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Abstract:
We construct a statistical model for term-structure of implied volatilities of currency options based on daily historical data for 13 currency pairs in a 19-month period. We examine the joint evolution of 1 month, 2 month, 3 month, 6 month and 1 year 50-delta options in all the currency pairs. We show that there exist three uncorrelated state variables (principal components) which account for the parallel movement, slope oscillation, and curvature of the term structure and which explain, on average, the movements of the term-structure of volatility to more than 95% in all cases. We test and construct an exponential ARCH, or E-ARCH, model for each state variable. One of the applications of this model is to produce confidence bands for the term- structure of volatility.

Abstract:
In this paper, we analyze the usefulness of technical analysis, specifically the widely used moving average trading rule, from an asset allocation perspective. We show that when stock returns are predictable, technical analysis adds value to commonly used allocation rules that invest fixed proportions of wealth in stocks. When there is uncertainty about predictability, the fixed allocation rules combined with technical analysis can outperform the prior-dependent optimal learning rule when the prior is not too informative. Moreover, the technical trading rules are robust to model specification, and they tend to substantially outperform the model-based optimal trading strategies when there is uncertainty about the model governing the stock price.

Technical analysis, trading rules, asset allocation, predictability, learning

Abstract:
Using no arbitrage principle, we derive a relationship between the drift term of risk-neutral dynamics for instantaneous variance and the term structure of forward variance curve. We show that the forward variance curve can be derived from options market. Based on the variance term structure, we derive a no arbitrage pricing model for VIX futures pricing. The model is the first no arbitrage model combining options market and VIX futures market. The model can be easily generalized to price other volatility derivatives.

Stochastic volatility, variance term structure, arbitrage-free model, volatility derivatives, VIX futures

Abstract:
In this paper we analyze CBOE VIX futures price time series data from Mar. 2004 to Feb. 2008. We derive a new pricing framework for VIX futures that is convenient to study variance term structure dynamics. Our main contribution to existing literature is the identification of the number of factors implicit in VIX futures term structure. We find that three-factor model is ideal to characterize the variance term structure. We further construct and estimate structured two- and three-factor models to identify the components and find similar results.

VIX futures, loglinear model, Kalman filter, Principal Component Analysis (PCA), variance term structure

Abstract:
In this paper, we extend the long-run risks model of Bansal and Yaron (BY, 2004) to allow both a long- and a short-run volatility component in consumption growth, long-run risks, and dividend growth. Our two volatility model better captures macroeconomic volatility than a single volatility model, and can reconcile simultaneously the large negative market variance risk premium, differing predictability in excess returns, consumption, dividends, and stock market volatility, all of which are difficult to explain previously by the BY model.

Long-run Risk, Equity Risk Premium, Predictability, Variance Risk Premium

Abstract:
In the recent financial crisis, the Dow Jones stock market index dropped about 54% from a high of 14164.53 on October 9, 2007 to a low of 6547.05 on March 9, 2009. Alan Greenspan calls this a 'once-in-a century' crisis. While we do not know how he drew his conclusion, we show that the probability of a stock market drop of 50% from its high within a century is about 90% based on the popular random walk model of the stock prices. With a broad market index of the S&P500 and a more sophisticated asset pricing model which captures more risks in the economy, the probability rises to above 99%. The message of this paper is that a market drop of 50% or more is very likely in long-run stock market investments, and the investors should be prepared for it.

financial crisis, Once-in-a-Century event, drawndown probability

Abstract:
VIX futures are exchange-traded contracts on a future volatility index level (VIX) derived from a basket of SPX stock index options. The paper posits a stochastic variance model of VIX time evolution, and develops an expression for VIX futures. Free parameters are estimated from market data over the past few years. It is found that the model with parameters estimated from the whole period from 1990 to 2005 overprices the futures contracts by 16-44%. But the discrepancy is dramatically reduced to 2-12% if the parameters are estimated from the most recent one-year period.

VIX, VIX futures, square root process

Abstract:
This paper serves two purposes. First, we provide an analytical approximate solution method to solve the optimal consumption and portfolio choice problem for an investor with recursive utility in a complete market. The investment opportunity set is stochastic over time. The problem is solved exactly for special case with unit elasticity of intertemporal substitution, and approximate solution is derived in closed form for more general cases. The solution method provides the same solutions for cases when there exist known analytical solutions. Second, as an important application as well as an illustration of the performance of the solution method, we solve in detail a practical example of Heston's (1993) stochastic volatility model. The market is complete with trading of derivatives, either through options, or pure volatility derivatives such as variance swap. We pay special attention to the new insights that the solution method provides. Specifically we discuss in detail the hedging demand for derivative securities due to the stochastic nature of price volatility. Previous solutions either exclude volatility trading, or assume the expected additive utility without intertemporal consumptions. We calibrate the model to S\&P 500 index and VIX index. The impact of elasticity of intertemporal substitution is separated from that of the risk aversion. We show, contrary to the existing literature on hedging demand of volatility, the effect of the elasticity of intertemporal substitution on hedging demand for derivatives is of first order importance. The investment horizon effect on portfolio choice is also examined.

Portfolio Choice, Recursive Utility, Martingale Approach, Stochastic Volatility