Ming-hua Hsieh

An Efficient Algorithm for Basket Default Swap Valuation

Monte Carlo simulation has become a workhorse of practical derivatives valuation because it is often impossible to construct theoretical pricing models for real world instruments in closed form. Credit derivatives are a case in point. But a major problem in the use of simulation methods for securities tied to default risk is that they are often written on baskets of individual credits, meaning that the number of correlated random variables to simulate can be very large, and the derivatives value depends heavily on the correlations among defaults. Individual defaults are very low probability events, so joint defaults are rare indeed. Often, a great many Monte Carlo paths must be simulated to produce even a single credit event for a contract like a k-th-to-default basket swap. Importance sampling, which focuses the simulation effort on the most important paths for a given problem, is a useful variance reduction technique. In this article, the authors present an importance sampling methodology that is easy to implement and guarantees variance reduction for k-th-to-default basket swaps. A set of numerical examples using 1st-, 2nd- and 3rd-to-default contracts based on industry portfolios, each containing five bonds, demonstrates how powerful the approach is. The amount of variance reduction depends on the level of correlation among the credits, which is substantial, and can be an order of magnitude, or even more, for highly correlated risks.

Confidence regions for stochastic approximation algorithms

In principle, known central limit theorems for stochastic approximation schemes permit the simulationist to provide confidence regions for both the optimum and optimizer of a stochastic optimization problem that is solved by means of such algorithms. Unfortunately, the covariance structure of the limiting normal distribution depends in a complex way on the problem data. In particular, the covariance matrix depends not only on variance constants but also on even more statistically challenging parameters (e.g. the Hessian of the objective function at the optimizer). In this paper, we describe an approach to producing such confidence regions that avoids the necessity of having to explicitly estimate the covariance structure of the limiting normal distribution. This procedure offers an easy way to provide confidence regions in the stochastic optimization setting.

Empirical performance of bias-reducing estimators for regenerative steady-state simulations

When simulating a stochastic system, simulationists often are interested in estimating various steady-state performance measures. The classical point estimator for such a measure involves simply taking the time average of an appropriate function of the process being simulated. Since the simulation can not be initiated with the (unknown) steady-state distribution, the classical point estimator is generally biased. In the context of regenerative steady-state simulation, a variety of other point estimators have been developed in an attempt to minimize the bias. In this paper, we provide an empirical comparison of these estimators in the context of four different continuous-time Markov chain models. The bias of the point estimators and the coverage probabilities of the associated confidence intervals are reported for the four models. Conclusions are drawn from this experimental work as to which methods are most effective in reducing bias.

Adaptive Monte Carlo methods for rare event simulations

We review two types of adaptive Monte Carlo methods for rare event simulations. These methods are based on importance sampling. The first approach selects importance sampling distributions by minimizing the variance of importance sampling estimator. The second approach selects importance sampling distributions by minimizing the cross entropy to the optimal importance sampling distribution. We also review the basic concepts of importance sampling in the rare event simulation context. To make the basic concepts concrete, we introduce these ideas via the study of rare events of M/M/1 queues.

Monte Carlo methods for valuation of ratchet Equity Indexed Annuities

Equity Indexed Annuities (EIAs) are popular insurance contracts. EIAs provide the insured with a guaranteed accumulation rate on their premium at maturity. In addition, the insured may receive extra benefit if the return of the linked index is high enough. There are a few variations of EIAs. We consider two types of EIAs: compound ratchet and simple ratchet. Under the geometric Brownian motion assumption for the equity index, plain compound ratchet options is known to have closed form solutions, but plain simple ratchet option is not. In this paper, we derive a closed form solution for plain simple ratchet option. For more exotic options, Monte Carlo methods are usually used for their valuation. To improve their efficiency, we propose two control variates based on the analytical solutions for the price of plain ratchet options. The effectiveness of the proposed control variates is examined via numerical examples of a typical contract.

A Simple Algorithm for Population Classification

A single-nucleotide polymorphism (SNP) is a variation in the DNA sequence that occurs when a single nucleotide in the genome differs across members of the same species. Variations in the DNA sequences of humans are associated with human diseases. This makes SNPs as a key to open up the door of personalized medicine. SNP(s) can also be used for human identification and forensic applications. Compared to short tandem repeat (STR) loci, SNPs have much lower statistical testing power for individual recognition due to the fact that there are only 3 possible genotypes for each SNP marker, but it may provide sufficient information to identify the population to which a certain samples may belong. In this report, using eight SNP markers for 641 samples, we performed a standard statistical classification procedure and found that 86% of the samples could be classified accurately under a two-population model. This study suggests the potential use of SNP(s) in population classification with a small number (n ≤ 8) of genetic markers for forensic screening, biodiversity and disaster victim controlling.

New estimators for parallel steady-state simulations

When estimating steady-state parameters in parallel discrete event simulation, initial transient is an important issue to consider. To mitigate the impact of initial condition on the quality of the estimator, we consider a class of estimators obtained by putting different weights on the sampling average across replications at selected time points. The weights are chosen to maximize their Gaussian likelihood. Then we apply model selection criterion due to Akaike and Schwarz to select two of them as our proposed estimators. In terms of relative root MSE, the proposed estimators compared favorably to the standard time average estimator in a typical test problem with significant initial transient.

Valuation of variable annuity contracts with cliquet options in Asia markets

Variable annuities are very appealing to the investor. For example, in United States, sales volume on variable annuities grew to a record 184 billion in calendar year 2006. However, due to their complicated payoff structure, their valuation and risk management are challenges to the insurers. In this paper, we study a variable annuity contract with cliquet options in Asia markets. The contact has quanto feature. We propose an efficient Monte Carlo method to value the contract. Numerical examples suggest our approach is quite efficient.

A Fast Monte Carlo Algorithm for Estimating Value at Risk and Expected Shortfall

Risk management today focuses heavily on estimating the location and conditional expectation of the left tail of the probability distribution for returns or portfolio value. The Holy Grail in derivatives pricing is a closed-form valuation equation such as in the Black–Scholes model, which takes a small number of input parameters and produces the exact arbitrage-free properties of the target portfolio, including value-at-risk (VaR) and expected shortfall (ES). But closed-form solutions are rare and largely limited to highly idealized markets. Lattice-based approximation techniques are available for more general settings, but they also have serious constraints. When all else fails, there is Monte Carlo simulation. Simulation always works, in principle, but the amount of calculation required in practice can be tremendous, which provides a strong incentive to find ways to speed up the process. Antithetic variates, control variates, and importance sampling are all helpful. In this article, the authors propose a new technique for estimating VaR and ES that is simple but remarkably powerful. Their first step is to determine which underlying risk factor is the most important. Next, for each simulated value of this primary factor, they simulate values for the remaining factors, requiring that every path generated exceed the VaR threshold. By not computing numerous paths that do not end up in the tail, the procedure can achieve the same accuracy as standard Monte Carlo simulation but several orders of magnitude faster.

Valuation of Ratchet Equity-Indexed Annuities

Ratchet EIAs are the most popular equity-indexed annuities (EIAs) because returns are credited periodically with a guaranteed minimum and the account value never decreases once the return is credited. Pricing ratchet EIAs, however, is challenging. This paper derives pricing formulas that cover more contract features of ratchet EIAs than any in the literature. We obtain closed-form solutions in the Black-Scholes framework for both compound and simple versions of annual-reset ratchet products that may have a return cap and employ two types of geometric return averaging. Our numerical results demonstrate the impacts of individual contract features on contract value and the interactions among these features.