Interpolation of missing swaption volatility data using variational autoencoders
Behaviormetrika - Trang 1-27 - 2023
Tóm tắt
Albeit of crucial interest for financial researchers, market-implied volatility data of European swaptions often exhibit large portions of missing quotes due to illiquidity of the underlying swaption instruments. In this case, standard stochastic interpolation tools like the common SABR model cannot be calibrated to observed volatility smiles, due to data being only available for the at-the-money quote of the respective underlying swaption. Here, we propose to infer the geometry of the full unknown implied volatility cube by learning stochastic latent representations of implied volatility cubes via variational autoencoders, enabling inference about the missing volatility data conditional on the observed data by an approximate Gibbs sampling approach. Up to our knowledge, our studies constitute the first-ever completely nonparametric approach to modeling swaption volatility using unsupervised learning methods while simultaneously tackling the issue of missing data. Since training data for the employed variational autoencoder model is usually sparsely available, we propose a novel method to generate synthetic swaption volatility data for training and afterwards test the robustness of our approach on real market quotes. In particular, we show that SABR interpolated volatilities calibrated to reconstructed volatility cubes with artificially imputed missing values differ by not much more than two basis points compared to SABR fits calibrated to the complete cube. Moreover, we demonstrate how the imputation can be used to successfully set up delta-neutral portfolios for hedging purposes.
Tài liệu tham khảo
Andreichenko P (2011) A parsimonious model for the joint evolution of yield vurces and the interest rate smile surface under the objective measure. Msc thesis, University of Oxford
Antonov A, Konikov M, Spector M (2019) Modern SABR analytics: formulas and insights for quants, former physicists and mathematicians. Springer
Arvanitidis G, Hansen LK, Hauberg S (2018) Latent space oddity: on the curvature of deep generative models. In: Proceedings of International Conference on Learning Representations
Bachman P, Precup D (2015) Data generation as sequential decision making. Advances in neural information processing systems. Springer, Cham
Bartlett B (2006) Hedging under the SABR model. Wilmott Mag 04(06):2–4
Brigo D, Mercurio F (2007) Interest rate models: theory and practice. Springer, Berlin
Burda Y, Grosse R, Salakhutdinov R (2016) Importance weighted autoencoders. In: 4th International Conference on Learning Representations (ICLR)
Camino R, Hammerschmidt C, State R (2019) Improving missing data imputation with deep generative models. arXiv:1902.10666
Collier M, Nazabal A, Williams C (2020) VAEs in the presence of missing data. arXiv:2006.05301
Crispoldi C, Wigger G, Larkin P (2016) SABR and SABR LIBOR market models in practice: with examples implemented in Python. Palgrave MacMillan, Hampshire
Dai B, Wang Y, Aston J, Wipf D (2018) Connections with robust PCA and the role of emergent sparsity in variational autoencoder models. J Mach Learn Res 19(1):1–42
Dimitroff G, de Kock J (2011) Calibrating and completing the volatility cube in the SABR model. In: Berichte des Fraunhofer-Instituts for Techno- und Wirtschaftsmathematik (ITWM Report 202)
Du C, Zhu J, Zhang B (2018) Learning deep generative models with doubly stochastic gradient MCMC. IEEE Trans Neural Netw Learn Syst 29(7):3084–3096
Dun T, Schlögl E, Barton G (2001) Simulated swaption delta-hedging in the lognormal forward LIBOR model. Int J Theor Appl Finance 4(4):677–709
Dutilleul P (1999) The MLE algorithm for the matrix normal distribution. J Stat Comput Simul 64(2):105–123
Flegal JM, Jones GL (2011) Implementing MCMC: estimating with confidence. In: Brooks S, Gelman A, Jones GL, Meng X (eds) Handbook of Markov Chain Monte Carlo. CHapman and Hall/CRC, New York, pp 175–197
Geman S, Geman D (1984) Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans Pattern Anal Mach Intell 6(6):721–741
Gondara L, Wang K (2018) Mida: multiple imputation using denoising autoencoders. Advances in knowledge discovery and data mining. Springer, Cham, pp 260–272
Goyal V et al (2019) Missing data imputation by principal component analysis (PCA) and fuzzy C means (FCM). Int J Control Autom 12(6):127–134
Hagan P, Kumar D, Lesniewski A, Woodward DE (2002) Managing smile risk. Wilmott Mag 01(02):84–108
Hagan P, Konikov M (2004) Interest rate volatility cube: construction and use. Technical Report, Bloomberg Technical Reports
Hauberg S (2018) Only Bayes should learn a manifold (on estimation of differential geometric structure from data). arXiv:1806.04994
Ipsen NB, Mattei PA, Frellsen J (2021) not-MIWAE: deep generative modelling with missing not at random data. In: ICLR 2021 International Conference on Learning Representations
Ivanov O, Figurnov M, Vetrov D (2019) Variational autoencoder with arbitrary conditioning. In: Proceedings of the 7th International Conference on Learning Representations, pp 1–25
Jäckel P, Rebonato (2000) Linking caplet and swaption volatilities in a GBM/J framework: approximate solutions. Quantitative Research Centre. The Royal Bank of Scotland
Johnson S, Nonas B (2009) Arbitrage-free construction of the swaption cube. Wilmott J 1(3):137–143
Kingma D, Welling M (2013) Auto-encoding variational Bayes. In: Paper presented at 2nd International Conference on Learning Representations, ICLR, Banff, AB, Canada, April 14-16. Technical Report
Kunsági-Máté S, Fáth G, Csabai I (2021) Analyizing the dynamics of the swaption market using neural networks. Eur J Econ 1(2):1–13
Le Floc’h F, Kennedy G (2014) Explicit SABR calibration through simple expansions. SSRN. https://doi.org/10.2139/ssrn.2467231
Lewis S, Matejovicova T, Li Y, Lamb A, Zaykov Y, Allamanis M, Zhang C (2021) Accurate imputation and efficient data acquisition with transformer-based vaes. In: NeurIPS 2021 Workshop on Deep Generative Models and Downstream Applications, 2021
Li C, Zhu J, Zhang B (2016) Learning to generate with memory. In: Proceedings of The 33rd International Conference on Machine Learning, pp 1177–1186
Magdon-Ismail M, Purnell JT (2010) Approximating the covariance matrix of GMMs with low-rank perturbations. Int Conf Intell Data Eng Autom Learn 2010:300–307
Ma C, Tschiatschek S, Hernandez-Lobato JM, Turner R, Zhang C (2020) Vaem: a deep generative model for heterogeneous mixed type data. In: 34th Conference on Neural Information Processing Systems (NeurIPS 2020), Vancouver, Canada
Mattei PA, Frellsen J (2018) Leveraging the exact likelihood of deep latent variable models. In: Proceedings of the 32nd International Conference on Neural Information Processing Systems, pp 3859–3870
Mattei PA, Frellsen J (2019) MIWAR: Deep generative modelling and imputation of incomplete data sets. Proc Int Conf Mach Learn 97:4413–4423
Ma C, Zhang C (2021) Identifiable generative models for missing not at random data imputation. In: 35th Conference on Neural Information Processing Systems (NeurIPS 2021)
Nazabal A, Olmos PM, Ghahramani Z, Valera I (2020) Handling incomplete heterogeneous data using vaes. Pattern Recognit 107:107501
Oblój J (2008) Fine-tune your simle: Correction to Hagan et al. Wilmott Magazine 01/08, pp 102- 104
Qiu YL, Zheng H, Gevaert O (2023) Genomic data imputation with variational auto-encoders. GigaScience 9(8):giaa082
Rebonato R, Pogudin A, White R (2008) Delta and vega hedging in the SABR and LMM-SABR models. Risk Magazine, December 2008
Rezende D, Mohamed S, Wierstra D (2014) Stochastic backpropagation and approximate inference in deep generative models. Proc Int Conf Mach Learn 32(2):1278–1286
Rezende D, Eslami SMA, Mohamed S, Battaglia P, Jaderberg M, Heess N (2016) Unsupervised learning of 3D structure from images. In: Advances in Neural Information Processing Systems, pp 4996–5004
Roberts GO, Smith AFM (1994) Simple conditions for the convergence of the Gibbs sampler and Metropolis-Hastings algorithms. Stoch Processes Appl 49(2):207–216
Roskams-Hieter B, Wells J, Wade S (2022) Leveraging variational autoencoders for multiple data imputation. arXiv:2209.15321
Skantzos N, Garston G (2019) The perfect smile. Filling the gaps in the swaption volatility cube. Deloitte Belgium. https://www2.deloitte.com/content/dam/Deloitte/be/Documents/risk/deloitte-be-the-perfect-smile.pdf. Accessed 19 April 2021
Sohl-Dickstein J, Weiss E, Maheswaranathan N, Ganguli S (2015) Deep unsupervised learning using nonequilibrium thermodynamics. In: Proceedings of the 32nd International Conference on Machine Learning 37, pp 2256–2265
Thorin H (2020) Artifical neural networks for SABR model calibration & hedging. Msc thesis, Imperial College London
Vincent P, Larochelle H, Bengio Y, Manzagol PA (2008) Extracting and composing robust features with denoising autoencoders. In: Proceedings of the 25th International Conference on Machine Learning, pp 1096–1103
Wang H, Chen H, Sudjianto A, Liu R, Shen Q (2018) Deep learning-based BSDE solver for Libor market model with application to Bermudan swaption pricing and hedging. arXiv:1807.06622
West G (2005) Calibration of the SABR model in illiquid markets. Appl Math Finance 12(4):371–385