scholarly journals From Bachelier to Dupire via optimal transport

2021 ◽  
Vol 26 (1) ◽  
pp. 59-84
Author(s):  
Mathias Beiglböck ◽  
Gudmund Pammer ◽  
Walter Schachermayer
Keyword(s):  
Optimal Transport ◽  
Formal Solution ◽  
Mathematical Finance ◽  
Forward Equation ◽  
Survey Article ◽  
Formal Derivation ◽  
Kolmogorov Forward Equation ◽  
Free Model ◽  
Volatility Surface ◽  
Phd Thesis

AbstractFamously, 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.

scholarly journals Approximation of SDE solutions using local asymptotic expansions

2021 ◽  
Author(s):  
A. M. Davie
Keyword(s):  
Asymptotic Expansion ◽  
Strong Approximation ◽  
Optimal Transport ◽  
Small Time ◽  
Degenerate System ◽  
Weak Approximation ◽  
Forward Equation ◽  
Smooth Coefficients ◽  
One Step ◽  
Edgeworth Type Expansion

AbstractWe develop an asymptotic expansion for small time of the density of the solution of a non-degenerate system of stochastic differential equations with smooth coefficients, and apply this to the stepwise approximation of solutions. The asymptotic expansion, which takes the form of a multivariate Edgeworth-type expansion, is obtained from the Kolmogorov forward equation using some standard PDE results. To generate one step of the approximation to the solution, we use a Cornish–Fisher type expansion derived from the Edgeworth expansion. To interpret the approximation generated in this way as a strong approximation we use couplings between the (normal) random variables used and the Brownian path driving the SDE. These couplings are constructed using techniques from optimal transport and Vaserstein metrics. The type of approximation so obtained may be regarded as intermediate between a conventional strong approximation and a weak approximation. In principle approximations of any order can be obtained, though for higher orders the algebra becomes very heavy. In order 1/2 the method gives the usual Euler approximation; in order 1 it gives a variant of the Milstein method, but which needs only normal variables to be generated. However the method is somewhat limited by the non-degeneracy requirement.

scholarly journals Multinomial approximation to the Kolmogorov Forward Equation for jump (population) processes

2018 ◽  
Vol 5 (1) ◽  
pp. 1556192 ◽  
Author(s):  
Mario A. Natiello ◽  
Raúl H. Barriga ◽  
Marcelo Otero ◽  
Hernán G. Solari ◽  
Yuriy Rogovchenko
Keyword(s):  
Forward Equation ◽  
Kolmogorov Forward Equation ◽  
Population Processes

A NEW REPRESENTATION OF THE LOCAL VOLATILITY SURFACE

2008 ◽  
Vol 11 (07) ◽  
pp. 691-703
Author(s):  
MARIANITO R. RODRIGO ◽  
ROGEMAR S. MAMON
Keyword(s):  
Explicit Solution ◽  
Implied Volatility ◽  
Option Prices ◽  
Strike Price ◽  
Local Volatility ◽  
Forward Equation ◽  
Volatility Surface ◽  
Volatility Function ◽  
Ansatz Approach ◽  
Local Volatility Surface

In this paper, we address the problem of recovering the local volatility surface from option prices consistent with observed market data. We revisit the implied volatility problem and derive an explicit formula for the implied volatility together with bounds for the call price and its derivative with respect to the strike price. The analysis of the implied volatility problem leads to the development of an ansatz approach, which is employed to obtain a semi-explicit solution of Dupire's forward equation. This solution, in turn, gives rise to a new expression for the volatility surface in terms of the price of a European call or put. We provide numerical simulations to demonstrate the robustness of our technique and its capability of accurately reproducing the volatility function.

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