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Abstract:
Typical value-at-risk (VaR) calculations involve the probabilities of extreme dollar losses, based on the statistical distributions of market prices. Such quantities do not account for the fact that the same dollar loss can have two very different economic valuations, depending on business conditions. We propose a nonparametric VaR measure that incorporates economic valuation according to the state-price density associated with the underlying price processes. The state-price density yields VaR values that are adjusted for risk aversion, time preferences, and other variations in economic valuation. In the context of a representative agent equilibrium model, we construct an estimator of the risk-aversion coefficient that is implied by the joint observations on the cross-section of option prices and time-series of underlying asset values.

Abstract:
We study asset allocation when the conditional moments of returns are partly predictable. Rather than first model the return distribution and subsequently characterize the portfolio choice, we determine directly the dependence of the optimal portfolio weights on the predictive variables. We combine the predictors into a single index that best captures time-variations in investment opportunities. This index helps investors determine which economic variables they should track and, more importantly, in what combination. We consider investors with both expected utility (mean-variance and CRRA) and non-expected utility (ambiguity aversion and prospect theory) objectives and characterize their market-timing, horizon effects, and hedging demands.

Abstract:
We develop a specification test for discretely-sampled jump-diffusions, based on a comparison of a nonparametric estimate of the transition density or distribution function to their corresponding parametric counterparts. As a special case, our method applies to pure diffusions. We propose three different discrepancy measures between the null and alternative transition density and distribution functions. We establish the asymptotic null distributions of proposed test statistics and compute their power functions. The finite sample properties are investigated via simulation studies and are compared with those of alternative tests.

Generalized likelihood ratio tests, local linear fit, null distribution, jump-diffusions, Markovian processes, power, specification tests, transition density

Abstract:
This paper evaluates the equity premium using novel data on the consumption of luxury goods. Specifying utility as a nonhomothetic function of both luxury and basic consumption goods, we derive pricing equations and evaluate the risk of holding equity. Household survey and national accounts data mostly reflect basic consumption and therefore overstate the risk aversion necessary to match the observed equity premium. The risk aversion implied by the consumption of luxury goods is more than an order of magnitude less than that implied by national accounts data. For the very rich, the equity premium is much less of a puzzle.

Asset pricing, Consumption, Equity premium puzzle, Portfolio choice

Abstract:
This paper evaluates the return on equity using novel data on the consumption of luxury goods. Specifying household utility as a nonhomothetic function of the consumption of both a luxury good and a basic good, we derive and evaluate the riskiness of equity in such a world. Household survey and national accounts consumption data overstate the risk aversion necessary to match the observed equity premium because they contain basic consumption goods. The risk aversion implied by equity returns and the consumption of luxury goods is more than an order of magnitude less than found using national accounts consumption data. For the very rich, the equity premium is much less of a puzzle.

Abstract:
When a continuous-time diffusion is observed only at discrete dates, not necessarily close together, the likelihood function of the observations is in most cases not explicitly computable. Researchers have relied on simulations of sample paths in between the observation points, or numerical solutions of partial differential equations, to obtain estimates of the function to be maximized. By contrast, we construct a sequence of fully explicit functions which we show converge under very general conditions, including non-ergodicity, to the true (but unknown) likelihood function of the discretely-sampled diffusion. We document that the rate of convergence of the sequence is extremely fast for a number of examples relevant in finance. We then show that maximizing the sequence instead of the true function results in an estimator which converges to the true maximum-likelihood estimator and shares its asymptotic properties of consistency, asymptotic normality and efficiency. Applications to the valuation of derivative securities are also discussed.

Abstract:
We compare the forecasts of Quadratic Variation given by the Realized Volatility (RV) and the Two Scales Realized Volatility (TSRV) computed from high frequency data in the presence of market microstructure noise, under several different dynamics for the volatility process and assumptions on the noise. We show that TSRV largely outperforms RV, whether looking at bias, variance, RMSE or out-of-sample forecasting ability. An empirical application to all DJIA stocks confirms the simulation results.

Market microstructure noise, high frequency data, measurement error, realized volatility, two scales realized volatility, out of sample forecasts.

Abstract:
We propose an empirical implementation of the consumption-investment problem using the martingale representation alternative to dynamic programming. Our method is based on the direct observation of state prices from options data. This greatly simplifies the investor's task of specifying the investment opportunity set and inherits the computational convenience of the martingale representation. Our method also makes explicit the economic trade-off between exploiting differences in state prices and probabilities, which generate variation in consumption, and the consumption smoothing induced by risk aversion. Using options-implied information, we find quantitatively different optimal consumption and portfolio policies than those implied by standard return dynamics.

Abstract:
We propose an empirical implementation of the consumption-investment problem using the martingale representation alternative to dynamic programming. Our method is based on the direct observation of state prices from options data. This greatly simplifies the investor's task of specifying the investment opportunity set and inherits the computational convenience of the martingale representation. Our method also makes explicit the economic trade-off between exploiting differences in state prices and probabilities, which generate variation in consumption, and the consumption smoothing induced by risk aversion. Using options-implied information, we find quantitatively different optimal consumption and portfolio policies than those implied by standard return dynamics.

Abstract:
We propose an empirical implementation of the consumption-investment problem using the martingale representation alternative to dynamic programming. Our method is based on the direct observation of state prices from options data. This greatly simplifies the investor's task of specifying the investment opportunity set and inherits the computational convenience of the martingale representation. Our method also makes explicit the economic trade-off between exploiting differences in state prices and probabilities, which generate variation in consumption, and the consumption smoothing induced by risk aversion. Using options-implied information, we find quantitatively different optimal consumption and portfolio policies than those implied by standard return dynamics.

Portfolio problem, martingale representation

Abstract:
We propose a new test to determine whether jumps are present in asset returns or other discretelly sampled processses. As the sampling interval tends to 0, our test statistic converges to 1 if there are jumps, and to another deterministic and known value (such as 2) if there are no jumps. The test is valid for all Itô semimartingales, depends neither on the law of the process nor on the coefficients of the equation which it solves, does not require a preliminary estimation of these coefficients, and when there are jumps the test is applicable whether jumps have finite or infinite activity and for an arbitrary Blumenthal-Getoor index. We finally implement the test on simulations and asset returns data.

jumps, test, discrete, sampling, high frequency

Abstract:
We develop and implement a technique for maximum likelihood estimation in closed-form of multivariate affine yield models of the term structure of interest rates. We derive closed-form approximations to the likelihood functions for all nine of the Dai and Singleton (2000) canonical affine models with one, two, or three underlying factors. Monte Carlo simulations reveal that this technique very accurately approximates true maximum likelihood, which is, in general, infeasible for affine models. We also apply the method to a dataset consisting of synthetic US Treasury strips, and find parameter estimates for nine different affine yield models, each using two different market price of risk specifications. One advantage of maximum likelihood estimation is the ability to compare non-nested models using likelihood ratio tests. We find, using these tests, that the choice of preferred canonical model can depend on the market price of riskspecification. Comparison to other approximation methods, Euler and QML, on both simulated and real data suggest that our approximation technique is much closer to true MLE than alternative methods.

Abstract:
We analyze the impact of time series dependence in market microstructure noise on the properties of estimators of the integrated volatility of an asset price based on data sampled at frequencies high enough for that noise to be a dominant consideration. We show that combining two time scales for that purpose will work even when the noise exhibits time series dependence, analyze in that context a refinement of this approach based on multiple time scales, and compare empirically our different estimators to the standard realized volatility.

Abstract:
We develop and implement a new method for maximum likelihood estimation in closed-form of stochastic volatility models. Using Monte Carlo simulations, we compare a full likelihood procedure, where an option price is inverted into the unobservable volatility state, to an approximate likelihood procedure where the volatility state is replaced by the implied volatility of a short dated at-the-money option. We find that the approximation results in a negligible loss of accuracy. We apply this method to market prices of index options for several stochastic volatility models, and compare the characteristics of the estimated models. The evidence for a general CEV model, which nests both the affine model of Heston (1993) and a GARCH model, suggests that the elasticity of variance of volatility lies between that assumed by the two nested models.

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Abstract:
This paper describes a simple yet powerful methodology to decompose asset returns sampled at high frequency into their base components (continuous, small jumps, large jumps), determine the relative magnitude of the components, and analyze the finer characteristics of these components such as the degree of activity of the jumps.

continuous-time models, semimartingales, jumps, volatility, spectrum, high frequency financial returns

Abstract:
Different continuous-time models for interest rates coexist in the literature. We test parametric models by comparing their implied parametric density to the same density estimated nonparametrically. We do not replace the continuous-time model by discrete approximations, even though the data are recorded at discrete intervals. The principal source of rejection of existing models is the strong nonlinearity of the drift. Around its mean, where the drift is essentially zero, the spot rate behaves like a random walk. The drift then mean-reverts strongly when far away from the mean. The volatility is higher when away from the mean.

Abstract:
Different continuous-time models for interest rates coexist in the literature. We test parametric models by comparing their implied parametric density to the same density estimated nonparametrically. We do not replace the continuous-time model by discrete approximations, even though the data are recorded at discrete intervals. The principal source of rejection of existing models is the strong nonlinearity of the drift. Around its mean, where the drift is essentially zero, the spot rate behaves like a random walk. The drift then mean-reverts strongly when far away from the mean. The volatility is higher when away from the mean.

Abstract:
Many models of the term structure of interest rates rely on a continuous-time specification of the short rate process as one of their factors. Different parametric specifications for this process, often arbitrary and mutually exclusive, coexist in the literature. It is important to specify this process correctly, because it is the input of most interest rate derivative pricing models. This paper proposes a simple testing procedure designed to assess the fit of any given specification of the drift and diffusion of a continuous- time model. The test is based on a comparison of the distribution of the interest rate data implied by the particular parametric model and the distribution estimated nonparametrically. The nonparametric estimate is robust to misspecification of the parametric model and will, therefore, be consistent under both the null and the alternative hypotheses, while the parametric density estimate is consistent under the null and inconsistent under the alternative. The basic idea is that a parametric model for the spot rate process is not rejected if its implied distribution is sufficiently close to the estimated nonparametric distribution, and conversely. The results suggest that the drift of the interest rate process tends to mean-revert more strongly when far away from the mean than predicted by commonly used parametrizations. This reflects a credible commitment by the Federal Reserve to return the short-term interest rate to less extreme levels. Moreover, the interest rate process behaves likes a random walk when close to its mean, with the drift being close to zero in that region. The diffusion function is larger when the interest rate is away from the central region, signaling a higher level of volatility.

Abstract:
Using recent advances in the econometrics literature, we disentangle from high frequency observations on the transaction prices of a large sample of NYSE stocks a fundamental component and a microstructure noise component. We then relate these statistical measurements of market microstructure noise to observable characteristics of the underlying stocks, and in particular to different financial measures of their liquidity. We find that more liquid stocks based on financial characteristics have lower noise and noise-to-signal ratio measured from their high frequency returns. We then examine whether there exists a common, market-wide, factor in high frequency stock-level measurements of noise, and whether that factor is priced in asset returns.

Abstract:
We study asset allocation when the conditional moments of returns are partly predictable. Rather than first model the return distribution and subsequently characterize the portfolio choice, we determine directly the dependence of the optimal portfolio weights on the predictive variables. We combine the predictors into a single index that best captures time-variations in investment opportunities. This index helps investors determine which economic variables they should track and, more importantly, in what combination. We consider investors with both expected utility (mean-variance and CRRA) and non-expected utility (ambiguity aversion and prospect theory) objectives and characterize their market-timing, horizon effects, and hedging demands.

Abstract:
We propose a nonparametric estimation procedure for continuous- time stochastic models. Because prices of derivative securities depend crucially on the form of the instantaneous volatility of the underlying process, we leave the volatility function unrestricted and estimate it nonparametrically. Only discrete data are used but the estimation procedure still does not rely on replacing the continuous- time model by some discrete approximation. Instead the drift and volatility functions are forced to match the densities of the process. We estimate the stochastic differential equation followed by the short term interest rate and compute nonparametric prices for bonds and bond options.

Abstract:
This paper derives a nonparametric estimation procedure for continuous-time models and provides the asymptotic distribution of the estimator. Since the pricing of derivative securities depends crucially on the form of the instantaneous volatility of the underlying asset, the diffusion function of the spot interest rate is unrestricted and estimated nonparametrically. The diffusion function is identified by requiring that the distribution of the process match that of the data. Even though only discrete data are used, the estimation procedure does not rely on replacing the continuous-time process by some discrete approximation. The continuous-time process followed by the short term interest rate is estimated. Nonparametric prices are computed for bonds, yielding the term structure of interest rates, and bond options.

Abstract:
Implicit in the prices of traded financial assets are Arrow- Debreu state prices or, in the continuous-state case, the state-price density (SPD). We construct an estimator for the SPD implicit in option prices and derive an asymptotic sampling theory for this estimator to gauge its accuracy. The SPD estimator provides an arbitrage-free method of pricing new, more complex, or less liquid securities while capturing those features of the data that are most relevant from an asset-pricing perspective, e.g., negative skewness and excess kurtosis for asset returns, volatility 'smiles' for option prices. We perform Monte Carlo simulation experiments to show that the SPD estimator can be successfully extracted from option prices and we present an empirical application using S&P 500 index options.

Abstract:
Implicit in the prices of traded financial assets are Arrow-Debreu prices or, with continuous states, the state-price density (SPD). We construct a nonparametric estimator for the SPD implicit in option prices and derive its asymptotic sampling theory. This estimator provides an arbitrage-free method of pricing new, complex, or illiquid securities while capturing those features of the data that are most relevant from an asset-pricing perspective, e.g.,negative skewness and excess kurtosis for asset returns, volatility "smiles" for option prices. We perform Monte Carlo experiments and extract the SPD from actual S&P 500 option prices.

Abstract:
Asset returns have traditionally been modeled in the literature as following continuous-time Markov processes, and in many cases diffusions. Can discretely sampled financial rate data help us decide which continuous-time models are sensible? Diffusion processes are characterized by the continuity of their sample paths. This cannot be verified from the discrete sample path: by nature, even if the underlying sample path were continuous, the discretely sampled data will always appear as a sequence of discrete jumps. Instead, this paper relies on a characterization of the transition density of the discrete data to determine whether the discontinuities observed in the discrete data are the result of the discreteness of sampling, or rather evidence of genuine jump dynamics for the underlying continuous-time process. I then focus on the implications of this approach for option pricing models.

Abstract:
Classical statistics suggest that for inference purposes one should always use as much data as is available. We study how the presence of market microstructure noise in high-frequency financial data can change that result. We show that the optimal sampling frequency at which to estimate the parameters of a discretely sampled continuous-time model can be finite when the observations are contaminated by market microstructure effects. We then address the question of what to do about the presence of the noise. We show that modelling the noise term explicitly restores the first order statistical effect that sampling as often as possible is optimal. But, more surprisingly, we also demonstrate that this is true even if one misspecifies the assumed distribution of the noise term. Not only is it still optimal to sample as often as possible, but the estimator has the same variance as if the noise distribution had been correctly specified, implying that attempts to incorporate the noise into the analysis cannot do more harm than good. Finally, we study the same questions when the observations are sampled at random time intervals, which are an essential feature of transaction-level data.

Abstract:
Typical value-at-risk (VAR) calculations involve the probabilities of extreme dollar losses, based on the statistical distributions of market prices. Such quantities do not account for the fact that the same dollar loss can have two very different economic valuations, depending on business conditions. We propose a nonparametric VAR measure that incorporates economic valuation according to the state-price density associated with the underlying price processes. The state-price density yields VAR values that are adjusted for risk aversion, time preferences, and other variations in economic valuation. In the context of a representative agent equilibrium model, we construct an estimator of the risk-aversion coefficient that is implied by the joint observations on the cross-section of option prices and time-series of underlying asset values. "

Abstract:
It is a common practice in finance to estimate volatility from the sum of frequently-sampled squared returns. However market microstructure poses challenges to this estimation approach, as evidenced by recent empirical studies in finance. This work attempts to lay out theoretical grounds that reconcile continuous-time modeling and discrete-time samples. We propose an estimation approach that takes advantage of the rich sources in tick-by-tick data while preserving the continuous-time assumption on the underlying returns. Under our framework, it becomes clear why and where the 'usual' volatility estimator fails when the returns are sampled at the highest frequency.

Abstract:
We develop and implement a technique for maximum likelihood estimation in closed-form of multivariate affine yield models of the term structure of interest rates. Affine yield models owe their popularity among both practitioners and academics to the fact that they allow for straightforward pricing of bonds and other interest rate derivatives. However, estimation still poses many challenging issues. Applying the method of Ait-Sahalia (2001), we derive closed-form approximations to the likelihood functions for all nine of the Dai and Singleton (2000) canonical affine models corresponding to dimensions 1, 2 and 3 of the state vector. Monte Carlo simulations reveal that our technique produces extremely accurate approximations of the exact likelihood function.

Abstract:
Realistic models for financial asset prices used in portfolio choice, option pricing or risk management include both a continuous Brownian and a jump components. This paper studies our ability to distinguish one from the other. I find that, surprisingly, it is possible to perfectly disentangle Brownian noise from jumps. This is true even if, unlike the usual Poisson jumps, the jump process exhibits an infinite number of small jumps in any finite time interval, which ought to be harder to distinguish from Brownian noise, itself made up of many small moves.

Abstract:
This paper examines the impact of macroeconomic and financial sector policy announcements in the United States, the United Kingdom, the euro area, and Japan during the recent crisis on interbank credit and liquidity risk premia. Announcements of interest rate cuts, liquidity support, liability guarantees, and recapitalization were associated with a reduction of interbank risk premia, albeit to a different degree during the subprime and global phases of the crisis. Decisions not to reduce interest rates and bail out individual banks in an ad hoc manner had adverse repercussions, both domestically and abroad. The results are robust to controlling for the surprise content of announcements and using alternative measures of financial distress.

Crisis, policy, announcement, event, financial, liquidity, monetary, fiscal, bank

Abstract:
The paper reviews the theoretical foundations of the use of forward interest rates to infer expected future rates of interest, inflation, currency depreciation and inflation differentials. Forward rates are related to these expected future variables via combinations of term, inflation and foreign exchange risk premia. A unified derivation, discussion and comparison of these premia is provided under both general and specific assumptions, as well as some comments on empirical estimation.

Abstract:
Frequently, economic theory places shape restrictions on functional relationships between economic variables. This paper develops a method to constrain the values of the first and second derivatives of nonparametric locally polynomial estimators. We apply this technique to estimate the state price density (SPD), or risk-neutral density, implicit in the market prices of options. The option pricing function must be monotonic and convex. Simulations demonstrate that nonparametric estimates can be quite feasible in the small samples relevant for day-to-day option pricing, once appropriate theory-motivated shape restrictions are imposed. Using S&P500 option prices, we show that unconstrained nonparametric estimators violate the constraints during more than half the trading days in 1999, unlike the constrained estimator we propose.

Abstract:
This paper provides closed-form expansions for the transition density and likelihood function of arbitrary multivariate diffusions. The expansions are based on a Hermite series, whose coefficients are calculated explicitly by exploiting the special structure afforded by the diffusion hypothesis. Because the transition function for most diffusion models is not known explicitly, the expansions of this paper can help make maximum-likelihood a practical estimation method for discretely sampled multivariate diffusions. Examples of interest in financial econometrics are included.

Abstract:
When a continuous-time diffusion is observed only at discrete dates, not necessarily close together, the likelihood function of the observations is in most cases not explicitly computable. Researchers have relied on simulations of sample paths in between the observations points, or numerical solutions of partial differential equations, to obtain estimates of the function to be maximized. By contrast, we construct a sequence of fully explicit functions which we show converge under very general conditions, including non-ergodicity, to the true (but unknown) likelihood function of the discretely-sampled diffusion. We document that the rate of convergence of the sequence is extremely fast for a number of examples relevant in finance. We then show that maximizing the sequence instead of the true function results in an estimator which converges to the true maximum-likelihood estimator and shares its asymptotic properties of consistency, asymptotic normality and efficiency. Applications to the valuation of derivative securities are also discussed.

Abstract:
This paper develops and estimates a continuous-time model of a financial market where investors' trading strategies and the specialist's rule of price adjustments are the best response to each other. We examine how far modeling market microstructure in a purely rational framework can go in explaining alleged asset pricing `anomalies.' The model produces some major findings of the empirical literature: excess volatility of the market price compared to the asset's fundamental value, serially correlated volatility, contemporaneous volume-volatility correlation, and excess kurtosis of price changes. We implement a nonlinear filter to estimate the unobservable fundamental value, and avoid the discretization bias by computing the exact conditional moments of the price and volume processes over time intervals of any length.

Abstract:
This paper develops and estimates a continuous-time model of a financial market where investors' trading strategies and the market maker's rule of price adjustments are best response to each other. There exists an equilibrium where the market maker finds it optimal to add volatility to the price s/he posts compared to the fundamental value of the asset. The model produces in a single and simple framework the two major findings of the empirical price-volume literature: serially correlated volatility and contemporaneous volume-volatility correlation. The fundamental price at every instant can be estimated based on the path of the actual market price, as well as transaction volume, up to that time. Nonlinear filters are derived to address the unobservability of the fundamental price.

Abstract:
This paper shows that the asymptotic normal approximation is often insufficiently accurate for volatility estimators based on high frequency data. To remedy this, we compute Edgeworth expansions for such estimators. Unlike the usual expansions, we have found that in order to obtain meaningful terms, one needs to let the size of the noise to go zero asymptotically. The results have application to Cornish-Fisher inversion and bootstrapping.

Abstract:
High-frequency financial data are not only discretely sampled in time but the time separating successive observations is often random. We analyze the consequences of this dual feature of the data when estimating a continuous-time model. In particular, we measure the additional effects of the randomness of the sampling intervals over and beyond those due to the discreteness of the data. We also examine the effect of simply ignoring the sampling randomness. We find that in many situations the randomness of the sampling has a larger impact than the discreteness of the data.

Abstract:
When a continuous-time diffusion is observed only at discrete dates, in most cases the transition distribution and hence the likelihood function of the observations is not explicitly computable. Using Hermite polynomials, I construct an explicit sequence of closed-form functions and show that it converges to the true (but unknown) likelihood function. I document that the approximation is very accurate and prove that maximizing the sequence results in an estimator that converges to the true maximum likelihood estimator and shares its asymptotic properties. Monte Carlo evidence reveals that this method outperforms other approximation schemes in situations relevant for financial models.