Pricing VIX options with realized volatility

Self Cite

As the volatility index (VIX) is nontradable, most investors use the exchange-traded VIX futures to hedge their exposures in VIX options. However, the information role of VIX futures in pricing VIX options is not fully explored empirically. This paper derives two types of VIX option pricing formula using VIX index and VIX futures as state variables accordingly based on a simple discrete-time VIX dynamics with long memory and asymmetric jumps. Empirical results show that models utilizing VIX futures significantly outperform competing models based on S&P 500 index (SPX) returns, realized volatility, or the VIX index itself. Our findings are robust in out-of-sample.

“…The VIX option price for GARV model is provided in the Proposition 1 in Tong and Huang (2021), where the price is an integration of the moment generation function of forward expected variance:

C_{t} = \frac{e^{- r (T - t)}}{2 \sqrt{π}} \int_{0}^{\infty} Re (][, e^{u a} φ_{t} (T, u)) \times \frac{1 - erf (K \sqrt{u}))}{{(\sqrt{u})}^{3}}) normald u_{I},

where

φ_{t} (T, u)

is the characteristic function of

y_{T} = b h_{T + 1}^{R} + c h_{T + 1}^{R V}

φ_{t} ()(T, u)) = E_{t}^{Q} ()(e^{u y_{T}})) = exp ()(, A (u, T) + B (u, T) h_{t + 1}^{R} + D (u, T) h_{t + 1}^{R V})

and

u

is a complex number denoted as

u = u_{R} + i u_{I}

, with

u_{R} > 0

and

u_{I} \in double-struckR

Re [\cdot]

stands for the real part of the complex number inside the square bracket.…”

Section: Competing Modelsmentioning

confidence: 99%

Section: Competing Modelsmentioning

confidence: 99%

“…Jarrow & Rudd (1982) proposed a second‐order approximation of

\overset{g}{true} (z)

using standard normal distribution and higher order moments of

z

\overset{g}{true} (z) \approx (][, 1 - \frac{κ_{3}}{6} H_{3} (z) + \frac{(κ_{4} - 3 false))}{24} H_{4} (z) + \frac{κ_{3}^{2}}{72} H_{6} (z)) ϕ (z),

where

ϕ (z)

is the density function of the standard normal distribution,

κ_{i}

is the

i

th moment of

z_{T}

, and

H_{n} (z)

is the

n

th order Hermite polynomial. Tong and Huang (2021) provide the expansion based pricing formula in their Proposition 2. The approximated European VIX call price equals:

C_{a p p r o x} = A_{0} - \frac{κ_{3}}{6} A_{3} + \frac{(κ_{4} - 3 false))}{24} A_{4} + \frac{κ_{3}^{2}}{72} A_{6},

where

A_{0}

A_{3}

A_{4}

A_{6}

are defined with moments of normal distribution and

κ_{i}

are defined as the standardized origin moments of

\log {VIX}_{T}

.…”

Section: Competing Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Do VIX futures contribute to the valuation of VIX options?

Tong

Huang

Wang

2021

Self Cite

“…The opposite is true with a regular S-shape. 3 A growing literature is exploring ways to utilize RV measures for derivatives pricing, see, for example, Christoffersen et al (2014), Corsi et al (2013), Huang et al (2019Huang et al ( , 2017, Majewski et al (2015), and Tong and Huang (2021). 4 Hansen and Tong (2021) also introduce an option-pricing model with time-varying volatility risk aversion.…”

mentioning

confidence: 99%

Option pricing with state‐dependent pricing kernel

Tong

Hansen

Huang

2022

Self Cite

We introduce a new volatility model for option pricing that combines Markov switching with the realized generalized autoregressive conditional heteroskedasticity (GARCH) framework. This leads to a novel pricing kernel with a state-dependent variance risk premium and a pricing formula for European options, which is derived with an analytical approximation method. We apply the Markov-switching Realized GARCH model to Standard and Poor's 500 index options from 1990 to 2019 and find that investors' aversion to volatility-specific risk is time-varying. The proposed framework outperforms competing models and reduces (in-sample and out-of-sample) option-pricing errors by 15% or more.

VIX option pricing through nonaffine GARCH dynamics and semianalytical formula

Liu,

Wang,

Zhang

2024

This paper develops analytical approximations for volatility index (VIX) option pricing under nonaffine generalized autoregressive conditional heteroskedasticity (GARCH) models as advocated by Christoffersen et al. We obtain the approximation formulae for pricing VIX options and then evaluate their performance with three expansions under four empirically well‐tested models. Our numerical experiments find that the weighted expansion generated by the fat‐tailed weighting kernel can significantly reduce approximation error over the Gram‐Charlier expansion; the Taylor expansion of conditional moments can lead to divergence for parameters with certain high persistence in the affine GARCH, nonlinear asymmetric GARCH, and Glosten‐Jagannathan‐Runkle GARCH models, while surviving during high persistence in the exponential GARCH.