Kiseop Lee

Insiders' hedging in a jump diffusion model

In this paper, we formulate the optimal hedging problem when the underlying stock price has jumps, especially for insiders who have more information than the general public. The jumps in the underlying price process depend on another diffusion process, which models a sequence of firm-specific information. This diffusion process is observed only by insiders. Nevertheless, the market is incomplete to insiders as well as to the general public. We use the local risk minimization method to find an optimal hedging strategy for insiders. We also numerically compare the value of the insiders hedging portfolio with the value of an honest traders hedging portfolio for a simulated sample path of a stock price.

t-Statistics for Weighted Means in Credit Risk Modelling

Purpose – The purpose of this paper is to describe a generalization of the familiar two-sample t-test for equality of means to the case where the sample values are to be given unequal weights. This is a natural situation in financial risk modeling when some samples are considered more reliable than others in predicting a common mean. We also describe an example with real credit data showing that ignoring this modification of the two-sample test can lead to the wrong statistical conclusion. Design/methodology/approach – We follow the analysis of the classical two-sample tests in the more general situation of weighted means. We also test our methods against some market data to assess the importance of the findings. Findings – We formulate some explicit test statistics that should be used when the sample values are to be assigned differing known weights. Different cases are presented depending on how much is known about the variances. In the most typical case (the unpooled two-sample test), we approximate the test statistic with a t-distribution. Proofs are given where possible. Research limitations/implications – In the unpooled case, we still only have an approximate t-distribution. This is related to the classical Behrens-Fisher problem, which is still not fully solved. We also focus on the case where the sample values are normally distributed. It would be valuable to see how far the discussion can be extended to non-normal distributions. Practical implications – Researchers should use the two-sample test statistics given in this paper instead of the standard ones when testing for equality of weighted means. Originality/value – Weighted means occur frequently in situations when the credibility or reliability of data vary. However, standard tests for equality of means do not take weights into account. These results will be of value to any researchers studying statistical means of data of varying reliability, such as corporate bond spreads.

Estimation of NIG and VG Models for High Frequency Financial Data

Numerous empirical studies have shown that certain exponential Levy models are able to fit the empirical distribution of daily financial returns quite well. By contrast, very few papers have considered intraday data in spite of their growing importance. In this paper, we fill this gap by studying the ability of the Normal Inverse Gaussian (NIG) and the Variance Gamma (VG) models to fit the statistical features of intraday data at different sampling frequencies. We propose to assess the suitability of the model by analyzing the signature plots of the point estimates at different sampling frequencies. Using high frequency transaction data from the U.S. equity market, we find the estimator of the volatility parameter to be quite stable at a wide range of intraday frequencies, in sharp contrast to the estimator of the kurtosis parameter, which is more sensitive to market microstructure effects. As a secondary contribution, we also assess the performance of the two most favored parametric estimation methods, the Method of Moments Estimators (MME) and the Maximum Likelihood Estimators (MLE), when dealing with high frequency observations. By Monte Carlo simulations, we show that neither high frequency sampling nor maxi- mum likelihood estimation significantly reduces the estimation error of the volatility parameter of the model. On the contrary, the estimation error of the parameter controlling the kurtosis of log returns can be significantly reduced by using MLE and high-frequency sampling. Both of these results appear to be new in the literature on statistical analysis of high frequency data.

Risk Minimization for a Filtering Micromovement Model of Asset Price

Abstract The classical option hedging problems have mostly been studied under continuous-time or equally spaced discrete-time models, which ignore two important components in the actual price: random trading times and market microstructure noise. In this paper, we study optimal hedging strategies for European derivatives based on a filtering micromovement model of asset prices with the two commonly ignored characteristics. We employ the local risk-minimization criterion to develop optimal hedging strategies under full information. Then, we project the hedging strategies on the observed information to obtain hedging strategies under partial information. Furthermore, we develop a related nonlinear filtering technique under the minimal martingale measure for the computation of such hedging strategies.

A mathematical model for multi-name credit based on community flocking

We present a new mathematical model for multi-name credit that employs stochastic flocking. Flocking mechanisms have been used in a variety of models of biological, sociological and physical aggregation phenomena. As a direct application of a flocking mechanism, we introduce a credit risk model based on community flocking for a credit worthiness index. Correlations between different credit worthiness indices are explained in terms of communication rates and coupling strengths from the flocking system. Based on the flocking model, we compute credit curves for individual names and default time distributions. We also apply the proposed model to the pricing of credit derivatives such as credit default swaps and collateralized debt obligations.

Information on jump sizes and hedging

We study a hedging problem in a market where traders have various levels of information. The exclusive information available only to informed traders is modelled by a diffusion process rather than discrete arrivals of new information. The asset price follows a jump–diffusion process and an information process affects jump sizes of the asset price. We find the local risk minimization hedging strategy of informed traders. Numerical examples as well as their comparison with the Black–Scholes strategy are provided via Monte Carlo.

Computation of estimates in segmented regression and a liquidity effect model

Weighted least squares (WLS) estimation in segmented regression with multiple change points is considered. A computationally efficient algorithm for calculating the WLS estimate of a single change point is derived. Then, iterative methods of approximating the global solution of the multiple change-point problem based on estimating change points one-at-a-time are discussed. It is shown that these results can also be applied to a liquidity effect model in finance with multiple change points. The liquidity effect model we consider is a generalization of one proposed by Cetin et al. [2006. Pricing options in an extended Black Scholes economy with illiquidity: theory and empirical evidence. Rev. Financial Stud. 19, 493-529], allowing that the magnitude of liquidity effect depends on the size of a trade. Two data sets are used to illustrate these methods.

Risk minimization under budget constraints

Purpose - The purpose of this paper is to find the optimal hedging strategy when an investor has budget constraints on both the initial capital and the future cash flow. Design/methodology/approach - The paper follows the utility minimization of the total cost, using convex utility functions on both initial capital and future cash flows. Findings - Closed-form solutions of optimal hedging strategies are found in some specific but popular cases. It is also found that this method corresponds to the local risk minimization method in quadratic hedging. Research limitations/implications - Hedging strategies are calculated for only two popular choices. One may want to calculate hedging strategies for other popular utility functions such as power utility or HARA utility. Practical implications - When a trader has some budget constraint in both initial capital and future cash flows, this paper gives a simple alternative. Originality/value - Budget constraints on both initial capital and future cash flow are new to this kind of study. Connection to the local risk minimization strategy is original too.

Asymptotic computation of Greeks under a stochastic volatility model

We study asymptotic expansion formulae for numerical computation of Greeks (i.e. sensitivity) in finance. Our approach is based on the integration-by-parts formula of the Malliavin calculus. We propose asymptotic expansion of Greeks for a stochastic volatility model using the Greeks formula of the Black-Scholes model. A singular perturbation method is applied to derive asymptotic Greeks formulae. We also provide numerical simulation of our method and compare it to the Monte Carlo finite difference approach.

Parameter estimation in the spatial auto-logistic model with working independent subblocks

We propose an approximation to the likelihood function with independent sub-blocks in the spatial auto-logistic model. The entire data is subdivided into many sub-blocks which are treated as independent from each other. The approximate maximum likelihood estimator, called maximum block independent likelihood estimator, is shown to have the same asymptotic distribution as that of the maximum likelihood estimator in the Ising model, a special case of the spatial auto-logistic model. The computational load for the proposed estimator is much lighter than that for the maximum likelihood estimator, and decreases geometrically as the size of a sub-block decreases. Also, limited simulation studies show that, in finite samples, the maximum block independent likelihood estimator performs as well as the maximum likelihood estimator in mean squared error. We apply our procedure to an estimation and a test of spatial dependence in the longleaf pine tree data in Cressie (1993) and the aerial image data in Pyun et al. (2007). Finally, we discuss the extension of the proposed estimator to other spatial auto-regressive models.