Additive normal tempered stable processes for equity derivatives and power-law scaling

Abstract

We introduce a simple additive process for equity index derivatives. The model generalizes Lévy Normal Tempered Stable processes (e.g. NIG and VG) with time-dependent parameters. It accurately fits the equity index volatility surfaces in the whole time range of quoted instruments, including options with small time-horizon (days) and long time-horizon (years). We introduce the model via its characteristic function. This allows using classical Fourier pricing techniques. We discuss the calibration issues in detail and we show that, in terms of mean squared error, calibration is on average two orders of magnitude better than both Lévy and Sato processes alternatives. We show that even if the model loses the classical stationarity property of Lévy processes, it presents interesting scaling properties for the calibrated parameters.

Keywords:

• Additive process
• Volatility surface
• Calibration

JEL Classification:

• C51
• G13

Acknowledgments

We thank P. Carr, J. Gatheral, and F. Benth for enlightening discussions on this topic. We thank also G. Guatteri, M.P. Gregoratti, J. Guyon, P. Spreij, and all participants to WSMF 2019 in Lunteren, to VCMF 2019 in Vienna, and to AFM 2020 in Paris. We are grateful to the Editor and the Referees for their useful comments and careful review. R.B. feels indebted to P. Laurence for several helpful and wise suggestions on the subject.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notation

Symbol	=	Description
$A_t$	=	diffusion term of the additive process $\{X_t{\}}_{t\ge 0}$
$B_T$	=	discount factor between value date and T
$\mathbb{B}\left(\mathbb{R}\right)$	=	Borel sigma algebra on $\mathbb{R}$
$C(T,K)$	=	call option price at value date with maturity T and strike K
$\{f_t{\}}_{t\ge 0}$	=	ATS process that models the forward exponent
$\{{\hat{f}}_{\theta }{\}}_{\theta \ge 0}$	=	re-scaled ATS process w.r.t. the time $\theta ={\sigma }_T^{2}T$
$F_t\left(T\right)$	=	price at time t of a Forward contract with maturity T
k	=	variance of jumps of LTS
$k_t$	=	variance of jumps of ATS
${\hat{k}}_{\theta }$	=	re-scaled variance of jumps of ATS
$\overline{k}$	=	constant part of the re-scaled variance of jumps of ATS ${\hat{k}}_{\theta }$
${\mathcal{L}}_t$	=	Laplace transform of the subordinator $S_t$ in (3(3) $\begin{array}{rl}& \ln {\mathcal{L}}_t(u;k,\alpha )\\ & :=\{\begin{array}{ll}{\displaystyle \frac{t}{k}{\displaystyle \frac{1-\alpha }{\alpha }\{1-{(1+\frac{uk}{1-\alpha })}^{\alpha }\}}}& \text{if }0<\alpha <1\\ {\displaystyle -\frac{t}{k}\ln (1+uk)}& \text{if }\alpha =0\end{array}.\end{array}$(3) )
$\{S_t{\}}_{t\ge 0}$	=	Lévy subordinator
T	=	option time to maturity
$W_t$	=	Brownian motion
x	=	option moneyness
α	=	Index of stability: tempered stable parameter of ATS, $\alpha \in [0,1)$]
β	=	scaling parameter of ${\hat{k}}_{\theta }$
${\gamma }_t$	=	drift term of additive process $\{X_t{\}}_{t\ge 0}$
$\mathrm{\Gamma }(\ast )$	=	Gamma function evaluated in $\ast$
δ	=	scaling parameter of ${\hat{\eta }}_{\theta }$
φ	=	deterministic drift term of LTS
${\phi }_t$	=	deterministic drift term of ATS
${\varphi }^c$	=	characteristic function of the forward exponent
η	=	skew parameter of LTS
${\eta }_t$	=	skew parameter of ATS
${\hat{\eta }}_{\theta }$	=	re-scaled skew parameter of ATS
$\overline{\eta }$	=	constant part of the re-scaled ATS skew parameter ${\hat{\eta }}_{\theta }$
${\nu }_t$	=	Lévy measure of the additive process $\{X_t{\}}_{t\ge 0}$
σ	=	diffusion parameter of LTS
${\sigma }_t$	=	diffusion parameter of ATS
${\hat{\sigma }}_{\theta }$	=	re-scaled diffusion parameter of ATS, equal to one
$\overline{\sigma }$	=	constant diffusion parameter of ATS
θ	=	re-scaled maturity, defined as ${\sigma }_T^{2}T$

Notes

1 The relevance of pure jump dynamics in the equity and commodity asset classes has been discussed in the recent literature, see, e.g. Ornthanalai 2014, Li and Linetsky 2014, Ballotta and Rayée 2018.

2 Gamma is the Greek measure that quantifies the amount of this hedging and, generally, it decreases with time-to-maturity.

3 A parametrization scheme of the drift in terms of η can be suitable for some applications: η controls the volatility skew. In particular, it can be proven that for $\eta =0$ the smile is symmetric, i.e. the implied volatility skew is zero see, e.g. Baviera (2007), Prop. p.21.

4 In this paper, the notation follows closely the one in Sato (1999).

5 We underline that, in both cases (LTS and Sato), model parameters are obtained through a global calibration of the whole volatility surface.

6 Calibrated model parameters are available upon request.

7 We have also considered a global calibration of the implied volatility surfaces considering the power-law scaling parameters in (7(7) $\{\begin{array}{l}{\hat{k}}_{\theta }=\overline{k}{\theta }^{\text{β}}\\ {\hat{\eta }}_{\theta }=\overline{\eta }{\theta }^{\delta }\end{array},$(7) ) with $\text{β}=1$ and $\delta =-0.5$. The results are of the same order of magnitude of table 1.

8 Proof available upon request.

9 These are the penultimate business days of November 2012, February 2013, August 2013, and November 2013.

10 Results for the 27th of February 2013 and the 30th of August 2013 are available upon request.