# American Institute of Mathematical Sciences

March  2022, 1(1): 99-135. doi: 10.3934/fmf.2021004

## Multilayer heat equations: Application to finance

 1 Tandon School of Engineering, New York University, 1 Metro Tech Center, 10th floor, Brooklyn NY, USA 2 Connection Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA 3 The Jerusalem School of Business Administration, The Hebrew University of Jerusalem, Jerusalem, Israel 4 Moscow State University, Moscow, Russia

* Corresponding author: Andrey Itkin

Received  February 2021 Revised  June 2021 Published  March 2022 Early access  June 2021

Fund Project: Dmitry Muravey acknowledges support by the Russian Science Foundation under the Grant number 20-68-47030

In this paper, we develop a Multilayer (ML) method for solving one-factor parabolic equations. Our approach provides a powerful alternative to the well-known finite difference and Monte Carlo methods. We discuss various advantages of this approach, which judiciously combines semi-analytical and numerical techniques and provides a fast and accurate way of finding solutions to the corresponding equations. To introduce the core of the method, we consider multilayer heat equations, known in physics for a relatively long time but never used when solving financial problems. Thus, we expand the analytic machinery of quantitative finance by augmenting it with the ML method. We demonstrate how one can solve various problems of mathematical finance by using our approach. Specifically, we develop efficient algorithms for pricing barrier options for time-dependent one-factor short-rate models, such as Black-Karasinski and Verhulst. Besides, we show how to solve the well-known Dupire equation quickly and accurately. Numerical examples confirm that our approach is considerably more efficient for solving the corresponding partial differential equations than the conventional finite difference method by being much faster and more accurate than the known alternatives.

Citation: Andrey Itkin, Alexander Lipton, Dmitry Muravey. Multilayer heat equations: Application to finance. Frontiers of Mathematical Finance, 2022, 1 (1) : 99-135. doi: 10.3934/fmf.2021004
##### References:
 [1] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover Publications, Inc., 1964. [2] L. Andersen and V. Piterbarg, Interest Rate Modeling, no. v. 2 in Interest Rate Modeling, Atlantic Financial Press, 2010. [3] A. Antonov and M. Spector, General short-rate analytics, Risk, 66–71. [4] M. Asvestas, A. G. Sifalakis, E. P. Papadopoulou and Y. G. Saridakis, Fokas method for a multi-domain linear reaction-diffusion equation with discontinuous diffusivity, Journal of Physics: Conference Series, 490 (2014), 012143. doi: 10.1088/1742-6596/490/1/012143. [5] N. Bacaër, A short history of mathematical population dynamics, Springer-Verlag, London, chapter 6, (2011), 35–39. doi: 10.1007/978-0-85729-115-8. [6] F. Black and P. Karasinski, Bond and option pricing when short rates are lognormal, Financial Analysts Journal, 47 (1991), 52-59.  doi: 10.2469/faj.v47.n4.52. [7] D. Brigo and F. Mercurio, Interest Rate Models – Theory and Practice with Smile, Inflation and Credit, 2nd edition, Springer Verlag, 2006. [8] P. Carr and A. Itkin, Geometric local variance gamma model, The Journal of Derivatives Winter, 27 (2019), 7-30.  doi: 10.3905/jod.2019.1.084. [9] P. Carr and A. Itkin, An expanded local variance gamma model. [10] P. Carr and A. Itkin, Semi-closed form solutions for barrier and American options written on a time-dependent Ornstein Uhlenbeck process, Journal of Derivatives, Fall. [11] P. Carr, A. Itkin and D. Muravey, Semi-closed form prices of barrier options in the time-dependent cev and cir models, Journal of Derivatives, 28 (2020), 26-50. [12] E. J. Carr and N. G. March, Semi-analytical solution of multilayer diffusion problems with time-varying boundary conditions and general interface conditions, Appl. Math. Comput., 333 (2018), 286-303.  doi: 10.1016/j.amc.2018.03.095. [13] P. Carr and S. Nadtochiy, Local variance gamma and explicit calibration to option prices, Math. Finance, 27 (2017), 151-193.  doi: 10.1111/mafi.12086. [14] M. Craddock, Fundamental solutions, transition densities and the integration of Lie symmetries, J. Differential Equations, 246 (2009), 2538-2560.  doi: 10.1016/j.jde.2008.10.017. [15] C. Dias, A method of recursive images to solve transient heat diffusionin multilayer materials, 85, 1075–1083. [16] B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20. [17] G. E. Fasshauer, A. Q. M. Khaliq and D. A. Voss, Using meshfree approximation for multi-asset American option problems, J. Chinese Inst. Engrs., 27 (2004), 563-571. [18] J.-S. Giet, P. Vallois and S. Wantz-Mézieres, The logistic S.D.E., Theory Stoch. Process., 20 (2015), 28-62. [19] Y. C. Hon and X. Z. Mao, A radial basis function method for solving options pricing model, Financial Engineering, 8 (1999), 31-49. [20] B. Horvath, A. Jacquier and C. Turfus, Analytic option prices for the black-karasinski short rate model, 2017, URL https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3253833, SSRN: 3253833. [21] J. Hull, Options, Futures, and Other Derivatives, 8th edition, Prentice Hall, 2011. [22] A. Itkin, Pricing Derivatives Under Lévy Models, 1st edition, no. 12 in Pseudo-Differential Operators, Birkhäuser, Basel, 2017. doi: 10.1007/978-1-4939-6792-6. [23] A. Itkin, Fitting Local Volatility: Analytic and Numerical Approaches in Black-Scholes and Local Variance Gamma Models, 11623, World Scientific Publishing Co. Pte. Ltd., 2020. doi: 10.1142/11623. [24] A. Itkin and A. Lipton, Filling the gaps smoothly, J. Comput. Sci., 24 (2018), 195-208.  doi: 10.1016/j.jocs.2017.02.003. [25] A. Itkin, A. Lipton and D. Muravey, From the black-karasinski to the verhulst model to accommodate the unconventional fed's policy, 2020, URL https://arXiv.org/abs/2006.11976. [26] A. Itkin and D. Muravey, Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit, 2020, URL https://arXiv.org/abs/2009.09342. [27] A. Itkin and D. Muravey, Semi-closed form prices of barrier options in the Hull-White model, Risk. [28] E. M. Kartashov, Analytical methods for solution of non-stationary heat conductance boundary problems in domains with moving boundaries, Izvestiya RAS, Energetika, 133–185. [29] E. Kartashov, Analytical Methods in the Theory of Heat Conduction in Solids, Vysshaya Shkola, Moscow, 2001. [30] A. Kuznetsov, On the convergence of the Gaver-Stehfest algorithm, SIAM J. Numer. Anal., 51 (2013), 2984-2998.  doi: 10.1137/13091974X. [31] P. Lançon, G. Batrouni, L. Lobry and N. Ostrowsky, Drift without flux: Brownian walker with a space-dependent diffusion coefficient, Europhysics Letters (EPL), 54 (2001), 28–34. [32] A. Lejay, On the constructions of the skew Brownian motion, Probab. Surv., 3 (2006), 413-466.  doi: 10.1214/154957807000000013. [33] J. Lienhard IV and J. Lienhard V, A Heat Transfer Textbook, 5th edition, Phlogiston Press, Cambridge, MA, 2019. [34] A. Lipton, Mathematical Methods for Foreign Exchange: A Financial Engineer's Approach, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/4694. [35] A. Lipton and M. L. de Prado, A closed-form solution for optimal mean-reverting trading strategies, Risk, Available at SSRN, (2020), 32pp. doi: 10.2139/ssrn.3534445. [36] A. Lipton and V. Kaushansky, On the first hitting time density for a reducible diffusion process, Quant. Finance, 20 (2020), 723-743.  doi: 10.1080/14697688.2020.1713394. [37] A. Lipton and V. Kaushansky, On three important problems in mathematical finance, The Journal of Derivatives. Special Issue, 28. [38] A. Lipton and A. Sepp, Filling the gaps, Risk Magazine, 86–91. [39] D. Mumford, C. M. M. Nori, E. Previato and M. Stillman, Tata Lectures on Theta, Progress in Mathematics, Birkhäuser Boston, 1984. doi: 10.1007/978-0-8176-4578-6. [40] O. A. Ole${{\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}}$nik and E. V. Radkevič, Second Order Equations with Non-Negative Characteristic Form, Plenum Press, New York-London, 1973. [41] U. Pettersson, E. Larsson, G. Marcusson and J. Persson, Improved radial basis function methods for multi-dimensional option pricing, J. Comput. Appl. Math., 222 (2008), 82-93.  doi: 10.1016/j.cam.2007.10.038. [42] A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC, 2002. [43] G. Pontrelli, M. Lauricella, J. A. Ferreira and G. Pena, Iontophoretic transdermal drug delivery: A multi-layered approach, Math. Med. Biol., 34 (2017), 559-576.  doi: 10.1093/imammb/dqw017. [44] B. Stehlíková and L. Capriotti, An Effective Approximation for Zero-Coupon Bonds and Arrow-Debreu Prices in the Black-Karasinski Model, Int. J. Theor. Appl. Finance, 17 (2014), 1450037, 16 pp. doi: 10.1142/S021902491450037X. [45] A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics, Pergamon Press, Oxford, 1963. [46] C. Turfus, Analytic swaption pricing in the black-karasinski model, 2020, URL https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3253866, SSRN: 3253866. [47] P. Verhulst, Notice sur la loi que la population suit dans son accroisseement, Correspondance Mathematique et Physique, 10 (1838), 113-121.

show all references

##### References:
 [1] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover Publications, Inc., 1964. [2] L. Andersen and V. Piterbarg, Interest Rate Modeling, no. v. 2 in Interest Rate Modeling, Atlantic Financial Press, 2010. [3] A. Antonov and M. Spector, General short-rate analytics, Risk, 66–71. [4] M. Asvestas, A. G. Sifalakis, E. P. Papadopoulou and Y. G. Saridakis, Fokas method for a multi-domain linear reaction-diffusion equation with discontinuous diffusivity, Journal of Physics: Conference Series, 490 (2014), 012143. doi: 10.1088/1742-6596/490/1/012143. [5] N. Bacaër, A short history of mathematical population dynamics, Springer-Verlag, London, chapter 6, (2011), 35–39. doi: 10.1007/978-0-85729-115-8. [6] F. Black and P. Karasinski, Bond and option pricing when short rates are lognormal, Financial Analysts Journal, 47 (1991), 52-59.  doi: 10.2469/faj.v47.n4.52. [7] D. Brigo and F. Mercurio, Interest Rate Models – Theory and Practice with Smile, Inflation and Credit, 2nd edition, Springer Verlag, 2006. [8] P. Carr and A. Itkin, Geometric local variance gamma model, The Journal of Derivatives Winter, 27 (2019), 7-30.  doi: 10.3905/jod.2019.1.084. [9] P. Carr and A. Itkin, An expanded local variance gamma model. [10] P. Carr and A. Itkin, Semi-closed form solutions for barrier and American options written on a time-dependent Ornstein Uhlenbeck process, Journal of Derivatives, Fall. [11] P. Carr, A. Itkin and D. Muravey, Semi-closed form prices of barrier options in the time-dependent cev and cir models, Journal of Derivatives, 28 (2020), 26-50. [12] E. J. Carr and N. G. March, Semi-analytical solution of multilayer diffusion problems with time-varying boundary conditions and general interface conditions, Appl. Math. Comput., 333 (2018), 286-303.  doi: 10.1016/j.amc.2018.03.095. [13] P. Carr and S. Nadtochiy, Local variance gamma and explicit calibration to option prices, Math. Finance, 27 (2017), 151-193.  doi: 10.1111/mafi.12086. [14] M. Craddock, Fundamental solutions, transition densities and the integration of Lie symmetries, J. Differential Equations, 246 (2009), 2538-2560.  doi: 10.1016/j.jde.2008.10.017. [15] C. Dias, A method of recursive images to solve transient heat diffusionin multilayer materials, 85, 1075–1083. [16] B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20. [17] G. E. Fasshauer, A. Q. M. Khaliq and D. A. Voss, Using meshfree approximation for multi-asset American option problems, J. Chinese Inst. Engrs., 27 (2004), 563-571. [18] J.-S. Giet, P. Vallois and S. Wantz-Mézieres, The logistic S.D.E., Theory Stoch. Process., 20 (2015), 28-62. [19] Y. C. Hon and X. Z. Mao, A radial basis function method for solving options pricing model, Financial Engineering, 8 (1999), 31-49. [20] B. Horvath, A. Jacquier and C. Turfus, Analytic option prices for the black-karasinski short rate model, 2017, URL https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3253833, SSRN: 3253833. [21] J. Hull, Options, Futures, and Other Derivatives, 8th edition, Prentice Hall, 2011. [22] A. Itkin, Pricing Derivatives Under Lévy Models, 1st edition, no. 12 in Pseudo-Differential Operators, Birkhäuser, Basel, 2017. doi: 10.1007/978-1-4939-6792-6. [23] A. Itkin, Fitting Local Volatility: Analytic and Numerical Approaches in Black-Scholes and Local Variance Gamma Models, 11623, World Scientific Publishing Co. Pte. Ltd., 2020. doi: 10.1142/11623. [24] A. Itkin and A. Lipton, Filling the gaps smoothly, J. Comput. Sci., 24 (2018), 195-208.  doi: 10.1016/j.jocs.2017.02.003. [25] A. Itkin, A. Lipton and D. Muravey, From the black-karasinski to the verhulst model to accommodate the unconventional fed's policy, 2020, URL https://arXiv.org/abs/2006.11976. [26] A. Itkin and D. Muravey, Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit, 2020, URL https://arXiv.org/abs/2009.09342. [27] A. Itkin and D. Muravey, Semi-closed form prices of barrier options in the Hull-White model, Risk. [28] E. M. Kartashov, Analytical methods for solution of non-stationary heat conductance boundary problems in domains with moving boundaries, Izvestiya RAS, Energetika, 133–185. [29] E. Kartashov, Analytical Methods in the Theory of Heat Conduction in Solids, Vysshaya Shkola, Moscow, 2001. [30] A. Kuznetsov, On the convergence of the Gaver-Stehfest algorithm, SIAM J. Numer. Anal., 51 (2013), 2984-2998.  doi: 10.1137/13091974X. [31] P. Lançon, G. Batrouni, L. Lobry and N. Ostrowsky, Drift without flux: Brownian walker with a space-dependent diffusion coefficient, Europhysics Letters (EPL), 54 (2001), 28–34. [32] A. Lejay, On the constructions of the skew Brownian motion, Probab. Surv., 3 (2006), 413-466.  doi: 10.1214/154957807000000013. [33] J. Lienhard IV and J. Lienhard V, A Heat Transfer Textbook, 5th edition, Phlogiston Press, Cambridge, MA, 2019. [34] A. Lipton, Mathematical Methods for Foreign Exchange: A Financial Engineer's Approach, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/4694. [35] A. Lipton and M. L. de Prado, A closed-form solution for optimal mean-reverting trading strategies, Risk, Available at SSRN, (2020), 32pp. doi: 10.2139/ssrn.3534445. [36] A. Lipton and V. Kaushansky, On the first hitting time density for a reducible diffusion process, Quant. Finance, 20 (2020), 723-743.  doi: 10.1080/14697688.2020.1713394. [37] A. Lipton and V. Kaushansky, On three important problems in mathematical finance, The Journal of Derivatives. Special Issue, 28. [38] A. Lipton and A. Sepp, Filling the gaps, Risk Magazine, 86–91. [39] D. Mumford, C. M. M. Nori, E. Previato and M. Stillman, Tata Lectures on Theta, Progress in Mathematics, Birkhäuser Boston, 1984. doi: 10.1007/978-0-8176-4578-6. [40] O. A. Ole${{\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}}$nik and E. V. Radkevič, Second Order Equations with Non-Negative Characteristic Form, Plenum Press, New York-London, 1973. [41] U. Pettersson, E. Larsson, G. Marcusson and J. Persson, Improved radial basis function methods for multi-dimensional option pricing, J. Comput. Appl. Math., 222 (2008), 82-93.  doi: 10.1016/j.cam.2007.10.038. [42] A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC, 2002. [43] G. Pontrelli, M. Lauricella, J. A. Ferreira and G. Pena, Iontophoretic transdermal drug delivery: A multi-layered approach, Math. Med. Biol., 34 (2017), 559-576.  doi: 10.1093/imammb/dqw017. [44] B. Stehlíková and L. Capriotti, An Effective Approximation for Zero-Coupon Bonds and Arrow-Debreu Prices in the Black-Karasinski Model, Int. J. Theor. Appl. Finance, 17 (2014), 1450037, 16 pp. doi: 10.1142/S021902491450037X. [45] A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics, Pergamon Press, Oxford, 1963. [46] C. Turfus, Analytic swaption pricing in the black-karasinski model, 2020, URL https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3253866, SSRN: 3253866. [47] P. Verhulst, Notice sur la loi que la population suit dans son accroisseement, Correspondance Mathematique et Physique, 10 (1838), 113-121.
Internal layers constructed for the given external boundaries $y_0(t)$ and $y_N(t)$, and the number of layers $N$, by using 3 points for each boundary $y_i(t)$ and polynomial curves
Comparison of the Analytic and ML solutions (a), and Analytic, ML and FD solutions (grid with $41\times 40$ nodes) (b) for $\sigma_i = 0.5, T = 1$. Here Analytic denotes the analytic solution of the problem, ILT - the ML solution, FD - the FD solution, DiffILT - the relative error of the ML solution with respect to the analytic one, DifFD - same for the FD method
Comparison of the Analytic, ML and FD solutions for $\sigma_i = 0.3, T = 0.5$. Here Analytic denotes the analytic solution of the problem, ILT - the ML solution, FD - the FD solution, DiffILT - the relative error of the ML solution with respect to the analytic one, DifFD - same for the FD method
Comparison of the ML and FD solutions for a piecewise constant $\sigma(x)$. Here ILT denotes the ML solution, FD - the FD solution, Dif - the relative error of the FD solution with respect to the ML one
Parameters of the test
 $y_0$ $y_N$ $\sigma$ $T$ $N$ $m$ -1.0 1.0 0.5 1.0 20 16
 $y_0$ $y_N$ $\sigma$ $T$ $N$ $m$ -1.0 1.0 0.5 1.0 20 16
Parameters of the second experiment
 $y_0$ $y_N$ $T$ $N$ $m$ M -1.0 4.0 2.0 50 16 100
 $y_0$ $y_N$ $T$ $N$ $m$ M -1.0 4.0 2.0 50 16 100
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