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91
Importance sampling for portfolio credit risk
 Management Science
"... Monte Carlo simulation is widely used to measure the credit risk in portfolios of loans, corporate bonds, and other instruments subject to possible default. The accurate measurement of credit risk is often a rareevent simulation problem because default probabilities are low for highly rated obligo ..."
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Cited by 70 (7 self)
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Monte Carlo simulation is widely used to measure the credit risk in portfolios of loans, corporate bonds, and other instruments subject to possible default. The accurate measurement of credit risk is often a rareevent simulation problem because default probabilities are low for highly rated obligors and because risk management is particularly concerned with rare but significant losses resulting from a large number of defaults. This makes importance sampling (IS) potentially attractive. But the application of IS is complicated by the mechanisms used to model dependence between obligors; and capturing this dependence is essential to a portfolio view of credit risk. This paper provides an IS procedure for the widely used normal copula model of portfolio credit risk. The procedure has two parts: one applies IS conditional on a set of common factors affecting multiple obligors, the other applies IS to the factors themselves. The relative importance of the two parts of the procedure is determined by the strength of the dependence between obligors. We provide both theoretical and numerical support for the method. 1
Portfolio ValueatRisk with HeavyTailed Risk Factors,” Mathematical Finance 12
, 2002
"... This paper develops efficient methods for computing portfolio valueatrisk (VAR) when the underlying risk factors have a heavytailed distribution. In modeling heavy tails, we focus on multivariate t distributions and some extensions thereof. We develop two methods for VAR calculation that exploit ..."
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Cited by 67 (2 self)
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This paper develops efficient methods for computing portfolio valueatrisk (VAR) when the underlying risk factors have a heavytailed distribution. In modeling heavy tails, we focus on multivariate t distributions and some extensions thereof. We develop two methods for VAR calculation that exploit a quadratic approximation to the portfolio loss, such as the deltagamma approximation. In the first method, we derive the characteristic function of the quadratic approximation and then use numerical transform inversion to approximate the portfolio loss distribution. Because the quadratic approximation may not always yield accurate VAR estimates, we also develop a low variance Monte Carlo method. This method uses the quadratic approximation to guide the selection of an effective importance sampling distribution that samples risk factors so that large losses occur more often. Variance is further reduced by combining the importance sampling with stratified sampling. Numerical results on a variety of test portfolios indicate that large variance reductions are typically obtained. Both methods developed in this paper overcome difficulties associated with VAR calculation with heavytailed risk factors. The Monte Carlo method also extends to the problem of estimating the conditional excess, sometimes known as the conditional VAR.
Variance reduction techniques for estimating ValueatRisk
 Management Science
, 2000
"... This paper describes, analyzes and evaluates an algorithm for estimating portfolio loss probabilities using Monte Carlo simulation. Obtaining accurate estimates of such loss probabilities is essential to calculating valueatrisk, which is a quantile of the loss distribution. The method employs a ..."
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Cited by 45 (7 self)
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This paper describes, analyzes and evaluates an algorithm for estimating portfolio loss probabilities using Monte Carlo simulation. Obtaining accurate estimates of such loss probabilities is essential to calculating valueatrisk, which is a quantile of the loss distribution. The method employs a quadratic ("deltagamma") approximation to the change in portfolio value to guide the selection of effective variance reduction techniques; specifically importance sampling and stratified sampling. If the approximation is exact, then the importance sampling is shown to be asymptotically optimal. Numerical results indicate that an appropriate combination of importance sampling and stratified sampling can result in large variance reductions when estimating the probability of large portfolio losses. 1 Introduction An important concept for quantifying and managing portfolio risk is valueatrisk (VAR) [17, 19]. VAR is defined as a quantile of the loss in portfolio value during a holding ...
A new PDE approach for pricing arithmetic average Asian options
, 2000
"... . In this paper, arithmetic average Asian options are studied. It is observed that the Asian option is a special case of the option on a traded account. The price of the Asian option is characterized by a simple onedimensional partial dierential equation which could be applied to both continuous an ..."
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Cited by 33 (1 self)
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. In this paper, arithmetic average Asian options are studied. It is observed that the Asian option is a special case of the option on a traded account. The price of the Asian option is characterized by a simple onedimensional partial dierential equation which could be applied to both continuous and discrete average Asian option. The article also provides numerical implementation of the pricing equation. The implementation is fast and accurate even for low volatility and/or short maturity cases. Key words: Asian options, Options on a traded account, Brownian motion, xed strike, oating strike. 1 Introduction Asian options are securities with payo which depends on the average of the underlying stock price over certain time interval. Since no general analytical solution for the price of the Asian option is known, a variety of techniques have been developed to analyze arithmetic average Asian options. A number of approximations that produce closed form expressions have appeared, se...
Variance reduction methods for simulation of densities on Wiener space
 SIAM J. Numer. Anal
, 2002
"... density estimation. ..."
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SPLITTING FOR RAREEVENT SIMULATION
, 2006
"... Splitting and importance sampling are the two primary techniques to make important rare events happen more frequently in a simulation, and obtain an unbiased estimator with much smaller variance than the standard Monte Carlo estimator. Importance sampling has been discussed and studied in several ar ..."
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Cited by 17 (1 self)
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Splitting and importance sampling are the two primary techniques to make important rare events happen more frequently in a simulation, and obtain an unbiased estimator with much smaller variance than the standard Monte Carlo estimator. Importance sampling has been discussed and studied in several articles presented at the Winter Simulation Conference in the past. A smaller number of WSC articles have examined splitting. In this paper, we review the splitting technique and discuss some of its strengths and limitations from the practical viewpoint. We also introduce improvements in the implementation of the multilevel splitting technique. This is done in a setting where we want to estimate the probability of reaching B before reaching (or returning to) A when starting from a fixed state x0 ∈ B, where A and B are two disjoint subsets of the state space and B is very rarely attained. This problem has several practical applications.
Partially exact and bounded approximations for arithmetic Asian options
"... This paper considers the pricing of European Asian options in the BlackScholes framework. All approaches we consider are readily extendable to the case of an Asian basket option. We consider three methods for evaluating the price of an Asian option, and contribute to all three. Firstly, we show the ..."
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Cited by 16 (2 self)
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This paper considers the pricing of European Asian options in the BlackScholes framework. All approaches we consider are readily extendable to the case of an Asian basket option. We consider three methods for evaluating the price of an Asian option, and contribute to all three. Firstly, we show the link between the approaches of Rogers and Shi [1995], Andreasen [1999], Hoogland and Neumann [2000] and Večeř [2001]. For the latter formulation we propose two reductions, which increase the numerical stability and reduce the calculation time. Secondly, we show how a closedform expression can be derived for Curran’s and Rogers and Shi’s lower bound for the general case of multiple underlyings. Thirdly, we considerably sharpen Thompson’s [1999a,b] upper bound such that it is tighter than all known upper bounds. Finally, we consider analytical approximations and combine the traditional moment matching approximations with Curran’s conditioning approach. The resulting class of partially exact and bounded approximations can be proven to lie between a sharp lower and upper bound. In numerical examples we demonstrate that they outperform all current stateoftheart bounds and approximations.
Importance Sampling in the HeathJarrowMorton Framework
, 1999
"... This paper develops a variance reduction technique for pricing derivatives in highdimensional multifactor models, with particular emphasis on term structure models formulated in the HeathJarrow Morton framework. A premise of this work is that the largest gains in simulation e ciency come from ..."
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Cited by 14 (6 self)
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This paper develops a variance reduction technique for pricing derivatives in highdimensional multifactor models, with particular emphasis on term structure models formulated in the HeathJarrow Morton framework. A premise of this work is that the largest gains in simulation e ciency come from taking advantage of the structure of both the cashows of a security and the model in which it is priced; for this to be feasible in practice requires that the identication and use of relevant structure be automated. We exploit model and payo structure through a combination of importance sampling and stratied sampling. The importance sampling applies a change of drift to the underlying factors; we select the drift by rst solving an optimization problem. We then identify a particularly eective direction for stratied sampling (which may be thought of as an approximate numerical integration) by solving an eigenvector problem. Examples illustrate that the combination of the methods can pro...
Unconstrained Recursive Importance Sampling
, 807
"... We propose an unconstrained stochastic approximation method of finding the optimal measure change (in an a priori parametric family) for Monte Carlo simulations. We consider different parametric families based on the Girsanov theorem and the Esscher transform (or exponentialtilting). In a multidimen ..."
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Cited by 14 (2 self)
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We propose an unconstrained stochastic approximation method of finding the optimal measure change (in an a priori parametric family) for Monte Carlo simulations. We consider different parametric families based on the Girsanov theorem and the Esscher transform (or exponentialtilting). In a multidimensional Gaussian framework, Arouna uses a projected RobbinsMonro procedure to select the parameter minimizing the variance (see [2]). In our approach, the parameter (scalar or process) is selected by a classical RobbinsMonro procedure without projection or truncation. To obtain this unconstrained algorithm we intensively use the regularity of the density of the law without assume smoothness of the payoff. We prove the convergence for a large class of multidimensional distributions and diffusion processes. We illustrate the effectiveness of our algorithm via pricing a Basket payoff under a multidimensional NIG distribution, and pricing a barrier options in different markets.