Local Risk-Minimization for Barndorff-Nielsen and Shephard Models with Volatility Risk Premium

Arai, Takuji

doi:10.1007/978-981-10-0476-6_1

Cited by 1 publication

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References 13 publications

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“…Next, we show θ + log(1 − θ) ∈ L 1,2 1 . Note that we can demonstrate θ ∈ L 1,2 1 in the same manner as in the proof of condition C1. Hence, we have only to see items (g) and (h) in the definition of L 1,2 1 .…”

Section: Proof Of Theorem 31mentioning

confidence: 73%

“…and (e) in the definition of L 1,2 1 are given by Lemmas A.10 and A.7, respectively. As for item (f), Lemmas A.9 and A.10 imply…”

Section: Proof Of Theorem 31mentioning

confidence: 99%

“…To the best of our knowledge, literature on LRM strategies for BNS models is very limited. Arai [1] studied this problem for a different setting from ours. In [1], the volatility risk premium is taken into account, but ρ is restricted to 0.…”

Section: Introductionmentioning

confidence: 99%

“…Arai [1] studied this problem for a different setting from ours. In [1], the volatility risk premium is taken into account, but ρ is restricted to 0. Hence, S is described as…”

Section: Introductionmentioning

confidence: 99%

“…Note that S is continuous. Formulating a Malliavin calculus under the MMM, [1] gave an explicit representation of LRM strategies. On the other hand, there is some previous research on mean-variance hedging, which is an alternative quadratic hedging method, for BNS models.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Proof Of Theorem 31mentioning

confidence: 73%

“…and (e) in the definition of L 1,2 1 are given by Lemmas A.10 and A.7, respectively. As for item (f), Lemmas A.9 and A.10 imply…”

Section: Proof Of Theorem 31mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Arai [1] studied this problem for a different setting from ours. In [1], the volatility risk premium is taken into account, but ρ is restricted to 0. Hence, S is described as…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations