Numerical Analysis on Local Risk-Minimization for Exponential Lévy Models

Arai, Takuji; Imai, Yuto; Suzuki, Ryoichi

doi:10.1142/s0219024916500084

Cited by 13 publications

(59 citation statements)

References 17 publications

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“…We use the Nikkei 225 index for March 2014, as in [1]. We need to set the log price L t := log(S t /S 0 ), where S 0 is the price on 28 February 2014, which is 14841.07.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…An analytic form of ϕ T −t was given in [1,Proposition 4.5], and that of ν The following theorem is the same estimation as Theorem 7.…”

Section: Variance Gamma Modelsmentioning

confidence: 91%

“…LRM is closely related to the equivalent martingale measure which is well known as the minimal martingale measure (MMM). For more details on LRM, see [1,2]. Delta hedging strategies, which are also well-known and often used by practitioners, are given by differentiating the option price under a certain martingale measure with respect to the underlying asset price.…”

Section: Introductionmentioning

confidence: 99%

“…Carr and Madan introduced a numerical method for valuing options based on the fast Fourier transform (FFT) in [3]. In [1], the authors adopted Carr and Madan's method to compute LRM of call options for exponential Lévy models. In particular, the authors discussed Merton models and variance Gamma (VG) models as typical examples of exponential Lévy models.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, we show that delta hedging strategies are easily calculated by using the numerical scheme developed in [1]. We give inequality estimations of the differences of LRM and delta hedging strategies for the typical exponential Lévy models, known as Merton models and VG models.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Comparison of local risk minimization and delta hedging strategy for exponential Lévy models

Imai

Arai

2015

JSIAM Letters

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We discuss the differences of local risk minimization (LRM) and delta hedging strategies, in exponential Lévy models, where delta hedging strategies in this paper are defined under the minimal martingale measures (MMM). First of all we give inequality estimations for the differences of LRM and delta hedging strategies, and then show numerical examples for the two typical exponential Lévy models, Merton models and variance Gamma (VG) models.

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“…We use the Nikkei 225 index for March 2014, as in [1]. We need to set the log price L t := log(S t /S 0 ), where S 0 is the price on 28 February 2014, which is 14841.07.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…An analytic form of ϕ T −t was given in [1,Proposition 4.5], and that of ν The following theorem is the same estimation as Theorem 7.…”

Section: Variance Gamma Modelsmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Comparison of local risk minimization and delta hedging strategy for exponential Lévy models

Imai

Arai

2015

JSIAM Letters

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Numerical Analysis on Quadratic Hedging Strategies for Normal Inverse Gaussian Models

Arai

Imai

Nakashima

2018

Advances in Mathematical Economics

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The authors aim to develop numerical schemes of the two representative quadratic hedging strategies: locally risk minimizing and mean-variance hedging strategies, for models whose asset price process is given by the exponential of a normal inverse Gaussian process, using the results of Arai et al. [2], and Arai and Imai [1]. Here normal inverse Gaussian process is a framework of Lévy processes frequently appeared in financial literature. In addition, some numerical results are also introduced.

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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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Numerical Analysis on Local Risk-Minimization for Exponential Lévy Models

Cited by 13 publications

References 17 publications

Comparison of local risk minimization and delta hedging strategy for exponential Lévy models

Comparison of local risk minimization and delta hedging strategy for exponential Lévy models

Numerical Analysis on Quadratic Hedging Strategies for Normal Inverse Gaussian Models

Local risk-minimization for Barndorff-Nielsen and Shephard models

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