# 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. Google Scholar [2] L. Andersen and V. Piterbarg, Interest Rate Modeling, no. v. 2 in Interest Rate Modeling, Atlantic Financial Press, 2010.   Google Scholar [3] A. Antonov and M. Spector, General short-rate analytics, Risk, 66–71. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] D. Brigo and F. Mercurio, Interest Rate Models – Theory and Practice with Smile, Inflation and Credit, 2nd edition, Springer Verlag, 2006.  Google Scholar [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.  Google Scholar [9] P. Carr and A. Itkin, An expanded local variance gamma model. Google Scholar [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. Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] C. Dias, A method of recursive images to solve transient heat diffusionin multilayer materials, 85, 1075–1083. Google Scholar [16] B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20.   Google Scholar [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.   Google Scholar [18] J.-S. Giet, P. Vallois and S. Wantz-Mézieres, The logistic S.D.E., Theory Stoch. Process., 20 (2015), 28-62.   Google Scholar [19] Y. C. Hon and X. Z. Mao, A radial basis function method for solving options pricing model, Financial Engineering, 8 (1999), 31-49.   Google Scholar [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. Google Scholar [21] J. Hull, Options, Futures, and Other Derivatives, 8th edition, Prentice Hall, 2011. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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. Google Scholar [27] A. Itkin and D. Muravey, Semi-closed form prices of barrier options in the Hull-White model, Risk. Google Scholar [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. Google Scholar [29] E. Kartashov, Analytical Methods in the Theory of Heat Conduction in Solids, Vysshaya Shkola, Moscow, 2001. Google Scholar [30] A. Kuznetsov, On the convergence of the Gaver-Stehfest algorithm, SIAM J. Numer. Anal., 51 (2013), 2984-2998.  doi: 10.1137/13091974X.  Google Scholar [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. Google Scholar [32] A. Lejay, On the constructions of the skew Brownian motion, Probab. Surv., 3 (2006), 413-466.  doi: 10.1214/154957807000000013.  Google Scholar [33] J. Lienhard IV and J. Lienhard V, A Heat Transfer Textbook, 5th edition, Phlogiston Press, Cambridge, MA, 2019.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [37] A. Lipton and V. Kaushansky, On three important problems in mathematical finance, The Journal of Derivatives. Special Issue, 28. Google Scholar [38] A. Lipton and A. Sepp, Filling the gaps, Risk Magazine, 86–91. Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [42] A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC, 2002.  Google Scholar [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.  Google Scholar [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.  Google Scholar [45] A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics, Pergamon Press, Oxford, 1963.   Google Scholar [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. Google Scholar [47] P. Verhulst, Notice sur la loi que la population suit dans son accroisseement, Correspondance Mathematique et Physique, 10 (1838), 113-121.   Google Scholar

show all references

##### References:
 [1] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover Publications, Inc., 1964. Google Scholar [2] L. Andersen and V. Piterbarg, Interest Rate Modeling, no. v. 2 in Interest Rate Modeling, Atlantic Financial Press, 2010.   Google Scholar [3] A. Antonov and M. Spector, General short-rate analytics, Risk, 66–71. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] D. Brigo and F. Mercurio, Interest Rate Models – Theory and Practice with Smile, Inflation and Credit, 2nd edition, Springer Verlag, 2006.  Google Scholar [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.  Google Scholar [9] P. Carr and A. Itkin, An expanded local variance gamma model. Google Scholar [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. Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] C. Dias, A method of recursive images to solve transient heat diffusionin multilayer materials, 85, 1075–1083. Google Scholar [16] B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20.   Google Scholar [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.   Google Scholar [18] J.-S. Giet, P. Vallois and S. Wantz-Mézieres, The logistic S.D.E., Theory Stoch. Process., 20 (2015), 28-62.   Google Scholar [19] Y. C. Hon and X. Z. Mao, A radial basis function method for solving options pricing model, Financial Engineering, 8 (1999), 31-49.   Google Scholar [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. Google Scholar [21] J. Hull, Options, Futures, and Other Derivatives, 8th edition, Prentice Hall, 2011. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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. Google Scholar [27] A. Itkin and D. Muravey, Semi-closed form prices of barrier options in the Hull-White model, Risk. Google Scholar [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. Google Scholar [29] E. Kartashov, Analytical Methods in the Theory of Heat Conduction in Solids, Vysshaya Shkola, Moscow, 2001. Google Scholar [30] A. Kuznetsov, On the convergence of the Gaver-Stehfest algorithm, SIAM J. Numer. Anal., 51 (2013), 2984-2998.  doi: 10.1137/13091974X.  Google Scholar [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. Google Scholar [32] A. Lejay, On the constructions of the skew Brownian motion, Probab. Surv., 3 (2006), 413-466.  doi: 10.1214/154957807000000013.  Google Scholar [33] J. Lienhard IV and J. Lienhard V, A Heat Transfer Textbook, 5th edition, Phlogiston Press, Cambridge, MA, 2019.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [37] A. Lipton and V. Kaushansky, On three important problems in mathematical finance, The Journal of Derivatives. Special Issue, 28. Google Scholar [38] A. Lipton and A. Sepp, Filling the gaps, Risk Magazine, 86–91. Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [42] A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC, 2002.  Google Scholar [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.  Google Scholar [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.  Google Scholar [45] A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics, Pergamon Press, Oxford, 1963.   Google Scholar [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. Google Scholar [47] P. Verhulst, Notice sur la loi que la population suit dans son accroisseement, Correspondance Mathematique et Physique, 10 (1838), 113-121.   Google Scholar
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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