Editorial

2 pages, 266 KiB

Open AccessEditorial

Special Issue “Computational Finance and Risk Analysis in Insurance”

by Ralf Korn

Risks 2022, 10(3), 50; https://doi.org/10.3390/risks10030050 - 28 Feb 2022

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This Special Issue focuses on the rapid development of computational finance as well as on classical risk analysis issues in insurance that also benefit from modern computational methods [...] Full article

(This article belongs to the Special Issue Computational Finance and Risk Analysis in Insurance)

Research

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38 pages, 1629 KiB

Open AccessArticle

The Weak Convergence Rate of Two Semi-Exact Discretization Schemes for the Heston Model

by Annalena Mickel and Andreas Neuenkirch

Risks 2021, 9(1), 23; https://doi.org/10.3390/risks9010023 - 12 Jan 2021

Cited by 5 | Viewed by 2731

Abstract

Inspired by the article Weak Convergence Rate of a Time-Discrete Scheme for the Heston Stochastic Volatility Model, Chao Zheng, SIAM Journal on Numerical Analysis 2017, 55:3, 1243–1263, we studied the weak error of discretization schemes for the Heston model, which are based [...] Read more.

Inspired by the article Weak Convergence Rate of a Time-Discrete Scheme for the Heston Stochastic Volatility Model, Chao Zheng, SIAM Journal on Numerical Analysis 2017, 55:3, 1243–1263, we studied the weak error of discretization schemes for the Heston model, which are based on exact simulation of the underlying volatility process. Both for an Euler- and a trapezoidal-type scheme for the log-asset price, we established weak order one for smooth payoffs without any assumptions on the Feller index of the volatility process. In our analysis, we also observed the usual trade off between the smoothness assumption on the payoff and the restriction on the Feller index. Moreover, we provided error expansions, which could be used to construct second order schemes via extrapolation. In this paper, we illustrate our theoretical findings by several numerical examples. Full article

(This article belongs to the Special Issue Computational Finance and Risk Analysis in Insurance)

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18 pages, 585 KiB

Open AccessEditor’s ChoiceArticle

A Deep Neural Network Algorithm for Semilinear Elliptic PDEs with Applications in Insurance Mathematics

by Stefan Kremsner, Alexander Steinicke and Michaela Szölgyenyi

Risks 2020, 8(4), 136; https://doi.org/10.3390/risks8040136 - 09 Dec 2020

Cited by 14 | Viewed by 3197

Abstract

In insurance mathematics, optimal control problems over an infinite time horizon arise when computing risk measures. An example of such a risk measure is the expected discounted future dividend payments. In models which take multiple economic factors into account, this problem is high-dimensional. [...] Read more.

In insurance mathematics, optimal control problems over an infinite time horizon arise when computing risk measures. An example of such a risk measure is the expected discounted future dividend payments. In models which take multiple economic factors into account, this problem is high-dimensional. The solutions to such control problems correspond to solutions of deterministic semilinear (degenerate) elliptic partial differential equations. In the present paper we propose a novel deep neural network algorithm for solving such partial differential equations in high dimensions in order to be able to compute the proposed risk measure in a complex high-dimensional economic environment. The method is based on the correspondence of elliptic partial differential equations to backward stochastic differential equations with unbounded random terminal time. In particular, backward stochastic differential equations—which can be identified with solutions of elliptic partial differential equations—are approximated by means of deep neural networks. Full article

(This article belongs to the Special Issue Computational Finance and Risk Analysis in Insurance)

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21 pages, 729 KiB

Open AccessArticle

Least-Squares Monte Carlo for Proxy Modeling in Life Insurance: Neural Networks

by Anne-Sophie Krah, Zoran Nikolić and Ralf Korn

Risks 2020, 8(4), 116; https://doi.org/10.3390/risks8040116 - 04 Nov 2020

Cited by 6 | Viewed by 2993

Abstract

The least-squares Monte Carlo method has proved to be a suitable approximation technique for the calculation of a life insurer’s solvency capital requirements. We suggest to enhance it by the use of a neural network based approach to construct the proxy function that [...] Read more.

The least-squares Monte Carlo method has proved to be a suitable approximation technique for the calculation of a life insurer’s solvency capital requirements. We suggest to enhance it by the use of a neural network based approach to construct the proxy function that models the insurer’s loss with respect to the risk factors the insurance business is exposed to. After giving a mathematical introduction to feed forward neural networks and describing the involved hyperparameters, we apply this popular form of neural networks to a slightly disguised data set from a German life insurer. Thereby, we demonstrate all practical aspects, such as the hyperparameter choice, to obtain our candidate neural networks by bruteforce, the calibration (“training”) and validation (“testing”) of the neural networks and judging their approximation performance. Compared to adaptive OLS, GLM, GAM and FGLS regression approaches, an ensemble built of the 10 best derived neural networks shows an excellent performance. Through a comparison with the results obtained by every single neural network, we point out the significance of the ensemble-based approach. Lastly, we comment on the interpretability of neural networks compared to polynomials for sensitivity analyses. Full article

(This article belongs to the Special Issue Computational Finance and Risk Analysis in Insurance)

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22 pages, 5313 KiB

Open AccessArticle

Good-Deal Bounds for Option Prices under Value-at-Risk and Expected Shortfall Constraints

by Sascha Desmettre, Christian Laudagé and Jörn Sass

Risks 2020, 8(4), 114; https://doi.org/10.3390/risks8040114 - 30 Oct 2020

Cited by 2 | Viewed by 2166

Abstract

In this paper, we deal with the pricing of European options in an incomplete market. We use the common risk measures Value-at-Risk and Expected Shortfall to define good-deals on a financial market with log-normally distributed rate of returns. We show that the pricing [...] Read more.

In this paper, we deal with the pricing of European options in an incomplete market. We use the common risk measures Value-at-Risk and Expected Shortfall to define good-deals on a financial market with log-normally distributed rate of returns. We show that the pricing bounds obtained from the Value-at-Risk admit a non-smooth behavior under parameter changes. Additionally, we find situations in which the seller’s bound for a call option is smaller than the buyer’s bound. We identify the missing convexity of the Value-at-Risk as main reason for this behavior. Due to the strong connection between good-deal bounds and the theory of risk measures, we further obtain new insights in the finiteness and the continuity of risk measures based on multiple eligible assets in our setting. Full article

(This article belongs to the Special Issue Computational Finance and Risk Analysis in Insurance)

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27 pages, 492 KiB

Open AccessArticle

Optimal Dividend Payment in De Finetti Models: Survey and New Results and Strategies

by Christian Hipp

Risks 2020, 8(3), 96; https://doi.org/10.3390/risks8030096 - 10 Sep 2020

Cited by 3 | Viewed by 2059

Abstract

We consider optimal dividend payment under the constraint that the with-dividend ruin probability does not exceed a given value

α .

This is done in most simple discrete De Finetti models. We characterize the value function

V (s, α)

for [...] Read more.

We consider optimal dividend payment under the constraint that the with-dividend ruin probability does not exceed a given value

α .

This is done in most simple discrete De Finetti models. We characterize the value function

V (s, α)

for initial surplus s of this problem, characterize the corresponding optimal dividend strategies, and present an algorithm for its computation. In an earlier solution to this problem, a Hamilton-Jacobi-Bellman equation for

V (s, α)

can be found which leads to its representation as the limit of a monotone iteration scheme. However, this scheme is too complex for numerical computations. Here, we introduce the class of two-barrier dividend strategies with the following property: when dividends are paid above a barrier

B,

i.e., a dividend of size 1 is paid when reaching

B + 1

from

B,

then we repeat this dividend payment until reaching a limit L for some

0 \leq L \leq B .

For these strategies we obtain explicit formulas for ruin probabilities and present values of dividend payments, as well as simplifications of the above iteration scheme. The results of numerical experiments show that the values

V (s, α)

obtained in earlier work can be improved, they are suboptimal. Full article

(This article belongs to the Special Issue Computational Finance and Risk Analysis in Insurance)

26 pages, 1640 KiB

Open AccessEditor’s ChoiceArticle

Nagging Predictors

by Ronald Richman and Mario V. Wüthrich

Risks 2020, 8(3), 83; https://doi.org/10.3390/risks8030083 - 04 Aug 2020

Cited by 33 | Viewed by 4560

Abstract

We define the nagging predictor, which, instead of using bootstrapping to produce a series of i.i.d. predictors, exploits the randomness of neural network calibrations to provide a more stable and accurate predictor than is available from a single neural network run. Convergence results [...] Read more.

We define the nagging predictor, which, instead of using bootstrapping to produce a series of i.i.d. predictors, exploits the randomness of neural network calibrations to provide a more stable and accurate predictor than is available from a single neural network run. Convergence results for the family of Tweedie’s compound Poisson models, which are usually used for general insurance pricing, are provided. In the context of a French motor third-party liability insurance example, the nagging predictor achieves stability at portfolio level after about 20 runs. At an insurance policy level, we show that for some policies up to 400 neural network runs are required to achieve stability. Since working with 400 neural networks is impractical, we calibrate two meta models to the nagging predictor, one unweighted, and one using the coefficient of variation of the nagging predictor as a weight, finding that these latter meta networks can approximate the nagging predictor well, only with a small loss of accuracy. Full article

(This article belongs to the Special Issue Computational Finance and Risk Analysis in Insurance)

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26 pages, 557 KiB

Open AccessArticle

Joshi’s Split Tree for Option Pricing

by Guillaume Leduc and Merima Nurkanovic Hot

Risks 2020, 8(3), 81; https://doi.org/10.3390/risks8030081 - 01 Aug 2020

Cited by 3 | Viewed by 3007

Abstract

In a thorough study of binomial trees, Joshi introduced the split tree as a two-phase binomial tree designed to minimize oscillations, and demonstrated empirically its outstanding performance when applied to pricing American put options. Here we introduce a “flexible” version of Joshi’s tree, [...] Read more.

In a thorough study of binomial trees, Joshi introduced the split tree as a two-phase binomial tree designed to minimize oscillations, and demonstrated empirically its outstanding performance when applied to pricing American put options. Here we introduce a “flexible” version of Joshi’s tree, and develop the corresponding convergence theory in the European case: we find a closed form formula for the coefficients of

1 / n

and

1 / n^{3 / 2}

in the expansion of the error. Then we define several optimized versions of the tree, and find closed form formulae for the parameters of these optimal variants. In a numerical study, we found that in the American case, an optimized variant of the tree significantly improved the performance of Joshi’s original split tree. Full article

(This article belongs to the Special Issue Computational Finance and Risk Analysis in Insurance)

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30 pages, 830 KiB

Open AccessFeature PaperArticle

Numerical Algorithms for Reflected Anticipated Backward Stochastic Differential Equations with Two Obstacles and Default Risk

by Jingnan Wang and Ralf Korn

Risks 2020, 8(3), 72; https://doi.org/10.3390/risks8030072 - 01 Jul 2020

Cited by 2 | Viewed by 3016

Abstract

We study numerical algorithms for reflected anticipated backward stochastic differential equations (RABSDEs) driven by a Brownian motion and a mutually independent martingale in a defaultable setting. The generator of a RABSDE includes the present and future values of the solution. We introduce two [...] Read more.

We study numerical algorithms for reflected anticipated backward stochastic differential equations (RABSDEs) driven by a Brownian motion and a mutually independent martingale in a defaultable setting. The generator of a RABSDE includes the present and future values of the solution. We introduce two main algorithms, a discrete penalization scheme and a discrete reflected scheme basing on a random walk approximation of the Brownian motion as well as a discrete approximation of the default martingale, and we study these two methods in both the implicit and explicit versions respectively. We give the convergence results of the algorithms, provide a numerical example and an application in American game options in order to illustrate the performance of the algorithms. Full article

(This article belongs to the Special Issue Computational Finance and Risk Analysis in Insurance)

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