Can standard preferences explain the prices of out-of-the-money S&P 500 put options?

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 shortmaturity 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.Money ; Macroeconomics ; Pricing

Portfolio Choice over the Life-Cycle in the Presence of 'Trickle Down' Labor Income

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.

Can Standard Preferences Explain the Prices of out of the Money S&P 500 Put Options

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 %u2018expensive%u2019 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 %u2018standard preferences%u2019 and the endowment is an i.i.d. process. Below, however, we use the insight of Bansal and Yaron (2004) to demonstrate that the %u2018volatility smirk%u2019 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.

Portfolio choice over the life-cycle when the stock and labor markets are cointegrated

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 effectively becomes.Portfolio management ; Stock market ; Labor market

Can Interest Rate Volatility be Extracted from the Cross Section of Bond Yields? An Investigation of Unspanned Stochastic Volatility

Most affine models of the term structure with stochastic volatility (SV) predict that the variance of the short rate is simultaneously a linear combination of yields and the quadratic variation of the spot rate. However, we find empirically that the A1(3) SV model generates a time series for the variance state variable that is strongly negatively correlated with a GARCH estimate of the quadratic variation of the spot rate process. We then investigate affine models that exhibit "unspanned stochastic volatility (USV)." Of the models tested, only the A1(4) USV model is found to generate both realistic volatility estimates and a good cross-sectional fit. Our findings suggests that interest rate volatility cannot be extracted from the cross-section of bond prices. Separately, we propose an alternative to the canonical representation of affine models introduced by Dai and Singleton (2001). This representation has several advantages, including: (I) the state variables have simple physical interpretations such as level, slope and curvature, (ii) their dynamics remain affine and tractable, (iii) the model is econometrically identifiable, (iv) model-insensitive estimates of the state vector process implied from the term structure are readily available, and (v) it isolates those parameters which are not identifiable from bond prices alone if the model is specified to exhibit USV.

On the relative pricing of long-maturity index options and collateralized debt obligations

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Is Credit Event Risk Priced? Modeling Contagion via the Updating of Beliefs.

Empirical tests of reduced form models of default attribute a large fraction of observed credit spreads to compensation for jump-to-default risk. However, these models preclude a "contagion-risk'' channel, where the aggregate corporate bond index reacts adversely to a credit event. In this paper, we propose a tractable model for pricing corporate bonds subject to contagion-risk. We show that when investors have fragile beliefs (Hansen and Sargent (2009)), contagion premia may be sizable even if P-measure contagion across defaults is small. We find empirical support for contagion in bond returns in response to large credit events. Model calibrations suggest that while contagion risk premia may be sizable, jump-to-default risk premia have an upper bound of a few basis points.

On the Relative Pricing of long Maturity S&P 500 Index Options and CDX Tranches

We investigate a structural model of market and firm-level dynamics in order to jointly price long-dated S&P 500 options and tranche spreads on the five-year CDX index. We demonstrate the importance of calibrating the model to match the entire term structure of CDX index spreads because it contains pertinent information regarding the timing of expected defaults and the specification of idiosyncratic dynamics. Our model matches the time series of tranche spreads well, both before and during the financial crisis, thus offering a resolution to the puzzle reported by Coval, Jurek and Stafford (2009).

Does the Timing of Money Matter? A Case Study of the Arkansas Academic Challenge Scholarship

This paper examines the effect of a state-financed merit-aid scholarship—the Arkansas Academic Challenge Scholarship (ACS)—on post-secondary outcomes at a large university in Arkansas. Exploiting scholarship eligibility requirements, we implement a fuzzy regression discontinuity design to identify the scholarship’s causal impacts on college outcomes. The analysis focuses on currently enrolled sophomores, juniors, and seniors who receive the scholarship to investigate the broad impacts of receiving money at nontraditional points in an individual’s college trajectory. Findings indicate small, negative impacts of scholarship receipt on short-run outcomes such as GPA and credit accumulation, but large statistically significant declines in the likelihood of graduating within four, five, or six years of matriculation. The youngest cohort, who began receiving funding during their sophomore year of enrollment, primarily drives these findings. However, cohort analysis also reveals that seniors who do not graduate on time are 54 percentage points more likely to graduate within 6 years of matriculation when they receive the scholarship. These results highlight the fact that the timing of receiving money may heavily influence student behavior and postsecondary outcomes