Pure jump models for pricing and hedging VIX derivatives

Li, Jing; Li, Lingfei; Zhang, Gongqiu

doi:10.1016/j.jedc.2016.11.001

Cited by 37 publications

(16 citation statements)

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“…We also extend our analysis to provide error estimates for the Markov chain approximation method in pricing European options for a class of jump models obtained from diffusions through subordination. These jump processes are known as subordinate diffusions and their applications in finance can be found in, e.g., Barndorff-Nielsen and Levendorskii (2001), Mendoza-Arriaga, Carr, and Linetsky (2010), Li and Linetsky (2014), Li and Mendoza-Arriaga (2013), Li, Li, and Mendoza-Arriaga (2016), and Li, Li, and Zhang (2017).…”

Section: Introductionmentioning

confidence: 99%

Error analysis of finite difference and Markov chain approximations for option pricing

Zhang

2017

Mathematical Finance

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Mijatović and Pistorius proposed an efficient Markov chain approximation method for pricing European and barrier options in general one‐dimensional Markovian models. However, sharp convergence rates of this method for realistic financial payoffs, which are nonsmooth, are rarely available. In this paper, we solve this problem for general one‐dimensional diffusion models, which play a fundamental role in financial applications. For such models, the Markov chain approximation method is equivalent to the method of lines using the central difference. Our analysis is based on the spectral representation of the exact solution and the approximate solution. By establishing the convergence rate for the eigenvalues and the eigenfunctions, we obtain sharp convergence rates for the transition density and the price of options with nonsmooth payoffs. In particular, we show that for call‐/put‐type payoffs, convergence is second order, while for digital‐type payoffs, convergence is generally only first order. Furthermore, we provide theoretical justification for two well‐known smoothing techniques that can restore second‐order convergence for digital‐type payoffs and explain oscillations observed in the convergence for options with nonsmooth payoffs. As an extension, we also establish sharp convergence rates for European options for a rich class of Markovian jump models constructed from diffusions via subordination. The theoretical estimates are confirmed using numerical examples.

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Section: Introductionmentioning

confidence: 99%

Error analysis of finite difference and Markov chain approximations for option pricing

Zhang

2017

Mathematical Finance

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“…To calculate the prices for dual exotics, we need to truncate the eigenfunction expansion after a finite number of terms. Following [14], we truncate the infinite series when a given error tolerance level is reached. In practice, we find the convergence of the expansion is rather fast.…”

Section: Numerical Analysismentioning

confidence: 99%

“…See its successful applications for callable and putable bonds in [7], K. Z. Tong et al Journal of Mathematical Finance commodities in [8], electricity in [9] and variance swaps in [10]. We also refer to [11] for a brief introduction of subordinate Markov processes and [12] [13] [14] [15] for their further applications.…”

Section: Introductionmentioning

confidence: 99%

The Pricing of Dual-Expiry Exotics with Mean Reversion and Jumps

Tong¹,

Hou²,

Guan³

2019

JMF

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This paper develops a new class of models for pricing dual-expiry options that are characterized by two expiry dates. The underlying asset price is modeled by a time changed exponential Ornstein Uhlenbeck (OU) process, where the time change process is a Lévy subordinator. The new models can capture both mean reversion and jumps often observed in various types of underlying assets of exotics. The pricing method exploits the observation that dual expiry options have payoffs that can be perfectly replicated by a particular set of first and second order binary options. The novelty of the paper is that we are able to derive the analytical solutions to the prices of these binaries through eigenfunction expansion method. Based on that, we can obtain the formulas for dual-expiry exotics through static replication. We also numerically investigate the sensitivities of prices of chooser, compound and extendable options with respect to the parameters of the models.

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“…The VG process has a finite variation with a moderately low activity rate of small jumps, whereas the NIG process has an infinite variation with a stable arrival rate of small jumps (Cont & Tankov, 2004). Our paper is different to Li, Li, and Zhang (2017), in which a pure jump semimartingale (one type of time-changed Lévy process in Carr & Wu, 2004) is generated by an additive subordination. Todorov and Tauchen (2011) find that VIX dynamics have infinite-activity jumps.…”

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confidence: 94%

“…Todorov and Tauchen (2011) find that VIX dynamics have infinite-activity jumps. Our paper is different to Li, Li, and Zhang (2017), in which a pure jump semimartingale (one type of time-changed Lévy process in Carr & Wu, 2004) is generated by an additive subordination. Our model considers the time-varying mean level of the log VIX and is estimated using a much longer data sample.…”

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confidence: 94%

Pricing VIX derivatives with infinite‐activity jumps

Cao

Ruan

et al. 2019

Journal of Futures Markets

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In this paper, we investigate a two-factor VIX model with infinite-activity jumps, which is a more realistic way to reduce errors in pricing VIX derivatives, compared with Mencía and Sentana (2013), J Financ Econ, 108, 367-391. Our two-factor model features central tendency, stochastic volatility and infiniteactivity pure jump Lévy processes which include the variance gamma (VG) and the normal inverse Gaussian (NIG) processes as special cases. We find empirical evidence that the model with infinite-activity jumps is superior to the models with finite-activity jumps, particularly in pricing VIX options. As a result, infinite-activity jumps should not be ignored in pricing VIX derivatives. K E Y W O R D S infinite-activity jumps, maximum log-likelihood estimation, unscented Kalman filter, VIX derivatives J E L C L A S S I F I C A T I O N G12, G13 | INTRODUCTIONTime-varying volatility is a key risk factor in financial markets; thus, understanding the dynamics of volatility variation is crucial for market practitioners and academic researchers. The Chicago Board Options Exchange (CBOE) Volatility Index, known by its ticker symbol VIX, is a risk-neutral gauge of the future implied volatility in the U.S. stock market. It is calculated by the S&P 500 index options market over the next 30-day period and indicates overall stock market fear. 1 VIX can be traded through the VIX derivatives. Because of a negative correlation between the return of VIX and the return of the S&P 500 index, market participants treat the VIX derivatives as an important trading vehicle for reducing exposure to risk. Reported by CBOE, the VIX derivatives include VIX futures and VIX options. In 2004, CBOE introduced VIX futures. According to the CBOE website, the average daily volume of VIX futures rose more than 135fold from 2004 to 2017, with a significant growth after 2010. 2 After the successful launch of VIX futures, CBOE introduced VIX options in 2006, which can provide an alternative way for hedging the market risk with relatively low costs. Since then, the VIX options have become popular financial tools in risk management. Not surprisingly, the average daily volume of VIX options expanded over 28 times between 2006 and 2017. 3 Therefore, it is important to find an accurate valuation model of the VIX derivatives.The methodologies for pricing VIX derivatives can be classified into two directions. In the first direction, the VIX derivatives pricing formulae are derived from the instantaneous volatilities of the S&P 500 index, which can provide a clear picture of the relationship among the S&P 500 index, VIX, and VIX derivatives. Researchers in this direction 1 See https://www.cboe.com/micro/vix/vixwhite.pdf. 2 See http://www.cboe.com/blogs/options-hub/2017/03/01/eight-charts-highlighting-growth-in-options-and-vix-futures. 3 See http://www.cboe.com/blogs/options-hub/2017/08/25/record-days-for-vix-futures-and-options-volume-and-open-interest-this-month.Note: The VG and NIG processes are summarized. The "Parameters" row shows the parameters in t...

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Pure jump models for pricing and hedging VIX derivatives

Cited by 37 publications

References 38 publications

Error analysis of finite difference and Markov chain approximations for option pricing

Error analysis of finite difference and Markov chain approximations for option pricing

The Pricing of Dual-Expiry Exotics with Mean Reversion and Jumps

Pricing VIX derivatives with infinite‐activity jumps

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