In this paper, we aim to obtain explicit representations of locally risk-minimizing by using Malliavin calculus for Lévy processes. For incomplete market models whose asset price is described by a solution to a stochastic differential equation driven by a Lévy process, we derive general formulas of locally risk-minimizing including Malliavin derivatives; and calculate its concrete expressions for call options, Asian options and lookback options.
We illustrate how to compute local risk minimization (LRM) of call options for exponential Lévy models. We have previously obtained a representation of LRM for call options; here we transform it into a form that allows use of the fast Fourier transform method suggested by Carr & Madan.In particular, we consider Merton jump-diffusion models and variance gamma models as concrete applications.and θ x := µ S (e x − 1) σ 2 + R 0 (e y − 1) 2 ν(dy) for x ∈ R 0 . In the development of our approach, we rely on the following: Assumption 1.1.1. R 0 (|x| ∨ x 2 )ν(dx) < ∞, and R 0 (e x − 1) n ν(dx) < ∞ for n = 2, 4.
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,
The Clark-Ocone formula is an explicit stochastic integral representation for random variables in terms of Malliavin derivatives. In this paper, we prove a Clark-Ocone type formula under change of measure (COCM) for Lévy processes with L 2 -Lévy measure.To show the COCM for L 2 -Lévy processes, we develop Malliavin calculus for Lévy processes, based on [11]. By using σ-finiteness of Lévy measure, we obtain a commutation formula for the Lebesgue integration and the Malliavin derivative and a chain rule for Malliavin derivative. These formulas derive the COCM. Finally, we obtain a log-Sobolev type formula for Lévy functionals.
In this paper, we derive a Clark-Ocone type formula under change of measure for multidimensional Lévy processes. This is a multidimensional version of [14, 15, 9]. By using it, we obtain explicit representations of locally risk-minimizing hedging strategy for markets driven by multidimensional Lévy processes. This is a generalization of [3].
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