A Clark-Ocone type formula under change of measure for Lévy processes with L^2-Lévy measure

Suzuki, Ryoichi

doi:10.31390/cosa.7.3.03

Cited by 9 publications

(5 citation statements)

References 13 publications

(20 reference statements)

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“…Because u s = f u (σ s ) and σ s ∈ D 1,2 , Proposition 2.6 in [21], together with Lemma A.3, implies u s ∈ D 1,2 and (A.6). In particular, we have D t,0 u s = f u (σ s )D t,0 σ s = 0.…”

Section: A2 Properties Of σ T and Related Malliavin Derivativesmentioning

confidence: 90%

Local risk-minimization for Barndorff-Nielsen and Shephard models

2017

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We obtain explicit representations of locally risk-minimizing strategies of call and put options for the Barndorff-Nielsen and Shephard models, which are Ornstein-Uhlenbeck-type stochastic volatility models. Using Malliavin calculus for Lévy processes, Arai and Suzuki [3] obtained a formula for locally risk-minimizing strategies for Lévy markets under many additional conditions. Supposing mild conditions, we make sure that the Barndorff-Nielsen and Shephard models satisfy all the conditions imposed in [3]. Among others, we investigate the Malliavin differentiability of the density of the minimal martingale measure. Moreover, some numerical experiments for locally risk-minimizing strategies are introduced.We consider a financial market model in which only one risky asset and one riskless asset are tradable. For simplicity, we assume the interest rate to be 0. Let T be a finite time horizon. The fluctuation of the risky asset is described as a process S given by (1.3). We adopt the same mathematical framework as in [3]. The structure of the underlying probability space (Ω, F , P) will be discussed in Subsection 2.3 below. Notice that the Poisson random measure N and the Lévy measure ν of J are defined on [0, T] × (0, ∞) and (0, ∞), respectively, and that ∞ 0 (x ∧ 1)ν(dx) < ∞ 2. As seen in Subsection 2.3 of [3], the so-called (SC) condition is satisfied under Assumption 2.2. For more details on the (SC) condition, see Schweizer [18], [19]. Moreover, Lemma 2.11 of [3] implies that E sup t∈[0,T] |S t | 2 < ∞. 3. By (A.2) in Appendix, item 2 ensures that α σ 2 t +C ρ > −1 for any t ∈ [0, T]. Remark 2.4 We state two important examples of σ 2 introduced in Nicolato and Venardos [15] that fulfill Assumption 2.2 under certain conditions on the involved parameters. For more details on this topic, see also Schoutens [17].T 0 (Z T D t,0 log Z T ) 2 dt < ∞ follows by Lemma A.7 and Proposition 2.7. Next, let Ψ t,z be the increment quoting operator defined in [22]. That is, for any random variable F, ω W ∈ Ω W and ω J = ((t 1 , z 1 ), . . . , (t n , z n )) ∈ Ω J , we definewhere ω t,z J := ((t, z), (t 1 , z 1 ), . . . , (t n , z n )). As Z T ∈ D 1,2 by Section 5, Proposition 5.4 of [22] yields that, for z > 0,

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Section: A2 Properties Of σ T and Related Malliavin Derivativesmentioning

confidence: 90%

Local risk-minimization for Barndorff-Nielsen and Shephard models

2017

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“…A filtration F = {F t } t≥0 denotes the canonical filtration completed for P. Although the results obtained in this paper are basically not depending on the structure of the underlying probability space, we choose the canonical Lévy space framework in order to simplify mathematical description and discussion. For example, it is possible to use results on the canonical Lévy space introduced in Arai and Suzuki [3], Delong and Imkeller [12] and Suzuki [29].…”

Section: Model Descriptionmentioning

confidence: 99%

“…Note that we can extend f to a C 1 -function on R with bounded derivative f . Thus, since σ 2 T ∈ D 1,2 and D t,x σ 2 T = e −λ(T −t) 1 {x>0} by Lemma A.2 of [2], Proposition 2.6 of [29] implies that V T ∈ D 1,2 ,…”

Section: Proofmentioning

confidence: 99%

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Pricing and Hedging of Vix Options for Barndorff-Nielsen and Shephard Models

Arai

2019

Int. J. Theor. Appl. Finan.

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The VIX call options for the Barndorff-Nielsen and Shephard models will be discussed. Derivatives written on the VIX, which is the most popular volatility measurement, have been traded actively very much. In this paper, we give representations of the VIX call option price for the Barndorff-Nielsen and Shephard models: non-Gaussian Ornstein-Uhlenbeck type stochastic volatility models. Moreover, we provide representations of the locally risk-minimizing strategy constructed by a combination of the underlying riskless and risky assets. Remark that the representations obtained in this paper are efficient to develop a numerical method using the fast Fourier transform. Thus, numerical experiments will be implemented in the last section of this paper.

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“…We precisely define notations and give sufficient conditions for this formula in section 4. There are many results of CO formulas (see introduction of [9,14,15] and [6]). Girsanov transformations versions of CO formulas were also studied by many people because many applications in mathematical finance require representation of random variables with respect to risk neutral martingale measure.…”

Section: Introductionmentioning

confidence: 99%