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We explore the impact of stochastic mortality, health, and parasites on animal-based commodities risk management, with a specific emphasis on salmon aquaculture. Our investigation delves into the stochastic nature of mortality, fish lice infestation, and treatment plans based on comprehensive historical data from Norway. Given that salmon lice pose a significant challenge to salmon aquaculture, with associated treatment costs estimated to be comparable to feeding expenses, lice removal is imperative to ensure the survival of the salmon and comply with the Norwegian government's stipulation of maintaining an upper threshold of 0.5 lice per fish. We propose a new model that considers the relationship between hosts and parasites and determines the number of treatments required as well as the overall cost of these treatments. An important aspect of our model is its incorporation of stochastic effectiveness for each removal. After calibrating the model to our dataset, our study examines the impact of the host-parasite relationship and the required interventions on the optimal harvesting decision and draws a comparison to models that make an assumption of deterministic mortality. Our results indicate that a gain of 1.5 % in farm value (per rotation) can be obtained by employing an optimal harvesting rule based on the stochastic host-parasite model.

We study the effect of stochastic feeding costs on animal-based commodities with particular focus on aquaculture. More specifically, we use soybean futures to infer on the stochastic behavior of salmon feed, which we assume to follow a Schwartz 2-factor model. We compare the decision of harvesting salmon using a decision rule assuming either deterministic or stochastic feeding costs. We identify cases, where accounting for stochastic feeding costs leads to significant improvements as well as cases where deterministic feeding costs are a good enough proxy. Nevertheless, in all of the cases, the newly derived rules show superior performance, while the additional computational costs are negligible. In conclusion, we recommend to use decision rules taking stochastic feeding costs into account.

In this paper, we show how the Itô-stochastic Magnus expansion can be used to efficiently solve stochastic partial differential equations (SPDE) with two space variables numerically. To this end, we will first discretize the SPDE in space only by utilizing finite difference methods and vectorize the resulting equation exploiting its sparsity.

As a benchmark, we will apply it to the case of the stochastic Langevin equation with constant coefficients, where an explicit solution is available, and compare the Magnus scheme with the Euler–Maruyama scheme. We will see that the Magnus expansion is superior in terms of both accuracy and especially computational time by using a single GPU and verify it in a variable coefficient case. Notably, we will see speed-ups of order ranging form 20 to 200 compared to the Euler–Maruyama scheme, depending on the accuracy target and the spatial resolution.

In this paper, we propose a new exogenous model to address the problem of negative interest rates that preserves the analytical tractability of the original Cox–Ingersoll–Ross (CIR) model with a perfect fit to the observed term-structure. We use the difference between two independent CIR processes and apply the deterministic-shift extension technique. To allow for a fast calibration to the market swaption surface, we apply the Gram–Charlier expansion to calculate the swaption prices in our model. We run several numerical tests to demonstrate the strengths of this model by using Monte-Carlo techniques. In particular, the model produces close Bermudan swaption prices compared to Bloomberg’s Hull–White one-factor model. Moreover, it finds constant maturity swap (CMS) rates very close to Bloomberg’s CMS rates.

In this paper, we propose a new model to address the problem of negative interest rates that preserves the analytical tractability of the original Cox–Ingersoll–Ross (CIR) model without introducing a shift to the market interest rates, because it is defined as the difference of two independent CIR processes. The strength of our model lies within the fact that it is very simple and can be calibrated to the market zero yield curve using an analytical formula. We run several numerical experiments at two different dates, once with a partially sub-zero interest rate and once with a fully negative interest rate. In both cases, we obtain good results in the sense that the model reproduces the market term structures very well. We then simulate the model using the Euler–Maruyama scheme and examine the mean, variance and distribution of the model. The latter agrees with the skewness and fat tail seen in the original CIR model. In addition, we compare the model’s zero coupon prices with market prices at different future points in time. Finally, we test the market consistency of the model by evaluating swaptions with different tenors and maturities.

We derive the stochastic version of the Magnus expansion for linear systems of stochastic differential equations (SDEs). The main novelty with respect to the related literature is that we consider SDEs in the Itô sense, with progressively measurable coefficients, for which an explicit Itô-Stratonovich conversion is not available. We prove convergence of the Magnus expansion up to a stopping time τ and provide a novel asymptotic estimate of the cumulative distribution function of τ. As an application, we propose a new method for the numerical solution of stochastic partial differential equations (SPDEs) based on spatial discretization and application of the stochastic Magnus expansion. A notable feature of the method is that it is fully parallelizable. We also present numerical tests in order to asses the accuracy of the numerical schemes.

In this paper, we introduce a novel methodology to model rating transitions with a stochastic process. To introduce stochastic processes, whose values are valid rating matrices, we noticed the geometric properties of stochastic matrices and its link to matrix Lie groups. We give a gentle introduction to this topic and demonstrate how Itô-SDEs in R will generate the desired model for rating transitions. To calibrate the rating model to historical data, we use a Deep-Neural-Network (DNN) called TimeGAN to learn the features of a time series of historical rating matrices. Then, we use this DNN to generate synthetic rating transition matrices. Afterwards, we fit the moments of the generated rating matrices and the rating process at specific time points, which results in a good fit. After calibration, we discuss the quality of the calibrated rating transition process by examining some properties that a time series of rating matrices should satisfy, and we will see that this geometric approach works very well.

In this paper, we model the rating process of an entity as a piecewise homogeneous continuous time Markov chain. We focus specifically on calibrating the model to both historical data (rating transition matrices) and market data (CDS quotes), relying on a simple change of measure to switch from the historical probability to the risk-neutral one. We overcome some of the imperfections of the data by proposing a novel calibration procedure, which leads to an improvement of the entire scheme. We apply our model to compute bilateral credit and debit valuation adjustments of a netting set under a CSA with thresholds depending on ratings of the two parties.

In this paper, we improve the performance of the large basket approximation developed by Reisinger et al. to calibrate Collateralized Debt Obligations (CDO) to iTraxx market data. The iTraxx tranches and index are computed using a basket of size K=125. In the context of the large basket approximation, it is assumed that this is sufficiently large to approximate it by a limit SPDE describing the portfolio loss of a basket with size K→∞. For the resulting SPDE, we show four different numerical methods and demonstrate how the Magnus expansion can be applied to efficiently solve the large basket SPDE with high accuracy. Moreover, we will calibrate a structural model to the available market data. For this, it is important to efficiently infer the so-called initial distances to default from the Credit Default Swap (CDS) quotes of the constituents of the iTraxx for the large basket approximation. We will show how Deep Learning techniques can help us to improve the performance of this step significantly. We will see in the end a good fit to the market data and develop a highly parallelizable numerical scheme using GPU and multithreading techniques.

In this paper, we model the rating process of an entity by using a geometrical approach. We model rating transitions as an SDE on a Lie group. Specifically, we focus on calibrating the model to both historical data (rating transition matrices) and market data (CDS quotes) and compare the most popular choices of changes of measure to switch from the historical probability to the risk-neutral one. For this, we show how the classical Girsanov theorem can be applied in the Lie group setting. Moreover, we overcome some of the imperfections of rating matrices published by rating agencies, which are computed with the cohort method, by using a novel Deep Learning approach. This leads to an improvement of the entire scheme and makes the model more robust for applications. We apply our model to compute bilateral credit and debit valuation adjustments of a netting set under a CSA with thresholds depending on ratings of the two parties.