The Measure Preserving Martingale Sinkhorn Algorithm (2310.13797v3)
Abstract: We contribute to the recent studies of the so-called Bass martingale. Backhoff-Veraguas et al. (2020) showed it is the solution to the martingale Benamou-Brenier (mBB) problem, i.e., among all martingales with prescribed initial and terminal distributions it is the one closest to the Brownian motion. We link it with semimartingale optimal transport and deduce an alternative way to derive the dual formulation recently obtained in Backhoff-Veraguas et al. (2023). We then consider computational methods to compute the Bass martingale. The dual formulation of the transport problem leads to an iterative scheme that mirrors to the celebrated Sinkhorn algorithm for entropic optimal transport. We call it the measure preserving martingale Sinkhorn (MPMS) algorithm. We prove that in any dimension, each step of the algorithm improves the value of the dual problem, which implies its convergence. Our MPMS algorithm is equivalent to the fixed-point method of Conze and Henry-Labordere (2021), studied in Acciaio et al. (2023), and performs very well on a range of examples, including real market data.
- Calibration of the bass local volatility model. arXiv preprint arXiv:2311.14567, 2023.
- Martingale Benamou–Brenier: a probabilistic perspective. Ann. Probab., 48(5):2258–2289, 2020.
- The Structure of martingale Benamou-Brenier in ℝdsuperscriptℝ𝑑\mathbb{R}^{d}blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT. arXiv preprint arXiv:2306.11019, 2023.
- The bass functional of martingale transport. arXiv preprint arXiv:2306.11019, 2023.
- Calibration of local-stochastic and path-dependent volatility models to vanilla and no-touch options. Journal of Computational Finance, 24(4), 2021.
- Richard F Bass. Skorokhod imbedding via stochastic integrals. Séminaire de probabilités de Strasbourg, 17:221–224, 1983.
- Second-order stochastic target problems with generalized market impact. SIAM J. Control Optim., 57(6):4125–4149, 2019.
- Almost-sure hedging with permanent price impact. Finance Stoch., 20(3):741–771, 2016.
- Hedging of covered options with linear market impact and gamma constraint. SIAM J. Control Optim., 55(5):3319–3348, 2017.
- Prices of state-contingent claims implicit in option prices. Journal of business, pages 621–651, 1978.
- Bass construction with multi-marginals: Lightspeed computation in a new local volatility model. Available at SSRN 3853085, 2021.
- B. Dupire. Pricing with a smile. RISK, January:18–20, 1994.
- Path dependent optimal transport and model calibration on exotic derivatives. Ann. Appl. Probab., 31(3):1232–1263, 2021.
- Local volatility calibration by optimal transport. In 2017 MATRIX annals, volume 2 of MATRIX Book Ser., pages 51–64. Springer, Cham, 2019.
- A benamou–brenier formulation of martingale optimal transport. Bernoulli, 25(4A):2729–2757, 2019.
- A fast approach to optimal transport: The back-and-forth method. Numerische Mathematik, 146(3):513–544, 2020.
- Christian Léonard. A survey of the Schrödinger problem and some of its connections with optimal transport. Discrete Contin. Dyn. Syst., 34(4):1533–1574, 2014.
- Gregoire Loeper. Option pricing with linear market impact and nonlinear Black-Scholes equations. Ann. Appl. Probab., 28(5):2664–2726, 2018.
- Marcel Nutz. Introduction to entropic optimal transport. 2022.
- Jan Obłój. The Skorokhod embedding problem and its offspring. Probability Surveys, 1:321–392, 2004.
- Richard Sinkhorn. A relationship between arbitrary positive matrices and doubly stochastic matrices. Ann. Math. Statist., 35:876–879, 1964.
- Optimal transportation under controlled stochastic dynamics. Ann. Probab., 41(5):3201–3240, 2013.
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