Adam W. Kolkiewicz

An improved simulation method for pricing high-dimensional American derivatives

In this paper, we propose an estimator for pricing high-dimensional American-style options and show that asymptotically its upper bias converges to zero. An advantage of the proposed estimator is that when combined with low discrepancy sequences, it exhibits a superior rate of convergence. Numerical examples are conducted to demonstrate its efficiency.

Semi-Static Hedging for GMWB in Variable Annuities

Abstract The Guaranteed Minimum Withdrawal Benefit (GMWB) is an option embedded in a variable annuity that guarantees the policyholder to get the initial investment back by making periodic withdrawals regardless of the impact of poor market performance. In the paper we discuss methods of pricing and hedging of some versions of GMWBs. In particular we develop a method of constructing semi-static hedging strategies that offer several advantages over dynamic hedging. The idea is to first find the closest path-independent option to the guarantee, or its liability part, and then to construct a portfolio of traded European options that approximates the optimal option. This strategy requires fewer portfolio adjustments than delta hedging and outperforms the latter when there are random jumps in the underlying price. We illustrate the proposed method with numerical examples.

Pricing American Derivatives using Simulation: A Biased Low Approach

Boyle et al. (2000) proposed a simulation method for pricing highdimensional American style derivatives. The method exploits the uniformity property of the low discrepancy sequences so that the resulting biased high estimator can achieve the higher rate of convergence of quasi-Monte Carlo method. In this paper, we extend this work by proposing another estimator that is biased low. It has the computational advantage that it can be obtained concurrently with the high-biased estimator using a recursive valuation approach. Numerical examples are provided to demonstrate its efficiency. We also show that further enhancement to the proposed estimator is possible by incorporating standard variance reduction techniques such as control variates.

A parallel quasi-Monte Carlo approach to pricing multidimensional American options

In this paper, we develop parallel algorithms for pricing American options on multiple assets. Our parallel methods are based on the Low Discrepancy (LD) mesh method which combines the quasi-Monte Carlo technique with the stochastic mesh method. We present two approaches to parallelise the backward recursion step, which is the most computational intensive part of the LD mesh method. We perform parallel run time analysis of the proposed methods and prove that both parallel approaches are scalable. The algorithms are implemented using MPI. The parallel efficiency of the methods are demonstrated by pricing several American options, and near optimal speedup results are presented.

Computation of Multivariate Barrier Crossing Probability and its Applications in Credit Risk Models

Abstract In this paper we consider computational methods of finding exit probabilities for a class of multivariate diffusion processes. Although there is an abundance of results for one-dimensional diffusion processes, for multivariate processes one has to rely on approximations or simulation methods. We adopt a Large Deviations approach to approximate barrier crossing probabilities of a multivariate Brownian Bridge. We use this approach in conjunction with simulation methods to develop an efficient method of obtaining barrier crossing probabilities of a multivariate Brownian motion. Using numerical examples, we demonstrate that our method works better than other existing methods. We mainly focus on a three-dimensional process, but our framework can be extended to higher dimensions. We present two applications of the proposed method in credit risk modeling. First, we show that we can efficiently estimate the default probabilities of several correlated credit risky entities. Second, we use this method to efficiently price a credit default swap (CDS) with several correlated reference entities. In a conventional approach one normally adopts an arbitrary copula to capture dependency among counterparties. The method we propose allows us to incorporate the instantaneous variance-covariance structure of the underlying process into the CDS prices.

Pricing Bermudan options using low-discrepancy mesh methods

Abstract This paper proposes a new simulation method for pricing Bermudan derivatives that is applicable to problems where the transition density of the underlying asset price process is known analytically. We assume that the owner can exercise the option at a finite, although possibly large, number of exercise dates. The method is computationally efficient for high-dimensional problems and is easy to apply. Its efficiency stems from our use of quasi-Monte Carlo techniques, which have proven effective in the case of European derivatives. The valuation of a Bermudan derivative hinges on the optimal exercise strategy. The optimal exercise decision can be reduced to the evaluation of a series of conditional expectations with respect to different distributions. These expectations can be approximated by sampling from just a single distribution at each exercise point. We provide a theoretical basis for the selection of this distribution and develop a simple approximation that has good convergence properties. We describe how to implement the method and confirm its efficiency using numerical examples involving Bermudan options written on multiple assets and options on a foreign asset with a stochastic interest rate.

Bayesian inference of asymmetric stochastic conditional duration models

This paper extends stochastic conditional duration (SCD) models for financial transaction data to allow for correlation between error processes and innovations of observed duration process and latent log duration process. Suitable algorithms of Markov Chain Monte Carlo (MCMC) are developed to fit the resulting SCD models under various distributional assumptions about the innovation of the measurement equation. Unlike the estimation methods commonly used to estimate the SCD models in the literature, we work with the original specification of the model, without subjecting the observation equation to a logarithmic transformation. Results of simulation studies suggest that our proposed models and corresponding estimation methodology perform quite well. We also apply an auxiliary particle filter technique to construct one-step-ahead in-sample and out-of-sample duration forecasts of the fitted models. Applications to the IBM transaction data allow comparison of our models and methods to those existing in the literature.

The impacts of financial crisis on sovereign credit risk analysis in Asia and Europe

In this paper, we investigate the nature of sovereign credit risk for selected Asian and European countries based on a set of sovereign CDS data over an eight-year period that includes the episode of the 2007–2008 global financial crisis. Our results indicate that there exists strong commonality in sovereign credit risk among the countries studied in this paper following the crisis. In addition, our results also show that commonality is importantly associated with both local and global financial and economic variables. However, there are markedly different impacts of the sovereign of credit risk in Asian and European countries. Specifically, we find that foreign reserve, global stock market, and volatility risk premium, affect Asian and European sovereign credit risks in the opposite direction. Lastly, we model the arrival rates of credit events as a square-root diffusion process from which a pricing model is constructed and estimated over pre- and post-crisis periods. Then the resulting model is used to decompose credit spreads into risk premium and credit-event components. For most countries in our study, credit-event components appear to weight more than risk-premiums.

Efficient Monte Carlo simulation for integral functionals of Brownian motion

Abstract In the paper, we develop a variance reduction technique for Monte Carlo simulations of integral functionals of a Brownian motion. The procedure is based on a new method of sampling the process, which combines the Brownian bridge construction with conditioning on integrals along paths of the process. The key element in our method is the identification of a low-dimensional vector of variables that reduces the dimension of the integration problem more effectively than the Brownian bridge. We illustrate the method by applying it in conjunction with low-discrepancy sequences to the problem of pricing Asian options.

Volatility Risk For Regime-Switching Models

Abstract Regime-switching models have proven to be well-suited for capturing the time series behavior of many financial variables. In particular, they have become a popular framework for pricing equity-linked insurance products. The success of these models demonstrates that realistic modeling of financial time series must allow for random changes in volatility. In the context of valuation of contingent claims, however, random volatility poses additional challenges when compared with the standard Black-Scholes framework. The main reason is the incompleteness of such models, which implies that contingent claims cannot be hedged perfectly and that a unique identification of the correct risk-neutral measure is not possible. The objective of this paper is to provide tools for managing the volatility risk. First we present a formula for the expected value of a shortfall caused by misspecification of the realized cumulative variance. This, in particular, leads to a closed-form expression for the expected shortfall for any strategy a hedger may use to deal with the stochastic volatility. Next we identify a method of selection of the initial volatility that minimizes the expected shortfall. This strategy is the same as delta hedging based on the cumulative volatility that matches the Black-Scholes model with the stochastic volatility model. We also discuss methods of managing the volatility risk under model uncertainty. In these cases, super-hedging is a possible strategy but it is expensive. The results presented enable a more accurate analysis of the trade-off between the initial cost and the risk of a shortfall.