Double-jump diffusion model for VIX: evidence from VVIX

This paper studies the continuous-time dynamics of VIX with stochastic volatility and jumps in VIX and volatility. Built on the general parametric affine model with stochastic volatility and jumps in the logarithm of VIX, we derive a linear relationship between the stochastic volatility factor and the VVIX index. We detect the existence of a co-jump of VIX and VVIX and put forward a double-jump stochastic volatility model for VIX through its joint property with VVIX. Using the VVIX index as a proxy for stochastic volatility, we use the MCMC method to estimate the dynamics of VIX. Comparing nested models of VIX, we show that the jump in VIX and the volatility factor are statistically significant. The jump intensity is also stochastic. We analyse the impact of the jump factor on VIX dynamics.     

Option pricing under stochastic volatility and tempered stable Lévy jumps

The purpose of this paper is to introduce a stochastic volatility model for option pricing that exhibits Lévy jump behavior. For this model, we derive the general formula for a European call option. A well known particular case of this class of models is the Bates model, for which the jumps are modeled by a compound Poisson process with normally distributed jumps. Alternatively, we turn our attention to infinite activity jumps produced by a tempered stable process. Then we empirically compare the estimated log-return probability density and the option prices produced from this model to both the Bates model and the Black–Scholes model. We find that the tempered stable jumps describe more precisely market prices than compound Poisson jumps assumed in the Bates model.     

Optimal portfolios when volatility can jump

We consider an asset allocation problem in a continuous-time model with stochastic volatility and jumps in both the asset price and its volatility. First, we derive the optimal portfolio for an investor with constant relative risk aversion. The demand for jump risk includes a hedging component, which is not present in models without volatility jumps. We further show that the introduction of derivative contracts can have substantial economic value. We also analyze the distribution of terminal wealth for an investor who uses the wrong model, either by ignoring volatility jumps or by falsely including such jumps, or who is subject to estimation risk. Whenever a model different from the true one is used, the terminal wealth distribution exhibits fatter tails and (in some cases) significant default risk.     

Time‐Varying Asset Volatility and the Credit Spread Puzzle

Most extant structural credit risk models underestimate credit spreads—a shortcoming known as the credit spread puzzle. We consider a model with priced stochastic asset risk that is able to fit medium‐ to long‐term spreads. The model, augmented by jumps to help explain short‐term spreads, is estimated on firm‐level data and identifies significant asset variance risk premia. An important feature of the model is the significant time variation in risk premia induced by the uncertainty about asset risk. Various extensions are considered, among them optimal leverage and endogenous default.     

Jump-diffusion volatility models for variance swaps: An empirical performance analysis

This paper studies a class of tractable jump-diffusion models, including stochastic volatility models with various specifications of jump intensity for stock returns and variance processes. We employ the Markov chain Monte Carlo (MCMC) method to implement model estimation, and investigate the performance of all models in capturing the term structure of variance swap rates and fitting the dynamics of stock returns. It is evident that the stochastic volatility models, equipped with self-exciting jumps in the spot variance and linearly-dependent jumps in the central-tendency variance, can produce consistent model estimates, aptly explain the stylized facts in variance swaps, and boost pricing performance. Moreover, our empirical results show that large self-exciting jumps in the spot variance, as an independent risk source, facilitate term structure modeling for variance swaps, whilst the central-tendency variance may jump with small sizes, but signaling substantial regime changes in the long run. Both types of jumps occur infrequently, and are more related to market turmoils over the period from 2008 to 2021.     

Empirical Performance of Alternative Option Pricing Models

Substantial progress has been made in developing more realistic option pricing models. Empirically, however, it is not known whether and by how much each generalization improves option pricing and hedging. We fill this gap by first deriving an option model that allows volatility, interest rates and jumps to be stochastic. Using S&P 500 options, we examine several alternative models from three perspectives: (1) internal consistency of implied parameters/volatility with relevant time-series data, (2) out-of-sample pricing, and (3) hedging. Overall, incorporating stochastic volatility and jumps is important for pricing and internal consistency. But for hedging, modeling stochastic volatility alone yields the best performance.     

Interest Rate Caps “Smile” Too! But Can the LIBOR Market Models Capture the Smile?

Using 3 years of interest rate caps price data, we provide a comprehensive documentation of volatility smiles in the caps market. To capture the volatility smiles, we develop a multifactor term structure model with stochastic volatility and jumps that yields a closed‐form formula for cap prices. We show that although a three‐factor stochastic volatility model can price at‐the‐money caps well, significant negative jumps in interest rates are needed to capture the smile. The volatility smile contains information that is not available using only at‐the‐money caps, and this information is important for understanding term structure models.     

Linear-quadratic term structure models – Toward the understanding of jumps in interest rates

We study linear-quadratic term structure models with random jumps in the short rate process where the jump arrival rate follows a stochastic process. Empirical results based on the US data show that incorporating stochastic jump intensity significantly improves model fit to the dynamics of both interest rate and volatility term structure. Our results also show that jump intensity is negatively correlated with interest rate changes and the average size is larger on the downside than upside. Examining the relation between jump intensity and macroeconomic shocks, we find that at monthly frequency, jumps are neither triggered by nor predictive of changes in macroeconomic variables. At daily frequency, however, we document interesting patterns for jumps associated with information shocks.     

Asset allocation: How much does model choice matter?

This paper analyzes the optimal portfolio decision of a CRRA investor in models with stochastic volatility and stochastic jumps. The investor follows a buy-and-hold strategy in the stock, the money market account, and one additional derivative. We show that both the type of the model and the structure of the risk premia have a significant impact on the optimal portfolio, on the utility gain from having access to derivatives, and on whether the investor prefers to trade OTM or ATM options. We also show that model mis-specification results in significant utility losses. Omitting jumps in volatility can be devastating, in particular if the investor chooses the seemingly optimal OTM put options. A misestimation of the structure of the risk premia has a less devastating effect, but can still lead to a loss of around 4% in the annual certainty equivalent return.     

Portfolio optimization for student t and skewed t returns

We consider the problem of valuing a European option written on an asset whose dynamics are described by an exponential Lévy-type model. In our framework, both the volatility and jump-intensity are allowed to vary stochastically in time through common driving factors—one fast-varying and one slow-varying. Using Fourier analysis we derive an explicit formula for the approximate price of any European-style derivative whose payoff has a generalized Fourier transform; in particular, this includes European calls and puts. From a theoretical perspective, our results extend the class of multiscale stochastic volatility models of Fouque et al. [Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives, 2011] to models of the exponential Lévy type. From a financial perspective, the inclusion of jumps and stochastic volatility allow us to capture the term-structure of implied volatility, as demonstrated in a calibration to S&;P500 options data.     

Earning the right premium on the right factor in portfolio planning

The optimal portfolio as well as the utility from trading stocks and derivatives depends on the risk factors and on their market prices of risk. We analyze this dependence for a CRRA investor in models with stochastic volatility, jumps in the stock price, and jumps in volatility. We find that the compartment of the total variance into diffusion risk and jump risk has a small impact on the utility in an incomplete market only. In contrast, the decomposition of the equity risk premium into a diffusion component and a jump risk component and the compartment of the latter into its various elements has a huge impact on the utility in a complete market. The more extreme the market prices of risk, i.e. the more they deviate from their equilibrium values, the larger the utility of the investor. Additionally, we show that the structure of the optimal exposures to jump risk crucially depends on which elements of jump risk are priced.     

Risk contagion among international stock markets

We develop a stochastic volatility model with jumps in returns and volatility to analyze the risk spillover from the U.S. market and the regional market to a number of European countries’ equity markets. The key advantage of this approach compared to the earlier approaches is that it enables us to identify jumps and investigate spillover of extreme events across borders. We find that a large part of the jumps in the local markets are due to the U.S. market and the regional market. The U.S. contribution to the variances is in general below the contribution from the regional market. In general, we observe an increasing integration during the last two decades, which, to some extent, can be related to the advancement of the European Union. Furthermore, we show that the identification of the jumps can be used as a useful signal for portfolio reallocation.     

The impact of co-jumps in the oil sector

We study the dynamics of the oil sector using a new multivariate stochastic volatility model with a structure of common factors subjected to jumps in mean and conditional variance. This model contributes to the literature allowing the estimation of spillover effects between assets in a multivariate framework through joint jumps (co-jumps), identifying the permanent and transitory effects through a structure defined by Bernoulli processes. The jump structure introduced in the article can be interpreted as a regime-switching model with an endogenous number of states, avoiding the difficulties associated with models with a fixed number of regimes. We apply the model to oil prices and stock prices of integrated oil companies. The jump structure allows dating the relevant events in the oil sector in the period 2000–2019. The period analyzed encompasses important events in the oil market such as the price escalation in 2008 and the falling prices in 2014. We also apply the model to estimate risk management measures and portfolio allocation and perform a comparison with other multivariate models of conditional volatility, showing the good properties of the model in these applications.     

Bayesian estimation of the stochastic volatility model with double exponential jumps

Review of Derivatives Research - This paper generalizes the stochastic volatility model to allow for the double exponential jumps. To derive the jumps and time-varying volatility in returns, we...     

Quadratic hedging in affine stochastic volatility models

We determine the variance-optimal hedge for a subset of affine processes including a number of popular stochastic volatility models. This framework does not require the asset to be a martingale. We obtain semiexplicit formulas for the optimal hedging strategy and the minimal hedging error by applying general structural results and Laplace transform techniques. The approach is illustrated numerically for a Lévy-driven stochastic volatility model with jumps as in Carr et al. (Math Finance 13:345–382, 2003).      

The role of stochastic volatility and return jumps: reproducing volatility and higher moments in the KOSPI 200 returns dynamics

This paper investigates the role of stochastic volatility and return jumps in reproducing the volatility dynamics and the shape characteristics of the Korean Composite Stock Price Index (KOSPI) 200 returns distribution. Using efficient method of moments and reprojection analysis, we find that stochastic volatility models, both with and without return jumps, capture return dynamics surprisingly well. The stochastic volatility model without return jumps, however, cannot fully reproduce the conditional kurtosis implied by the data. Return jumps successfully complement this gap. We also find that return jumps are essential in capturing the volatility smirk effects observed in short-term options.

Geometric Asian option pricing in general affine stochastic volatility models with jumps

In this paper, we present some results on Geometric Asian option valuation for affine stochastic volatility models with jumps. We shall provide a general framework into which several different valuation problems based on some average process can be cast, and we shall obtain closed form solutions for some relevant affine model classes.     

Value at risk forecasts by extreme value models in a conditional duration framework

The analysis of extremes in financial return series is often based on the assumption of independent and identically distributed observations. However, stylized facts such as clustered extremes and serial dependence typically violate the assumption of independence. This has been the main motivation to propose an approach that is able to overcome these difficulties by considering the time between extreme events as a stochastic process. One of the advantages of the method consists in its capability to capture the short-term behavior of extremes without involving an arbitrary stochastic volatility model or a prefiltration of the data, which would certainly affect the estimate. We make use of the proposed model to obtain an improved estimate for the value at risk (VaR). The model is then compared to various competing approaches such as Engle and Marianelli's CAViaR and the GARCH-EVT model. Finally, we present a comparative empirical illustration with transaction data from Bayer AG, a typical blue chip stock from the German stock market index DAX, the DAX index itself and a hypothetical portfolio of international equity indexes already used by other authors.     

An empirical investigation of multiperiod tail risk forecasting models

In the context of multiperiod tail risk (i.e., VaR and ES) forecasting, we provide a new semiparametric risk model constructed based on the forward-looking return moments estimated by the stochastic volatility model with price jumps and the Cornish–Fisher expansion method, denoted by SVJCF. We apply the proposed SVJCF model to make multiperiod ahead tail risk forecasts over multiple forecast horizons for S&P 500 index, individual stocks and other representative financial instruments. The model performance of SVJCF is compared with other classical multiperiod risk forecasting models via various backtesting methods. The empirical results suggest that SVJCF is a valid alternative multiperiod tail risk measurement; in addition, the tail risk generated by the SVJCF model is more stable and thus should be favored by risk managers and regulatory authorities.