Geometric step options and Lévy models: Duality, PIDEs, and semi-analytical pricing

Walter Farkas; Ludovic Mathys

doi:10.3934/fmf.2021001

Article Contents

2022, Volume 1, Issue 1: 1-51. Doi: 10.3934/fmf.2021001 open access

This issue Previous Article Next Article Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit

Geometric step options and Lévy models: Duality, PIDEs, and semi-analytical pricing

Walter Farkas^1,2,3, , and
Ludovic Mathys^1,2,3,

1.
Department of Banking and Finance, University of Zurich, Switzerland
2.
Department of Mathematics, ETH Zurich, Switzerland
3.
Swiss Finance Institute, Switzerland

^* Corresponding author: Walter Farkas (walter.farkas@bf.uzh.ch)
^* Corresponding author: Walter Farkas (walter.farkas@bf.uzh.ch)

Received: December 31, 2020

Revised: March 31, 2021

Published: March 2022

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Abstract

The present article studies geometric step options in exponential Lévy markets. Our contribution is manifold and extends several aspects of the geometric step option pricing literature. First, we provide symmetry and duality relations and derive various characterizations for both European-type and American-type geometric double barrier step options. In particular, we are able to obtain a jump-diffusion disentanglement for the early exercise premium of American-type geometric double barrier step contracts and its maturity-randomized equivalent as well as to characterize the diffusion and jump contributions to these early exercise premiums separately by means of partial integro-differential equations and ordinary integro-differential equations. As an application of our characterizations, we derive semi-analytical pricing results for (regular) European-type and American-type geometric down-and-out step call options under hyper-exponential jump-diffusion models. Lastly, we use the latter results to discuss the early exercise structure of geometric step options once jumps are added and to subsequently provide an analysis of the impact of jumps on the price and hedging parameters of (European-type and American-type) geometric step contracts.

Keywords:

Mathematics Subject Classification: 91-08, 91B25, 91B70, 91G20, 91G60, 91G80.

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Figure 1. Relative early exercise contribution, diffusion contribution to the early exercise premium, and absolute early exercise premium of the geometric down-and-out step call as functions of the underlying price $ S_{0} \in [85,115] $ and the knock-out rate $ \rho_{L} \in [-1000,0] $, when the remaining parameters are chosen as: $ \mathcal{T} = 1.0 $, $ \sigma_{X} = 0.2 $, $ r = 0.05 $, $ \delta = 0.07 $, $ K = 100 $, $ L = 95 $, $ \lambda = 5 $, $ p = 0.5 $, $ \xi = 25 $, $ \eta = 50 $

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Figure 2. Difference in the prices, deltas, and gammas for the geometric down-and-out step calls with and without jumps as functions of the underlying price $ S_{0} \in [85.115] $ and the intensity parameter $ \lambda \in [0,20] $, when the remaining parameters are chosen as: $ \mathcal{T} = 1.0 $, $ \sigma_{X} = 0.2 $, $ r = 0.05 $, $ \delta = 0.07 $, $ K = 100 $, $ L = 95 $, $ \rho_{L} = -26.34 $, $ p = 0.5 $, $ \xi = 25 $, $ \eta = 50 $

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Figure 3. Difference in the prices, deltas, and gammas for the geometric down-and-out step calls with and without jumps as functions of the underlying price $ S_{0} \in [85.115] $ and the positive jump parameter $ \xi \in [5,100] $, when the remaining parameters are chosen as: $ \mathcal{T} = 1.0 $, $ \sigma_{X} = 0.2 $, $ r = 0.05 $, $ \delta = 0.07 $, $ K = 100 $, $ L = 95 $, $ \rho_{L} = -26.34 $, $ \lambda = 5 $, $ p = 0.5 $, $ \eta = 50 $

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Table 1. Theoretical (down-and-out) call values and diffusion contributions to the early exercise premium for $r = 0.05$, $\delta = 0.07$, $S_{0} = 100$, $K = 100$, $L = 95$, $\rho_{L} = -26.34$, $p = 0.7$, $\xi = 25$ and $\eta = 50$

(Down-and-Out) Call Option Prices
Parameters		Standard Call Price			Step Call Price			Barrier Call Price
	$\lambda$	Euro	Amer	DC (%)	Euro	Amer	DC (%)	Euro	Amer	DC (%)

	$1$	$6.833$	$7.040$	$91.52 \%$	$4.596$	$4.789$	$91.71 \% $	$3.374$	$3.551$	$91.88 \%$
$S_{0} = 100$	$0.1$	$6.622$	$6.822$	$99.07 \%$	$4.519$	$4.706$	$99.09 \%$	$3.338$	$3.514$	$99.12 \%$
$\sigma_{X}=0.2$	$0.01$	$6.600$	$6.800$	$99.91 \% $	$4.511$	$4.698$	$99.91 \%$	$3.334$	$3.510$	$99.91 \%$
$\mathcal{T} = 1.0$	$0.001$	$6.598$	$6.797$	$99.99 \%$	$4.510$	$4.697$	$99.99 \%$	$3.333$	$3.509$	$99.99 \%$
	$0.0001$	$6.598$	$6.797$	$100.00 \%$	$4.510$	$4.697$	$100.00\% $	$3.333$	$3.509$	$100.00 \% $
B & S Values	-	6.598	6.885	-	4.511	4.745	-	3.332	3.529	-
Rel. Error (%)	-	0.001%	-1.277%	-	0.015%	-1.025%	-	0.025%	-0.568%	-

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Table 2. Theoretical American-type (down-and-out) call values and structure of the early exercise premium for $\delta = 0.05$, $K = 100$, $\sigma_{X} = 0.2$, and $\mathcal{T} = 0.25$

(Down-and-Out) Barrier Call Option Prices
Parameters					Barrier Option Values
	$\lambda$	$p$	$\xi$	$\eta$	FST	Amer	DC (%)

	$3.029$	$0.381$	$26$	$24$	$1.016$	$1.017$	$80.08\%$
	$3.051$	$0.386$	$51$	$24$	$0.920$	$0.921$	$96.04\%$
(1)	$2.991$	$0.386$	$26$	$49$	$0.934$	$0.934$	$78.03\%$
$S_{0}=90$	$3.013$	$0.390$	$51$	$49$	$0.837$	$0.838$	$95.22\%$
$r = 0.07$	$7.067$	$0.381$	$26$	$24 $	$1.337$	$1.338$	$69.81\%$
$L = 78.261$	$7.120$	$0.386$	$51$	$24 $	$1.126$	$1.127$	$93.44\%$
	$6.978$	$0.386$	$26$	$49 $	$1.141$	$1.141$	$67.16\%$
	$7.031$	$0.390$	$51$	$49 $	$0.924$	$0.924$	$92.04\%$

	$3.029$	$0.381$	$26$	$24$	$3.850$	$3.850$	$85.02\%$
	$3.051$	$0.386$	$51$	$24$	$3.741$	$3.728$	$96.73\%$
(1)	$2.991$	$0.386$	$26$	$49$	$3.693$	$3.677$	$84.91\%$
$S_{0}=100$	$3.013$	$0.390$	$51$	$49$	$3.580$	$3.555$	$96.66\%$
$r = 0.00$	$7.067$	$0.381$	$26$	$24 $	$4.316$	$4.366$	$72.44\%$
$L = 86.957$	$7.120$	$0.386$	$51$	$24 $	$4.089$	$4.103$	$93.13\%$
	$6.978$	$0.386$	$26$	$49 $	$3.970$	$3.977$	$71.33\%$
	$7.031$	$0.390$	$51$	$49 $	$3.719$	$3.699$	$92.77\%$

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Table 3. Theoretical (down-and-out) call values and structure of the early exercise premium for $r = 0.05$, $\delta = 0.07$, $K = 100$, $L = 95$, $\rho_{L} = -26.34$, $p = 0.5$, $\xi = 50$ and $\eta = 25$

(Down-and-Out) Call Option Prices
Parameters		Standard Call Price				Step Call Price				Barrier Call Price
	$S_{0}$	Euro	EEP	EEP (%)	DC (%)	Euro	EEP	EEP (%)	DC (%)	Euro	EEP	EEP (%)	DC (%)

	$90$	$3.500$	$0.062$	$1.74 \%$	$94.20 \%$	$0.268$	$0.009$	$3.07 \% $	$94.32 \%$	$0$	$0$	-	-
(1)	$95$	$5.241$	$0.112$	$2.09 \%$	$94.27 \%$	$1.757$	$0.059$	$3.23 \%$	$94.33 \% $	$0$	$0$	-	-
$\sigma_{X}=0.2$	$100$	$7.416$	$0.190$	$2.50 \%$	$94.34 \%$	$4.992$	$0.178$	$3.45 \%$	$94.36 \%$	$3.686$	$0.165$	$4.28 \%$	$94.37 \%$
$\lambda = 5.0$	$105$	$10.011$	$0.305$	$2.96 \% $	$94.40 \% $	$8.309$	$0.330$	$3.82 \%$	$94.39 \%$	$7.305$	$0.353$	$4.61 \%$	$94.40 \%$
$\mathcal{T} = 1.0$	$110$	$12.992$	$0.469$	$3.48 \% $	$94.46 \%$	$11.804$	$0.535$	$4.34 \%$	$94.44 \%$	$11.037$	$0.597$	$5.13 \%$	$94.44 \%$
	$115$	$16.314$	$0.691$	$4.07 \% $	$94.52 \%$	$15.492$	$0.811$	$4.98 \%$	$94.50 \% $	$14.914$	$0.920$	$5.81 \%$	$94.54 \% $

	$90$	$4.098$	$0.065$	$1.57 \%$	$89.68 \%$	$0.344$	$0.010$	$2.79 \% $	$89.87 \%$	$0$	$0$	-	-
(2)	$95$	$5.933$	$0.113$	$1.87 \%$	$89.80 \%$	$2.012$	$0.061$	$2.93 \%$	$89.89 \% $	$0$	$0$	-	-
$\sigma_{X}=0.2$	$100$	$8.169$	$0.186$	$2.22 \%$	$89.90 \%$	$5.413$	$0.175$	$3.12 \%$	$89.93 \%$	$3.990$	$0.161$	$3.88 \%$	$89.95 \%$
$\lambda = 10.0$	$105$	$10.791$	$0.290$	$2.62 \% $	$90.00 \% $	$8.791$	$0.314$	$3.44 \%$	$89.99 \%$	$7.683$	$0.334$	$4.17 \%$	$89.99 \%$
$\mathcal{T} = 1.0$	$110$	$13.767$	$0.435$	$3.06 \% $	$90.08 \%$	$12.313$	$0.497$	$3.88 \%$	$90.05 \%$	$11.442$	$0.552$	$4.60 \%$	$90.05 \%$
	$115$	$17.056$	$0.628$	$3.55 \% $	$90.17 \%$	$16.004$	$0.738$	$4.41 \%$	$90.12 \% $	$15.325$	$0.835$	$5.16 \%$	$90.13 \% $

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Table 4. Theoretical (down-and-out) call values and structure of the early exercise premium for $r = 0.05$, $\delta = 0.07$, $K = 100$, $L = 95$, $\rho_{L} = -26.34$, $p = 0.5$, $\xi = 50$ and $\eta = 50$

(Down-and-Out) Call Option Prices
Parameters		Standard Call Price				Step Call Price				Barrier Call Price
	$S_{0}$	Euro	EEP	EEP (%)	DC (%)	Euro	EEP	EEP (%)	DC (%)	Euro	EEP	EEP (%)	DC (%)

	$90$	$3.163$	$0.064$	$1.98 \%$	$93.97 \%$	$0.232$	$0.008$	$3.46 \% $	$94.10 \%$	$0$	$0$	-	-
(1)	$95$	$4.835$	$0.117$	$2.37 \%$	$94.05 \%$	$1.588$	$0.060$	$3.63 \%$	$94.12 \% $	$0$	$0$	-	-
$\sigma_{X}=0.2$	$100$	$6.958$	$0.202$	$2.82 \%$	$94.12 \%$	$4.679$	$0.188$	$3.87 \%$	$94.14 \%$	$3.432$	$0.174$	$4.82\%$	$94.15 \%$
$\lambda = 5.0$	$105$	$9.523$	$0.328$	$3.33 \% $	$94.19 \% $	$7.949$	$0.355$	$4.28 \%$	$94.18 \%$	$6.983$	$0.382$	$5.18 \%$	$94.18 \%$
$\mathcal{T} = 1.0$	$110$	$12.498$	$0.509$	$3.91 \% $	$94.25 \%$	$11.430$	$0.583$	$4.85 \%$	$94.23 \%$	$10.702$	$0.654$	$5.76 \%$	$94.24 \%$
	$115$	$15.835$	$0.758$	$4.57 \% $	$94.33 \%$	$15.122$	$0.891$	$5.57 \%$	$94.31 \% $	$14.586$	$1.017$	$6.52 \%$	$94.43 \% $

	$90$	$3.441$	$0.068$	$1.94 \%$	$88.95 \%$	$0.268$	$0.009$	$3.40 \% $	$89.16 \%$	$0$	$0$	-	-
(2)	$95$	$5.155$	$0.121$	$2.30 \%$	$89.08 \%$	$1.685$	$0.062$	$3.55 \%$	$89.19 \% $	$0$	$0$	-	-
$\sigma_{X}=0.2$	$100$	$7.303$	$0.204$	$2.72 \%$	$89.20 \%$	$4.836$	$0.190$	$3.77 \%$	$89.23 \%$	$3.522$	$0.174$	$4.71 \%$	$89.25 \%$
$\lambda = 10.0$	$105$	$9.875$	$0.325$	$3.19 \% $	$89.30 \% $	$8.138$	$0.352$	$4.14 \%$	$89.29 \%$	$7.107$	$0.377$	$5.04 \%$	$89.29 \%$
$\mathcal{T} = 1.0$	$110$	$12.839$	$0.497$	$3.72 \% $	$89.40 \%$	$11.636$	$0.569$	$4.66 \%$	$89.36 \%$	$10.845$	$0.638$	$5.56 \%$	$89.37 \%$
	$115$	$16.152$	$0.729$	$4.32 \% $	$89.53 \%$	$15.330$	$0.859$	$5.31 \%$	$89.46 \% $	$14.737$	$0.981$	$6.24 \%$	$89.57 \% $

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Table 5. Theoretical (down-and-out) call values and structure of the early exercise premium for $r = 0.05$, $\delta = 0.07$, $K = 100$, $L = 95$, $\rho_{L} = -26.34$, $p = 0.5$, $\xi = 25$ and $\eta = 50$

(Down-and-Out) Call Option Prices
Parameters		Standard Call Price				Step Call Price				Barrier Call Price
	$S_{0}$	Euro	EEP	EEP (%)	DC (%)	Euro	EEP	EEP (%)	DC (%)	Euro	EEP	EEP (%)	DC (%)

	$90$	$3.645$	$0.080$	$2.15 \%$	$75.53 \%$	$0.294$	$0.012$	$3.75 \% $	$76.36 \%$	$0$	$0$	-	-
(1)	$95$	$5.362$	$0.137$	$2.49 \%$	$75.97 \%$	$1.685$	$0.067$	$3.82 \%$	$76.45 \% $	$0$	$0$	-	-
$\sigma_{X}=0.2$	$100$	$7.501$	$0.222$	$2.88 \%$	$76.37 \%$	$4.854$	$0.202$	$3.99 \%$	$76.61 \%$	$3.506$	$0.182$	$4.94\%$	$76.81 \%$
$\lambda = 5.0$	$105$	$10.054$	$0.345$	$3.31 \% $	$76.71 \% $	$8.177$	$0.368$	$4.31 \%$	$76.94 \%$	$7.110$	$0.391$	$5.21 \%$	$77.25 \%$
$\mathcal{T} = 1.0$	$110$	$12.994$	$0.514$	$3.80 \% $	$76.98 \%$	$11.685$	$0.585$	$4.77 \%$	$77.55 \%$	$10.861$	$0.652$	$5.67 \%$	$78.24 \%$
	$115$	$16.279$	$0.740$	$4.35 \% $	$77.14 \%$	$15.381$	$0.870$	$5.35 \%$	$78.79 \% $	$14.759$	$0.989$	$6.28 \%$	$80.55 \% $

	$90$	$4.347$	$0.096$	$2.16 \%$	$62.58 \%$	$0.391$	$0.015$	$3.78 \% $	$63.44 \%$	$0$	$0$	-	-
(2)	$95$	$6.141$	$0.155$	$2.45 \%$	$63.00 \%$	$1.865$	$0.074$	$3.82 \%$	$63.47 \% $	$0$	$0$	-	-
$\sigma_{X}=0.2$	$100$	$8.321$	$0.238$	$2.78 \%$	$63.38 \%$	$5.152$	$0.212$	$3.94 \%$	$63.56 \%$	$3.649$	$0.188$	$4.89\%$	$63.68 \%$
$\lambda = 5.0$	$105$	$10.878$	$0.354$	$3.15 \% $	$63.72 \% $	$8.549$	$0.374$	$4.19 \%$	$63.76 \%$	$7.328$	$0.393$	$5.09 \%$	$63.90 \%$
$\mathcal{T} = 1.0$	$110$	$13.788$	$0.508$	$3.55 \% $	$64.01 \%$	$12.099$	$0.577$	$4.55 \%$	$64.10 \%$	$11.126$	$0.640$	$5.44 \%$	$64.46 \%$
	$115$	$17.019$	$0.709$	$4.00 \% $	$64.18 \%$	$15.809$	$0.834$	$5.01 \%$	$64.74 \% $	$15.047$	$0.946$	$5.91 \%$	$65.93 \% $

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Table 6. Theoretical (down-and-out) call values and structure of the early exercise premium for $r = 0.05$, $\delta = 0.07$, $K = 100$, $L = 95$, $\rho_{L} = -26.34$, $p = 0.5$, $\xi = 25$ and $\eta = 25$

(Down-and-Out) Call Option Prices
Parameters		Standard Call Price				Step Call Price				Barrier Call Price
	$S_{0}$	Euro	EEP	EEP (%)	DC (%)	Euro	EEP	EEP (%)	DC (%)	Euro	EEP	EEP (%)	DC (%)

	$90$	$3.966$	$0.077$	$1.91 \%$	$76.61 \%$	$0.330$	$0.012$	$3.37 \% $	$77.40 \%$	$0$	$0$	-	-
(1)	$95$	$5.745$	$0.131$	$2.23 \%$	$77.04 \%$	$1.845$	$0.066$	$3.44 \%$	$77.48 \% $	$0$	$0$	-	-
$\sigma_{X}=0.2$	$100$	$7.931$	$0.210$	$2.58 \%$	$77.42 \%$	$5.151$	$0.192$	$3.60 \%$	$77.62 \%$	$3.748$	$0.174$	$4.44\%$	$77.79 \%$
$\lambda = 5.0$	$105$	$10.514$	$0.323$	$2.98 \% $	$77.75 \% $	$8.516$	$0.346$	$3.90 \%$	$77.89 \%$	$7.415$	$0.366$	$4.70 \%$	$78.11 \%$
$\mathcal{T} = 1.0$	$110$	$13.463$	$0.479$	$3.43 \% $	$78.01 \%$	$12.037$	$0.544$	$4.32 \%$	$78.35 \%$	$11.178$	$0.603$	$5.12 \%$	$78.81 \%$
	$115$	$16.740$	$0.685$	$3.93 \% $	$78.17 \%$	$15.730$	$0.803$	$4.85 \%$	$79.21 \% $	$15.069$	$0.907$	$5.68 \%$	$80.34 \% $

	$90$	$4.950$	$0.091$	$1.81 \%$	$64.97 \%$	$0.468$	$0.016$	$3.20 \% $	$65.77 \%$	$0$	$0$	-	-
(2)	$95$	$6.842$	$0.144$	$2.07 \%$	$65.37 \%$	$2.166$	$0.073$	$3.25 \%$	$65.81 \% $	$0$	$0$	-	-
$\sigma_{X}=0.2$	$100$	$9.098$	$0.220$	$2.36 \%$	$65.74 \%$	$5.678$	$0.198$	$3.37 \%$	$65.91 \%$	$4.077$	$0.177$	$4.16\%$	$66.01 \%$
$\lambda = 5.0$	$105$	$11.704$	$0.322$	$2.68 \% $	$66.08 \% $	$9.138$	$0.341$	$3.60 \%$	$66.09 \%$	$7.852$	$0.358$	$4.35 \%$	$66.17 \%$
$\mathcal{T} = 1.0$	$110$	$14.634$	$0.458$	$3.03 \% $	$66.37 \%$	$12.709$	$0.518$	$3.92 \%$	$66.34 \%$	$11.668$	$0.571$	$4.67 \%$	$66.50 \%$
	$115$	$17.857$	$0.632$	$3.42 \% $	$66.60 \%$	$16.419$	$0.741$	$4.32 \%$	$66.73 \% $	$15.583$	$0.834$	$5.08 \%$	$67.20 \% $

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References

[1]	J. Abate and W. Whitt, A unified framework for numerically inverting Laplace transforms, INFORMS Journal on Computing, 18 (2006), 408-421. doi: 10.1287/ijoc.1050.0137.
[2]	D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.
[3]	O. E. Barndorff-Nielsen, Normal inverse Gaussian distributions and stochastic volatility modelling, Scandinavian Journal of Statistics, 24 (1997), 1-13. doi: 10.1111/1467-9469.00045.
[4]	F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654. doi: 10.1086/260062.
[5]	M. Boyarchenko and S. Boyarchenko, Double barrier options in regime-switching hyper-exponential jump-diffusion models, International Journal of Theoretical and Applied Finance, 14 (2011), 1005-1043. doi: 10.1142/S0219024911006620.
[6]	S. Boyarchenko and S. Levendorskii, Efficient Laplace inversion, Wiener-Hopf factorization and pricing lookbacks, International Journal of Theoretical and Applied Finance, 16 (2013), 1350011. doi: 10.1142/S0219024913500118.
[7]	M. Boyarchenko and S. Levendorskii, Static and semi-static hedging as contrarian or conformist bets, preprint, arXiv: 1902.02854. doi: 10.1111/mafi.12240.
[8]	N. Cai, On first passage times of a hyper-exponential jump diffusion process, Operations Reseearch Letters, 37 (2009), 127-134. doi: 10.1016/j.orl.2009.01.002.
[9]	N. Cai, N. Chen and X. Wan, Pricing double-barrier options under a flexible jump diffusion model, Operations Research Letters, 37 (2009), 163-167. doi: 10.1016/j.orl.2009.02.006.
[10]	N. Cai, N. Chen and X. Wan, Occupation times of jump-diffusion processes with double exponential jumps and the pricing of options, Mathematics of Operations Research, 35 (2010), 412-437. doi: 10.1287/moor.1100.0447.
[11]	N. Cai and S. G. Kou, Option pricing under a mixed-exponential jump diffusion model, Management Science, 57 (2011), 2067-2081. doi: 10.1287/opre.1110.1006.
[12]	N. Cai and S. G. Kou, Pricing Asian options under a hyper-exponential jump-diffusion model, Operations Research, 60 (2012), 64-77. doi: 10.1287/opre.1110.1006.
[13]	G. Campolieti, R. N. Makarov and K. Wouterloot, Pricing step options under the CEV and other solvable diffusion models, International Journal of Theoretical and Applied Finance, 16 (2013), 1350027. doi: 10.1142/S0219024913500271.
[14]	P. P. Carr, Randomization and the American put, The Review of Financial Studies, 11 (1998), 597-626.
[15]	P. P. Carr, H. Geman, D. B. Madan and M. Yor, The fine structure of asset returns: An empirical investigation, Journal of Business, 75 (2002), 305-332.
[16]	M. Chesney and N. Vasiljevic, Parisian options with jumps: A maturity-excursion randomization approach, Quant. Finance, 18 (2018), 1887-1908. doi: 10.1080/14697688.2018.1444785.
[17]	Y. Chuancun, S. Ying and W. Yuzhen, Exit problems for jump processes with applications to dividend problems, Journal of Computational and Applied Mathematics, 245 (2013), 30-52. doi: 10.1016/j.cam.2012.12.004.
[18]	D. Davydov and V. Linetsky, Structuring, pricing and hedging double barrier step options, Journal of Computational Finance, 5 (2002), 55-87.
[19]	J. Detemple, A. S. Laminou and F. Moraux, American step options, European Journal of Operational Research, 282 (2020), 363-385. doi: 10.1016/j.ejor.2019.09.009.
[20]	J. Fajardo and E. Mordecki, Symmetry and duality in Lévy markets, Quant. Finance, 6 (2006), 219-227. doi: 10.1080/14697680600680068.
[21]	J. Fajardo and E. Mordecki, Skewness premium with Lévy processes, Quant. Finance, 14 (2014), 1619-1616. doi: 10.1080/14697688.2011.618809.
[22]	W. Farkas, L. Mathys and N. Vasiljević, Intra-Horizon expected shortfall and risk structure in models with jumps, Mathematical Finance, 31 (2021), 772-823. doi: 10.1111/mafi.12302.
[23]	M. B. Garman and S. W. Kohlhagen, Foreign currency option values, Journal of International Money and Finance, 2 (1983), 231-237.
[24]	H. U. Gerber and E. S. W. Shiu, Option pricing by Esscher transforms, Transactions of the Society of Actuaries, 46 (1994), 99-191.
[25]	M. J. Harrison and S. R. Pliska, Martingales and stochastic integrals in the theory of continous trading, Stochastic Process. Appl., 11 (1981), 215-260. doi: 10.1016/0304-4149(81)90026-0.
[26]	M. Hofer and P. Mayer, Pricing and hedging of lookback options in hyperexponential jump diffusion models, Applied Mathematical Finance, 20 (2013), 489-511. doi: 10.1080/1350486X.2013.774985.
[27]	K. R. Jackson, S. Jaimungal and V. Surkov, Fourier space time-stepping for option pricing with Lévy models, Journal of Computational Finance, 12 (2008), 1-29. doi: 10.21314/JCF.2008.178.
[28]	M. Jeanblanc and M. Chesney, Pricing American currency options in an exponential Lévy model, Applied Mathematical Finance, 11 (2004), 207-225.
[29]	M. Jeanblanc, M. Yor and M. Chesney, Mathematical Methods for Financial Markets, Springer Finance, Springer, Berlin, 2006. doi: 10.1007/978-1-84628-737-4.
[30]	T. Kimura, Alternative randomization for valuing American options, Asia-Pacific Journal of Operational Research, 27 (2010), 167-187. doi: 10.1142/S0217595910002624.
[31]	S. G. Kou, A jump-diffusion model for option pricing, Management Science, 48 (2002), 1086-1101.
[32]	S. G. Kou and H. Wang, First passage times of a jump diffusion process, Advances in Applied Probability, 35 (2003), 504-531. doi: 10.1239/aap/1051201658.
[33]	S. G. Kou and H. Wang, Option pricing under a double exponential jump diffusion model, Management Science, 50 (2004), 1178-1192. doi: 10.1287/opre.1110.1006.
[34]	A. Kuznetsov, On the convergence of the Gaver-Stehfest algorithm, Siam Journal on Numerical Analysis, 51 (2013), 2984-2998. doi: 10.1137/13091974X.
[35]	D. Lamberton and M. Mikou, The smooth-fit property in an exponential Lévy model, Journal of Applied Probability, 49 (2011), 137-149. doi: 10.1017/S0021900200008901.
[36]	M. Leippold and N. Vasiljević, Pricing and disentanglement of American puts in the hyper-exponential jump-diffusion model, Journal of Banking and Finance, 77 (2017), 78-94.
[37]	M. Leippold and N. Vasiljević, Option-Implied intra-horizon value-at-risk, Management Science, 66 (2019), 397-414. doi: 10.1287/mnsc.2018.3157.
[38]	S. Levendorskii, Pricing of the American put under Lévy processes, International Journal of Theoretical and Applied Finance, 7 (2004), 303-335. doi: 10.1142/S0219024904002463.
[39]	J. Lin and K. Palmer, Convergence of barrier option prices in the binomial model, Mathematical Finance, 23 (2013), 318-338. doi: 10.1111/j.1467-9965.2011.00501.x.
[40]	V. Linetsky, Step options, Mathematical Finance, 9 (1999), 55-96. doi: 10.1111/1467-9965.00063.
[41]	D. B. Madan, P. P. Carr and E. C. Chang, The variance gamma process and option pricing, European Finance Review, 2 (1998), 79-105.
[42]	D. B. Madan and E. Seneta, The variance gamma model for share market returns, The Journal of Business, 63 (1990), 511-524.
[43]	L. Mathys, Valuing tradeability in exponential Lévy models, Quantitative Finance and Economics, 4 (2020), 459-488. doi: 10.3934/QFE.2020021.
[44]	L. Mathys, On extensions of the Barone-Adesi & Whaley method to price American-type options, Journal of Computational Finance, 24 (2020), 33-76. doi: 10.21314/JCF.2020.397.
[45]	J. P. V. Nunes, J. P. Ruas and J. C. Dias, Early exercise boundaries for American-style knock-out options, European Journal of Operational Research, 285 (2020), 753-766. doi: 10.1016/j.ejor.2020.02.006.
[46]	G. Peskir and A. N. Shiryaev, Optimal Stopping and Free-Boundary Problems, Lectures in Mathematics, ETH Zürich, Bikhäuser, 2006.
[47]	N. Rodosthenous and H. Zhang, Beating the omega clock: An optimal stopping problem with random time-horizon under spectrally negative Lévy models, Annals of Applied Probability, 28 (2018), 2105-2140. doi: 10.1214/17-AAP1322.
[48]	K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1999.
[49]	L. Trigeorgis and A. E. Tsekrekos, Real options in operations research: A review, European Journal of Operational Research, 270 (2018), 1-24. doi: 10.1016/j.ejor.2017.11.055.
[50]	P. P. Valko and J. Abate, Comparison of sequence accelerators for the Gaver method of numerical Laplace transform inversion, Computers & Mathematics with Applications, 48 (2004), 629-636. doi: 10.1016/j.camwa.2002.10.017.
[51]	H. Y. Wong and J. Zhao, Valuing American options under the CEV model by Laplace-Carson transforms, Operations Research Letters, 38 (2010), 474-481. doi: 10.1016/j.orl.2010.07.006.
[52]	L. Wu and J. Zhou, Occupation times of hyper-exponential jump diffusion processes with application to price step options, Journal of Computational and Applied Mathematics, 294 (2016), 251-274. doi: 10.1016/j.cam.2015.09.001.
[53]	L. Wu, J. Zhou and Y. Bai, Occupation times of Lévy-driven Ornstein-Uhlenbeck processes with two-sided exponential jumps and applications, Statistics and Probability Letters, 125 (2017), 80-90. doi: 10.1016/j.spl.2017.01.021.
[54]	X. Xing and H. Yang, American type geometric step options, Journal of Industrial and Management Optimization, 9 (2013), 549-560. doi: 10.3934/jimo.2013.9.549.