Can time-homogeneous diffusions produce any distribution?

Ekström, Erik; Hobson, David; Janson, Svante; Tysk, Johan

doi:10.1007/s00440-011-0405-0

Cited by 16 publications

(22 citation statements)

References 15 publications

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“…A continuous time and state version of the Rubinstein result can be found in Carr and Madan [5]. Other methods of calibrating to a single smile are presented in Cox, Hobson and Ob lój [7], [9], [17], and to multiple smiles, in Madan and Yor [16].…”

Section: Introductionmentioning

confidence: 99%

Local Variance Gamma and Explicit Calibration to Option Prices

Carr

Nadtochiy

2014

Mathematical Finance

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In some options markets (e.g., commodities), options are listed with only a single maturity for each underlying. In others (e.g., equities, currencies), options are listed with multiple maturities. In this paper, we analyze a special class of pure jump Markov martingale models and provide an algorithm for calibrating such models to match the market prices of European options with multiple strikes and maturities. This algorithm matches option prices exactly and only requires solving several one‐dimensional root‐search problems and applying elementary functions. We show how to construct a time‐homogeneous process which meets a single smile, and a piecewise time‐homogeneous process which can meet multiple smiles.

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Section: Introductionmentioning

confidence: 99%

Local Variance Gamma and Explicit Calibration to Option Prices

Carr

Nadtochiy

2014

Mathematical Finance

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“…Then (b, µ) satisfies: (7) and b is defined by (1). Here the maximum is taken over sequences of length n for all n ∈ N.…”

Section: Results and Methods Of Proofmentioning

confidence: 99%

Time homogeneous diffusion with drift and killing to meet a given marginal

Noble

2015

Stochastic Processes and their Applications

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In this article, it is proved that for any probability law µ over R and a drift field b : R → R and killing field k : R → R + which satisfy hypotheses stated in the article and a given terminal time t > 0, there exists a string m, an α ∈ (0, 1], an initial condition x 0 ∈ R and a process X with infinitesimal generator  1 2Firstly, it is shown the problem with drift and without killing can be accommodated, after a simple coordinate change, entirely by the proof in Noble (2013). The killing field presents additional problems and the proofs follow the lines of Noble (2013) with additional arguments.

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“…Just to name a few, let us mention the explicit construction of peacocks (see [HPRY11,Chapter 6]), the construction of (martingale) diffusions matching prescribed marginal distributions at given (random) times (see e.g. [CHO11] and [EHJT13]) or Dupire's formula in mathematical finance (see [Dup94] and [CGMY04]). Further potential applications include probabilistic representations of the solution to irregular porous media type equations [BRR11] and measure-valued martingales [VBB + 17].…”

Section: Introductionmentioning

confidence: 99%

Existence and Uniqueness Results for Time-Inhomogeneous Time-Change Equations and Fokker–Planck Equations

et al. 2019

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We prove existence and uniqueness of solutions to Fokker-Planck equations associated to Markov operators multiplicatively perturbed by degenerate time-inhomogeneous coefficients. Precise conditions on the time-inhomogeneous coefficients are given. In particular, we do not necessarily require the coefficients to be neither globally bounded nor bounded away from zero. The approach is based on constructing random time-changes and studying related martingale problems for Markov processes with values in locally compact, complete and separable metric spaces.

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Can time-homogeneous diffusions produce any distribution?

Abstract: Abstract. Given a centred distribution, can one find a time-homogeneous martingale diffusion starting at zero which has the given law at time 1? We answer the question affirmatively if generalized diffusions are allowed.

Cited by 16 publications

References 15 publications

Local Variance Gamma and Explicit Calibration to Option Prices

Local Variance Gamma and Explicit Calibration to Option Prices

Time homogeneous diffusion with drift and killing to meet a given marginal

Existence and Uniqueness Results for Time-Inhomogeneous Time-Change Equations and Fokker–Planck Equations

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