Arbitrage-free market models for option prices: the multi-strike case

Schweizer, Martin; Wissel, Johannes

doi:10.1007/s00780-008-0068-6

Cited by 49 publications

(34 citation statements)

References 32 publications

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“…The idea is to treat the dynamics of the whole call price surface as fundamental, rather than derived from the dynamics of the underlying asset price. See the articles of Carmona and Nadtochiy [3], [4], Kallsen and Krühner [13], and Schweizer and Wissel [20], [21] for various partial implementations of this approach.…”

Section: Theorem 12 Let S Be a Positive Continuous Martingale With mentioning

confidence: 98%

On the Uniqueness of Martingales with Certain Prescribed Marginals

Tehranchi¹

2013

J. Appl. Probab.

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This note contains two main results. (i) (Discrete time) Suppose that S is a martingale whose marginal laws agree with a geometric simple random walk. (In financial terms, let S be a risk-neutral asset price and suppose that the initial option prices agree with the Cox-Ross-Rubinstein binomial tree model.) Then S is a geometric simple random walk. (ii) (Continuous time) Suppose that S = S 0 e σ X−σ 2 X /2 is a continuous martingale whose marginal laws agree with a geometric Brownian motion. (In financial terms, let S be a risk-neutral asset price and suppose that the initial option prices agree with the Black-Scholes model with volatility σ > 0.) Then there exists a Brownian motion W such that X t = W t + o(t 1/4+ε ) as t ↑ ∞ for any ε > 0.

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Section: Theorem 12 Let S Be a Positive Continuous Martingale With mentioning

confidence: 98%

On the Uniqueness of Martingales with Certain Prescribed Marginals

Tehranchi¹

2013

J. Appl. Probab.

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“…for some fixed T > t (case considered by Schweizer and Wissel in [13]) For the rest of this paper we concentrate on the market given by the last example:…”

Section: Tangent Models and Calibrationmentioning

confidence: 99%

Tangent Models as a Mathematical Framework for Dynamic Calibration

Carmona¹,

Nadtochiy²

2012

Finance at Fields

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ABSTRACT. Motivated by the desire to integrate repeated calibration procedures into a single dynamic market model, we introduce the notion of tangent market model in an abstract set up, and we show that this new mathematical paradigm accommodates all the recent attempts to study consistency and absence of arbitrage in market models. For the sake of illustration, we concentrate on equity models and we assume that market quotes provide the prices of European call options for a specific set of strikes and maturities. While reviewing our recent results on dynamic local volatility and tangent L'evy models, we provide new results on the short time-to-maturity asymptotics which shed new light on the dichotomy between these two disjoint classes of models, with and without jumps, helping choose in practice, which class of models is most appropriate to the market characteristics at hand.

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“…All the results obtained in this chapter can be extended to include the above specification, with the only difference that we would have to study the dynamics of two functionsκ + andκ − instead of a single one. However, for notational convenience, we will restrict ourselves to specification (34).…”

Section: Choosing the Right Functional Subspacesmentioning

confidence: 99%

“…This idea was then developed more thoroughly in the works of Schönbucher [33], Schweizer and Wissel [35] and Jacod and Protter [21], but the recent works of Schweizer and Wissel [34] and Carmona and Nadtochiy [3], [2] are more in the spirit of the market model approach that we advocate here.…”

Section: Introductionmentioning

confidence: 99%

Tangent Lévy market models

Carmona

Nadtochiy

2011

Finance Stoch

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In this paper, we introduce a new class of models for the time evolution of the prices of call options of all strikes and maturities. We capture the information contained in the option prices in the density of some time-inhomogeneous Lévy measure (an alternative to the implied volatility surface), and we set this static code-book in motion by means of stochastic dynamics of Itôs type in a function space, creating what we call a tangent Lévy model. We then provide the consistency conditions, namely, we show that the call prices produced by a given dynamic code-book (dynamic Lévy density) coincide with the conditional expectations of the respective payoffs if and only if certain restrictions on the dynamics of the code-book are satisfied (including a drift conditionà la HJM). We then provide an existence result, which allows us to construct a large class of tangent Lévy models, and describe a specific example for the sake of illustration.

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Arbitrage-free market models for option prices: the multi-strike case

Abstract: Option prices, Market model, Implied volatility, Static arbitrage, Dynamic arbitrage, Drift restrictions, Existence result, 60H10, 91B28, C60, G13,

Cited by 49 publications

References 32 publications

On the Uniqueness of Martingales with Certain Prescribed Marginals

On the Uniqueness of Martingales with Certain Prescribed Marginals

Tangent Models as a Mathematical Framework for Dynamic Calibration

Tangent Lévy market models

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