|
[1]
|
E. Abi Jaber and O. El Euch, Multifactor approximation of rough volatility models, SIAM Journal on Financial Mathematics, 10 (2019), 309-349.
doi: 10.1137/18M1170236.
|
|
[2]
|
L. Andersen, A simple approach to the pricing of Bermudan swaptions in the multi-factor Libor Market model, Mathematical Finance, 3 (1999), 5-32.
|
|
[3]
|
A. Bakan, Representation of measures with polynomial denseness in ${L}_p(\mathbb{R}, d \mu)$, $ 0 < p < \infty$, and its application to determinate moment problems, Proceedings of the American Mathematical Society, 136 (2008), 3579-3589.
doi: 10.1090/S0002-9939-08-09418-5.
|
|
[4]
|
V. Bally and G. Pagès, A quantization algorithm for solving multi-dimensional discrete-time optimal stopping problems, Bernoulli, 9 (2003), 1003-1049.
doi: 10.3150/bj/1072215199.
|
|
[5]
|
V. Bally, G. Pagès and J. Printems, A quantization tree method for pricing and hedging multidimensional American options, Mathematical Finance, 15 (2005), 119-168.
doi: 10.1111/j.0960-1627.2005.00213.x.
|
|
[6]
|
P. Bank and D. Besslich, On Lenglart's theory of Meyer-sigma-fields and El Karoui's theory of optimal stopping, arXiv: 1810.08485, Preprint, 2019.
|
|
[7]
|
J. Barraquand and D. Martineau, Numerical valuation of high dimensional multivariate American securities, The Journal of Financial and Quantitative Analysis, 30 (1995), 383-405.
|
|
[8]
|
C. Bayer, M. Eigel, L. Sallandt and P. Trunschke, Pricing high-dimensional Bermudan options with hierarchical tensor formats, SIAM Journal on Financial Mathematics, 14 (2023), 383-406.
doi: 10.1137/21M1402170.
|
|
[9]
|
C. Bayer, P. Friz and J. Gatheral, Pricing under rough volatility, Quantitative Finance, 16 (2016), 887-904.
doi: 10.1080/14697688.2015.1099717.
|
|
[10]
|
S. Becker, P. Cheridito and A. Jentzen, Deep optimal stopping, Journal of Machine Learning Research, 20 (2019), Paper No. 74, 25 pp.
|
|
[11]
|
S. Becker, P. Cheridito and A. Jentzen, Pricing and hedging American-style options with deep learning, Journal of Risk and Financial Management, 13 (2020), 158.
doi: 10.3390/jrfm13070158.
|
|
[12]
|
D. P. Bertsekas and J. N. Tsitsiklis, Neuro-Dynamic Programming, Athena Scientific, 1996.
|
|
[13]
|
B. Bouchard and X. Warin, Monte-Carlo valuation of American options: Facts and new algorithms to improve existing methods, in Proceedings of the Numerical Methods in Finance: Bordeaux, June 2010, Springer, 2012,215-255.
doi: 10.1007/978-3-642-25746-9_7.
|
|
[14]
|
P. P. Boyle, A. W. Kolkiewicz and K. S. Tan, An improved simulation method for pricing high-dimensional American derivatives, Mathematics and Computers in Simulation, 62 (2003), 315-322.
doi: 10.1016/S0378-4754(02)00248-3.
|
|
[15]
|
M. Broadie and P. Glasserman, A stochastic mesh method for pricing high-dimensional American options, Journal of Computational Finance, 7 (2004), 35-72.
doi: 10.21314/JCF.2004.117.
|
|
[16]
|
W. Cao, X. Wang, Z. Ming and J. Gao, A review on neural networks with random weights, Neurocomputing, 275 (2018), 278-287.
doi: 10.1016/j.neucom.2017.08.040.
|
|
[17]
|
J. F. Carriere, Valuation of the early-exercise price for options using simulations and nonparametric regression, Insurance: Mathematics and Economics, 19 (1996), 19-30.
doi: 10.1016/S0167-6687(96)00004-2.
|
|
[18]
|
S. Chen, A. M. Devraj, A. Bušić and S. Meyn, Zap Q-learning for optimal stopping, in Proceedings of the 2020 American Control Conference (ACC), IEEE, 2020, 3920-3925.
doi: 10.23919/ACC45564.2020.9147481.
|
|
[19]
|
E. Chevalier, S. Pulido and E. Zúñiga, American options in the Volterra Heston model, SIAM Journal on Financial Mathematics, 13 (2022), 426-458.
doi: 10.1137/21M140674X.
|
|
[20]
|
E. Clément, D. Lamberton and P. Protter, An Analysis of the Longstaff-Schwartz Algorithm for American Option Pricing, Technical report, Cornell University Operations Research and Industrial Engineering, 2001.
|
|
[21]
|
J. C. Cox, S. A. Ross and M. Rubinstein, Option pricing: A simplified approach, Journal of Financial Economics, 7 (1979), 229-263.
doi: 10.1016/0304-405X(79)90015-1.
|
|
[22]
|
M. de Bellefroid, The Derivatives Academy, 2022., https://bookdown.org/maxime_debellefroid/MyBook/.
|
|
[23]
|
D. Egloff, Monte Carlo algorithms for optimal stopping and statistical learning, The Annals of Applied Probability, 15 (2005), 1396-1432.
doi: 10.1214/105051605000000043.
|
|
[24]
|
D. Egloff, M. Kohler and N. Todorovic, A dynamic look-ahead Monte Carlo algorithm for pricing Bermudan options, The Annals of Applied Probability, 17 (2007), 1138-1171.
doi: 10.1214/105051607000000249.
|
|
[25]
|
O. El Euch, M. Fukasawa and M. Rosenbaum, The microstructural foundations of leverage effect and rough volatility, Finance and Stochastics, 22 (2018), 241-280.
doi: 10.1007/s00780-018-0360-z.
|
|
[26]
|
O. El Euch, J. Gatheral and M. Rosenbaum, Roughening Heston, Risk, (2019), 84-89.
doi: 10.2139/ssrn.3116887.
|
|
[27]
|
O. El Euch and M. Rosenbaum, Perfect hedging in rough Heston models, The Annals of Applied Probability, 28 (2018), 3813-3856.
doi: 10.1214/18-AAP1408.
|
|
[28]
|
N. El Karoui, Les aspects probabilistes du controle stochastique, in Proceedings of the École d'été de Probabilités de Saint-Flour IX-1979, Springer, 1981, 73-238.
|
|
[29]
|
H. Föllmer and A. Schied, Stochastic Finance: An Introduction in Discrete Time, De Gruyter, 2016.
doi: 10.1515/9783110463453.
|
|
[30]
|
C. Gallicchio, A. Micheli and L. Pedrelli, Deep reservoir computing: A critical experimental analysis, Neurocomputing, 268 (2017), 87-99.
doi: 10.1016/j.neucom.2016.12.089.
|
|
[31]
|
D. García, Convergence and biases of Monte Carlo estimates of American option prices using a parametric exercise rule, Journal of Economic Dynamics and Control, 27 (2003), 1855-1879.
doi: 10.1016/S0165-1889(02)00086-6.
|
|
[32]
|
J. Gatheral, T. Jaisson and M. Rosenbaum, Volatility is rough, Quantitative Finance, 18 (2018), 933-949.
doi: 10.1080/14697688.2017.1393551.
|
|
[33]
|
J. Gatheral, P. Jusselin and M. Rosenbaum, The quadratic rough heston model and the joint S&P 500/VIX smile calibration problem, arXiv: 2001.01789, Preprint, 2020.
|
|
[34]
|
E. Gobet, J.-P. Lemor and X. Warin, A regression-based Monte Carlo method to solve backward stochastic differential equations, The Annals of Applied Probability, 15 (2005), 2172-2202.
doi: 10.1214/105051605000000412.
|
|
[35]
|
L. Gonon and J.-P. Ortega, Reservoir computing universality with stochastic inputs, IEEE Transactions on Neural Networks and Learning Systems, 31 (2020), 100-112.
doi: 10.1109/TNNLS.2019.2899649.
|
|
[36]
|
A. N. Gorban, I. Y. Tyukin, D. V. Prokhorov and K. I. Sofeikov, Approximation with random bases: Pro et contra, Information Sciences, 364 (2016), 129-145.
doi: 10.1016/j.ins.2015.09.021.
|
|
[37]
|
H. Hanbali and D. Linders, American-type basket option pricing: A simple two-dimensional partial differential equation, Quantitative Finance, 19 (2019), 1689-1704.
doi: 10.1080/14697688.2019.1588987.
|
|
[38]
|
M. B. Haugh and L. Kogan, Pricing American options: A duality approach, Operations Research, 52 (2004), 258-270.
doi: 10.1287/opre.1030.0070.
|
|
[39]
|
C. Herrera, F. Krach and J. Teichmann, Estimating full Lipschitz constants of deep neural networks, arXiv: 2004.13135, Preprint, 2020.
|
|
[40]
|
S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies, 6 (1993), 327-343.
doi: 10.1093/rfs/6.2.327.
|
|
[41]
|
K. Hornik, Approximation capabilities of multilayer feedforward networks, Neural Networks, 4 (1991), 251-257.
doi: 10.1016/0893-6080(91)90009-T.
|
|
[42]
|
G.-B. Huang, L. Chen, C. K. Siew, et al., Universal approximation using incremental constructive feedforward networks with random Hidden nodes, IEEE Transactions on Neural Networks, 17 (2006), 879-892.
|
|
[43]
|
S. Jain and C. W. Oosterlee, The stochastic grid bundling method: Efficient pricing of Bermudan options and their Greeks, Applied Mathematics and Computation, 269 (2015), 412-431.
doi: 10.1016/j.amc.2015.07.085.
|
|
[44]
|
L. P. Kaelbling, M. L. Littman and A. W. Moore, Reinforcement learning: A survey, Journal of Artificial Intelligence Research, 4 (1996), 237-285.
doi: 10.1613/jair.301.
|
|
[45]
|
M. Kohler, A. Krzyżak and N. Todorovic, Pricing of high-dimensional American options by neural networks, Mathematical Finance, 20 (2010), 383-410.
doi: 10.1111/j.1467-9965.2010.00404.x.
|
|
[46]
|
A. Kolodko and J. Schoenmakers, Iterative construction of the optimal Bermudan stopping time, Finance and Stochastic, 10 (2006), 27-49.
doi: 10.1007/s00780-005-0168-5.
|
|
[47]
|
B. Lapeyre and J. Lelong, Neural network regression for Bermudan option pricing, Monte Carlo Methods and Applications, 27 (2021), 227-247.
doi: 10.1515/mcma-2021-2091.
|
|
[48]
|
P. Letourneau and L. Stentoft, Simulated Greeks for American options, Quantitative Finance, 23 (2023), 653-676.
doi: 10.1080/14697688.2022.2159869.
|
|
[49]
|
Y. Li, C. Szepesvari and D. Schuurmans, Learning exercise policies for American options, in Proceedings of the Twelth International Conference on Artificial Intelligence and Statistics, PMLR, 2009,352-359.
|
|
[50]
|
G. Livieri, S. Mouti, A. Pallavicini and M. Rosenbaum, Rough volatility: Evidence from option prices, IISE Transactions, 50 (2018), 767-776.
|
|
[51]
|
F. A. Longstaff and E. S. Schwartz, Valuing American options by simulation: A simple least-squares approach, The Review of Financial Studies, 14 (2001), 113-147.
doi: 10.1093/rfs/14.1.113.
|
|
[52]
|
M. Lukoševičius and H. Jaeger, Reservoir computing approaches to recurrent neural network training, Computer Science Review, 3 (2009), 127-149.
|
|
[53]
|
G. Pagès, Numerical Probability: An Introduction with Applications to Finance, 1st edition, Springer, 2018.
doi: 10.1007/978-3-319-90276-0.
|
|
[54]
|
H. Pham, Optimal stopping, free boundary, and American option in a jump-diffusion model, Applied Mathematics and Optimization, 35 (1997), 145-164.
|
|
[55]
|
L. C. G. Rogers, Monte Carlo valuation of American options, Mathematical Finance, 12 (2002), 271-286.
doi: 10.1111/1467-9965.02010.
|
|
[56]
|
L. C. G. Rogers, Dual valuation and hedging of Bermudan options, SIAM Journal on Financial Mathematics, 1 (2010), 604-608.
doi: 10.1137/090772198.
|
|
[57]
|
A. M. Schäfer and H. G. Zimmermann, Recurrent neural networks are universal approximators, in Proceedings of the 16th International Conference on Artificial Neural Networks–ICANN 2006: Athens, Greece, Springer, 2006,632-640.
|
|
[58]
|
B. Schrauwen, D. Verstraeten and J. V. Campenhout, An overview of reservoir computing: Theory, applications and implementations, in Proceedings of the 15th European Symposium on Artificial Neural Networks, 2007,471-482.
|
|
[59]
|
M. Schweizer, On Bermudan options, in Advances in Finance and Stochastics: Essays in Honour of Dieter Sondermann, Springer, Berlin, Heidelberg, 2002,257-270.
|
|
[60]
|
L. Stentoft, Convergence of the least squares Monte Carlo approach to American option valuation, Management Science, 50 (2004), 1193-1203.
doi: 10.1287/mnsc.1030.0155.
|
|
[61]
|
R. S. Sutton and A. G. Barto, Reinforcement Learning: An Introduction, MIT press, 2018.
|
|
[62]
|
J. A. Tilley, Valuing American options in a path simulation model, Insurance Mathematics and Economics, 2 (1995), 169.
|
|
[63]
|
J. N. Tsitsiklis and B. Van Roy, Optimal stopping of Markov processes: Hilbert space theory, approximation algorithms, and an application to pricing high-dimensional financial derivatives, IEEE Transactions on Automatic Control, 44 (1999), 1840-1851.
doi: 10.1109/9.793723.
|
|
[64]
|
J. N. Tsitsiklis and B. Van Roy, Regression methods for pricing complex American-style options, IEEE Transactions on Neural Networks, 12 (2001), 694-703.
doi: 10.1109/72.935083.
|
|
[65]
|
D. Verstraeten, B. Schrauwen, M. D'Haene and D. Stroobandt, An experimental unification of reservoir computing methods, Neural Networks, 20 (2007), 391-403.
doi: 10.1016/j.neunet.2007.04.003.
|
|
[66]
|
H. Yu and D. P. Bertsekas, Q-learning algorithms for optimal stopping based on least squares, in Proceedings of the 2007 European Control Conference (ECC), IEEE, 2007, 2368-2375.
doi: 10.23919/ECC.2007.7068523.
|
|
[67]
|
D. Z. Zanger, Convergence of a least-squares Monte Carlo algorithm for bounded approximating sets, Applied Mathematical Finance, 16 (2009), 123-150.
doi: 10.1080/13504860802516881.
|
|
[68]
|
D. Z. Zanger, Quantitative error estimates for a least-squares Monte Carlo algorithm for American option pricing, Finance and Stochastics, 17 (2013), 503-534.
doi: 10.1007/s00780-013-0204-9.
|
|
[69]
|
D. Z. Zanger, Convergence of a least-squares Monte Carlo algorithm for American option pricing with dependent sample data, Mathematical Finance, 28 (2018), 447-479.
doi: 10.1111/mafi.12125.
|
|
[70]
|
D. Z. Zanger, General error estimates for the Longstaff-Schwartz least-squares Monte Carlo algorithm, Mathematics of Operations Research, 45 (2020), 923-946.
doi: 10.1287/moor.2019.1017.
|
|
[71]
|
R. Zhang, Y. Lan, G.-B. Huang and Z.-B. Xu, Universal approximation of extreme learning machine with adaptive growth of Hidden nodes, IEEE Transactions on Neural Networks and Learning Systems, 23 (2012), 365-371.
doi: 10.1109/TNNLS.2011.2178124.
|
Download:
{{journal.titleEn}} 
Open Access Under a Creative Commons 
