Much research has investigated the differences between option implied volatilities and econometric model-based forecasts Implied volatility is a market determined forecast, in contrast to model-based forecasts that employ some degree of smoothing of past volatility to generate forecasts Implied volatility has the potential to reflect information that a model-based forecast could not This paper considers two issues relating to the informational content of the S&P 500 VIX implied volatility index First, whether it subsumes information on how historical jump activity contributed to the price volatility, followed by whether the VIX reflects any incremental information pertaining to future jump activity relative to model-based forecasts It is found that the VIX index both subsumes information relating to past jump contributions to total volatility and reflects incremental information pertaining to future jump activity This issue has not been examined previously and expands our understanding of how option markets form their volatility forecasts

/pdf/the-jump-component-of-s-p-500-volatility-and-the-vix-index-2dopiy3hnl.pdf

The jump component of S&P 500 volatility and the VIX index

Financial risk measurement relies on models of prices and other market variables, but models inevitably rely on imperfect assumptions and estimates, creating model risk. Moreover, optimization decisions, such as portfolio selection, amplify the effect of model error. In this work, we develop a framework for quantifying the impact of model error and for measuring and minimizing risk in a way that is robust to model error. This robust approach starts from a baseline model and finds the worst-case error in risk measurement that would be incurred through a deviation from the baseline model, given a precise constraint on the plausibility of the deviation. Using relative entropy to constrain model distance leads to an explicit characterization of worst-case model errors; this characterization lends itself to Monte Carlo simulation, allowing straightforward calculation of bounds on model error with very little computational effort beyond that required to evaluate performance under the baseline nominal model. This approach goes well beyond the effect of errors in parameter estimates to consider errors in the underlying stochastic assumptions of the model and to characterize the greatest vulnerabilities to error in a model. We apply this approach to problems of portfolio risk measurement, credit risk, delta hedging, and counterparty risk measured through credit valuation adjustment.

Robust Risk Measurement and Model Risk

We determine the optimal portfolio for an ambiguity averse investor who has access to stock and derivatives markets. The stock price follows a stochastic volatility jump-diffusion process and the investor can have different levels of uncertainty about the diffusion parts of the stock and its volatility. We find strong evidence that the optimal exposures to stock and volatility risks are significantly affected by the ambiguity aversion to the corresponding risk factor only. We also show that volatility ambiguity has a smaller impact in incomplete markets. Investors who ignore jump risk/model uncertainty/derivatives always incur welfare losses. In our numerical example, the loss from neglecting model uncertainty can be almost as much as the loss from not trading the derivatives.

Robust portfolio choice with derivative trading under stochastic volatility

Volatility jumps and their determinants in REIT returns

We use S&P 500 index return data for the time period 1985-2012 to evaluate the performance of portfolio insurance strategies. We shed light on the question if the performance of a constant proportion portfolio insurance (CPPI) strategy can be improved by means of a time-varying multiplier which depends on the estimated future volatility. Neglecting any inter-temporal hedging demand, the theoretical foundation of the strategies is given by maximizing the expected utility of a HARA investor in a diffusion model setup. If the risk premium is assumed to be proportional to the variance, the optimal strategy is a CPPI strategy. Otherwise, the multiple is time-varying (PPI). It turns out that even time-varying multiple strategies based on a rolling window of historical volatility estimates give a significant improvement of CPPI strategies. The out-performance is robust w.r.t. alternative performance measures and is also true for proportional transaction costs and adequate trigger trading.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID2345441_code1939527.pdf?abstractid=2183110&mirid=1

Performance Evaluation of Optimized Portfolio Insurance Strategies

We consider an asset allocation problem in a continuous-time model with stochastic volatility and (possibly correlated) jumps in both the asset price and its volatility. First, we derive the optimal portfolio for an investor with constant relative risk aversion. One main finding is that the demand for jump risk now also includes a hedging component, which is not present in models without jumps in volatility. Second, we show in a partial equilibrium framework that the introduction of non-linear derivative contracts can have a substantial economic value. Third, we analyze the distribution of terminal wealth for an investor who uses the wrong model when making portfolio choices, either by ignoring volatility jumps or by falsely including such jumps although they are not present in the true model. In both cases the terminal wealth distribution exhibits fatter tails than under the correctly specified model, as well as significant default risk. Volatility jumps are thus an important risk factor in portfolio planning.

/pdf/optimal-portfolios-when-volatility-can-jump-1ohaddb37v.pdf

Optimal Portfolios When Volatility Can Jump

It is often difficult to distinguish among different option pricing models that consider stochastic volatility and/or jumps based on a cross-section of European option prices. This can result in model misspecification. We analyze the hedging error induced by model misspecification and show that it can be economically significant in the cases of a delta hedge, a minimum variance hedge, and a delta-vega hedge. Furthermore, we explain the surprisingly good performance of a simple ad-hoc Black-Scholes hedge. We compare realized hedging errors (an incorrect hedge model is applied) and anticipated hedging errors (the hedge model is the true one) and find that there are substantial differences between the two distributions, particularly depending on whether stochastic volatility is included in the hedge model. Therefore, hedging errors can be useful for identifying model misspecification. Furthermore, model risk has severe implications for risk measurement and can lead to a significant misestimation, specifically underestimation, of the risk to which a hedged position is exposed.

Hedging Under Model Misspecification: All Risk Factors are Equal, But Some are More Equal than Others ...

In order to use an option pricing model for dynamic hedging an investor will have to calibrate it to a cross-section of option prices. Microstructural noise in option prices results in a set of indistinguishable parametrizations which may give rise to different hedging errors. In our simulation study for the Heston (1993) model, we identify the parameters most important for hedging and show which set of strikes and time to maturity is relevant for the identification of certain parameters. In our empirical study we show that different but indistinguishable parametrizations w.r.t. prices may induce large differences in hedging performance.

Hedging Options in the Presence of Microstructural Noise

In a recent paper, Duan and Wei (2007) find that the higher the proportion of systematic risk the higher will be the level and the slope of the implied volatility curve. We show that this result can be explained in a variety of continuous - time option pricing models and explicitly point out the transmission mechanisms that lead to an impact of systematic risk on option prices. Most importantly we show that an investor who uses the structurally correct model but ignores the proportion of systematic risk in the underlying would still price options correctly.

Systematic Risk and Option Prices

Option pricing models that include stochastic volatility and/or stochastic jumps are often hard to distinguish from each other based on the prices of European plain-vanilla options only, where one reason are rather high bid-ask spreads. We flrst analyze the hedging error induced by this model mis-speciflcation. This impact of model risk is economically signiflcant. We flnd that it is largest for delta-vega and smallest for minimum variance hedges. Furthermore, we show that a simple BS ad-hoc hedge is biased towards a minimum-variance hedge, which explains its surprisingly good performance. Second, we analyze whether hedging errors can help in identifying model mis-speciflcation and ideally also the correct model. We show that there are substantial difierences between realized hedging errors (if the incorrect hedge model is applied to the true data-generating process) and anticipated hedging errors (if the hedge model were applied to prices generated by the hedge model), in particular depending on whether or not stochastic volatility is included in the hedge model. Hedging errors can thus provide useful support in model identiflcation.

Which Risk Factors Should You Not Forget

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