Monte Carlo Simulation

Stochastic Calculus for Option Pricing with Convex Duality, Logistic Model, and Numerical Examination ArXiv ID: 2408.05672 “View on arXiv” Authors: Unknown Abstract This thesis explores the historical progression and theoretical constructs of financial mathematics, with an in-depth exploration of Stochastic Calculus as showcased in the Binomial Asset Pricing Model and the Continuous-Time Models. A comprehensive survey of stochastic calculus principles applied to option pricing is offered, highlighting insights from Peter Carr and Lorenzo Torricelli’s ``Convex Duality in Continuous Option Pricing Models". This manuscript adopts techniques such as Monte-Carlo Simulation and machine learning algorithms to examine the propositions of Carr and Torricelli, drawing comparisons between the Logistic and Bachelier models. Additionally, it suggests directions for potential future research on option pricing methods. ...

CVA Sensitivities, Hedging and Risk ArXiv ID: 2407.18583 “View on arXiv” Authors: Unknown Abstract We present a unified framework for computing CVA sensitivities, hedging the CVA, and assessing CVA risk, using probabilistic machine learning meant as refined regression tools on simulated data, validatable by low-cost companion Monte Carlo procedures. Various notions of sensitivities are introduced and benchmarked numerically. We identify the sensitivities representing the best practical trade-offs in downstream tasks including CVA hedging and risk assessment. ...

On Deep Learning for computing the Dynamic Initial Margin and Margin Value Adjustment ArXiv ID: 2407.16435 “View on arXiv” Authors: Unknown Abstract The present work addresses the challenge of training neural networks for Dynamic Initial Margin (DIM) computation in counterparty credit risk, a task traditionally burdened by the high costs associated with generating training datasets through nested Monte Carlo (MC) simulations. By condensing the initial market state variables into an input vector, determined through an interest rate model and a parsimonious parameterization of the current interest rate term structure, we construct a training dataset where labels are noisy but unbiased DIM samples derived from single MC paths. A multi-output neural network structure is employed to handle DIM as a time-dependent function, facilitating training across a mesh of monitoring times. The methodology offers significant advantages: it reduces the dataset generation cost to a single MC execution and parameterizes the neural network by initial market state variables, obviating the need for repeated training. Experimental results demonstrate the approach’s convergence properties and robustness across different interest rate models (Vasicek and Hull-White) and portfolio complexities, validating its general applicability and efficiency in more realistic scenarios. ...

The $κ$-generalised Distribution for Stock Returns ArXiv ID: 2405.09929 “View on arXiv” Authors: Unknown Abstract Empirical evidence shows stock returns are often heavy-tailed rather than normally distributed. The $κ$-generalised distribution, originated in the context of statistical physics by Kaniadakis, is characterised by the $κ$-exponential function that is asymptotically exponential for small values and asymptotically power law for large values. This proves to be a useful property and makes it a good candidate distribution for many types of quantities. In this paper we focus on fitting historic daily stock returns for the FTSE 100 and the top 100 Nasdaq stocks. Using a Monte-Carlo goodness of fit test there is evidence that the $κ$-generalised distribution is a good fit for a significant proportion of the 200 stock returns analysed. ...

Cost-Benefit Analysis using Modular Dynamic Fault Tree Analysis and Monte Carlo Simulations for Condition-based Maintenance of Unmanned Systems ArXiv ID: 2405.09519 “View on arXiv” Authors: Unknown Abstract Recent developments in condition-based maintenance (CBM) have helped make it a promising approach to maintenance cost avoidance in engineering systems. By performing maintenance based on conditions of the component with regards to failure or time, there is potential to avoid the large costs of system shutdown and maintenance delays. However, CBM requires a large investment cost compared to other available maintenance strategies. The investment cost is required for research, development, and implementation. Despite the potential to avoid significant maintenance costs, the large investment cost of CBM makes decision makers hesitant to implement. This study is the first in the literature that attempts to address the problem of conducting a cost-benefit analysis (CBA) for implementing CBM concepts for unmanned systems. This paper proposes a method for conducting a CBA to determine the return on investment (ROI) of potential CBM strategies. The CBA seeks to compare different CBM strategies based on the differences in the various maintenance requirements associated with maintaining a multi-component, unmanned system. The proposed method uses modular dynamic fault tree analysis (MDFTA) with Monte Carlo simulations (MCS) to assess the various maintenance requirements. The proposed method is demonstrated on an unmanned surface vessel (USV) example taken from the literature that consists of 5 subsystems and 71 components. Following this USV example, it is found that selecting different combinations of components for a CBM strategy can have a significant impact on maintenance requirements and ROI by impacting cost avoidances and investment costs. ...

Deep Joint Learning valuation of Bermudan Swaptions ArXiv ID: 2404.11257 “View on arXiv” Authors: Unknown Abstract This paper addresses the problem of pricing involved financial derivatives by means of advanced of deep learning techniques. More precisely, we smartly combine several sophisticated neural network-based concepts like differential machine learning, Monte Carlo simulation-like training samples and joint learning to come up with an efficient numerical solution. The application of the latter development represents a novelty in the context of computational finance. We also propose a novel design of interdependent neural networks to price early-exercise products, in this case, Bermudan swaptions. The improvements in efficiency and accuracy provided by the here proposed approach is widely illustrated throughout a range of numerical experiments. Moreover, this novel methodology can be extended to the pricing of other financial derivatives. ...

Prediction of Cryptocurrency Prices through a Path Dependent Monte Carlo Simulation ArXiv ID: 2405.12988 “View on arXiv” Authors: Unknown Abstract In this paper, our focus lies on the Merton’s jump diffusion model, employing jump processes characterized by the compound Poisson process. Our primary objective is to forecast the drift and volatility of the model using a variety of methodologies. We adopt an approach that involves implementing different drift, volatility, and jump terms within the model through various machine learning techniques, traditional methods, and statistical methods on price-volume data. Additionally, we introduce a path-dependent Monte Carlo simulation to model cryptocurrency prices, taking into account the volatility and unexpected jumps in prices. ...

Optimizing Neural Networks for Bermudan Option Pricing: Convergence Acceleration, Future Exposure Evaluation and Interpolation in Counterparty Credit Risk ArXiv ID: 2402.15936 “View on arXiv” Authors: Unknown Abstract This paper presents a Monte-Carlo-based artificial neural network framework for pricing Bermudan options, offering several notable advantages. These advantages encompass the efficient static hedging of the target Bermudan option and the effective generation of exposure profiles for risk management. We also introduce a novel optimisation algorithm designed to expedite the convergence of the neural network framework proposed by Lokeshwar et al. (2022) supported by a comprehensive error convergence analysis. We conduct an extensive comparative analysis of the Present Value (PV) distribution under Markovian and no-arbitrage assumptions. We compare the proposed neural network model in conjunction with the approach initially introduced by Longstaff and Schwartz (2001) and benchmark it against the COS model, the pricing model pioneered by Fang and Oosterlee (2009), across all Bermudan exercise time points. Additionally, we evaluate exposure profiles, including Expected Exposure and Potential Future Exposure, generated by our proposed model and the Longstaff-Schwartz model, comparing them against the COS model. We also derive exposure profiles at finer non-standard grid points or risk horizons using the proposed approach, juxtaposed with the Longstaff Schwartz method with linear interpolation and benchmark against the COS method. In addition, we explore the effectiveness of various interpolation schemes within the context of the Longstaff-Schwartz method for generating exposures at finer grid horizons. ...

Option pricing for Barndorff-Nielsen and Shephard model by supervised deep learning ArXiv ID: 2402.00445 “View on arXiv” Authors: Unknown Abstract This paper aims to develop a supervised deep-learning scheme to compute call option prices for the Barndorff-Nielsen and Shephard model with a non-martingale asset price process having infinite active jumps. In our deep learning scheme, teaching data is generated through the Monte Carlo method developed by Arai and Imai (2024). Moreover, the BNS model includes many variables, which makes the deep learning accuracy worse. Therefore, we will create another input variable using the Black-Scholes formula. As a result, the accuracy is improved dramatically. ...

Fast and General Simulation of Lévy-driven OU processes for Energy Derivatives ArXiv ID: 2401.15483 “View on arXiv” Authors: Unknown Abstract Lévy-driven Ornstein-Uhlenbeck (OU) processes represent an intriguing class of stochastic processes that have garnered interest in the energy sector for their ability to capture typical features of market dynamics. However, in the current state of play, Monte Carlo simulations of these processes are not straightforward for two main reasons: i) algorithms are available only for some specific processes within this class; ii) they are often computationally expensive. In this paper, we introduce a new simulation technique designed to address both challenges. It relies on the numerical inversion of the characteristic function, offering a general methodology applicable to all Lévy-driven OU processes. Moreover, leveraging FFT, the proposed methodology ensures fast and accurate simulations, providing a solid basis for the widespread adoption of these processes in the energy sector. Lastly, the algorithm allows an optimal control of the numerical error. We apply the technique to the pricing of energy derivatives, comparing the results with the existing benchmarks. Our findings indicate that the proposed methodology is at least one order of magnitude faster than the existing algorithms, while maintaining an equivalent level of accuracy. ...