The Short-Time Behaviour of VIX Implied Volatilities in a Multifactor Stochastic Volatility Framework PDF Download
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Author: Andrea Barletta Publisher: ISBN: Category : Languages : en Pages : 31
Book Description
We consider a modelling setup where the VIX index dynamics are explicitly computable as a smooth transformation of a purely diffusive, multidimensional Markov process. The framework is general enough to embed many popular stochastic volatility models. We develop closed-form expansions and sharp error bounds for VIX futures, options and implied volatilities. In particular, we derive exact asymptotic results for VIX implied volatilities, and their sensitivities, in the joint limit of short time-to-maturity and small log-moneyness. The obtained expansions are explicit, based on elementary functions and they neatly uncover how the VIX skew depends on the specific choice of the volatility and the vol-of-vol processes. Our results are based on perturbation techniques applied to the infinitesimal generator of the underlying process. This methodology has been previously adopted to derive approximations of equity (SPX) options. However, the generalizations needed to cover the case of VIX options are by no means straightforward as the dynamics of the underlying VIX futures are not explicitly known. To illustrate the accuracy of our technique, we provide numerical implementations for a selection of model specifications.
Author: Andrea Barletta Publisher: ISBN: Category : Languages : en Pages : 31
Book Description
We consider a modelling setup where the VIX index dynamics are explicitly computable as a smooth transformation of a purely diffusive, multidimensional Markov process. The framework is general enough to embed many popular stochastic volatility models. We develop closed-form expansions and sharp error bounds for VIX futures, options and implied volatilities. In particular, we derive exact asymptotic results for VIX implied volatilities, and their sensitivities, in the joint limit of short time-to-maturity and small log-moneyness. The obtained expansions are explicit, based on elementary functions and they neatly uncover how the VIX skew depends on the specific choice of the volatility and the vol-of-vol processes. Our results are based on perturbation techniques applied to the infinitesimal generator of the underlying process. This methodology has been previously adopted to derive approximations of equity (SPX) options. However, the generalizations needed to cover the case of VIX options are by no means straightforward as the dynamics of the underlying VIX futures are not explicitly known. To illustrate the accuracy of our technique, we provide numerical implementations for a selection of model specifications.
Author: Neil Shephard Publisher: Oxford University Press, USA ISBN: 0199257205 Category : Business & Economics Languages : en Pages : 534
Book Description
Stochastic volatility is the main concept used in the fields of financial economics and mathematical finance to deal with time-varying volatility in financial markets. This work brings together some of the main papers that have influenced this field, andshows that the development of this subject has been highly multidisciplinary.
Author: Matthew Lorig Publisher: ISBN: Category : Languages : en Pages : 36
Book Description
We consider an asset whose risk-neutral dynamics are described by a general class of local-stochastic volatility models and derive a family of asymptotic expansions for European-style option prices and implied volatilities. Our implied volatility expansions are explicit; they do not require any special functions nor do they require numerical integration. To illustrate the accuracy and versatility of our method, we implement it under five different model dynamics: CEV local volatility, quadratic local volatility, Heston stochastic volatility, 3/2 stochastic volatility, and SABR local-stochastic volatility.
Author: Chloe Lacombe Publisher: ISBN: Category : Languages : en Pages : 28
Book Description
We present small-time implied volatility asymptotics for Realised Variance (RV) and VIX options for a number of (rough) stochastic volatility models via large deviations principle. We provide numerical results along with efficient and robust numerical recipes to compute the rate function; the backbone of our theoretical framework. Based on our results, we further develop approximation schemes for the density of RV, which in turn allows to express the volatility swap in close-form. Lastly, we investigate different constructions of multi-factor models and how each of them affects the convexity of the implied volatility smile. Interestingly, we identify the class of models that generate non-linear smiles around-the-money.
Author: Mikkel Bennedsen Publisher: ISBN: Category : Languages : en Pages : 46
Book Description
We study the empirical properties of realized volatility of the E-mini S&P 500 futures contract at various time scales, ranging from a few minutes to one day. Our main finding is that intraday volatility is remarkably rough and persistent. What is more, by further studying daily realized volatility measures of close to two thousand individual US equities, we find that both roughness and persistence appear to be universal properties of volatility. Inspired by the empirical findings, we introduce a new class of continuous-time stochastic volatility models, capable of decoupling roughness (short-term behavior) from long memory and persistence (long-term behavior) in a simple and parsimonious way, which allows us to successfully model volatility at all intraday time scales. Our prime model is based on the so-called Brownian semistationary process and we derive a number of theoretical properties of this process, relevant to volatility modeling. As an illustration of the usefulness our new models, we conduct an extensive forecasting study; we find that the models proposed in this paper outperform a wide array of benchmarks considerably, indicating that it pays off to exploit both roughness and persistence in volatility forecasting.
Author: Andrew Papanicolaou Publisher: ISBN: Category : Languages : en Pages : 31
Book Description
This paper shows how to recover stochastic volatility models (SVMs) from market models for the VIX futures term structure. Market models have more flexibility for fitting of curves than do SVMs, and therefore they are better-suited for pricing VIX futures and derivatives. But the VIX itself is a derivative of the S&P500 (SPX) and it is common practice to price SPX derivatives using an SVM. Hence, a consistent model for both SPX and VIX derivatives would be one where the SVM is obtained by inverting the market model. This paper's main result is a method for the recovery of a stochastic volatility function as the output of an inverse problem, with the inputs given by a VIX futures market model. Analysis will show that some conditions need to be met in order for there to not be any inter-model arbitrage or mis-priced derivatives. Given these conditions the inverse problem can be solved. Several models are analyzed and explored numerically to gain a better understanding of the theory and its limitations.
Author: Mikhail Chernov Publisher: ISBN: Category : Languages : en Pages : 32
Book Description
The unbiasedness tests of implied volatility as a forecast of future realized volatility have found implied volatility to be a biased predictor. We explain this puzzle by recognizing that option prices contain a market risk premium not only on the asset itself, but also on its volatility. Hull and White (1987) show using a stochastic volatility model that a call option price can be represented as an expected value of the Black-Scholes formula evaluated at the average integrated volatility. If we allow volatility risk to be priced, this expectation should be taken under the risk-neutral probability measure, and can be decomposed into the expectation with respect to the physical measure and the risk-premium term. This term is just a linear function of the unobservable spot volatility. The decomposition explains the bias documented in the empirical literature and shows that the realized and historical volatility, which are used in the tests, are in fact the estimates of the unobserved quadratic variation and spot volatility of the stock-return generating process. Therefore, the use of these estimates generates the error-in-the-variables problem. We generalize the above results from a stochastic volatility model to a model with multiple volatility and jump factors. We provide an empirical illustration based on two US equity indices and three foreign currency rates. We find, that when we take into an account the risk-premium and use efficient methods to estimate volatility, the unbiasedness hypothesis can not be rejected, and the point estimate of the loading on the implied volatility in the traditional regression is equal to 1.
Author: Andrea Barletta
Author: Andrea Barletta
Author: Neil Shephard
Author: Alexey Medvedev
Author: Elisa Alós
Author: Matthew Lorig
Author: Elisa Alós
Author: Chloe Lacombe
Author: Mikkel Bennedsen
Author: Andrew Papanicolaou
Author: Mikhail Chernov