論文

We focus on a backward induction of the q-optimal martingale measure for discrete-time models, where 1 < q < infinity. As for the bounded asset price process case, the same backward induction has been obtained by Grandits (Bernoulli, 5:225-247, 1999). To remove the boundedness, we shall discuss a sufficient condition under which there exists a signed martingale measure whose density is in the L-q-space, which topic is our second aim.

© 2019 Elsevier B.V. This paper proposes the notion of optimal initial capital (OIC) induced by the optimized certainty equivalent (OCE), as discussed in Ben-Tal and Teboulle (1986) and Ben-Tal and Teboulle (2007). It also investigates the properties of the OIC with various types of utility functions. It is shown that the OIC can be a monetary utility function (negative value of risk measure) for future payoffs with the decision maker's concrete criteria in the background.

We obtain explicit representations of locally risk-minimizing strategies for call and put options in Barndorff-Nielsen and Shephard models, which are Ornstein-Uhlenbeck-type stochastic volatility models. Using Malliavin calculus for L,vy processes, Arai and Suzuki (Int. J. Financ. Eng. 2:1550015, 2015) obtained a formula for locally risk-minimizing strategies for L,vy markets under many additional conditions. Supposing mild conditions, we make sure that the Barndorff-Nielsen and Shephard models satisfy all the conditions imposed in (Arai and Suzuki in Int. J. Financ. Eng. 2:1550015, 2015). Among others, we investigate the Malliavin differentiability of the density of the minimal martingale measure. Moreover, we introduce some numerical experiments for locally risk-minimizing strategies.

We derive representations of locally risk-minimizing strategies of call and put options for Barndorff-Nielsen and Shephard models: jump type stochastic volatility models whose squared volatility process is given by a non-Gaussian Ornstein-Uhlenbeck process. The general form of Barndorff-Nielsen and Shephard models includes two parameters: volatility risk premium beta and leverage effect rho. Arai and Suzuki (Local risk minimization for Barndorff-Nielsen and Shephard models. submitted. Available at http://arxiv.org/pdf/1503.08589v2) dealt with the same problem under constraint. beta = -1/2. In this paper, we relax the restriction on beta; and restrict rho to 0 instead. We introduce a Malliavin calculus under the minimal martingale measure to solve the problem.

We illustrate how to compute local risk minimization (LRM) of call options for exponential Levy models. Here, LRM is a popular hedging method through a quadratic criterion for contingent claims in incomplete markets. Arai & Suzuki (2015) 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 (FFT) method suggested by Carr & Madan (1999). Considering Merton jump-diffusion models and variance gamma models as typical examples of exponential Levy models, we provide the forms for the FFT explicitly; and compute the values of LRM numerically for given parameter sets. Furthermore, we illustrate numerical results for a variance gamma model with estimated parameters from the Nikkei 225 index.

In this paper, we aim to obtain explicit representations of locally risk-minimizing by using Malliavin calculus for Levy processes. For incomplete market models whose asset price is described by a solution to a stochastic differential equation driven by a Levy 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.

The VIX call options for the Barndorff-Nielsen and Shephard models will be discussed. Derivatives written on the VIX, which is the most popular volatility measurement, have been traded actively very much. In this paper, we give representations of the VIX call option price for the Barndorff-Nielsen and Shephard models: non-Gaussian Ornstein-Uhlenbeck type stochastic volatility models. Moreover, we provide representations of the locally risk-minimizing strategy constructed by a combination of the underlying riskless and risky assets. Remark that the representations obtained in this paper are efficient to develop a numerical method using the fast Fourier transform. Thus, numerical experiments will be implemented in the last section of this paper.

Mean-variance hedging is well-known as one of hedging methods for incomplete markets. Our end is leading to mean-variance hedging strategy for incomplete market models whose asset price process is given by a discontinuous semimartingale and whose mean-variance trade-off process is not deterministic. In this paper, on account, we focus on this problem under the following assumptions: (1) the local martingale part of the stock price process is a process with independent increments; (2) a certain condition restricting the number and the size of jumps of the asset price process is satisfied; (3) the mean-variance trade-off process is uniformly bounded; (4) the minimal martingale measure coincides with the variance-optimal martingale measure. © 2002 by the University of Notre Dame. All rights reserved.

We consider an incomplete market model whose stock price fluctuation is given by a jump diffusion process. For this model, we calculate the density process of the minimal martingale measure. Also, we state the relation to a locally risk-minimizing strategy.

The Barndorff–Nielsen and Shephard (BNS) model is a representative jump-type stochastic volatility model. Still, no method exists to compute option prices numerically for the non-martingale case with infinite active jumps. In this paper, selecting the minimal martingale measure (MMM) as a representative martingale measure, we develop two simulation methods for the BNS model under the MMM. The first method simulates the asset price at maturity and the Radon–Nikodym density of the MMM separately. On the other hand, the second method directly computes the asset price distribution under the MMM. In addition, we implement some numerical experiments to evaluate the performance of our simulation methods.

We discuss the difference between locally risk-minimizing and delta hedging strategies for exponential L,vy models, where delta hedging strategies in this paper are defined under the minimal martingale measure. We give firstly model-independent upper estimations for the difference. In addition we show numerical examples for two typical exponential L,vy models: Merton models and variance gamma models.

We focus on properties of the variance-optimal martingale measure for discontinuous semimartingales. In particular, we give sufficient conditions for the variance-optimal martingale measure to be a probability measure, and for the density process of the variance-optimal martingale measure to satisfy the reverse Hölder inequality, respectively. Moreover, we study relationship with mean-variance hedging. © 2005 Elsevier B.V. All rights reserved.

Mean-variance hedging for the discontinuous semimartingale case is obtained under some assumptions related to the variance-optimal martingale measure. In the present paper, two remarks on it are discussed. One is an extension of Hou-Karatzas' duality approach from the continuous case to discontinuous. Another is to prove that there is the consistency with the case where the mean-variance trade-off process is continuous and deterministic. In particular, one-dimensional jump diffusion models are discussed as simple examples.

We discuss the p-optimal martingale measure for p∈(1,∞) in continuous incomplete markets whose stock price is fluctuated by a d-dimensional continuous semimartingale. In this paper, we treat two simple models. One is a model where the mean-variance trade-off process is deterministic. Another is a model where the minimal martingale measure coincides with the minimal entropy martingale measure. In these models, we prove that the p-optimal martingale measure coincides with the minimal martingale measure under some conditions. © 2001 Elsevier Science B.V.

We consider incomplete markets, where each risky asset fluctuation is a continuous semimartingale, and study a subset of Equivalent Local Martingale Measures in which Minimal Martingale Measure minimizes relative entropy. We also discuss, as special cases, some models with the risky asset fluctuation represented as a solution of some stochastic differential equations. Finally, we mention that the predictable representation property is essential in order that Minimal Martingale Measure coincides with Minimal Entropy Martingale Measure. © 2001 Kluwer Academic Publishers.