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Abstract:
Realized volatility is a nonparametric ex-post estimate of the return variation. The most obvious realized volatility measure is the sum of finely-sampled squared return realizations over a fixed time interval. In a frictionless market the estimate achieves consistency for the underlying quadratic return variation when returns are sampled at increasingly higher frequency. We begin with an account of how and why the procedure works in a simplified setting and then extend the discussion to a more general framework. Along the way we clarify how the realized volatility and quadratic return variation relate to the more commonly applied concept of conditional return variance. We then review a set of related and useful notions of return variation along with practical measurement issues (e.g., discretization error and microstructure noise) before briefly touching on the existing empirical applications.

Realized Volatility, Stochastic Volatility, Quadratic Variation, Bipower Variation, Variance Swap, Impled Volatility

Abstract:
Given the importance of return volatility on a number of practical financial management decisions, the efforts to provide good real-time estimates and forecasts of current and future volatility have been extensive. The main framework used in this context involves stochastic volatility models. In a broad sense, this model class includes GARCH, but we focus on a narrower set of specifications in which volatility follows its own random process, as is common in models originating within financial economics. The distinguishing feature of these specifications is that volatility, being inherently unobservable and subject to independent random shocks, is not measurable with respect to observable information. In what follows, we refer to these models as genuine stochastic volatility models.

Much modern asset pricing theory is built on continuous-time models. The natural concept of volatility within this setting is that of genuine stochastic volatility. For example, stochastic-volatility (jump-)diffusions have provided a useful tool for a wide range of applications, including the pricing of options and other derivatives, the modeling of the term structure of risk-free interest rates, and the pricing of foreign currencies and defaultable bonds. The increased use of intraday transaction data for construction of so-called realized volatility measures provides additional impetus for considering genuine stochastic volatility models. As we demonstrate below, the realized volatility approach is closely associated with the continuous-time stochastic volatility framework of financial economics.

There are some unique challenges in dealing with genuine stochastic volatility models. For example, volatility is truly latent and this feature complicates estimation and inference. Further, the presence of an additional state variable - volatility - renders the model less tractable from an analytic perspective. We review how such challenges have been addressed through development of new estimation methods and imposition of model restrictions allowing for closed-form solutions while remaining consistent with the dominant empirical features of the data.

Stochastic Volatility, Realized Volatility, Impled Volatility, Options, Smirk, Smile, Term Structure of Interest Rates, Affine Models

Abstract:
We investigate whether bonds span the volatility risk in the U.S. Treasury market, as predicted by most "affine" term structure models. To this end, we construct powerful and model-free empirical measures of the quadratic yield variation for a cross-section of fixed-maturity zero-coupon bonds ("realized yield volatility") through the use of high-frequency data. We find that the yield curve fails to span yield volatility, as the systematic volatility factors are largely unrelated to the cross-section of yields. We conclude that a broad class of affine diffusive, Gaussian-quadratic and affine jump-diffusive models is incapable of accommodating the observed yield volatility dynamics. An important implication is that the bond markets per se are incomplete and yield volatility risk cannot be hedged by taking positions solely in the Treasury bond market. We also advocate using the empirical realized yield volatility measures more broadly as a basis for specification testing and (parametric) model selection within the term structure literature.

Abstract:
We investigate whether bonds span the volatility risk in the U.S. Treasury market, as predicted by most `affine' term structure models. To this end, we construct powerful and model-free empirical measures of the quadratic yield variation for a cross-section of fixed-maturity zero-coupon bonds (`realized yield volatility') through the use of high-frequency data. We find that the yield curve fails to span yield volatility, as the systematic volatility factors are largely unrelated to the cross-section of yields. We conclude that a broad class of affine diffusive, Gaussian-quadratic and affine jump-diffusive models is incapable of accommodating the observed yield volatility dynamics. An important implication is that the bond markets per se are incomplete and yield volatility risk cannot be hedged by taking positions solely in the Treasury bond market. We also advocate using the empirical realized yield volatility measures more broadly as a basis for specification testing and (parametric) model selection within the term structure literature.

Interest Rate Volatility, Hedging, Volatility Risk, Unspanned Stochastic Volatility, Affine Models, Term Structure Models

Abstract:
We study portfolio choice when labor income and dividends are cointegrated. Economically plausible calibrations suggest young investors should take substantial short positions in the stock market. Because of cointegration the young agent's human capital electively becomes stock-like. However, for older agents with shorter times - to - retirement, cointegration does not have sufficient time to act, and thus their human capital becomes more bond-like. Together, these exects create hump - shaped life - cycle portfolio holdings, consistent with empirical observation. These results hold even when asset return predictability is accounted for.

Human Capital, Risky Labor Income, Limited Stock Market Participation, Portfolio Choice, Life Cycle

Abstract:
Prior to the stock market crash of 1987, Black-Scholes implied volatilities of S&P 500 index options were relatively constant across moneyness. Since the crash, however, deep out-of-the-money S&P 500 put options have become 'expensive' relative to the Black-Scholes benchmark. Many researchers (e.g., Liu, Pan and Wang (2005)) have argued that such prices cannot be justified in a general equilibrium setting if the representative agent has 'standard preferences' and the endowment is an i.i.d. process. Below, however, we use the insight of Bansal and Yaron (2004) to demonstrate that the 'volatility smirk' can be rationalized if the agent is endowed with Epstein-Zin preferences and if the aggregate dividend and consumption processes are driven by a persistent stochastic growth variable that can jump. We identify a realistic calibration of the model that simultaneously matches the empirical properties of dividends, the equity premium, the prices of both at-the-money and deep out-of-the-money puts, and the level of the risk-free rate.

A more challenging question (that to our knowledge has not been previously investigated) is whether one can explain within a standard preference framework the stark regime change in the volatility smirk that has maintained since the 1987 market crash. To this end, we extend the model to a Bayesian setting in which the agent updates her beliefs about the average jump size in the event of a jump. Note that such beliefs only update at crash dates, and hence can explain why the volatility smirk has not diminished over the last eighteen years. We find that the model can capture the shape of the implied volatility curve both pre- and post-crash while maintaining reasonable estimates for expected returns, price-dividend ratios, and risk-free rates.

Volatility smile, volatility smirk, implied volatility, option pricing, portfolio insurance, market risk

Abstract:
We examine the long-term return performance and fundamental valuation of IPOs underwritten by relationship banks. We find that over one- to three-year horizons these IPOs yield returns similar to those on IPOs underwritten by non-relationship banks. Moreover, we show that there is selection among firms that go public with their relationship bank. Investors value these new issues higher than similar IPOs underwritten by non-relationship banks. These findings support the certification role of relationship banks and suggest that, in this respect, the effect of the 1999 repeal of Sections 20 and 32 of the Glass-Steagall Act has not been negative.

Conflicts of interest, lending relationships, IPOs, Glass Steagall Act

Abstract:
Empirical evidence shows that changes in aggregate labor income and stock market returns exhibit only weak correlation at short horizons. As we document below, however, this correlation increases substantially at longer horizons, which provides at least suggestive evidence that stock returns and labor income are cointegrated. In this paper, we investigate the implications of such a cointegrated relation for life-cycle optimal portfolio and consumption decisions of an agent whose non-tradable labor income faces permanent and temporary idiosyncratic shocks. We find that, under economically plausible calibrations, the optimal portfolio choice for the young investor is to take a substantial {\em short} position in the risky portfolio, in spite of the large risk premium associated with it. Intuitively, this occurs because the cointegration effect makes the present value of future labor income flows `stock-like' for the young agent. However, for older agents who have shorter times-to-retirement, the cointegration effect does not have sufficient time to act, and the remaining human capital becomes more `bond-like.' Together, these effects create a hump-shaped optimal portfolio decision for the agent over the life cycle, consistent with empirical observation.

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Abstract:
Empirical evidence shows that changes in aggregate labor income and stock market returns exhibit only weak correlation at short horizons. As we document below, however, this correlation increases substantially at longer horizons, which provides at least suggestive evidence that stock returns and labor income are cointegrated. In this paper, we investigate the implications of such a cointegrated relation for life-cycle optimal portfolio and consumption decisions of an agent whose non-tradable labor income faces permanent and temporary idiosyncratic shocks. We find that, under economically plausible calibrations, the optimal portfolio choice for the young investor is to take a substantial short position in the risky portfolio, in spite of the large risk premium associated with it. Intuitively, this occurs because the cointegration effect makes the present value of future labor income flows 'stock-like' for the young agent. However, for older agents who have shorter times-to-retirement, the cointegration effect does not have sufficient time to act, and the remaining human capital becomes more 'bond-like.' Together, these effects create a hump-shaped optimal portfolio decision for the agent over the life cycle, consistent with empirical observation.

Labor Income Risk, Optimal Portfolio Choice, Limited Stock Market Participation, Human Capital

Abstract:
The 1987 stock market crash occurred with minimal impact on observable economic variables (e.g., consumption), yet dramatically and permanently changed the shape of the implied volatility curve for equity index options. Here, we propose a general equilibrium model that captures many salient features of the U.S. equity and options markets before, during, and after the crash. The representative agent is endowed with Epstein-Zin preferences and the aggregate dividend and consumption processes are driven by a persistent stochastic growth variable that can jump. In reaction to a market crash, the agent updates her beliefs about the distribution of the jump component. We identify a realistic calibration of the model that matches the prices of short-maturity at-the-money and deep out-of-the-money S&P 500 put options, as well as the prices of individual stock options. Further, the model generates a steep shift in the implied volatility 'smirk' for S&P 500 options after the 1987 crash. This 'regime shift' occurs in spite of a minimal impact on observable macroeconomic fundamentals. Finally, the model's implications are consistent with the empirical properties of dividends, the equity premium, as well as the level and standard deviation of the risk-free rate. Overall, our findings show that it is possible to reconcile the stylized properties of the equity and option markets in the framework of rational expectations, consistent with the notion that these two markets are integrated.

Volatility Smile, Volatility Smirk, Implied Volatility, Option Pricing, Portfolio Insurance, Market Risk, Individual Stock Options

Abstract:
Many financial advisors and much of the academic literature often argue that young people should place most of their savings in stocks. In contrast, a significant fraction of U.S. households do not hold stocks. Investors typically hold very little in stocks when they are young, progressively increase their holdings as they age, and decrease their exposure to stock market risk when they approach retirement. The authors show how long-run labor income risk helps explain this evidence. Moreover, they discuss the effect of long-run labor income risk on the valuation of pension plan obligations, their funding, and the allocation of pension assets across different investment classes.

Abstract:
Many financial advisors and much of the academic literature often argue that young people should place most of their savings in stocks. In contrast, a significant fraction of U.S. households do not hold stocks. Moreover, life-cycle stock holdings are 'hump shaped:' Young investors typically hold very little in stocks, progressively increase their holdings as they age, and decrease their exposure to stock market risk when they approach retirement. In this article, we show how long-run labor income risk helps explain this evidence. Next, we discuss recent developments in the literature that has studied the effect of long-run labor income risk on the valuation of pension fund obligations, their funding, and the allocation of pension assets across different investment classes.

Human Capital, Labor Income Risk, Long-Run Risk, Limited Stock Market Participation, Portfolio Choice, Life Cycle

Abstract:
This paper extends the class of stochastic volatility diffusions for asset returns to encompass Poisson jumps of time-varying intensity. We find that any reasonably descriptive continuous-time model for equity-index returns must allow for discrete jumps as well as stochastic volatility with a pronounced negative relationship between return and volatility innovations. We also find that the dominant empirical characteristics of the return process appear to be priced by the option market. Our analysis indicates a general correspondence between the evidence extracted from daily equity-index returns and the stylized features of the corresponding options market prices.

Abstract:
This paper extends the class of stochastic volatility diffusions for asset returns to encompass Poisson jumps of time-varying intensity. We find that any reasonably descriptive continuous-time model for equity-index returns must allow for discrete jumps as well as stochastic volatility with a pronounced negative relationship between return and volatility innovations. We also find that the dominant empirical characteristics of the return process appear to be priced by the option market. Our analysis indicates a general correspondence between the evidence extracted from daily equity-index returns and the stylized features of the corresponding options market prices.

Abstract:
We examine the long-term return performance of U.S. IPOs underwritten by relationship banks. We show that, over one- to three-year horizons, IPOs managed by relationship banks experience buy-and-hold benchmark-adjusted returns that are similar to those observed for a matching sample of stocks managed by non-relationship underwriters. This result holds even when the returns' skewness and cross-sectional correlation is accounted for. Further, we examine the calendar-time returns on a portfolio that is long the stocks underwritten by relationship banks and short ex-ante similar stocks taken public by non-relationship institutions. Again, we conclude that the two groups of IPOs yield similar long-run returns. These findings support the certification role of relationship banks and suggest that, in this respect, the effect of the 1999 repeal of Sections 20 and 32 of the Glass-Steagall Act has not been negative.

Glass-Steagall Act, IPOs, Certification, Conflict of Interest, Lending Relationships

Abstract:
The 1987 market crash was associated with minimal changes in observable macro-economic fundamentals. Yet, subsequently, the implied volatility curve for equity index options dramatically and permanently steepened. We explain this evidence in a general equilibrium framework in which expected growth in fundamentals and economic uncertainty are subject to jumps. The jumps are rare events which occur with a frequency unknown to agents. The 1987 crash and steepening in smile can be explained by such a (small) jump and the associated updating of beliefs about the probability of future jumps. Our framework also simultaneously captures salient features of individual option prices, stock returns, and interest rates.

Volatility Smile, Volatility Smirk, Implied Volatility, Option Pricing, Portfolio Insurance, Market Risk

Abstract:
Prior to the stock market crash of 1987, Black-Scholes implied volatilities of S&P 500 index options were relatively constant across moneyness. Since the crash, however, deep out-of-the-money S&P 500 put options have become 'expensive' relative to the Black-Scholes benchmark. Many researchers (e.g., Liu, Pan and Wang (2005)) have argued that such prices cannot be justified in a general equilibrium setting if the representative agent has 'standard preferences' and the endowment is an i.i.d. process. Below, however, we use the insight of Bansal and Yaron (2004) to demonstrate that the 'volatility smirk' can be rationalized if the agent is endowed with Epstein-Zin preferences and if the aggregate dividend and consumption processes are driven by a persistent stochastic growth variable that can jump. We identify a realistic calibration of the model that simultaneously matches the empirical properties of dividends, the equity premium, the prices of both at-the-money and deep out-of-the-money puts, and the level of the risk-free rate. A more challenging question (that to our knowledge has not been previously investigated) is whether one can explain within a standard preference framework the stark regime change in the volatility smirk that has maintained since the 1987 market crash. To this end, we extend the model to a Bayesian setting in which the agent updates her beliefs about the average jump size in the event of a jump. Note that such beliefs only update at crash dates, and hence can explain why the volatility smirk has not diminished over the last eighteen years. We find that the model can capture the shape of the implied volatility curve both pre- and post-crash while maintaining reasonable estimates for expected returns, price-dividend ratios, and risk-free rates.