Top

Review of Quantitative Finance and Accounting

Published in:

10-09-2023 | Original Research

Non-linear volatility with normal inverse Gaussian innovations: ad-hoc analytic option pricing

Authors: Sharif Mozumder, Bakhtear Talukdar, M. Humayun Kabir, Bingxin Li

Published in: Review of Quantitative Finance and Accounting | Issue 1/2024

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

This paper proposes an approximate closed-form option-pricing model based on a non-linear GARCH process with Normal Inverse Gaussian (NIG) Lévy innovations. We develop the mathematical framework and demonstrate how to obtain a closed-form solution to the option price when the return dynamics are characterized by NIG innovations for volatility that follow a non-linear GARCH process. Using a sample of S&P 500 index options, we calibrate the proposed model alongside popular existing models. Overall, from a unified comparison of various analytic pricing approaches, we find that our model performs significantly better than existing models, both in-sample and out-of-sample.

previous article The market price to embedded value gap: an analysis of European life insurers

next article Chasing noise in the stock market: an inquiry into the dynamics of investor sentiment and asset pricing

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

Kim et al. (2008) propose the “KR distribution” as one such subclass of the tempered stable distribution, and empirically test and compare it with CGMY and the modified tempered stable (MTS) distribution. Kim et al. (2010) introduce the rapidly decreasing tempered stable (RDTS) GARCH model with an infinitely divisible distributed innovation and compare the findings based on the classical tempered stable (CTS) GARCH model.

For example, it can model both positive and negative shocks.

Moreover, the approach can yield hedge ratios, delta, and gamma, analytically (up to a numerical integration) using the recursions defining the characteristic function of the GARCH-NIG process. In the same way, delta and gamma can be obtained analytically from Heston and Nandi (2000) model premised on the GARCH-Normal process, see Rouah and Vainberg (2007).

The moments of X_t ~ NIG (α, β, δt) are shown in Appendix 1 for t = 1.

En passant, we note that our solution can be extended to other Lévy innovations such as Variance Gamma (VG) and CGMY processes.

Note that here the notation \({\sigma }_{t}\) is for conditional variance, not conditional volatility.

This introduction of non-linearity does affect the equivalent martingale relationships among the parameters and is demonstrated in Appendix 3.

A similar problem was encountered in Ornthanalai (2010) GARCH-Lévy dynamics for asset pricing. He concluded that, for Lévy innovations capable of exhibiting both positive and negative jumps, there is no alternative to the Monte-Carlo valuation of derivatives. However, the problem with Monte Carlo is that it requires a long time to price even a single option. To attempt such pricing, we need to consider a large number of simulations—at least 5000 trials are needed—at the expense of huge computational time that renders quick calibration practically infeasible.

The one-day-ahead GARCH variance, \(\sigma_{t + 1}\), can also be treated as a parameter. However treating variance as a parameter is only manageable for a short sample because each new day creates a new parameter to be estimated. So, for calibrations using option records over a long period, we need to directly feed the one-day-ahead variance in a dynamic manner. Heston and Nandi (2000) input these through a GARCH process that forces the calibration to rely heavily on a long time series of asset returns, in addition to the market price of options. We implement the same approach using GARCH-NIG closed-form volatility dynamics.

We implement the constrained optimization using the MATLAB function “fmincon”.

There are many jump diffusion (JD) models in the literature, and we focus on one based on double exponential (DE) jump sizes for positive and negative unsystematic (jump) movements in the market as it incorporates asymmetry, see Kou (2002).

CGMY stands for Carr et al. (2002).

We tried the GARCH-TS model of Mercuri (2008), as well, but this model performs very similar to Christoffersen’s (2006) GARCH-IG and improves only marginally compared to Heston and Nandi’s (2000) model. So instead of keeping several similar models, we drop GARCH-TS and keep it for broad comparison, with a number of other Lévy innovations to the GARCH model, in future work.

We discuss different aggregation schemes and corresponding in-sample and out-of-sample data requirements in Sect. 5.

Dumas, Fleming, and Whaley (1998) discuss the advantages of Wednesday data in detail.

If options do appear to have exactly same moneyness despite having different stock price on different days then if those options have same maturity we just pick up the first option and remove others.

We survey up to 2021 market data; but these more recent markets are heavily perturbed by major events like COVID’19 and in a decade leading up to the inception of COVID, the period 2012–2014 is found to be relatively calm.

Other investigations toward establishing robustness include calibrating all the models over a number of cross-sections of data ranging over the first 6 months of a specific year, e.g., the first 6 months of 2015. To save space, these results are not presented and can be provided on request.

That’s because its detailed discussion and extended analysis require huge space, which can potentially comprise a whole new paper.

Deferring the simulation-based approximation issues in pricing for a different new paper.

Moreover, in a calm market, whether JD-DE and BS can perform similarly or not is not confirmed or well-documented in the literature, except that a model with more parameters is highly likely to perform better than its competitors with fewer parameters.

But not as directly as in Heston-Nandi (2000) type non-linear GARCH dynamics, which we consider in our GARCH-NIG model through an approximation.

This relative outperformance deserves a theoretical justification and hence investigation, too.

With the obvious fact that the unconstrained version of Heston-Nandi’s (2000) model, HN (U), infuses extra flexibility to perform better than its restricted version, HN (R).

There are other simulation-based studies of GARCH option pricing with NIG innovations, e.g., Moolman (2008), Chorro and Zazaravaka (2020), and Stentoft (2008); however, they do not use the Heston-Nandi type non-linear GARCH particularly with NIG innovations.

This deserves further analysis and could be explored with other Lévy innovations under similar approximate analytic pricing in future work.

Here, we note that GARCH-IG models the positive and negative shocks by standardizing a distribution that has positive support (so standardizing only positive shocks); whereas GARCH-NIG models shocks with the standardization of a distribution whose support is the set of real numbers (so standardizing both positive and negative shocks).

This would imply, for example, that when it comes to out-of-sample fit, a stochastic volatility model with multiple parameters is not guaranteed to have a significant advantage over the single-parameter BS model.

Christoffersen et al. (2006) GARCH-IG model is different in which only skewness has time-varying characterization.

Between calibrations with minimal information content in options traded on one volatile day and those using 6-months traded options, we find the one with minimal information content is more interesting to present, because it differs from the main calibration with two years’ information content more poignantly than what calibrations with 6 months’ information do.

2015 is a period outside our original calibration period, with S&P 500 fluctuating substantially.

However, this additional flexibility in Scott (1997) model comes with three additional parameters compared to Bates (1996) model; thus, the big margin out-performance should not be deemed implausible.

However, this inconsistency is observed only between our minimal information calibration with one-day traded options and the main calibration results discussed in Sect. 6.1; in a number of calibration experiments over different six-month windows that we do not report in the paper, we find HN (U) often (not always) beats the Christoffersen GARCH-IG model.

Christoffersen et al. (2006) doesn’t give any confirmation as to the consistently better performance of the GARCH-IG model compared to Heston and Nandi (2000).

This could be done quite distinctly from the studies which have already tried VG (Kao 2012), alpha-stable (Menn 2005), and even NIG innovations (Stentoft 2008). However, none of those studies considered either Heston and Nandi (2000) type non-linearity in GARCH characterization or analytic pricing.

Badescu AM, Kulperger RJ (2008) GARCH option pricing: a semiparametric approach. Insur Math Econ 43(1):69–84. https://doi.org/10.1016/j.insmatheco.2007.09.011CrossRef

Badescu A, Elliott RJ, Oretga JP (2015) Non-Gaussian GARCH option pricing models and their diffusion limits. Eur J Oper Res 247:820–830CrossRef

Bakshi G, Cao C, Chen Z (1997) Empirical performance of alternative option pricing models. J Financ 52(5):2003–2049. https://doi.org/10.1111/j.1540-6261.1997.tb02749.xCrossRef

Barone-Adesi G, Engle R, Mancini L (2008) A GARCH option pricing model with filtered historical simulation. Rev Financ Stud 21:1223–1258CrossRef

Bates DS (1996) Jumps and stochastic volatility: exchange rate processes implicit in Deutsche Mark options. Rev Financ Stud 9:69–107CrossRef

Bates DS (2003) Empirical option pricing: a retrospection. J Econom 116:387–404CrossRef

Black F, Scholes M (1973) The pricing of options and corporate liabilities. J Polit Econ 81:637–659CrossRef

Bollerslev T (1986) Generalized autoregressive conditional heteroeskedasticity. J Econom 31:307–327CrossRef

Carr P, Wu L (2004) Time-changed Lévy processes and option pricing. J Financ Econ 71:113–141CrossRef

Carr P, Geman H, Madan DB, Yor M (2002) The affine structure of asset returns: an empirical investigation. J Bus 75:305–332CrossRef

Chorro C, Zazaravaka RHF (2020) Discriminating between GARCH models for option pricing by their ability to compute accurate VIX measures, Working paper, University Paris 1 Panthéon-Sorbonne, Centre d’Économie de la Sorbonne, France

Chourdakis K (2004) Option pricing using the fractional FFT. J Comput Financ 8(2):1–18CrossRef

Christoffersen P, Heston SL, Jacobs K (2006) Option valuation with conditional skewness. J Econom 131:253–284CrossRef

Christoffersen P, Elkamhi R, Feunou B, Jacobs K (2010) Option valuation with conditional heteroskedasticity and nonnormality. Rev Financ Stud 23(5):2139–2183CrossRef

Christoffersen P, Jacobs K, Ornthanalaia C (2012) Dynamic jump intensities and risk premiums: evidence from S&P500 returns and options. J Financ Econ 106(3):447–472CrossRef

Dingec KD, Hormann W (2012) A general control variate method for option pricing under Lévy processes. Eur J Oper Res 221:368–377CrossRef

Duan JC (1995) The GARCH option pricing model. Math Financ 5:13–32CrossRef

Dumas B, Fleming J, Whaley R (1998) Implied volatility functions: empirical tests. J Financ 53:2059–2106CrossRef

Engle R (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflations. Econometrica 50:987–1008CrossRef

Geman H (2002) Pure Jump Lévy processes for asset price modelling. J Bank Financ 26:1297–1316CrossRef

Geman H, Madan D, Yor M (2001) Time changes for Lévy processes. Math Financ 11:79–96CrossRef

Gerber HU, Shiu ESW (1994) Option pricing by Esscher transforms. Trans Soc Actuar 46:99–191

He X-J, Lin S (2023) Analytically pricing exchange options with stochastic liquidity and regime switching. J Fut Mark. https://doi.org/10.1002/fut.22403CrossRef

Heston S (1993) A closed form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 6:327–343CrossRef

Heston SL, Nandi S (2000) A closed form GARCH option valuation model. Rev Financ Stud 13:585–625CrossRef

Huang JZ, Wu L (2004) Specification analysis of option pricing models based on time-changed Lévy processes. J Financ 59:1405–1439CrossRef

Hull J, White A (1987) The pricing of options on assets with stochastic volatilities. J Financ 42:281–300CrossRef

Kao L (2012) Locally risk-neutral valuation of options in GARCH models based on variance gamma process. Int J Theor Appl Financ 15(2):1250015CrossRef

Kim KS, Rachev ST, Bianchi ML, Fabozzi FJ (2008) Financial market models with Lévy processes and time-varying volatility. J Bank Financ 32:1363–1378CrossRef

Kim KS, Rachev ST, Bianchi ML, Fabozzi FJ (2010) Tempered stable and tempered infinitely divisible GARCH models. J Bank Financ 34:2096–2109CrossRef

Kou S (2002) A jump diffusion model for option pricing. Manag Sci 48:1086–1101CrossRef

Madan D, Carra P, Chang E (1998) The variance gamma process and option pricing model. Eur Financ Rev 2:79–105CrossRef

Menn C, Rachev ST (2005) A GARCH option pricing model with α-stable innovations. Eur J Oper Res 163(1):201–209CrossRef

Mercuri L (2008) Option pricing in a GARCH model with tempered stable innovations. Financ Res Lett 5:172–182CrossRef

Merton R (1976) Option pricing when underlying stock returns are discontinuous. J Financ Econ 3:125–144CrossRef

Moolman GP (2008) Option pricing: a GARCH model with Lévy innovations. University of Johannesburg.

Mozumder S, Sorwar G, Dowd K (2013) Option pricing under non-normality: a comparative analysis. Rev Quant Financ Acc 40(2):273–292CrossRef

Mozumder S, Sorwar G, Dowd K (2015) Revisiting variance gamma pricing: an application to S&P500 index options. Int J Financ Eng 2(2):1550022CrossRef

Muroi Y, Suda S (2023) Lattice approach for option pricing under Lévy processes. J Deriv. https://doi.org/10.3905/jod.2023.1.185CrossRef

Oh DH, Park YH (2023) GARCH option pricing with volatility derivatives. J Bank Financ 146:106718CrossRef

Ornthanalai C (2014) Lévy jump risk: Evidence from options and returns. J Financ Econ 112:69–90CrossRef

Ornthanalai C (2010) A new class of asset pricing models with Lévy processes: theory and applications. Working Paper, Georgia Institute of Technology

Psychoyios D, Dotsis G, Markellos RN (2010) A jump diffusion model for VIX volatility options and futures. Rev Quant Financ Acc 35:245–269CrossRef

Rouah FD, Vainberg G (2007) Option pricing models & volatility using excel-VBA. Wiley, Hoboken

Schouten W (2003) Lévy processes in finance: pricing financial derivatives. Wiley, HobokenCrossRef

Scott LO (1997) Pricing stock options in a jump-diffusion model with stochastic volatility and interest rates: application of Fourier inversion methods. Math Financ 7:413–426CrossRef

Shiu TK, Tong H, Yang H (2004) On pricing derivatives under GARCH models: a dynamic Gerber-Shiu approach. N Am Actuar J 8:17–31CrossRef

Stentoft L (2008) American option pricing using GARCH models and the normal inverse gaussian distribution. J Financ Economet 6(4):540–582CrossRef

Title: Non-linear volatility with normal inverse Gaussian innovations: ad-hoc analytic option pricing
Authors: Sharif Mozumder
Bakhtear Talukdar
M. Humayun Kabir
Bingxin Li
Publication date: 10-09-2023
Publisher: Springer US
Published in: Review of Quantitative Finance and Accounting / Issue 1/2024
Print ISSN: 0924-865X
Electronic ISSN: 1573-7179
DOI: https://doi.org/10.1007/s11156-023-01195-8

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Other articles of this Issue 1/2024

CFO overconfidence and conditional accounting conservatism

The impact of social media on venture capital financing: evidence from Twitter interactions

Chasing noise in the stock market: an inquiry into the dynamics of investor sentiment and asset pricing

The dynamic relation between board gender diversity and firm performance: the moderating role of shareholder activism

The market price to embedded value gap: an analysis of European life insurers

Liquidity difference between non-U.S. and U.S. IPOs on the NYSE listings