Markov-Komposition und eine Anwendung auf Martingale

Kellerer, Hans

doi:10.1007/bf01432281

Cited by 154 publications

(146 citation statements)

References 4 publications

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“…In other words, for each of the maturities T i , prices of calls are only available for a finite set K i1 < K i2 < · · · < K in i of strikes. This more realistic form of the set-up has been considered, starting with the work of Laurent and Leisen [24], followed by the recent technical reports by Cousot [9] and Buehler [5] who use Kellerer [22] theorem, and by the recent work of Davis and Hobson [10] which relies instead on the Sherman-Stein-Blackwell theorem [35,36,2].…”

Section: Introduction and Notationmentioning

confidence: 99%

Local volatility dynamic models

Carmona

Nadtochiy

2008

Finance Stoch

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ABSTRACT. This paper is concerned with the characterization of arbitrage free dynamic stochastic models for the equity markets when Itô stochastic differential equations are used to model the dynamics of a set of basic instruments including, but not limited to, the underliers. We study these market models in the framework of the HJM philosophy originally articulated for Treasury bond markets. The approach to dynamic equity models which we follow was originally advocated by Derman and Kani in a rather informal way. The present paper can be viewed as a rigorous development of this program, with explicit formulae, rigorous proofs and numerical examples.

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Section: Introduction and Notationmentioning

confidence: 99%

Local volatility dynamic models

Carmona

Nadtochiy

2008

Finance Stoch

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“…The reasoning leading from (107) to (108) is known (see [15] or Appendix 1 in [8]). By Kellerer's theorem (Theorem 3 in [27]), (108) and (27) imply the existence of a filtered probability space (Ω, F, F t , P * ) and of a Markov (F t , P * )-martingale Y such that the distribution of Y T coincides with the measure ν T for every T ≥ 0. Now put X T = e rT Y T , T ≥ 0.…”

Section: Stock Price Distribution Densities and Pricing Functions In mentioning

confidence: 99%

“…Next, we see that condition 4 implies μ 0 = δ x 0 , and hence X 0 = x 0 P * -a.s. Taking into account inequality (27), we define the following function:…”

Section: Stock Price Distribution Densities and Pricing Functions In mentioning

confidence: 99%

Asymptotic Formulas with Error Estimates for Call Pricing Functions and the Implied Volatility at Extreme Strikes

Gulisashvili¹

2010

SIAM J. Finan. Math.

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Abstract. In this paper, we obtain asymptotic formulas with error estimates for the implied volatility associated with a European call pricing function. We show that these formulas imply Lee's moment formulas for the implied volatility and the tail-wing formulas due to Benaim and Friz. In addition, we analyze Pareto-type tails of stock price distributions in uncorrelated Hull-White, Stein-Stein, and Heston models and find asymptotic formulas with error estimates for call pricing functions in these models.

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“…Kellerer [18] extended this to the uncountable setting, by showing that a collection of probability distributions parametrised by t ∈ R + satisfies μ s ≤ cx μ t (resp. μ s ≤ icx μ t ) for all s ≤ t if and only if there exists a martingale (resp.…”

Section: Related Workmentioning

confidence: 99%

Conditional convex orders and measurable martingale couplings

Leskelä¹,

Vihola

2017

Bernoulli

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Strassen's classical martingale coupling theorem states that two random vectors are ordered in the convex (resp. increasing convex) stochastic order if and only if they admit a martingale (resp. submartingale) coupling. By analysing topological properties of spaces of probability measures equipped with a Wasserstein metric and applying a measurable selection theorem, we prove a conditional version of this result for random vectors conditioned on a random element taking values in a general measurable space. We provide an analogue of the conditional martingale coupling theorem in the language of probability kernels, and discuss how it can be applied in the analysis of pseudo-marginal Markov chain Monte Carlo algorithms. We also illustrate how our results imply the existence of a measurable minimiser in the context of martingale optimal transport.

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Markov-Komposition und eine Anwendung auf Martingale

Cited by 154 publications

References 4 publications

Local volatility dynamic models

Local volatility dynamic models

Asymptotic Formulas with Error Estimates for Call Pricing Functions and the Implied Volatility at Extreme Strikes

Conditional convex orders and measurable martingale couplings

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