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Table of Links Abstract and 1. Introduction 1.1 Option Pricing 1.2 Asymptotic Notation (Big O) 1.3 Finite Difference 1.4 The Black-Schole Model 1.5 Monte Carlo Simulation and Variance Reduction Techniques 1.6 Our Contribution Literature Review
Methodology 3.1 Model Assumption 3.2 Theorems and Model Discussion Result Analysis
Conclusion and References 5. CONCLUSION The research findings have demonstrated that finite difference methods offer a powerful tool for approximating option prices and hedging parameters, especially in cases where closed-form solutions are unavailable or impractical. The asymptotic analysis conducted in this study has provided valuable insights into the convergence properties and accuracy of finite difference approximations, shedding light on the optimal choice of discretization schemes and grid sizes for different option contracts. Furthermore, the combination of finite difference methods and Monte Carlo simulation with variance reduction techniques offers a comprehensive approach to hedge error analysis in option pricing. By leveraging the strengths of both methodologies, financial practitioners can obtain more robust and reliable estimates of option prices and hedge parameters, thereby enhancing their risk management capabilities and decision-making processes. It is important to acknowledge the limitations of this study and areas for future research. While the asymptotic analysis provides valuable theoretical insights, further empirical validation is warranted to assess the robustness of the findings across different market conditions and asset classes. Additionally, exploring alternative variance reduction techniques and incorporating more sophisticated models of market dynamics could yield further improvements in hedge error analysis and option pricing accuracy. this study has advanced our understanding of the factors influencing hedge errors and provided practical tools for mitigating their impact on derivative pricing and risk management. As financial markets continue to evolve, the insights gained from this research will remain valuable for academics, practitioners, and policymakers striving to enhance the efficiency and stability of global financial systems. REFERENCES [1] C. W. Smith, “Option pricing. A review,” J. financ. econ., vol. 3, no. 1–2, pp. 3–51, 1976, doi: 10.1016/0304-405X(76)90019-2. [2] S. Guernsey, “an Introduction To the Black-Scholes Pde Model,” 2013. [3] B. Peeters, C. L. Dert, and A. Lucas, “Black Scholes for Portfolios of Options in Discrete Time,” SSRN Electron. J., no. 03, 2005, doi: 10.2139/ssrn.469022. [4] J. He, “A Study on Analytical and Numerical Solutions of Three Types of Black-Scholes Models,” Int. J. Trade, Econ. Financ., vol. 13, no. 2, 2022, doi: 10.18178/ijtef.2022.13.2.720. [5] A. S. Shinde and K. C. Takale, “Study of BlackScholes model and its applications,” Procedia Eng., vol. 38, pp. 270–279, 2012, doi: 10.1016/j.proeng.2012.06.035. [6] Y. Zhang, “The value of Monte Carlo model-based variance reduction technology in the pricing of financial derivatives,” PLoS One, vol. 15, no. 2, pp. 1–13, 2020, doi: 10.1371/journal.pone.0229737. [7] P. L. Bonate, “A brief introduction to Monte Carlo simulation,” Clin. Pharmacokinet., vol. 40, no. 1, pp. 15–22, 2001, doi: 10.2165/00003088- 200140010-00002. [8] D. R. Tobergte and S. Curtis, “Importance sampling for high speed statistical Monte-Carlo simulations,” J. Chem. Inf. Model., vol. 53, no. 9, pp. 1689–1699, 2013. [9] E. C. Anderson and M. Stephens, “Importance Sampling and (MC)^3,” Sisg 2020, vol. 2, no. July, pp. 1–29, 2020. [10] I. N. Medvedev, “Vector estimators of the Monte Carlo method with a finite variance,” Russ. J. Numer. Anal. Math. Model., vol. 28, no. 3, pp. 231– 244, 2013, doi: 10.1515/rnam-2013-0014. [11] E. Saliby and R. J. Paul, “A farewell to the use of antithetic variates in monte carlo simulation,” J. Oper. Res. Soc., vol. 60, no. 7, pp. 1026–1035, 2009, doi: 10.1057/palgrave.jors.2602645. [12] P. W. Glynn and R. Szechtman, “Some New Perspectives on the Method of Control Variates,” Monte Carlo Quasi-Monte Carlo Methods 2000, pp. 27–49, 2002, doi: 10.1007/978-3-642-56046-0_3. [13] W. Gustafsson, “Evaluating the Longstaff-Schwartz method for pricing of American options,” 2015. [14] L. Andersen and M. Broadie, “Primal-dual simulation algorithm for pricing multidimensional American options,” Manage. Sci., vol. 50, no. 9, pp. 1222–1234, 2004, doi: 10.1287/mnsc.1040.0258. [15] E. Wipplinger, Jim Gatheral: The volatility surface, a practitioner’s guide, vol. 22, no. 1. 2008. doi: 10.1007/s11408-007-0072-4. [16] M. Avellaneda and S. Stoikov, “High-frequency trading in a limit order book,” Quant. Financ., vol. 8, no. 3, pp. 217–224, 2008, doi: 10.1080/14697680701381228. [17] S. Mayoral, D. Moreno, and A. Zareei, “Using a hedging network to minimize portfolio risk,” Financ. Res. Lett., vol. 44, no. April, p. 102044, 2022, doi: 10.1016/j.frl.2021.102044. [18] S. Gayathri Devi, K. Selvam, and S. P. Rajagopalan, “An abstract to calculate big o factors of time and space complexity of machine code,” IET Conf. Publ., vol. 2011, no. 583 CP, pp. 844–847, 2011, doi: 10.1049/cp.2011.0483. [19] M. S. H. Mojumder, M. N. Haque, and M. J. Alam, “Efficient Finite Difference Methods for the Numerical Analysis of One-Dimensional Heat Equation,” J. Appl. Math. Phys., vol. 11, no. 10, pp. 3099–3123, 2023, doi: 10.4236/jamp.2023.1110204. [20] I. R. Khan and R. Ohba, “Taylor series based finite difference approximations of higher-degree derivatives,” J. Comput. Appl. Math., vol. 154, no. 1, pp. 115–124, 2003, doi: 10.1016/S0377- 0427(02)00816-6. [21] S. Danho, “Pricing Financial Derivatives with the Finite Difference Method,” p. 237, 2017. [22] G. Connor and T. Lasarte, “An Introduction to Hedge Funds Strategies. Introductory Guide,” pp. 1–16, 2004, [Online]. Available: http://eprints.lse.ac.uk/24675/1/dp477.pdf [23] M. Nurul Anwar and L. Sazzad Andallah, “A Study on Numerical Solution of Black-Scholes Model,” J. Math. Financ., vol. 08, no. 02, pp. 372–381, 2018, doi: 10.4236/jmf.2018.82024. [24] J. Akahori, F. Barsotti, and Y. Imamura, “Hedging error as generalized timing risk,” Quant. Financ., vol. 23, no. 4, pp. 693–703, 2023, doi: 10.1080/14697688.2022.2154255. [25] R. C. A. Oomen and G. J. Jiang, “Hedging Derivatives Risks - A Simulation Study,” SSRN Electron. J., no. May, 2005, doi: 10.2139/ssrn.302525. [26] K. A. Froot, “Hedging Portfolios with Real Assets,” J. Portf. Manag., vol. 21, no. 4, pp. 60–77, 1995, doi: 10.3905/jpm.1995.409527. [27] H. E. LELAND, “Option Pricing and Replication with Transactions Costs,” J. Finance, vol. 40, no. 5, pp. 1283–1301, 1985, doi: 10.1111/j.1540- 6261.1985.tb02383.x. [28] T. Hayashi and P. A. Mykland, “Evaluating hedging errors: An asymptotic approach,” Math. Financ., vol. 15, no. 2, pp. 309–343, 2005, doi: 10.1111/j.0960-1627.2005.00221.x. [29] J.-L. Prigent, R. Hentati, and A. Kaffel, “Dynamic versus static optimization of hedge fund portfolios: the relevance of performance measures,” Int. J. Bus., vol. 15, no. 156, pp. 1–17, 2010. [30] R. A. Jarrow, D. Lando, and F. Yu, “Default Risk and Diversification : Theory and Empirical Implications * Default Risk and Diversification: Theory and Empirical Implications Abstract,” no. 607, 2003. [31] S. Kim, D. Jeong, C. Lee, and J. Kim, “Finite difference method for the multi-asset black-scholes equations,” Mathematics, vol. 8, no. 3, pp. 1–17, 2020, doi: 10.3390/math8030391. [32] D. Jeong, M. Yoo, and J. Kim, “Finite Difference Method for the Black–Scholes Equation Without Boundary Conditions,” Comput. Econ., vol. 51, no. 4, pp. 961–972, 2018, doi: 10.1007/s10614-017- 9653-0. [33] T. Guo, Black-Scholes Process and Monte Carlo Simulation-Based Options Pricing. Atlantis Press International BV, 2023. doi: 10.2991/978-94-6463- 010-7_75. [34] P. Glasserman, “Monte Carlo Methods in Financial Engineerring,” J. Chem. Inf. Model., vol. 53, no. 9, pp. 1689–1699, 2013. [35] Q. Jiang*, “Comparison of Black–Scholes Model and Monte-Carlo Simulation on Stock Price Modeling,” vol. 109, no. Icemci, pp. 135–137, 2020, doi: 10.2991/aebmr.k.191217.025. [36] E. Derman and E. K, “Stochastic Caclulus for Finance,” pp. 1–63, 2000. Authors:
(1) Agni Rakshit, Department of Mathematics, National Institute of Technology, Durgapur, Durgapur, India (spiritualagnimath.statml@gmail.com);
(2) Gautam Bandyopadhyay, Department of Management Studies, National Institute of Technology, Durgapur, Durgapur, India (gbandyopadhyay.dms@nitdgp.ac.in);
(3) Tanujit Chakraborty, Department of Science and Engineering & Sorbonne Center for AI, Sorbonne University, Abu Dhabi, United Arab Emirates (tanujit.chakraborty@sorbonne.ae). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Abstract and 1. Introduction 1.1 Option Pricing 1.1 Option Pricing 1.2 Asymptotic Notation (Big O) 1.2 Asymptotic Notation (Big O) 1.3 Finite Difference 1.3 Finite Difference 1.4 The Black-Schole Model 1.4 The Black-Schole Model 1.5 Monte Carlo Simulation and Variance Reduction Techniques 1.5 Monte Carlo Simulation and Variance Reduction Techniques 1.6 Our Contribution 1.6 Our Contribution Literature Review Methodology Literature Review Literature Review Literature Review Methodology Methodology 3.1 Model Assumption 3.1 Model Assumption 3.2 Theorems and Model Discussion 3.2 Theorems and Model Discussion Result Analysis Conclusion and References Result Analysis Result Analysis Result Analysis Conclusion and References Conclusion and References Conclusion and References 5. CONCLUSION The research findings have demonstrated that finite difference methods offer a powerful tool for approximating option prices and hedging parameters, especially in cases where closed-form solutions are unavailable or impractical. The asymptotic analysis conducted in this study has provided valuable insights into the convergence properties and accuracy of finite difference approximations, shedding light on the optimal choice of discretization schemes and grid sizes for different option contracts. Furthermore, the combination of finite difference methods and Monte Carlo simulation with variance reduction techniques offers a comprehensive approach to hedge error analysis in option pricing. By leveraging the strengths of both methodologies, financial practitioners can obtain more robust and reliable estimates of option prices and hedge parameters, thereby enhancing their risk management capabilities and decision-making processes. It is important to acknowledge the limitations of this study and areas for future research. While the asymptotic analysis provides valuable theoretical insights, further empirical validation is warranted to assess the robustness of the findings across different market conditions and asset classes. Additionally, exploring alternative variance reduction techniques and incorporating more sophisticated models of market dynamics could yield further improvements in hedge error analysis and option pricing accuracy. this study has advanced our understanding of the factors influencing hedge errors and provided practical tools for mitigating their impact on derivative pricing and risk management. As financial markets continue to evolve, the insights gained from this research will remain valuable for academics, practitioners, and policymakers striving to enhance the efficiency and stability of global financial systems. REFERENCES [1] C. W. Smith, “Option pricing. A review,” J. financ. econ., vol. 3, no. 1–2, pp. 3–51, 1976, doi: 10.1016/0304-405X(76)90019-2. [2] S. Guernsey, “an Introduction To the Black-Scholes Pde Model,” 2013. [3] B. Peeters, C. L. Dert, and A. Lucas, “Black Scholes for Portfolios of Options in Discrete Time,” SSRN Electron. J., no. 03, 2005, doi: 10.2139/ssrn.469022. [4] J. He, “A Study on Analytical and Numerical Solutions of Three Types of Black-Scholes Models,” Int. J. Trade, Econ. Financ., vol. 13, no. 2, 2022, doi: 10.18178/ijtef.2022.13.2.720. [5] A. S. Shinde and K. C. Takale, “Study of BlackScholes model and its applications,” Procedia Eng., vol. 38, pp. 270–279, 2012, doi: 10.1016/j.proeng.2012.06.035. [6] Y. Zhang, “The value of Monte Carlo model-based variance reduction technology in the pricing of financial derivatives,” PLoS One, vol. 15, no. 2, pp. 1–13, 2020, doi: 10.1371/journal.pone.0229737. [7] P. L. Bonate, “A brief introduction to Monte Carlo simulation,” Clin. Pharmacokinet., vol. 40, no. 1, pp. 15–22, 2001, doi: 10.2165/00003088- 200140010-00002. [8] D. R. Tobergte and S. Curtis, “Importance sampling for high speed statistical Monte-Carlo simulations,” J. Chem. Inf. Model., vol. 53, no. 9, pp. 1689–1699, 2013. [9] E. C. Anderson and M. Stephens, “Importance Sampling and (MC)^3,” Sisg 2020, vol. 2, no. July, pp. 1–29, 2020. [10] I. N. Medvedev, “Vector estimators of the Monte Carlo method with a finite variance,” Russ. J. Numer. Anal. Math. Model., vol. 28, no. 3, pp. 231– 244, 2013, doi: 10.1515/rnam-2013-0014. [11] E. Saliby and R. J. Paul, “A farewell to the use of antithetic variates in monte carlo simulation,” J. Oper. Res. Soc., vol. 60, no. 7, pp. 1026–1035, 2009, doi: 10.1057/palgrave.jors.2602645. [12] P. W. Glynn and R. Szechtman, “Some New Perspectives on the Method of Control Variates,” Monte Carlo Quasi-Monte Carlo Methods 2000, pp. 27–49, 2002, doi: 10.1007/978-3-642-56046-0_3. [13] W. Gustafsson, “Evaluating the Longstaff-Schwartz method for pricing of American options,” 2015. [14] L. Andersen and M. Broadie, “Primal-dual simulation algorithm for pricing multidimensional American options,” Manage. Sci., vol. 50, no. 9, pp. 1222–1234, 2004, doi: 10.1287/mnsc.1040.0258. [15] E. Wipplinger, Jim Gatheral: The volatility surface, a practitioner’s guide, vol. 22, no. 1. 2008. doi: 10.1007/s11408-007-0072-4. [16] M. Avellaneda and S. Stoikov, “High-frequency trading in a limit order book,” Quant. Financ., vol. 8, no. 3, pp. 217–224, 2008, doi: 10.1080/14697680701381228. [17] S. Mayoral, D. Moreno, and A. Zareei, “Using a hedging network to minimize portfolio risk,” Financ. Res. Lett., vol. 44, no. April, p. 102044, 2022, doi: 10.1016/j.frl.2021.102044. [18] S. Gayathri Devi, K. Selvam, and S. P. Rajagopalan, “An abstract to calculate big o factors of time and space complexity of machine code,” IET Conf. Publ., vol. 2011, no. 583 CP, pp. 844–847, 2011, doi: 10.1049/cp.2011.0483. [19] M. S. H. Mojumder, M. N. Haque, and M. J. Alam, “Efficient Finite Difference Methods for the Numerical Analysis of One-Dimensional Heat Equation,” J. Appl. Math. Phys., vol. 11, no. 10, pp. 3099–3123, 2023, doi: 10.4236/jamp.2023.1110204. [20] I. R. Khan and R. Ohba, “Taylor series based finite difference approximations of higher-degree derivatives,” J. Comput. Appl. Math., vol. 154, no. 1, pp. 115–124, 2003, doi: 10.1016/S0377- 0427(02)00816-6. [21] S. Danho, “Pricing Financial Derivatives with the Finite Difference Method,” p. 237, 2017. [22] G. Connor and T. Lasarte, “An Introduction to Hedge Funds Strategies. Introductory Guide,” pp. 1–16, 2004, [Online]. Available: http://eprints.lse.ac.uk/24675/1/dp477.pdf http://eprints.lse.ac.uk/24675/1/dp477.pdf [23] M. Nurul Anwar and L. Sazzad Andallah, “A Study on Numerical Solution of Black-Scholes Model,” J. Math. Financ., vol. 08, no. 02, pp. 372–381, 2018, doi: 10.4236/jmf.2018.82024. [24] J. Akahori, F. Barsotti, and Y. Imamura, “Hedging error as generalized timing risk,” Quant. Financ., vol. 23, no. 4, pp. 693–703, 2023, doi: 10.1080/14697688.2022.2154255. [25] R. C. A. Oomen and G. J. Jiang, “Hedging Derivatives Risks - A Simulation Study,” SSRN Electron. J., no. May, 2005, doi: 10.2139/ssrn.302525. [26] K. A. Froot, “Hedging Portfolios with Real Assets,” J. Portf. Manag., vol. 21, no. 4, pp. 60–77, 1995, doi: 10.3905/jpm.1995.409527. [27] H. E. LELAND, “Option Pricing and Replication with Transactions Costs,” J. Finance, vol. 40, no. 5, pp. 1283–1301, 1985, doi: 10.1111/j.1540- 6261.1985.tb02383.x. [28] T. Hayashi and P. A. Mykland, “Evaluating hedging errors: An asymptotic approach,” Math. Financ., vol. 15, no. 2, pp. 309–343, 2005, doi: 10.1111/j.0960-1627.2005.00221.x. [29] J.-L. Prigent, R. Hentati, and A. Kaffel, “Dynamic versus static optimization of hedge fund portfolios: the relevance of performance measures,” Int. J. Bus., vol. 15, no. 156, pp. 1–17, 2010. [30] R. A. Jarrow, D. Lando, and F. Yu, “Default Risk and Diversification : Theory and Empirical Implications * Default Risk and Diversification: Theory and Empirical Implications Abstract,” no. 607, 2003. [31] S. Kim, D. Jeong, C. Lee, and J. Kim, “Finite difference method for the multi-asset black-scholes equations,” Mathematics, vol. 8, no. 3, pp. 1–17, 2020, doi: 10.3390/math8030391. [32] D. Jeong, M. Yoo, and J. Kim, “Finite Difference Method for the Black–Scholes Equation Without Boundary Conditions,” Comput. Econ., vol. 51, no. 4, pp. 961–972, 2018, doi: 10.1007/s10614-017- 9653-0. [33] T. Guo, Black-Scholes Process and Monte Carlo Simulation-Based Options Pricing. Atlantis Press International BV, 2023. doi: 10.2991/978-94-6463- 010-7_75. [34] P. Glasserman, “Monte Carlo Methods in Financial Engineerring,” J. Chem. Inf. Model., vol. 53, no. 9, pp. 1689–1699, 2013. [35] Q. Jiang*, “Comparison of Black–Scholes Model and Monte-Carlo Simulation on Stock Price Modeling,” vol. 109, no. Icemci, pp. 135–137, 2020, doi: 10.2991/aebmr.k.191217.025. [36] E. Derman and E. K, “Stochastic Caclulus for Finance,” pp. 1–63, 2000. Authors: (1) Agni Rakshit, Department of Mathematics, National Institute of Technology, Durgapur, Durgapur, India (spiritualagnimath.statml@gmail.com); (2) Gautam Bandyopadhyay, Department of Management Studies, National Institute of Technology, Durgapur, Durgapur, India (gbandyopadhyay.dms@nitdgp.ac.in); (3) Tanujit Chakraborty, Department of Science and Engineering & Sorbonne Center for AI, Sorbonne University, Abu Dhabi, United Arab Emirates (tanujit.chakraborty@sorbonne.ae). Authors: Authors: (1) Agni Rakshit, Department of Mathematics, National Institute of Technology, Durgapur, Durgapur, India (spiritualagnimath.statml@gmail.com); (2) Gautam Bandyopadhyay, Department of Management Studies, National Institute of Technology, Durgapur, Durgapur, India (gbandyopadhyay.dms@nitdgp.ac.in); (3) Tanujit Chakraborty, Department of Science and Engineering & Sorbonne Center for AI, Sorbonne University, Abu Dhabi, United Arab Emirates (tanujit.chakraborty@sorbonne.ae). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. available on arxiv available on arxiv

Harnessing Finite Difference Methods for Enhanced Option Pricing

Table of Links Abstract and 1. Introduction 1.1 Option Pricing 1.2 Asymptotic Notation (Big O) 1.3 Finite Difference 1.4 The Black-Schole Model 1.5 Monte Carlo Simulation and Variance Reduction Techniques 1.6 Our Contribution Literature Review
Methodology 3.1 Model Assumption 3.2 Theorems and Model Discussion Result Analysis
Conclusion and References 1.3 FINITE DIFFERENCE Finite difference methods are numerical techniques used to approximate derivatives and solve differential equations by discretizing the domain into a grid of points. This approach is based on Taylor series expansions, where derivatives are expressed as combinations of function values at nearby points. Consider a function 𝑓(𝑥) and its derivative 𝑓′(𝑥) then finite difference approximation of the derivative at a point 𝑥𝑖 is given by: In this plot, the tangent line represents the true derivative 𝑓 ′ (𝑥0) , Meanwhile, the secant line represents the finite difference approximation. As the step size ℎ decreases, the secant line becomes closer to the tangent line, demonstrating the convergence of the finite difference approximation to the true derivative as ℎ approaches zero. The application of finite difference methods in option pricing allows for more accurate simulations of market dynamics and estimation of option prices. These methods enable practitioners to account for various factors affecting option values, such as changes in asset prices, volatility, and interest rates. Finite difference methods offer flexibility and scalability, making them suitable for pricing various types of options and constructing hedging strategies. Despite their computational complexity, advancements in numerical algorithms and computing power have made finite difference methods increasingly accessible and efficient for option pricing applications [4]. Authors:
(1) Agni Rakshit, Department of Mathematics, National Institute of Technology, Durgapur, Durgapur, India (spiritualagnimath.statml@gmail.com);
(2) Gautam Bandyopadhyay, Department of Management Studies, National Institute of Technology, Durgapur, Durgapur, India (gbandyopadhyay.dms@nitdgp.ac.in);
(3) Tanujit Chakraborty, Department of Science and Engineering & Sorbonne Center for AI, Sorbonne University, Abu Dhabi, United Arab Emirates (tanujit.chakraborty@sorbonne.ae). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Abstract and 1. Introduction 1.1 Option Pricing 1.1 Option Pricing 1.2 Asymptotic Notation (Big O) 1.2 Asymptotic Notation (Big O) 1.3 Finite Difference 1.3 Finite Difference 1.4 The Black-Schole Model 1.4 The Black-Schole Model 1.5 Monte Carlo Simulation and Variance Reduction Techniques 1.5 Monte Carlo Simulation and Variance Reduction Techniques 1.6 Our Contribution 1.6 Our Contribution Literature Review Methodology Literature Review Literature Review Literature Review Methodology Methodology 3.1 Model Assumption 3.1 Model Assumption 3.2 Theorems and Model Discussion 3.2 Theorems and Model Discussion Result Analysis Conclusion and References Result Analysis Result Analysis Result Analysis Conclusion and References Conclusion and References Conclusion and References 1.3 FINITE DIFFERENCE Finite difference methods are numerical techniques used to approximate derivatives and solve differential equations by discretizing the domain into a grid of points. This approach is based on Taylor series expansions, where derivatives are expressed as combinations of function values at nearby points. Consider a function 𝑓(𝑥) and its derivative 𝑓′(𝑥) then finite difference approximation of the derivative at a point 𝑥𝑖 is given by: In this plot, the tangent line represents the true derivative 𝑓 ′ (𝑥0) , Meanwhile, the secant line represents the finite difference approximation. As the step size ℎ decreases, the secant line becomes closer to the tangent line, demonstrating the convergence of the finite difference approximation to the true derivative as ℎ approaches zero. The application of finite difference methods in option pricing allows for more accurate simulations of market dynamics and estimation of option prices. These methods enable practitioners to account for various factors affecting option values, such as changes in asset prices, volatility, and interest rates. Finite difference methods offer flexibility and scalability, making them suitable for pricing various types of options and constructing hedging strategies. Despite their computational complexity, advancements in numerical algorithms and computing power have made finite difference methods increasingly accessible and efficient for option pricing applications [4]. Authors: (1) Agni Rakshit, Department of Mathematics, National Institute of Technology, Durgapur, Durgapur, India (spiritualagnimath.statml@gmail.com); (2) Gautam Bandyopadhyay, Department of Management Studies, National Institute of Technology, Durgapur, Durgapur, India (gbandyopadhyay.dms@nitdgp.ac.in); (3) Tanujit Chakraborty, Department of Science and Engineering & Sorbonne Center for AI, Sorbonne University, Abu Dhabi, United Arab Emirates (tanujit.chakraborty@sorbonne.ae). Authors: Authors: (1) Agni Rakshit, Department of Mathematics, National Institute of Technology, Durgapur, Durgapur, India (spiritualagnimath.statml@gmail.com); (2) Gautam Bandyopadhyay, Department of Management Studies, National Institute of Technology, Durgapur, Durgapur, India (gbandyopadhyay.dms@nitdgp.ac.in); (3) Tanujit Chakraborty, Department of Science and Engineering & Sorbonne Center for AI, Sorbonne University, Abu Dhabi, United Arab Emirates (tanujit.chakraborty@sorbonne.ae). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. available on arxiv available on arxiv

Finite Difference Methods: A Numerical Approach to Option Pricing and Derivatives

Authors:
(1) Revathy B S, Raman Research Institute, Bengaluru, 560080, Karnataka, India and Corresponding author;
(2) Victor Mukherjee, Department of Physical Sciences, Indian Institute of Science Education and Research Berhampur, Berhampur, 760010, Odisha, India;
(3) Uma Divakaran, Department of Physics, Indian Institute of Technology Palakkad, Palakkad, 678623, Kerala, India. Table of Links Abstract and 1 Introduction 2 Free fermionic model 3 Many body quantum Otto cycle 4 Universal scalings in work output 5 Transverse Ising model as working medium 6 Conclusion and References 6 Conclusion We construct a many body quantum Otto cycle with a WM that undergoes a quantum phase transition. The non-unitary strokes of the cycle are powered by finite temperature baths, while the unitary strokes involve driving the WM close to the critical point. This driving leads to nonadiabatic excitations which can be quantified using relative excess energy that follows universal scalings with the rate of driving as well as the temperature of the cold bath. The excess energy can be linked to the output work of the engine which thus manifests the universal scalings shown by the excess energy. Notably, we show that higher values of the cold bath temperature TC allows one to operate the engine close to the adiabatic limit for lower values of τ2 ≈ τmin, which further follows universal scaling relations. This raises interesting questions regarding the importance of control methods such as shortcuts to adiabaticity [41], or bath engineering [23], for finite temperature quantum heat engines. Furthermore, our results for one-dimensional transverse Ising model WM suggest the existence of an optimal value of the cold bath temperature TC > 0, for operating the QHE with high work output at high power. These counterintuitive results stem from the dominance of thermal fluctuations over quantum fluctuations in finite-temperature quantum critical heat engines, for higher bath temperatures. Acknowledgements. R.B.S. and U.D. acknowledge the use of HPC facility Chandra at IIT Palakkad. U.D. acknowledges support from SERB (SPG/2022/000708). V.M. acknowledges support from SERB through MATRICS (Project No. MTR/2021/000055) and a Seed Grant from IISER Berhampur. Data Availability Statement. Any data that support the findings of this study are included within the article. References [1] Jochen Gemmer, Mathias Michel, and G¨unter Mahler. Quantum thermodynamics: Emergence of thermodynamic behavior within composite quantum systems, volume 784. Springer, 2009. [2] Robert Alicki and Ronnie Kosloff. Introduction to Quantum Thermodynamics: History and Prospects. Springer International Publishing, 2018. [3] Sai Vinjanampathy and Janet Anders. Quantum thermodynamics. Contemporary Physics, 57:545, 2016. [4] Johannes Roßnagel, Samuel T. Dawkins, Karl N. Tolazzi, Obinna Abah, Eric Lutz, Ferdinand Schmidt-Kaler, and Kilian Singer. A single-atom heat engine. Science, 352(6283):325–329, 2016. [5] D. von Lindenfels, O. Grab, C. T. Schmiegelow, V. Kaushal, J. Schulz, Mark T. Mitchison, John Goold, F. Schmidt-Kaler, and U. G. Poschinger. Spin heat engine coupled to a harmonic-oscillator flywheel. Phys. Rev. Lett., 123:080602, Aug 2019. [6] John P. S. Peterson, Tiago B. Batalh˜ao, Marcela Herrera, Alexandre M. Souza, Roberto S. Sarthour, Ivan S. Oliveira, and Roberto M. Serra. Experimental characterization of a spin quantum heat engine. Phys. Rev. Lett., 123:240601, Dec 2019. [7] James Klatzow, Jonas N. Becker, Patrick M. Ledingham, Christian Weinzetl, Krzysztof T. Kaczmarek, Dylan J. Saunders, Joshua Nunn, Ian A. Walmsley, Raam Uzdin, and Eilon Poem. Experimental demonstration of quantum effects in the operation of microscopic heat engines. Phys. Rev. Lett., 122:110601, Mar 2019. [8] Obinna Abah and Eric Lutz. Optimal performance of a quantum otto refrigerator. Europhysics Letters, 113(6):60002, apr 2016. [9] Andreas Hartmann, Victor Mukherjee, Glen Bigan Mbeng, Wolfgang Niedenzu, and Wolfgang Lechner. Multi-spin counterdiabatic driving in many-body quantum otto refrigerators. Quantum, 4:377, 2020. [10] Thao P. Le, Jesper Levinsen, Kavan Modi, Meera M. Parish, and Felix A. Pollock. Spinchain model of a many-body quantum battery. Phys. Rev. A, 97:022106, Feb 2018. [11] Francesco Campaioli, Felix A. Pollock, and Sai Vinjanampathy. Quantum batteries - review chapter. arXiv: Quantum Physics, 2018. [12] Victor Montenegro, Marco G. Genoni, Abolfazl Bayat, and Matteo G. A. Paris. Quantum metrology with boundary time crystals. Communications Physics, 6(1):304, Oct 2023. [13] Karl Joulain, Jeremie Drevillon, Younes Ezzahri, and Jose Ordonez-Miranda. Quantum thermal transistor. Phys. Rev. Lett., 116:200601, May 2016. [14] Nikhil Gupt, Srijan Bhattacharyya, Bikash Das, Subhadeep Datta, Victor Mukherjee, and Arnab Ghosh. Floquet quantum thermal transistor. Phys. Rev. E, 106:024110, Aug 2022. [15] H. T. Quan, Yu-xi Liu, C. P. Sun, and Franco Nori. Quantum thermodynamic cycles and quantum heat engines. Phys. Rev. E, 76:031105, Sep 2007. [16] Ronnie Kosloff and Yair Rezek. The quantum harmonic otto cycle. Entropy, 19(4), 2017. [17] M. Campisi and R. Fazio. The power of a critical heat engine. Nat. Commun., 7:11895, 2016. [18] Thomas Fogarty and Thomas Busch. A many-body heat engine at criticality. Quantum Science and Technology, 6(1):015003, nov 2020. [19] Giulia Piccitto, Michele Campisi, and Davide Rossini. The ising critical quantum otto engine. New Journal of Physics, 2022. [20] Yu-Han Ma, Shan-He Su, and Chang-Pu Sun. Quantum thermodynamic cycle with quantum phase transition. Phys. Rev. E, 96:022143, Aug 2017. [21] Watanabe G. Yu YC Guan XW Chen, YY. and del Campo. An interaction-driven many particles quantum heat engine and its universal behavior. npj Quantum Inf, 5, 2019. [22] Revathy B.S, Victor Mukherjee, Uma Divakaran, and Adolfo del Campo. Universal finite-time thermodynamics of many-body quantum machines from kibble-zurek scaling. Phys. Rev. Res., 2:043247, 2020. [23] Revathy B.S, Victor Mukherjee, and Uma Divakaran. Bath engineering enhanced quantum critical engines. Entropy, 24(10), 2022. [24] Revathy B.S, Harsh Sharma, and Uma Divakaran. Improving performance of quantum heat engines by free evolution. arXiv preprint arXiv:2302.07003, 2023. [25] Jacek Dziarmaga. Dynamics of a quantum phase transition: Exact solution of the quantum ising model. Phys. Rev. Lett., 95:245701, Dec 2005. [26] Victor Mukherjee, Uma Divakaran, Amit Dutta, and Diptiman Sen. Quenching dynamics of a quantum xy spin- 1 2 chain in a transverse field. Phys. Rev. B, 76:174303, Nov 2007. [27] K. Sengupta, Diptiman Sen, and Shreyoshi Mondal. Exact results for quench dynamics and defect production in a two-dimensional model. Phys. Rev. Lett., 100:077204, Feb 2008. [28] Maximilian Keck, Simone Montangero, Giuseppe E Santoro, Rosario Fazio, and Davide Rossini. Dissipation in adiabatic quantum computers: lessons from an exactly solvable model. New Journal of Physics, 19(11):113029, nov 2017. [29] Souvik Bandyopadhyay, Sudarshana Laha, Utso Bhattacharya, and Amit Dutta. Exploring the possibilities of dynamical quantum phase transitions in the presence of a markovian bath. Scientific Reports, 8(1):11921, 2018. [30] Bogdan Damski and Wojciech H. Zurek. Adiabatic-impulse approximation for avoided level crossings: From phase-transition dynamics to landau-zener evolutions and back again. Phys. Rev. A, 73:063405, Jun 2006. [31] Anatoli Polkovnikov, Krishnendu Sengupta, Alessandro Silva, and Mukund Vengalattore. Colloquium: Nonequilibrium dynamics of closed interacting quantum systems. Rev. Mod. Phys., 83:863–883, Aug 2011. [32] Adolfo Del Campo and Wojciech H Zurek. Universality of phase transition dynamics: Topological defects from symmetry breaking. International Journal of Modern Physics A, 29(08):1430018, 2014. [33] Amit Dutta, Gabriel Aeppli, Bikas K Chakrabarti, Uma Divakaran, Thomas F Rosenbaum, and Diptiman Sen. Quantum phase transitions in transverse field spin models: from statistical physics to quantum information. Cambridge University Press, 2015. [34] A. Polkovnikov and V. Gritsev. Breakdown of the adiabatic limit in low-dimensional gapless systems. Nature Physics, 4:477, June 2008. [35] C. De Grandi, V. Gritsev, and A. Polkovnikov. Quench dynamics near a quantum critical point: Application to the sine-gordon model. Phys. Rev. B, 81:224301, Jun 2010. [36] C. De Grandi, V. Gritsev, and A. Polkovnikov. Quench dynamics near a quantum critical point. Phys. Rev. B, 81:012303, Jan 2010. [37] Shusa Deng, Gerardo Ortiz, and Lorenza Viola. Dynamical critical scaling and effective thermalization in quantum quenches: Role of the initial state. Phys. Rev. B, 83:094304, 2011. [38] Pierre Pfeuty. The one-dimensional ising model with a transverse field. Annals of Physics, 57(1):79 – 90, 1970. [39] J. E. Bunder and Ross H. McKenzie. Effect of disorder on quantum phase transitions in anisotropic xy spin chains in a transverse field. Phys. Rev. B, 60:344–358, Jul 1999. [40] Elliott Lieb, Theodore Schultz, and Daniel Mattis. Two soluble models of an antiferromagnetic chain. Annals of Physics, 16(3):407 – 466, 1961. [41] Andreas Hartmann, Victor Mukherjee, Wolfgang Niedenzu, and Wolfgang Lechner. Many-body quantum heat engines with shortcuts to adiabaticity. Phys. Rev. Res., 2:023145, May 2020. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Revathy B S, Raman Research Institute, Bengaluru, 560080, Karnataka, India and Corresponding author; (2) Victor Mukherjee, Department of Physical Sciences, Indian Institute of Science Education and Research Berhampur, Berhampur, 760010, Odisha, India; (3) Uma Divakaran, Department of Physics, Indian Institute of Technology Palakkad, Palakkad, 678623, Kerala, India. Authors: Authors: (1) Revathy B S, Raman Research Institute, Bengaluru, 560080, Karnataka, India and Corresponding author; (2) Victor Mukherjee, Department of Physical Sciences, Indian Institute of Science Education and Research Berhampur, Berhampur, 760010, Odisha, India; (3) Uma Divakaran, Department of Physics, Indian Institute of Technology Palakkad, Palakkad, 678623, Kerala, India. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Free fermionic model 2 Free fermionic model 3 Many body quantum Otto cycle 3 Many body quantum Otto cycle 4 Universal scalings in work output 4 Universal scalings in work output 5 Transverse Ising model as working medium 5 Transverse Ising model as working medium 6 Conclusion and References 6 Conclusion and References 6 Conclusion We construct a many body quantum Otto cycle with a WM that undergoes a quantum phase transition. The non-unitary strokes of the cycle are powered by finite temperature baths, while the unitary strokes involve driving the WM close to the critical point. This driving leads to nonadiabatic excitations which can be quantified using relative excess energy that follows universal scalings with the rate of driving as well as the temperature of the cold bath. The excess energy can be linked to the output work of the engine which thus manifests the universal scalings shown by the excess energy. Notably, we show that higher values of the cold bath temperature TC allows one to operate the engine close to the adiabatic limit for lower values of τ2 ≈ τmin, which further follows universal scaling relations. This raises interesting questions regarding the importance of control methods such as shortcuts to adiabaticity [41], or bath engineering [23], for finite temperature quantum heat engines. Furthermore, our results for one-dimensional transverse Ising model WM suggest the existence of an optimal value of the cold bath temperature TC > 0, for operating the QHE with high work output at high power. These counterintuitive results stem from the dominance of thermal fluctuations over quantum fluctuations in finite-temperature quantum critical heat engines, for higher bath temperatures. Acknowledgements . R.B.S. and U.D. acknowledge the use of HPC facility Chandra at IIT Palakkad. U.D. acknowledges support from SERB (SPG/2022/000708). V.M. acknowledges support from SERB through MATRICS (Project No. MTR/2021/000055) and a Seed Grant from IISER Berhampur. Acknowledgements Data Availability Statement. Any data that support the findings of this study are included within the article. Data Availability Statement. References [1] Jochen Gemmer, Mathias Michel, and G¨unter Mahler. Quantum thermodynamics: Emergence of thermodynamic behavior within composite quantum systems, volume 784. Springer, 2009. [2] Robert Alicki and Ronnie Kosloff. Introduction to Quantum Thermodynamics: History and Prospects. Springer International Publishing, 2018. [3] Sai Vinjanampathy and Janet Anders. Quantum thermodynamics. Contemporary Physics, 57:545, 2016. [4] Johannes Roßnagel, Samuel T. Dawkins, Karl N. Tolazzi, Obinna Abah, Eric Lutz, Ferdinand Schmidt-Kaler, and Kilian Singer. A single-atom heat engine. Science, 352(6283):325–329, 2016. [5] D. von Lindenfels, O. Grab, C. T. Schmiegelow, V. Kaushal, J. Schulz, Mark T. Mitchison, John Goold, F. Schmidt-Kaler, and U. G. Poschinger. Spin heat engine coupled to a harmonic-oscillator flywheel. Phys. Rev. Lett., 123:080602, Aug 2019. [6] John P. S. Peterson, Tiago B. Batalh˜ao, Marcela Herrera, Alexandre M. Souza, Roberto S. Sarthour, Ivan S. Oliveira, and Roberto M. Serra. Experimental characterization of a spin quantum heat engine. Phys. Rev. Lett., 123:240601, Dec 2019. [7] James Klatzow, Jonas N. Becker, Patrick M. Ledingham, Christian Weinzetl, Krzysztof T. Kaczmarek, Dylan J. Saunders, Joshua Nunn, Ian A. Walmsley, Raam Uzdin, and Eilon Poem. Experimental demonstration of quantum effects in the operation of microscopic heat engines. Phys. Rev. Lett., 122:110601, Mar 2019. [8] Obinna Abah and Eric Lutz. Optimal performance of a quantum otto refrigerator. Europhysics Letters, 113(6):60002, apr 2016. [9] Andreas Hartmann, Victor Mukherjee, Glen Bigan Mbeng, Wolfgang Niedenzu, and Wolfgang Lechner. Multi-spin counterdiabatic driving in many-body quantum otto refrigerators. Quantum, 4:377, 2020. [10] Thao P. Le, Jesper Levinsen, Kavan Modi, Meera M. Parish, and Felix A. Pollock. Spinchain model of a many-body quantum battery. Phys. Rev. A, 97:022106, Feb 2018. [11] Francesco Campaioli, Felix A. Pollock, and Sai Vinjanampathy. Quantum batteries - review chapter. arXiv: Quantum Physics, 2018. [12] Victor Montenegro, Marco G. Genoni, Abolfazl Bayat, and Matteo G. A. Paris. Quantum metrology with boundary time crystals. Communications Physics, 6(1):304, Oct 2023. [13] Karl Joulain, Jeremie Drevillon, Younes Ezzahri, and Jose Ordonez-Miranda. Quantum thermal transistor. Phys. Rev. Lett., 116:200601, May 2016. [14] Nikhil Gupt, Srijan Bhattacharyya, Bikash Das, Subhadeep Datta, Victor Mukherjee, and Arnab Ghosh. Floquet quantum thermal transistor. Phys. Rev. E, 106:024110, Aug 2022. [15] H. T. Quan, Yu-xi Liu, C. P. Sun, and Franco Nori. Quantum thermodynamic cycles and quantum heat engines. Phys. Rev. E, 76:031105, Sep 2007. [16] Ronnie Kosloff and Yair Rezek. The quantum harmonic otto cycle. Entropy, 19(4), 2017. [17] M. Campisi and R. Fazio. The power of a critical heat engine. Nat. Commun., 7:11895, 2016. [18] Thomas Fogarty and Thomas Busch. A many-body heat engine at criticality. Quantum Science and Technology, 6(1):015003, nov 2020. [19] Giulia Piccitto, Michele Campisi, and Davide Rossini. The ising critical quantum otto engine. New Journal of Physics, 2022. [20] Yu-Han Ma, Shan-He Su, and Chang-Pu Sun. Quantum thermodynamic cycle with quantum phase transition. Phys. Rev. E, 96:022143, Aug 2017. [21] Watanabe G. Yu YC Guan XW Chen, YY. and del Campo. An interaction-driven many particles quantum heat engine and its universal behavior. npj Quantum Inf, 5, 2019. [22] Revathy B.S, Victor Mukherjee, Uma Divakaran, and Adolfo del Campo. Universal finite-time thermodynamics of many-body quantum machines from kibble-zurek scaling. Phys. Rev. Res., 2:043247, 2020. [23] Revathy B.S, Victor Mukherjee, and Uma Divakaran. Bath engineering enhanced quantum critical engines. Entropy, 24(10), 2022. [24] Revathy B.S, Harsh Sharma, and Uma Divakaran. Improving performance of quantum heat engines by free evolution. arXiv preprint arXiv:2302.07003, 2023. [25] Jacek Dziarmaga. Dynamics of a quantum phase transition: Exact solution of the quantum ising model. Phys. Rev. Lett., 95:245701, Dec 2005. [26] Victor Mukherjee, Uma Divakaran, Amit Dutta, and Diptiman Sen. Quenching dynamics of a quantum xy spin- 1 2 chain in a transverse field. Phys. Rev. B, 76:174303, Nov 2007. [27] K. Sengupta, Diptiman Sen, and Shreyoshi Mondal. Exact results for quench dynamics and defect production in a two-dimensional model. Phys. Rev. Lett., 100:077204, Feb 2008. [28] Maximilian Keck, Simone Montangero, Giuseppe E Santoro, Rosario Fazio, and Davide Rossini. Dissipation in adiabatic quantum computers: lessons from an exactly solvable model. New Journal of Physics, 19(11):113029, nov 2017. [29] Souvik Bandyopadhyay, Sudarshana Laha, Utso Bhattacharya, and Amit Dutta. Exploring the possibilities of dynamical quantum phase transitions in the presence of a markovian bath. Scientific Reports, 8(1):11921, 2018. [30] Bogdan Damski and Wojciech H. Zurek. Adiabatic-impulse approximation for avoided level crossings: From phase-transition dynamics to landau-zener evolutions and back again. Phys. Rev. A, 73:063405, Jun 2006. [31] Anatoli Polkovnikov, Krishnendu Sengupta, Alessandro Silva, and Mukund Vengalattore. Colloquium: Nonequilibrium dynamics of closed interacting quantum systems. Rev. Mod. Phys., 83:863–883, Aug 2011. [32] Adolfo Del Campo and Wojciech H Zurek. Universality of phase transition dynamics: Topological defects from symmetry breaking. International Journal of Modern Physics A, 29(08):1430018, 2014. [33] Amit Dutta, Gabriel Aeppli, Bikas K Chakrabarti, Uma Divakaran, Thomas F Rosenbaum, and Diptiman Sen. Quantum phase transitions in transverse field spin models: from statistical physics to quantum information. Cambridge University Press, 2015. [34] A. Polkovnikov and V. Gritsev. Breakdown of the adiabatic limit in low-dimensional gapless systems. Nature Physics, 4:477, June 2008. [35] C. De Grandi, V. Gritsev, and A. Polkovnikov. Quench dynamics near a quantum critical point: Application to the sine-gordon model. Phys. Rev. B, 81:224301, Jun 2010. [36] C. De Grandi, V. Gritsev, and A. Polkovnikov. Quench dynamics near a quantum critical point. Phys. Rev. B, 81:012303, Jan 2010. [37] Shusa Deng, Gerardo Ortiz, and Lorenza Viola. Dynamical critical scaling and effective thermalization in quantum quenches: Role of the initial state. Phys. Rev. B, 83:094304, 2011. [38] Pierre Pfeuty. The one-dimensional ising model with a transverse field. Annals of Physics, 57(1):79 – 90, 1970. [39] J. E. Bunder and Ross H. McKenzie. Effect of disorder on quantum phase transitions in anisotropic xy spin chains in a transverse field. Phys. Rev. B, 60:344–358, Jul 1999. [40] Elliott Lieb, Theodore Schultz, and Daniel Mattis. Two soluble models of an antiferromagnetic chain. Annals of Physics, 16(3):407 – 466, 1961. [41] Andreas Hartmann, Victor Mukherjee, Wolfgang Niedenzu, and Wolfgang Lechner. Many-body quantum heat engines with shortcuts to adiabaticity. Phys. Rev. Res., 2:023145, May 2020. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Quantum Critical Engine at Finite Temperatures: Conclusion and References

Authors:
(1) Revathy B S, Raman Research Institute, Bengaluru, 560080, Karnataka, India and Corresponding author;
(2) Victor Mukherjee, Department of Physical Sciences, Indian Institute of Science Education and Research Berhampur, Berhampur, 760010, Odisha, India;
(3) Uma Divakaran, Department of Physics, Indian Institute of Technology Palakkad, Palakkad, 678623, Kerala, India. Table of Links Abstract and 1 Introduction 2 Free fermionic model 3 Many body quantum Otto cycle 4 Universal scalings in work output 5 Transverse Ising model as working medium 6 Conclusion and References 5 Transverse Ising model as working medium We demonstrate the results derived in the previous section using the prototypical model of transverse Ising model (TIM) as the WM of the Otto cycle. The Hamiltonian of TIM is given by When written in momentum (k) space using the basis |0, 0⟩, |k, 0⟩, |0, −k⟩, |k, −k⟩, the Hamiltonian takes the form During the unitary strokes of the QOC, the transverse field is changed from h1 to h2 in the B → C stroke using the driving protocol and vice versa in the D → A stroke using the protocol For TIM, the value of the critical exponents are ν = z = 1 which gives One can also obtain the expression for excess defects or excess energy by integrating the analytical expression given in Eq. 18 where pk is given by the Landau Zener probability which in our case takes the form This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Revathy B S, Raman Research Institute, Bengaluru, 560080, Karnataka, India and Corresponding author; (2) Victor Mukherjee, Department of Physical Sciences, Indian Institute of Science Education and Research Berhampur, Berhampur, 760010, Odisha, India; (3) Uma Divakaran, Department of Physics, Indian Institute of Technology Palakkad, Palakkad, 678623, Kerala, India. Authors: Authors: (1) Revathy B S, Raman Research Institute, Bengaluru, 560080, Karnataka, India and Corresponding author; (2) Victor Mukherjee, Department of Physical Sciences, Indian Institute of Science Education and Research Berhampur, Berhampur, 760010, Odisha, India; (3) Uma Divakaran, Department of Physics, Indian Institute of Technology Palakkad, Palakkad, 678623, Kerala, India. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Free fermionic model 2 Free fermionic model 3 Many body quantum Otto cycle 3 Many body quantum Otto cycle 4 Universal scalings in work output 4 Universal scalings in work output 5 Transverse Ising model as working medium 5 Transverse Ising model as working medium 6 Conclusion and References 6 Conclusion and References 5 Transverse Ising model as working medium We demonstrate the results derived in the previous section using the prototypical model of transverse Ising model (TIM) as the WM of the Otto cycle. The Hamiltonian of TIM is given by When written in momentum (k) space using the basis |0, 0⟩, |k, 0⟩, |0, −k⟩, |k, −k⟩, the Hamiltonian takes the form During the unitary strokes of the QOC, the transverse field is changed from h1 to h2 in the B → C stroke using the driving protocol and vice versa in the D → A stroke using the protocol For TIM, the value of the critical exponents are ν = z = 1 which gives One can also obtain the expression for excess defects or excess energy by integrating the analytical expression given in Eq. 18 where pk is given by the Landau Zener probability which in our case takes the form This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Quantum Critical Engine at Finite Temperatures: Transverse Ising Model as Working Medium

Authors:
(1) Revathy B S, Raman Research Institute, Bengaluru, 560080, Karnataka, India and Corresponding author;
(2) Victor Mukherjee, Department of Physical Sciences, Indian Institute of Science Education and Research Berhampur, Berhampur, 760010, Odisha, India;
(3) Uma Divakaran, Department of Physics, Indian Institute of Technology Palakkad, Palakkad, 678623, Kerala, India. Table of Links Abstract and 1 Introduction 2 Free fermionic model 3 Many body quantum Otto cycle 4 Universal scalings in work output 5 Transverse Ising model as working medium 6 Conclusion and References 4 Universal scalings in work output The two unitary strokes of the quantum Otto cycle involve driving the Hamiltonian of the WM from one parameter to another. During this driving, the quantum critical point may or may not be crossed. Let us quickly revisit universal scalings in the non-equilibrium dynamics of a quantum system which is initially prepared in the ground state of the Hamiltonian, and is driven through the critical point linearly with a speed 1/τ . The diverging relaxation time close to the CP results in loss of adiabaticity, and thus generation of defects (excitations) no matter how slowly the CP is crossed [30–33]. The density of such defects nex follows a universal power law with the rate of driving where the power is determined by the critical exponents and dimensionality of the system, and is given by Here nex denotes the defect density, ν is the exponent associated with correlation length and z is the dynamical exponent with d being the dimensionality of the system. This scaling between the defect density and the rate of driving is called the Kibble-Zurek scaling which connects the equilibrium critical exponents with the non-equilibrium dynamics. However, this scaling gets modified when the driving starts from a thermal equilibrium state as opposed to the ground state of the system; In Refs. [34–37], the authors consider the case when the driving starts from the critical point and obtain a scaling of defects as a function of temperature and the driving rate, which we present below. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Revathy B S, Raman Research Institute, Bengaluru, 560080, Karnataka, India and Corresponding author; (2) Victor Mukherjee, Department of Physical Sciences, Indian Institute of Science Education and Research Berhampur, Berhampur, 760010, Odisha, India; (3) Uma Divakaran, Department of Physics, Indian Institute of Technology Palakkad, Palakkad, 678623, Kerala, India. Authors: Authors: (1) Revathy B S, Raman Research Institute, Bengaluru, 560080, Karnataka, India and Corresponding author; (2) Victor Mukherjee, Department of Physical Sciences, Indian Institute of Science Education and Research Berhampur, Berhampur, 760010, Odisha, India; (3) Uma Divakaran, Department of Physics, Indian Institute of Technology Palakkad, Palakkad, 678623, Kerala, India. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Free fermionic model 2 Free fermionic model 3 Many body quantum Otto cycle 3 Many body quantum Otto cycle 4 Universal scalings in work output 4 Universal scalings in work output 5 Transverse Ising model as working medium 5 Transverse Ising model as working medium 6 Conclusion and References 6 Conclusion and References 4 Universal scalings in work output The two unitary strokes of the quantum Otto cycle involve driving the Hamiltonian of the WM from one parameter to another. During this driving, the quantum critical point may or may not be crossed. Let us quickly revisit universal scalings in the non-equilibrium dynamics of a quantum system which is initially prepared in the ground state of the Hamiltonian, and is driven through the critical point linearly with a speed 1/τ . The diverging relaxation time close to the CP results in loss of adiabaticity, and thus generation of defects (excitations) no matter how slowly the CP is crossed [30–33]. The density of such defects n ex follows a universal power law with the rate of driving where the power is determined by the critical exponents and dimensionality of the system, and is given by ex Here n ex denotes the defect density, ν is the exponent associated with correlation length and z is the dynamical exponent with d being the dimensionality of the system. This scaling between the defect density and the rate of driving is called the Kibble-Zurek scaling which connects the equilibrium critical exponents with the non-equilibrium dynamics. However, this scaling gets modified when the driving starts from a thermal equilibrium state as opposed to the ground state of the system; In Refs. [34–37], the authors consider the case when the driving starts from the critical point and obtain a scaling of defects as a function of temperature and the driving rate, which we present below. ex This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Quantum Critical Engine at Finite Temperatures: Universal Scalings in Work Output

Authors:
(1) Revathy B S, Raman Research Institute, Bengaluru, 560080, Karnataka, India and Corresponding author;
(2) Victor Mukherjee, Department of Physical Sciences, Indian Institute of Science Education and Research Berhampur, Berhampur, 760010, Odisha, India;
(3) Uma Divakaran, Department of Physics, Indian Institute of Technology Palakkad, Palakkad, 678623, Kerala, India. Table of Links Abstract and 1 Introduction 2 Free fermionic model 3 Many body quantum Otto cycle 4 Universal scalings in work output 5 Transverse Ising model as working medium 6 Conclusion and References 3 Many body quantum Otto cycle We now describe the quantum Otto cycle (QOC) which consist of four strokes (also shown in Fig.1): (ii) Stroke B → C: The WM is disconnected from the hot bath and α is changed from α1 to α2 using the driving protocol, The evolution being a unitary evolution is given by the von-Neumann equation of motion: In this work, we shall focus on α2 = αc, the critical value, for the reasons that will be explained later. (iii) Stroke C → D: The WM with α = α2 is next connected to the cold bath at a temperature TC till τC so that it reaches the thermal state at D given by (iv) Stroke D → A: In this last stroke, the WM is disconnected from the cold bath, and α is changed back to α1 from α2 using to reach A through unitary dynamics and thus the cycle repeats. Energies at the end of each stroke i is calculated using the equation We characterize the engine performance using the quantities efficiency and power which are computed as This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Revathy B S, Raman Research Institute, Bengaluru, 560080, Karnataka, India and Corresponding author; (2) Victor Mukherjee, Department of Physical Sciences, Indian Institute of Science Education and Research Berhampur, Berhampur, 760010, Odisha, India; (3) Uma Divakaran, Department of Physics, Indian Institute of Technology Palakkad, Palakkad, 678623, Kerala, India. Authors: Authors: (1) Revathy B S, Raman Research Institute, Bengaluru, 560080, Karnataka, India and Corresponding author; (2) Victor Mukherjee, Department of Physical Sciences, Indian Institute of Science Education and Research Berhampur, Berhampur, 760010, Odisha, India; (3) Uma Divakaran, Department of Physics, Indian Institute of Technology Palakkad, Palakkad, 678623, Kerala, India. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Free fermionic model 2 Free fermionic model 3 Many body quantum Otto cycle 3 Many body quantum Otto cycle 4 Universal scalings in work output 4 Universal scalings in work output 5 Transverse Ising model as working medium 5 Transverse Ising model as working medium 6 Conclusion and References 6 Conclusion and References 3 Many body quantum Otto cycle We now describe the quantum Otto cycle (QOC) which consist of four strokes (also shown in Fig.1): (ii) Stroke B → C: The WM is disconnected from the hot bath and α is changed from α1 to α2 using the driving protocol, The evolution being a unitary evolution is given by the von-Neumann equation of motion: In this work, we shall focus on α2 = αc, the critical value, for the reasons that will be explained later. (iii) Stroke C → D: The WM with α = α2 is next connected to the cold bath at a temperature TC till τC so that it reaches the thermal state at D given by (iv) Stroke D → A: In this last stroke, the WM is disconnected from the cold bath, and α is changed back to α1 from α2 using to reach A through unitary dynamics and thus the cycle repeats. Energies at the end of each stroke i is calculated using the equation We characterize the engine performance using the quantities efficiency and power which are computed as This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Quantum Critical Engine at Finite Temperatures: Many Body Quantum Otto Cycle

Authors:
(1) Revathy B S, Raman Research Institute, Bengaluru, 560080, Karnataka, India and Corresponding author;
(2) Victor Mukherjee, Department of Physical Sciences, Indian Institute of Science Education and Research Berhampur, Berhampur, 760010, Odisha, India;
(3) Uma Divakaran, Department of Physics, Indian Institute of Technology Palakkad, Palakkad, 678623, Kerala, India. Table of Links Abstract and 1 Introduction 2 Free fermionic model 3 Many body quantum Otto cycle 4 Universal scalings in work output 5 Transverse Ising model as working medium 6 Conclusion and References Abstract We construct a quantum critical Otto engine that is powered by finite temperature baths. We show that the work output of the engine shows universal power law behavior that depends on the critical exponents of the working medium, as well as on the temperature of the cold bath. Furthermore, higher temperatures of the cold bath allows the engine to approach the limit of adiabatic operation for smaller values of the time period, while the corresponding power shows a maximum at an intermediate value of the cold bath temperature. These counterintuitive results stems from thermal excitations dominating the dynamics at higher temperatures. Keywords: Quantum heat engines, quantum phase transitions, Kibble Zurek scaling 1 Introduction Quantum thermodynamics is a rapidly progressing field aimed at understanding the thermodynamics at the quantum level [1, 2]. The field is getting much attention from the scientific community not just because it provides a link between quantum mechanics and thermodynamics but also because it aids in the development of nanoscale devices that aims to harness the potential benefits of the “quantumness” in their working medium (WM) [3]. The recent advances in the experiments such as trapped ions or ultracold atoms [4, 5], using NMR techniques [6], nitrogen vacancy centres in diamond [7] have made the realization of quantum devices possible. Among the various quantum devices studied, for example, quantum refrigerators [8, 9], quantum batteries [10, 11], quantum sensors [12], quantum thermal transistors [13, 14], etc., our work focusses on quantum heat engines that follow the quantum Otto cycle [15, 16]. The effect of phase transitions in quantum engines has been studied in Refs. [17–24]. While some of these studies concentrated on how to improve the performance of quantum engines with respect to its efficiency and power [23, 24], some focused on showcasing universality in their working [21, 22]. For instance, in Ref. [22], the authors showed that the work output of the engine up to an additive constant follows a universal power law governed by the driving speed with which the quantum critical points are crossed, where the exponent of the power law is determined by the critical points (CP) crossed. However, they considered the case of relaxing bath that takes the WM close to its ground state so that Kibble Zurek mechanism becomes relevant. On the other hand, in this paper, we specifically use thermal baths at different temperatures and study the effect of thermal excitations on the scalings of the work output. As discussed before, we prepare a many body quantum heat engine using a free fermionic model as WM that undergoes a quantum phase transition. We describe the details of the free fermionic model in Section 2, elaborate on the many body quantum Otto cycle that the engine follows in Section 3, followed by the universal scalings shown by the engine in Section 4. We demonstrate our results using the transverse field Ising model in Section 5 and finally conclude in Section 6. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Revathy B S, Raman Research Institute, Bengaluru, 560080, Karnataka, India and Corresponding author; (2) Victor Mukherjee, Department of Physical Sciences, Indian Institute of Science Education and Research Berhampur, Berhampur, 760010, Odisha, India; (3) Uma Divakaran, Department of Physics, Indian Institute of Technology Palakkad, Palakkad, 678623, Kerala, India. Authors: Authors: (1) Revathy B S, Raman Research Institute, Bengaluru, 560080, Karnataka, India and Corresponding author; (2) Victor Mukherjee, Department of Physical Sciences, Indian Institute of Science Education and Research Berhampur, Berhampur, 760010, Odisha, India; (3) Uma Divakaran, Department of Physics, Indian Institute of Technology Palakkad, Palakkad, 678623, Kerala, India. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Free fermionic model 2 Free fermionic model 3 Many body quantum Otto cycle 3 Many body quantum Otto cycle 4 Universal scalings in work output 4 Universal scalings in work output 5 Transverse Ising model as working medium 5 Transverse Ising model as working medium 6 Conclusion and References 6 Conclusion and References Abstract We construct a quantum critical Otto engine that is powered by finite temperature baths. We show that the work output of the engine shows universal power law behavior that depends on the critical exponents of the working medium, as well as on the temperature of the cold bath. Furthermore, higher temperatures of the cold bath allows the engine to approach the limit of adiabatic operation for smaller values of the time period, while the corresponding power shows a maximum at an intermediate value of the cold bath temperature. These counterintuitive results stems from thermal excitations dominating the dynamics at higher temperatures. Keywords : Quantum heat engines, quantum phase transitions, Kibble Zurek scaling Keywords 1 Introduction Quantum thermodynamics is a rapidly progressing field aimed at understanding the thermodynamics at the quantum level [1, 2]. The field is getting much attention from the scientific community not just because it provides a link between quantum mechanics and thermodynamics but also because it aids in the development of nanoscale devices that aims to harness the potential benefits of the “quantumness” in their working medium (WM) [3]. The recent advances in the experiments such as trapped ions or ultracold atoms [4, 5], using NMR techniques [6], nitrogen vacancy centres in diamond [7] have made the realization of quantum devices possible. Among the various quantum devices studied, for example, quantum refrigerators [8, 9], quantum batteries [10, 11], quantum sensors [12], quantum thermal transistors [13, 14], etc., our work focusses on quantum heat engines that follow the quantum Otto cycle [15, 16]. The effect of phase transitions in quantum engines has been studied in Refs. [17–24]. While some of these studies concentrated on how to improve the performance of quantum engines with respect to its efficiency and power [23, 24], some focused on showcasing universality in their working [21, 22]. For instance, in Ref. [22], the authors showed that the work output of the engine up to an additive constant follows a universal power law governed by the driving speed with which the quantum critical points are crossed, where the exponent of the power law is determined by the critical points (CP) crossed. However, they considered the case of relaxing bath that takes the WM close to its ground state so that Kibble Zurek mechanism becomes relevant. On the other hand, in this paper, we specifically use thermal baths at different temperatures and study the effect of thermal excitations on the scalings of the work output. As discussed before, we prepare a many body quantum heat engine using a free fermionic model as WM that undergoes a quantum phase transition. We describe the details of the free fermionic model in Section 2, elaborate on the many body quantum Otto cycle that the engine follows in Section 3, followed by the universal scalings shown by the engine in Section 4. We demonstrate our results using the transverse field Ising model in Section 5 and finally conclude in Section 6. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Quantum Critical Engine at Finite Temperatures: Abstract and Introduction

Why Is the AI Surge Boosting HDD Stocks? Finis Conner Says: Drives Are to AI as Batteries Are to EVs

This paper is available on arxiv under CC 4.0 license. Authors: (1) Vianney Brouard, ENS de Lyon, UMPA, CNRS UMR 5669, 46 All´ee d’Italie, 69364 Lyon Cedex 07, France; E-mail: vianney.brouard@ens-lyon.f. Table of Links Abstract & Introduction and presentation of the model Main results and biological interpretation First-order asymptotics of the mutant sub-populations for an infinite mono-directional grap First-order asymptotics of the mutant sub-populations for a general finite trait space (Theorem 2.1) Convergence for the stochastic exponents (Theorem 2.2) Acknowledgements & References 4 First-order asymptotics of the mutant sub-populations for a general finite trait space (Theorem 2.1) Notice that with such a construction it immediately follows the monotone coupling In the next definition, we are going to introduce an equivalence relation on Γ(V ). Two paths are said to be equivalent if they are the same up to cycles (in particular cycles formed by backward mutations are taken into account). More precisely, that there exists a minimal path, from which the two previous paths are using all the edges, but potentially also some other edges forming cycles. The aim of this equivalence relation is to say that among one class of equivalence, only the path with the minimal length may contribute for the asymptotic of the mutant sub-population sizes Now we have all the preliminary results and definitions to prove Theorem 2.1. We show Equation (11). The proof of Equation (12) is similar and is left to the reader. Proof of Theorem 2.1. And the last term of the r.h.s converges to 0 because

First-order Asymptotics of the Mutant Sub-populations for a General Finite Trait Space (Theorem 2.1)

As a product designer navigating the ever-evolving tech landscape, I've seen firsthand how the surge in complex products keeps us on our toes. It's not just about crafting pretty interfaces; it's about diving deep into the and joining forces with tech wizards and coders to bring those concepts to life. user experience With this piece, I want to share the core principles and tactics I've leaned on to design interfaces for intricate systems. Plus, I'll dive into the critical role we designers play from start to finish in the creative process. Jumping into the development fray right from the get-go, I've found, utterly transforms the creation of complex products. Suddenly, those daunting interfaces become more like friendly chats. Working shoulder to shoulder with developers, I gain an insider's view of the technical maze - its limits and possibilities. This insight is priceless. It arms me with the knowledge to weave together design and functionality, tackling intricate challenges with solutions that not only work but resonate with users on a human level. Kicking things off, I want to dive into why bringing designers into the project fold early is crucial. When you're crafting complex systems, it's not just about the nuts and bolts; it's about getting to the heart of what users need (and hey, remember, ). Leveraging time-tested design principles dramatically enhances the user experience. It's about striking that sweet spot where functionality meets simplicity, making the system a breeze to navigate and a joy to use. users are real people with real frustrations and desires So, what are these design principles? Simplifying Complex Tasks As a designer, my mission is to cut through the complexity of interfaces. It's about reducing the steps needed to get something done, organizing features logically, and giving users crystal-clear guidance. Embracing the mantra of " ," I aim to declutter the space where users interact, making everything feel just right. less is more Making Sense of Data When it comes to handling dense packs of data, good design can light the way. I lean on charts, diagrams, and infographics to break down and simplify the avalanche of information. But it's not just about visuals; it's also about . Through design metaphors, I bridge the gap between the unfamiliar and the familiar, guiding users on how to engage with the system naturally. speaking a language users already know Tailoring the Experience Customizing the interface to fit the unique preferences or behaviors of users takes their experience from good to great. Whether it's fine-tuning the system to align with specific tasks, interests, or levels of expertise, the goal is to make every interaction with the system not just efficient, but also enjoyable. An exciting leap in design innovation I've come across is crafting admin panels that understand plain English. Imagine this: an administrator types their needs straight into a search-like bar, and voila, an AI-driven system gets the gist and jumps into action. This breakthrough slashes the steep learning curve typically associated with new interfaces, making the system a breeze to navigate. If you're already on board with the idea that , let me show you what effective collaboration looks like designers are crucial Bringing Designers in Early Right from the get-go, having designers work closely with developers is key. It's all about ensuring that the user interface clicks perfectly with the system's inner workings, making everything flow just right. Using Design Systems Design systems are like the secret sauce for consistency. They give us a blueprint of standards and elements to craft interfaces that not only look cohesive but also provide a seamless user experience across the board. The Magic of Prototyping Sketching up prototypes and then putting them through their paces with actual users is a game-changer. It lets us spot the snags early, fine-tuning everything before we even think about hitting the big "launch" button. Feedback Nothing beats regular catch-ups and brainstorming sessions between designers and developers. It's about making sure everyone's singing from the same hymn sheet and squashing any bugs in the plan pronto. Welcoming designers into the heart of the development process isn't just good for the end result; it nurtures a vibe of mutual respect and teamwork. For me, that's the bedrock of successfully tackling those complex challenges. When it comes to designing intricate systems, I've got a couple of go-to tools that make all the difference 📈 For slicing through complex data and presenting it in a way that's straightforward and engaging, I lean on Tableau and D3.js. These powerhouses enable me to whip up interactive and lively visualizations that bring dense data to life, making it accessible to anyone. 💠 Then there's the game-changer: . It's reshaping how I approach design, from conjuring up layouts, with a snap of my fingers (figuratively speaking), based on the content to fine-tuning user interfaces by analyzing how people interact with them. AI even helps me craft interfaces that adapt on the fly to user preferences and actions. artificial intelligence When designing complex systems, I'm always caught between making things simple and adding loads of features. It's a delicate balance: too simple, and it might lack functionality, too feature-packed, and it becomes overwhelming. This balancing act is crucial because getting it right means creating something that's both powerful and easy to use, which is essential for keeping users engaged and driving sales. Essentially, the project's success hinges on finding that sweet spot. My own story about Complex Systems The Observability Platform project at Company X truly opened my eyes to the impact of integrating design early on. From day one, involving our design team was a game-changer, allowing us to not only match the technical prowess of tools like Jaeger and Grafana but also to create a platform that was a breeze for users across the board. This move drastically improved how we managed incidents and accessed critical data, boosting our efficiency across the board. Our dive into user research, with feedback from over 200 people, was enlightening. It showed us the critical role of alerts and the value of seamless navigation between tools, guiding us towards an iframe integration approach that balanced cost and functionality beautifully. The collaboration between our designers and developers crafted a platform that wasn't just technically sound but also a hit with our users due to its intuitive design. Looking back, embracing design thinking from the start was a game-changer. It proved that focusing on the user experience, alongside leveraging tech like AI, leads to solutions that are not just smart but also straightforward and engaging. The success of the project wasn't just in meeting our technical goals but also in fostering a culture of innovation and continuous improvement. Key Traits of Effective Designers Begin with the user in mind Iterate and validate constantly Team up Stay curious about new tech Peering into what's ahead, it's crystal clear that artificial intelligence and cutting-edge tech are set to deeply influence how we craft user experiences, pushing the boundaries of what's expected and truly addressing what users need.

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