Incremental computation of block triangular matrix exponentials with application to option pricing

Kreßner, Daniel; Luce, Robert; Statti, Francesco

doi:10.1553/etna_vol47s57

Cited by 5 publications

(6 citation statements)

References 13 publications

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“…Once the blocks of U θ are known, the exponential θ (x) is efficiently evaluated from (3.4) using the block triangular structure of U θ (see e.g. Kressner et al [14]).…”

Section: Ruin Time and Probability Of Ruinmentioning

confidence: 99%

Ruin problems for epidemic insurance

Lefèvre

Simon

2021

Adv. Appl. Probab.

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The paper discusses the risk of ruin in insurance coverage of an epidemic in a closed population. The model studied is an extended susceptible–infective–removed (SIR) epidemic model built by Lefèvre and Simon (Methodology Comput. Appl. Prob.22, 2020) as a block-structured Markov process. A fluid component is then introduced to describe the premium amounts received and the care costs reimbursed by the insurance. Our interest is in the risk of collapse of the corresponding reserves of the company. The use of matrix-analytic methods allows us to determine the distribution of ruin time, the probability of ruin, and the final amount of reserves. The case where the reserves are subjected to a Brownian noise is also studied. Finally, some of the results obtained are illustrated for two particular standard SIR epidemic models.

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“…Once the blocks of U θ are known, the exponential θ (x) is efficiently evaluated from (3.4) using the block triangular structure of U θ (see e.g. Kressner et al [14]).…”

Section: Ruin Time and Probability Of Ruinmentioning

confidence: 99%

Ruin problems for epidemic insurance

Lefèvre

Simon

2021

Adv. Appl. Probab.

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“…Once the blocks of U θ are known, the blocks of its exponential Φ θ pxq can be computed efficiently using the special structure of U θ by following for instance Kressner et al [14] where the authors propose an efficient incremental procedure to compute the (blocktriangular) square matrix constituted by the first n block lines and columns of e U θ x , from the (previously obtained) square matrix constituted by the first n ´1 block lines and columns of e U θ x . Particular case where A k " A and D k " D for all k ě 1.…”

Section: 11)mentioning

confidence: 99%

Ruin problems for risk processes with dependent phase-type claims

Peralta,

Simon

2020

Preprint

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We consider continuous time risk processes in which the claim sizes are dependent and non-identically distributed phase-type distributions. The class of distributions we propose is easy to characterize and allows to incorporate the dependence between claims in a simple and intuitive way. It is also designed to facilitate the study of the risk processes by using a Markov-modulated fluid embedding technique. Using this technique, we obtain simple recursive procedures to determine the joint distribution of the time of ruin, the deficit at ruin and the number of claims before the ruin. We also obtain some bounds for the ultimate ruin probability. Finally, we provide a few examples of multivariate phase-type distributions and use them for numerical illustration.

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“…As long as the square roots in (18) are positive the Euler-Maruyama scheme is well-defined. In our numerical experiments this was the case.…”

Section: Call Option In Stochastic Volatility Modelsmentioning

confidence: 99%

“…Finally, G n is the matrix representation of the action of the generator of (V t , X t ) restricted to the space Pol n (R 2 ). Note that the matrix G n can be constructed as explained in [18], with respect to the monomial basis.…”

Section: Call Option In Stochastic Volatility Modelsmentioning

confidence: 99%

Weighted Monte Carlo with Least Squares and Randomized Extended Kaczmarz for Option Pricing

et al. 2019

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We propose a methodology for computing single and multi-asset European option prices, and more generally expectations of scalar functions of (multivariate) random variables. This new approach combines the ability of Monte Carlo simulation to handle high-dimensional problems with the efficiency of function approximation. Specifically, we first generalize the recently developed method for multivariate integration in [arXiv:1806.05492] to integration with respect to probability measures. The method is based on the principle "approximate and integrate" in three steps i) sample the integrand at points in the integration domain, ii) approximate the integrand by solving a least-squares problem, iii) integrate the approximate function. In high-dimensional applications we face memory limitations due to large storage requirements in step ii). Combining weighted sampling and the randomized extended Kaczmarz algorithm we obtain a new efficient approach to solve large-scale least-squares problems. Our convergence and cost analysis along with numerical experiments show the effectiveness of the method in both low and high dimensions, and under the assumption of a limited number of available simulations.

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Incremental computation of block triangular matrix exponentials with application to option pricing

Cited by 5 publications

References 13 publications

Ruin problems for epidemic insurance

Ruin problems for epidemic insurance

Ruin problems for risk processes with dependent phase-type claims

Weighted Monte Carlo with Least Squares and Randomized Extended Kaczmarz for Option Pricing

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