This paper analyzes exponentially affine and non-affine stochastic volatility models with jumps in returns, and jumps in returns and volatility. One of the main research topics in finance is to find models which fully capture the statistical properties of asset returns. In particular, the model framework forstock prices has evolved dramatically over the past decades. Staring from the fairly simpleassumption of constant volatility in the Black-Scholes-Merton framework, model complexity and sophistication have increased in order to capture stylized facts of the data. These include for examplecrash events in stock markets, volatility clustering, smile patterns in options data, and the leverage effect.
Including further risk factors such as stochastic volatility, jumps in returns and jumps in volatility allows the modeling of these stylized facts in a comprehensive way. A general framework that allows for jumps in returns, stochastic volatility, and jumps in returns and volatility can be found in Duffie, Pan and Singleton (2000). Thismodel nests the specifications analyzed by Merton (1976), Heston (1993), and Bates (1996).
Empirical testing of these continuous time stock price processes has been carried out in severalstudies. Stochastic volatility models were analyzed in Jacquier, Polson, and Rossi (1994, 2004). Models with jumps in returns and stochastic volatility can be found in Pan (2002), Bakshi, Cao and Chen (1997) and Chernov, Gallant and Ghysels (2003). Conclusions drawn from these studies are thatthe Heston (1993) type stochastic volatility model is severely misspecified. Although models including additionally jumps in the return process have a better empirical performance compared to stochastic volatility models, it has been found by,e.g., Bakshi et al. (1997), Bates (2000), and Pan (2002) that these models are still potentially misspecified. Recent work by Eraker, Johannes and Polson (2003) and Christoffersen et al. (2008) test two approaches to overcome the problem of misspecification. The first study uses the framework laid out by Duffie et al. (2000) and analyzes severalmodels using the S&P 500 and the NASDAQ 100 stock market indices. The models under consideration are the stochastic volatility model, the stochastic volatility model containing jumps in returns, and twostochastic volatility models with jumps in returns and volatility. The first stochastic volatility model assumes that jumps in returns and volatility are correlated, whereas the second modelsassumes jump components to be independent. Eraker et al. (2003) find significant better performance of models including jumps in volatility compared to models without jumps in volatility. The second approach proposed by Christoffersen et al. (2008) suggests to leave the class of the exponentially affine models by investigating some alternative non-affine specifications for the volatility process without considering jumps. They test several alternative specifications for the stochastic volatility process using the data of S&P 500 index returns. In particular, they change thespeed of mean reversion and the diffusion term in the specification of the stochastic volatility process. They find significant improvements resultingin less misspecification and better empirical performance of stochastic volatility models when leaving the class of affine models.
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Stochastic Volatility and Jumps
