Local risk-minimization for Barndorff-Nielsen and Shephard models

Abstract

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,

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

“…In [1] dealing with the case of β = − 1 2 , ∞ 0 e 2B(T)x ν(dx) < ∞ is assumed in their Assumption 2.2, which is almost same as the above (A2) for β = − 1 2 . 2.…”

“…Our purpose is to obtain representations of LRM of call and put options for BNS models under constraint ρ = 0 and no constraint on β. On the other hand, Arai and Suzuki [1] studied the same problem under constraint β = − 1 2 and no constraint on ρ. That is, they dealt with the case where volatility risk premium is not taken into account.…”

“…When β = − 1 2 , the density process Z of the MMM is given as a solution to an SDE with the Lipschitz continuity. Thus, as shown in [1], Z has the Malliavin differentiability, which played a vital role in [1]. However, this property is not generalized to the case of β = − 1 2 .…”

“…However, this property is not generalized to the case of β = − 1 2 . Hence, we need to take a different approach from [1]. In order to overcome this difficulty, making the best of the fact that the Lévy property of H is preserved, we innovate a Malliavin calculus under the MMM.…”

“…To our best knowledge, except for [1], there is only one preceding research on LRM for BNS models: Wang, Qian and Wang [13]. Besides they treated the problem under the same parameter restrictions as ours, although they did not use Malliavin calculus.…”