From Bachelier to Dupire via optimal transport

被引：1

作者：

Beiglboeck, Mathias ^{[1
]}

Pammer, Gudmund ^{[2
]}

Schachermayer, Walter ^{[1
]}

机构：

[1] Univ Wien, Dept Math, Oskar Morgensternpl 1, A-1090 Vienna, Austria

[2] Swiss Fed Inst Technol, Dept Math, Ramistr 101, CH-8092 Zurich, Switzerland

来源：

FINANCE AND STOCHASTICS | 2022年 / 26卷 / 01期

关键词：

Bachelier; Dupire's formula; Kellerer's theorem; Optimal transport; Martingales; Peacocks; FULL MARGINALS; MARTINGALE; MIMICKING; EXISTENCE; PRICE;

D O I：

10.1007/s00780-021-00466-3

中图分类号：

F8 [财政、金融];

学科分类号：

0202 ;

摘要：

Famously, mathematical finance was started by Bachelier in his 1900 PhD thesis where - among many other achievements - he also provided a formal derivation of the Kolmogorov forward equation. This also forms the basis for Dupire's (again formal) solution to the problem of finding an arbitrage-free model calibrated to a given volatility surface. The latter result has rigorous counterparts in the theorems of Kellerer and Lowther. In this survey article, we revisit these hallmarks of stochastic finance, highlighting the role played by some optimal transport results in this context.

引用

页码：59 / 84

页数：26

共 56 条

[21] Dynkin E., 1956, Theory of Probability and its Applications, V1, P149
[22] Mimicking self-similar processes
Fan, Jie Yen
Hamza, Kais
Klebaner, Fima
[J]. BERNOULLI, 2015, 21 (03) : 1341 - 1360
[23] Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients
Figalli, Alessio
[J]. JOURNAL OF FUNCTIONAL ANALYSIS, 2008, 254 (01) : 109 - 153
[24] On weak Brownian motions of arbitrary order
Föllmer, H
Wu, CT
Yor, M
[J]. ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES, 2000, 36 (04): : 447 - 487
[25] Guyon J., 2012, RISK, V25, P88
[26] Guyon J., 2011, SMILE CALIBRATION PR
[27] Guyon J., 2013, Nonlinear Option Pricing
[28] MIMICKING THE ONE-DIMENSIONAL MARGINAL DISTRIBUTIONS OF PROCESSES HAVING AN ITO DIFFERENTIAL
GYONGY, I
[J]. PROBABILITY THEORY AND RELATED FIELDS, 1986, 71 (04) : 501 - 516
[29] Hamza K., 2007, J APPL MATH STOCH AN, V2007
[30] An explicit martingale version of the one-dimensional Brenier's Theorem with full marginals constraint
Henry-Labordere, Pierre
Tan, Xiaolu
Touzi, Nizar
[J]. STOCHASTIC PROCESSES AND THEIR APPLICATIONS, 2016, 126 (09) : 2800 - 2834

← 1 2 3 4 5 6 →