International Journal of Theoretical and Applied Finance

We illustrate how to compute local risk minimization (LRM) of call options for exponential Lévy 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 by Carr & Madan (1999). Considering Merton jump-diffusion models and variance gamma models as typical examples of exponential Lévy 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.

We investigate a utility maximization problem in the presence of asset price bubbles. At random times, the investor receives warnings that a bubble has formed in the market which may lead to a crash in the risky asset. We propose a regime-switching model for the warnings and we make no assumptions about the distribution of the timing and the size of the crashes. Instead, we assume that the investor takes a worst-case perspective towards their impacts, i.e. the investor maximizes her expected utility under the worst-case crash scenario. We characterize the value function by a system of Hamilton–Jacobi–Bellman equations and derive a coupled system of ordinary differential equations for the optimal strategies. Numerical examples are provided.

We propose a model of a financial market with multiple assets that takes into account the impact of a large institutional investor rebalancing its positions so as to maintain a fixed allocation in each asset. We show that feedback effects can lead to significant excess realized correlation between asset returns and modify the principal component structure of the (realized) correlation matrix of returns. Our study naturally links, in a quantitative manner, the properties of the realized correlation matrix — correlation between assets, eigenvectors and eigenvalues — to the sizes and trading volumes of large institutional investors. In particular, we show that even starting with uncorrelated “fundamentals”, fund rebalancing endogenously generates a correlation matrix of returns with a first eigenvector with positive components, which can be associated to the market, as observed empirically. Finally, we show that feedback effects flatten the differences between the expected returns of assets and tend to align them with the returns of the institutional investor’s portfolio, making this benchmark fund more difficult to beat, not because of its strategy but precisely because of its size and market impact.

We propose robust numerical algorithms for pricing variance options and volatility swaps on discrete realized variance under general time-changed Lévy processes. Since analytic pricing formulas of these derivatives are not available, some of the earlier pricing methods use the quadratic variation approximation for the discrete realized variance. While this approximation works quite well for long-maturity options on discrete realized variance, numerical accuracy deteriorates for options with low frequency of monitoring or short maturity. To circumvent these shortcomings, we construct numerical algorithms that rely on the computation of the Laplace transform of the discrete realized variance under time-changed Lévy processes. We adopt the randomization of the Laplace transform of the discrete log return with a standard normal random variable and develop a recursive quadrature algorithm to compute the Laplace transform of the discrete realized variance. Our pricing approach is rather computationally efficient when compared with the Monte Carlo simulation and works particularly well for discrete realized variance and volatility derivatives with low frequency of monitoring or short maturity. The pricing behaviors of variance options and volatility swaps under various time-changed Lévy processes are also investigated.

This paper develops simple and efficient tree approaches for option pricing in switching jump diffusion models where the rates of switching are assumed to depend on the underlying asset price process. The models generalize many existing models in the literature and in particular, the Markovian regime-switching models with jumps. The proposed trees grow linearly as the number of tree steps increases. Conditions on the choices of key parameters for the tree design are provided that guarantee the positivity of branch probabilities. Numerical results are provided and compared with results reported in the literature for the Markovian regime-switching cases. The reported numerical results for the state-dependent switching models are new and can be used for comparison in the future.

When the discretely adjusted option hedges are constructed by the continuous-time Black–Scholes delta, then the hedging errors appear. The first objective of the paper is to consider a discrete-time adjusted delta, such that the hedging error can be reduced. Consequently, a partial differential equation for option valuation associated with the problem is derived and solved.

The second objective is to compare the obtained results with the results given by the Black–Scholes formula. The obtained option values may be higher than those given by the Black–Scholes formula, however, unless the option is near expiry, the difference is relatively small.

In this paper, a class of generalized Barndorff-Nielsen and Shephard (BN–S) models is investigated from the viewpoint of derivative asset analysis. Incompleteness of this type of markets is studied in terms of equivalent martingale measures (EMM). Variance process is studied in details for the case of Inverse-Gaussian distribution. Various structure preserving subclasses of EMMs are derived. The model is then effectively used for pricing European style options and fitting implied volatility smiles.