Kyoung-Kuk Kim

Gamma Expansion of the Heston Stochastic Volatility Model

We derive an explicit representation of the transitions of the Heston stochastic volatility model and use it for fast and accurate simulation of the model. Of particular interest is the integral of the variance process over an interval, conditional on the level of the variance at the endpoints. We give an explicit representation of this quantity in terms of infinite sums and mixtures of gamma random variables. The increments of the variance process are themselves mixtures of gamma random variables. The representation of the integrated conditional variance applies the Pitman–Yor decomposition of Bessel bridges. We combine this representation with the Broadie–Kaya exact simulation method and use it to circumvent the most time-consuming step in that method.

Moment Explosions and Stationary Distributions in Affine Diffusion Models

Many of the most widely used models in finance fall within the affine family of diffusion processes. The affine family combines modeling flexibility with substantial tractability, particularly through transform analysis; these models are used both for econometric modeling and for pricing and hedging of derivative securities. We analyze the tail behavior, the range of finite exponential moments, and the convergence to stationarity in affine models, focusing on the class of canonical models defined by Dai and Singleton (2000). We show that these models have limiting stationary distributions and characterize these limits. We show that the tails of both the transient and stationary distributions of these models are necessarily exponential or Gaussian; in the non-Gaussian case, we characterize the tail decay rate for any linear combination of factors. We also give necessary and sufficient conditions for a linear combination of factors to be Gaussian. Our results follow from an investigation into the stability properties of the systems of ordinary differential equations associated with affine diffusions.

Stochastic kriging with biased sample estimates

Stochastic kriging has been studied as an effective metamodeling technique for approximating response surfaces in the context of stochastic simulation. In a simulation experiment, an analyst typically needs to estimate relevant metamodel parameters and further do prediction; therefore, the impact of parameter estimation on the performance of the metamodel-based predictor has drawn some attention in the literature. However, how the standard stochastic kriging predictor is affected by the presence of bias in finite-sample estimates has not yet been fully investigated. In this article, we study the predictive performance and investigate optimal budget allocation rules subject to a fixed computational budget constraint. Furthermore, we extend the analysis to two-level or nested simulation, which has been recently documented in the risk management literature, with biased estimators.

Building metamodels for quantile-based measures using sectioning

Simulation metamodeling has been used as an effective tool in predicting the mean performance of complex systems, reducing the computational burden of costly and time-consuming simulation runs. One of the successful metamodeling techniques developed is the recently proposed stochastic kriging. However, standard stochastic kriging is confined to the case where the sample averages and sample variances of the simulation outputs at design points are the main building blocks for creating a metamodel. In this paper, we show that if each simulation output is further comprised of i.i.d. observations, then it is possible to extend the original framework into a more general one. Such a generalization enables us to utilize estimation methods including sectioning for obtaining point and interval estimates in constructing stochastic kriging metamodels for performance measures such as quantiles and tail conditional expectations. We demonstrate the superior performance of stochastic kriging metamodels under the generalized framework through some examples.

Transferring and sharing exchange-rate risk in a risk-averse supply chain of a multinational firm

This paper analyzes risk management contracts used to handle currency risk in a decentralized supply chain that consists of risk-averse divisions in a multinational firm. Particular contracts of interest involve transferring risk to a third party by using risk-transfer contracts such as currency options and re-arranging risk between supply chain members using risk-sharing contracts. Due to decentralization, operational and risk management decisions are made locally; however, a headquarter who is interested in total supply chain profit has some controllability over those activities. We question if each kind of risk management contract can improve the utility of all supply chain members compared to the utility without any of those, and how the conditions to achieve such improvements are different. Further structural differences are investigated via sensitivity analysis with respect to the transfer price, the variability of exchange rates, and the location of the headquarter. We also find that using the two kinds of contracts jointly does not necessarily result in better outcomes.

Simulation of Tempered Stable Lévy Bridges and Its Applications

We consider tempered stable Levy subordinators and develop a bridge sampling method. An approximate conditional probability density function (PDF) given the terminal values is derived with stable index less than one, using the double saddlepoint approximation. We then propose an acceptance-rejection algorithm based on the existing gamma bridge and the inverse Gaussian bridge as proposal densities. Its performance is comparable to existing sequential sampling methods such as Devroye (2009) [Devroye L (2009) Random variate generation for exponentially and ploynomially tilted stable distributions. ACM Trans. Modeling Comput. Simulation 19(4):18:1–20.] and Hofert (2011) [Hofert M (2011) Sampling exponentially tilted stable distributions. ACM Trans. Modeling Comput. Simulation 22(1):3:1–11.] when generating a fixed number of observations. As applications, we consider option pricing problems in Levy models. First, we demonstrate the effectiveness of bridge sampling when combined with adaptive sampling under finite-variance CGMY processes. Second, further efficiency gain is achieved in terms of variance reduction via stratified sampling.

Computing lower bounds on basket option prices by discretizing semi-infinite linear programming

The problem of finding static-arbitrage bounds on basket option prices has received a growing attention in the literature. In this paper, we focus on the lower bound case and propose a novel efficient solution procedure that is based on the separation problem. The computational burden of the proposed method is polynomial in the input data size. We also discuss the case of possibly negative weight vectors which can be applied to spread options.

Dynamic pricing with ‘BOGO’ promotion in revenue management

We consider a dynamic pricing problem when a seller, facing uncertain demands, sells a single product in a finite horizon. The seller actively adopts dynamic pricing and quantity discount schemes. The proposed model is based on the assumption that each customer has random reservation prices and the purchase size depends on the posted price and discount. We particularly focus on the widely adopted promotional schemes ‘buy one get one free’ and ‘50% off’ and study the optimal strategic choices of the seller. Analytical results together with numerical experiments are presented to help us obtain managerial insights. Additional numerical results for a generalised model are provided so as to examine the effectiveness of promotional schemes.

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.

Analysis and Design of Microfinance Services: A Case of ROSCA

ABSTRACT Rotating savings and credit association (ROSCA) is a well-known microfinance association widely used in many countries around the world with long histories. By considering extra profits that such a system can provide when compared to banking transactions, we develop optimization problems to achieve an optimal design of a ROSCA. We find that ROSCAs might attract investors when deposit and loan rates from formal banking systems are not favorable. Furthermore, optimal rates and optimal orders to maximize system outputs are reported.