Valuation of VIX Derivatives

In this paper, we study the VXX Exchange Traded Note (ETN), that has been actively traded in recent years, but has lost 99.84% of its value. Using Zhang's formula for VIX futures prices, we develop the first theoretical model for the VXX that links the SPX, VIX, and VXX. We show that the roll yield of VIX futures drives the difference between the VXX and VIX returns. The roll yield is a mostly negative process. We then provide a simple yet robust estimation of the market price of variance risk using VXX and VIX futures prices. The model can be used to price VXX options.

Section: Introductionmentioning

confidence: 99%

Modeling VXX

Gehricke

Zhang

2018

“…Let

(normalΩ, scriptMJX-tex-caligraphicF, double-struckQ)

be a complete probability space, equipped with an information flow

{F_{t}}_{t \geq 0}

, where

double-struckQ

is a risk‐neutral probability measure. We build a model to describe the evolution process of the logarithmic VIX, as there is an empirical finding that modeling the logarithm of the VIX is superior to modeling its level directly (see Kaeck & Alexander, ; Mencía & Sentana, ; Psychoyios et al, ). As discussed before, we introduce a Hawkes process to model the dynamics of the jump component.…”

Section: Modelmentioning

confidence: 99%

“…Therefore, we also allow for stochastic volatility. Unlike the stochastic volatility process driven by jumps in Mencía and Sentana (), in our setting, it is constructed by the CIR process, which is popular in financial modeling. On the probability space

(normalΩ, scriptMJX-tex-caligraphicF, double-struckQ)

, the dynamics of the logarithmic VIX, denoted by X t = ln VIX t , evolve as follows:

\begin{matrix} true0.33em \\ right d X_{t} & center = & left κ false(θ - X_{t} false) d t + \sqrt{V_{t}} d W_{t}^{1} + d true({true \sum}_{i = 1}^{N_{t}} Y_{i} true) - λ_{t} δ d t, \\ right d V_{t} & center = & left κ_{v} false(θ_{v} - V_{t} false) d t + σ \sqrt{V_{t}} d W_{t}^{2}, \\ right d λ_{t} & center = & left η false(\bar{λ} - λ_{t} false) d t + ϵ d N_{t}, \end{matrix}

where

W_{t}^{1}

and

W_{t}^{2}

are correlated standard Brownian motions with

d W_{t}^{1} d W_{t}^{2} = ρ d t

; N t is a Hawkes process with stochastic intensity λ t ; κ and θ represent the mean‐reversion speed and the long‐run mean level of X t , respectively; κ v , θ v , and σ capture the mean‐reversion speed, the long‐run mean level and the volatility of the instantaneous variance V t ; and

{Y_{i}}_{i \geq 1}

is a family of independent and identically distributed (i.i.d) random variables with mean δ and probability density function (p.d.f.)…”

Section: Modelmentioning

confidence: 99%

“…For example, Detemple and Osakwe () valued volatility options when volatility follows an exponential Ornstein‐Uhlenbeck (OU) process. Mencía and Sentana () extended the model of Detemple and Osakwe () by allowing for a time‐varying central tendency and unspanned stochastic volatility. Park () proposed a class of models with asymmetric volatility and jumps for the valuation of VIX derivatives.…”

Section: Introductionmentioning

confidence: 99%

“…This critical pattern is modeled by introducing a self‐exciting Hawkes process with stochastic intensity in the dynamics of the logarithm of the VIX. Previous empirical studies have indicated that the introduction of stochastic volatility is necessary when pricing VIX options (e.g., Kaeck & Alexander, ; Mencía & Sentana, ; Park, ). Therefore, we also incorporate a stochastic volatility component into the underlying dynamics.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Pricing VIX options with volatility clustering

Jing

2020

We investigate the valuation of volatility index (VIX) options by developing a model with a self-exciting Hawkes process that allows for clustering in the VIX. In the proposed framework, we find semianalytical expressions for the characteristic function and forward characteristic function, and then we solve the pricing problem of standard-start and forward-start options via the fast Fourier transform. The empirical results provide evidence to support the significance of accounting for volatility clustering when pricing VIX options. K E Y W O R D Sfast Fourier transform, self-exciting Hawkes process, VIX options, volatility clustering

The Distribution of Uncertainty: Evidence from the VIX Options Market

Völkert

2014

This article investigates the informational content implied in the risk‐neutral distribution of the VIX, a leading barometer of economic uncertainty. We extract the risk‐neutral distribution from VIX option prices over the sample period from 2006 to 2011 using a non‐parametric approach. We analyze the time‐series behavior of the option‐implied moments and assess whether the information implied in the risk‐neutral distribution has predictive power. The risk‐neutral distribution considerably changed shape during the financial crisis. Furthermore, risk‐neutral moments contain useful information with respect to the likelihood of upward jumps in volatility. Consistent with investors disliking high levels of economic uncertainty, we find that the overall shape of the estimated volatility pricing kernel is increasing. For certain periods, there is a puzzling U‐shape. The behavior of the volatility pricing kernel over time reveals that the financial crisis has affected investors’ attitudes towards risk. © 2014 Wiley Periodicals, Inc. Jrl Fut Mark 35:597–624, 2015