Abstract
We propose a new method to jointly estimate volatility risk and two-tail risk price with state-dependent features. Rather than assuming a constant risk price, as in existing models, this new method estimates an extended pricing kernel with macro-state-dependent risk prices. In contrast to the widely accepted constant risk price assumption, we find that the prices for equity, volatility, positive jump, and negative jump risks are strongly dependent on economic conditions. The empirical evidence shows that this new estimation for the macro-state-dependent property adds new pricing information that existing constant risk-price models do not provide. The estimation of macro-state-dependent property has important economic implications for the underlying dynamics and derivative markets. State-dependent risk prices substantially improve the explanation of the dynamic link between the underlying and option markets, and are important factors in the option market. With the out-of-sample test, the new method provides a stable estimation of the risk price dynamics.
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Notes
When the extreme value distribution is of the Frechet type, the rate of decay of the tails is approximately the same as that of the power law. The double exponential tails decay at the same rate as the power speed. Thus, we can approximately represent a general type of positive and negative jumps by an exponential distribution when the jumps are large enough.
We adopt a similar specification as that in Bollerslev et al. (2011), where we use the AR(1) process to estimate the dynamic of the volatility premium. This specification simplifies our estimation while maintaining a parsimonious autocorrelation structure.
This result is based on results in Bollerslev et al. (2011).
We can compute the variance of jumps explicitly based on the jump parameters:\( {\text{var}}\left( {\mathop \int \limits_{t}^{t + \Delta } \mathop \int \limits_{{\mathbb{R}}} \left( {e^{x} - 1} \right)\tilde{\mu }_{J}^{Q} \left( {{\text{d}}x,{\text{d}}t} \right)} \right) = \frac{{2\psi_{t}^{ + } \Delta }}{{\alpha_{t}^{ + ,Q} \left( {\alpha_{t}^{ + ,Q} - 1} \right)\left( {\alpha_{t}^{ + ,Q} - 2} \right)}} + \frac{{2\psi_{t}^{ - } \Delta }}{{\alpha_{t}^{ - ,Q} \left( {\alpha_{t}^{ - ,Q} + 1} \right)\left( {\alpha_{t}^{ - ,Q} + 2} \right)}} \). Bollerslev and Todorov (2014) and Bollerslev et al. (2015) estimate the jump parameters \( \alpha_{t}^{ \pm ,Q} {\text{and}} \psi_{t}^{ \pm } \). We use their approach to estimate these jump parameters for the moment conditions in the GMM.
We estimate the covariance matrix using heteroskedastic and autocorrelation-robust approaches.
The estimation uses the five-minute S&P 500 time series. According to Lee and Mykland (2008), the likelihood of misclassification by spurious detection of jumps and a failure to detect actual jumps is negligible at price data frequencies of 15 min or higher.
In an unreported result, we use a regression with the overlapping one-year sample period to examine the consistency of the high-frequency time series. The result confirms that the two types of the five-minute S&P 500 time series are extremely close, with a correlation more than 0.99.
When we calculate the model-implied premia, we estimate the Q-measure expectations using the option data; that is \( \widehat{IV}_{t} \) and \( \hat{\alpha }_{t}^{ \pm ,Q} \), and calculate the P-measure expectations based on the state-dependent risk price model in Sect. 2.
The details of the estimation of the moments in the Q-measure return distribution can be found in Bakshi et al. (2003), who established the model-free method using derivatives market data.
A more comprehensive question is what is the best selection of variables, though we do not discuss this issue here because an analysis needs at least \( 2^{28} \left( {4 {\text{risk prices}} \times 7 {\text{variables}}} \right) \) times likelihood maximization to select the most relevant variables in the model under the AIC or BIC framework.
Although the out-of-sample periods (52 weeks and 104 weeks) seem much shorter than the in-sample period, it contains a time interval of around one to two years. Because we use seven state variables, we need to estimate 40 parameters. A decrease in the in-sample period length can significantly reduce the accuracy of this estimation.
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Chen, S., Gu, Y. Joint estimation of volatility risk and tail risk premia with time-varying macro-state-dependent property. Rev Quant Finan Acc 56, 1357–1397 (2021). https://doi.org/10.1007/s11156-020-00925-6
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DOI: https://doi.org/10.1007/s11156-020-00925-6