Valuation of VIX and target volatility options with affine GARCH models

Cao, Hongkai; Badescu, Alexandru; Cui, Zhenyu; Jayaraman, Sarath Kumar

doi:10.1002/fut.22157

Cited by 17 publications

(24 citation statements)

References 55 publications

(65 reference statements)

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“…When combined with Equation (1), the European VIX call option price

C

is given by

C_{t} = e^{- r MathClass-open(T - t MathClass-close)} {double-struckE}_{t}^{double-struckQ} MathClass-open(\max MathClass-open({VIX}_{T} - K, 0 MathClass-close) MathClass-close) = e^{- r MathClass-open(T - t MathClass-close)} {double-struckE}_{t}^{double-struckQ} MathClass-open(\max MathClass-open(\sqrt{a + y_{T}} - K, 0 MathClass-close) MathClass-close)

where

T

is the date of maturity,

{VIX}_{T}

is the terminal VIX price, and

K

is the strike price. Due to the affine structure of the GARV model, a closed‐form VIX option price is available, which can be easily developed from a similar approach adopted by Lian and Zhu (2013) and Cao, Badescu, et al (2020). We illustrate our pricing formula in the following proposition.…”

Section: The Garv Modelmentioning

confidence: 99%

“…We adopt the variance‐dependent pricing kernel introduced by Christoffersen, Heston, et al (2013) to risk neutralize the HNG model, which takes the variance premium into account. The analytical VIX option pricing formula for the HNG model is provided by Cao, Badescu, et al (2020). In fact, as we have mentioned above, the HNG model is a special case of the GARV model, in which the weight on the volatility component of return is one (i.e.,

ξ = 1

).…”

Section: Model Comparisonmentioning

confidence: 99%

“…Building on the concept of the instantaneous squared VIX (ISVIX), Luo et al (2019) modeled the ISVIX as a mean‐reverting jump‐diffusion process with a stochastic long‐term mean, and thus obtained analytical formulas for VIX options and futures. For the discrete‐time models, Cao, Badescu, et al (2020) proposed a (semi) closed‐form solution for the price of VIX options by using the (affine) Heston–Nandi GARCH (HNG) models. The discrete‐time models considered in our paper belong to this strand of literature.…”

Section: Introductionmentioning

confidence: 99%

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Pricing VIX options with realized volatility

Tong

Huang

2021

Journal of Futures Markets

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We investigate the role of realized volatility in pricing VIX options by using the generalized affine realized volatility (GARV) model, and the Realized generalized autoregressive conditionally heteroscedastic (GARCH) model. We develop a closed-form pricing formula for the (affine) GARV model. For the (nonaffine) log-linear Realized GARCH model, we introduce a novel approximation approach to derive its analytical pricing formula. Empirical results show that models with realized volatility significantly outperform competing models based on daily returns only, both in-and out-of-sample. The Realized GARCH model offers the best pricing performance, as it has fewer constraints and a more flexible modeling structure.

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“…When combined with Equation (1), the European VIX call option price

C

is given by

C_{t} = e^{- r MathClass-open(T - t MathClass-close)} {double-struckE}_{t}^{double-struckQ} MathClass-open(\max MathClass-open({VIX}_{T} - K, 0 MathClass-close) MathClass-close) = e^{- r MathClass-open(T - t MathClass-close)} {double-struckE}_{t}^{double-struckQ} MathClass-open(\max MathClass-open(\sqrt{a + y_{T}} - K, 0 MathClass-close) MathClass-close)

where

T

is the date of maturity,

{VIX}_{T}

is the terminal VIX price, and

K

Section: The Garv Modelmentioning

confidence: 99%

ξ = 1

).…”

Section: Model Comparisonmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Pricing VIX options with realized volatility

Tong

Huang

2021

Journal of Futures Markets

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show abstract

“…First, Cao et al (2021) deal with the Heston-Nandi (HN) and inverse Gaussian (IG) GARCH models, while we utilize the two most important GARCH models, namely G11 and GJR. Second, Cao et al (2021) suffer the same problems mentioned previously for integrals of complex variables under continuous-time volatility models. Worse, the generalized Fourier transforms in Cao et al (2021) depend on recursive formulas for HN and IG and are therefore more complicated than those of continuous-time volatility models.…”

mentioning

confidence: 99%

GARCH pricing and hedging of VIX options

Liu

Jiao

Guo

2022

Journal of Futures Markets

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We are the first to study the pricing and hedging of VIX options via Monte Carlo (MC) under GARCH(1,1) and Glosten–Jagannathan–Runkle GARCH(1,1) models. Our pricing is ab initio and out‐of‐sample and can be implemented in real time. Importantly, we propose the so‐called single‐option hedge error, a better measure than that of Bakshi et al., and suggest several techniques to expedite MC by over 1000 times, which rivals the potential speedup of quantum computers. Empirically, our proposed approach outperforms two types of benchmarks; further, the asymmetric Glosten–Jagannathan–Runkle outperforms the symmetric GARCH. Overall, our paper has both theoretical and practical implications.

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“…Figure17reports a modified version of Figure5, showing volatility inputs for the three VolTarget strategies. We also added the VIXindex CBOE (2019), tge real-time market index representing the market's expectations for volatility of the S&P500 index over the coming 30 daysCao et al (2020), as a reference for comparison purposes. The VIX index is clearly higher since it is calculated as the 30 day expectation of volatility.…”

mentioning

confidence: 99%

Forward-Looking Volatility Estimation for Risk-Managed Investment Strategies during the COVID-19 Crisis

Persio

Garbelli

Wallbaum³

2021

Risks

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Under the impact of both increasing credit pressure and low economic returns characterizing developed countries, investment levels have decreased over recent years. Moreover, the recent turbulence caused by the COVID-19 crisis has accelerated the latter process. Within this scenario, we consider the so-called Volatility Target (VolTarget) strategy. In particular, we focus our attention on estimating volatility levels of a risky asset to perform a VolTarget simulation over two different time horizons. We first consider a 20 year period, from January 2000 to January 2020, then we analyse the last 12 months to emphasize the effects related to the COVID-19 virus’s diffusion. We propose a hybrid algorithm based on the composition of a GARCH model with a Neural Network (NN) approach. Let us underline that, as an alternative to standard allocation methods based on realized and backward oriented volatilities, we exploited an innovative forward-looking estimation process exploiting a Machine Learning (ML) solution. Our solution provides a more accurate volatility estimation, allowing us to derive an effective investor risk-return profile during market crisis periods. Moreover, we show that, via a forward-looking VolTarget strategy while using an ML-based prediction as the input, the average outcome for an investment in a drawdown plan is more sustainable while representing an efficient risk-control solution for long time period investments.

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Valuation of VIX and target volatility options with affine GARCH models

Cited by 17 publications

References 55 publications

Pricing VIX options with realized volatility

Pricing VIX options with realized volatility

GARCH pricing and hedging of VIX options

Forward-Looking Volatility Estimation for Risk-Managed Investment Strategies during the COVID-19 Crisis

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