstract:
This paper develops in a new framework, an equilibrium model of the U.S. Treasury Securities auction market under both discriminatory and uniform price auction rules. In the model, traders participate in forward trading of Treasury securities in a "when-issued" market before the Treasury auction. Next, bidding takes place in a Treasury auction and securities are delivered. Last, a resale market occurs where "short-cornering" and market manipulating "squeezes" can happen with rational traders. These manipulations occur in auctions by dealers who, participating in the when-issued market, use their knowledge of the net order flow to bid aggressively in the auction in order to corner the market and squeeze the shorts (from the when-issued market). This equilibrium is shown to be consistent with the abnormal price behavior of Treasury securities surrounding dealer induced squeezes. Our analysis also shows that manipulations will not occur in long-run equilibrium under uniform price auctions. Under this long-run equilibrium condition, the uniform price auction enhances social welfare by yielding greater revenue to the Treasury than does the discriminatory auction.

Abstract:
This paper models the U.S. Treasury securities auction market and demonstrates that market manipulation can occur in a rational equilibrium. It is a dynamic model with traders participating in a "when-issued" market, a Treasury auction, and a resale mark et. Manipulations occur when dealers in the when-issued market use their knowledge of the net order flow in order to corner the auction and squeeze the shorts (from the when-issued market). This manipulation equilibrium generates bubbles in Treasury sec urity prices and specials in repo rates. We also compare discriminatory and uniform price auction rules with respect to manipulation. Our analysis shows that manipulations can occur in long-run equilibrium under discriminatory price auctions, but not under uniform price auctions.

Abstract:
This paper formally incorporates parameter uncertainty and model error into the estimation of contingent claim models and the formulation of forecasts. This allows inference on functions of interest (option values, bias functions, hedge ratios) consistent with uncertainty in both parameters and models. We show how to recover the exact posterior distributions of the parameters or any function of the parameters. Exact posterior or predictive densities are crucial because a frequent updating setup results in small samples and requires the incorporation of specific prior information. Markov Chain Monte Carlo estimators are developed to solve this estimation problem. Within sample and predictive model specifications tests are provided which can be used in dynamic testing (or trading systems) making use of cross-sectional and time series options data. Finally, we discuss several generalizations of the error structure. These new techniques are applied to equity options using the Black-Scholes model. When model error is taken into account, Black-Scholes appears very robust, in contrast with previous studies which at best only incorporated parameter uncertainty. We extend the Black-Scholes model by adding polynomial functions of its inputs. This allows for intuitive specification tests. Although these simple extended models improve the in-sample error properties of the Black-Scholes, they do not result in major improvements in out of sample predictions. The differences between these models are important, however, because they produce different hedge ratios and posterior probabilities of mispricing.

Abstract:
Taking the term structure of Treasury securities and Eurodollar rates as exogenous, this paper provides an integrated approach to the pricing and hedging of LIBOR derivatives. Our approach allows the spread between Eurodollar and Treasury rates to reflect both the credit risk in holding Eurodollar deposits and a convenience yield from holding Treasury securities. This integrated approach includes the models of Babbs [1991], Grinblatt [1994], and Jarrow and Turnbull [1995] as special cases.

Abstract:
This paper provides a Markov model for the term structure of credit risk spreads. The model is based on Jarrow and Turnbull (1995) with the bankruptcy process following a discrete state space Markov chain in credit ratings. The parameters of this process are easily estimated using observable data. This model is useful for pricing and hedging corporate debt with imbedded options, for pricing and hedging OTC derivatives with counterparty risk, for pricing and hedging (foreign) government bonds subject to default risk (e.g., municipal bonds), for pricing and hedging credit derivatives, and for risk management.

Abstract:
This paper provides a Markov Model for the term structure of credit risk spreads. The model is based on Jarrow and Turnbull (1995) with the bankruptcy process following a discrete state space Markov chain in credit ratings. The parameters of this process are easily estimated using observable data. This model is useful for pricing and hedging corporate debt with imbedded options, for pricing and hedging OTC derivatives with counterparty risk, for pricing and hedging (foreign) government bonds subject to default risk (e.g. municipal bonds), for pricing and hedging credit derivatives, and for risk management.