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,

We focus on mean-variance hedging problem for models whose asset price follows an exponential additive process. Some representations of mean-variance hedging strategies for jump type models have already been suggested, but none is suited to develop numerical methods of the values of strategies for any given time up to the maturity. In this paper, we aim to derive a new explicit closed-form representation, which enables us to develop an efficient numerical method using the fast Fourier transforms. Note that our representation is described in terms of Malliavin derivatives. In addition, we illustrate numerical results for exponential Lévy models. strategies. Note that our representation is a closed-form one obtained by means of Malliavin calculus for Lévy processes. In addition, we develop a numerical method using the fast Fourier transforms (FFT); and show numerical results for exponential Lévy models.We consider throughout an incomplete financial market in which one risky asset and one riskless asset are tradable. Let T > 0 be the maturity of our market, and suppose that the interest rate of the riskless asset is 0 for sake of simplicity. The risky asset price process, denoted by S, is given as a solution to the following stochastic differential equation:, W is a one-dimensional standard Brownian motion, and N is the compensated version of a homogeneous Poisson random measure N.Here, α and β are deterministic measurable functions on [0, T], and γ is also deterministic and jointly measurable on [0, T] × R 0 . We assume γ > −1, which ensures the positivity of S. Then, S is given as an exponential of an additive process, that is, log(S) is continuous in probability and has independent increments. In addition, when α and β are given by a real number and a nonnegative real number, respectively, and γ t,z = e z − 1, we call S an exponential Lévy process. Let H be a square integrable random variable. We consider its value as the payoff of a contingent claim at the maturity T. In principle, since our market is incomplete, we cannot find a replicating strategy for H, that is, there is no pair (c, ϑ) ∈ R × Θ satisfyingwhere Θ is a set of predictable processes, which is considered as the set of all admissible strategies in some sense, and G(ϑ) denotes the gain process induced by ϑ, that is, G(ϑ) := · 0 ϑ u dS u . Note that each pair (c, ϑ) ∈ R × Θ represents a self-financing strategy. Instead of finding the replicating strategy, we consider the following minimization problem:and call its solution ( c H , ϑ H ) ∈ R × Θ the MVH strategy for claim H if it exists. In other words, the MVH strategy is defined as the self-financing strategy minimizing the corresponding L 2 -hedging error over R × Θ. Remark that c H gives the initial cost, which is regarded as the corresponding price of H; and ϑ H t represents the number of shares of the risky asset in the strategy at time t.In addition to MVH strategy, locally risk-minimizing (LRM) strategy has been studied well as alternative hedging method in quadratic way. Being different from ...

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.

An affine Kac–Moody algebra is a central extension of the Lie algebra of smooth mappings from S1 to the complexification of a Lie algebra. In this paper, we shall introduce a central extension of the Lie algebra of smooth mappings from S3 to the quaternization of a Lie algebra and investigate its root space decomposition. We think this extension of current algebra might give a mathematical tool for four-dimensional conformal field theory as Kac–Moody algebras give it for two-dimensional conformal field theory.