Tangent Lévy market models

Carmona, René; Nadtochiy, Sergey

doi:10.1007/s00780-011-0158-8

“…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: 97%

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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“…From the analytic representation of (9), provided in Section 4 (and discussed in more detail in [2]), it is not hard to see that the above mapping from (s,κ) to ("value of the underlying", "prices of the call options") is invertible, thus, producing a code-book. As before, if at a given moment of time t there exists a value of the code-book (s,κ), which reproduces the true market prices of call options and the underlying, then the model given by (s,κ) is a tangent Lévy model at time t.…”

Section: Tangent Models and Calibrationmentioning

confidence: 99%

“…Our work [2] on tangent Lévy models was a natural attempt to relax the assumption that S t is an Itô's process, and introduce jumps in its dynamics, the relevant question being: " What is the right substitute for the local volatility code? "…”

Section: Dynamic Tangent Lévy Modelsmentioning

confidence: 99%

Tangent Models as a Mathematical Framework for Dynamic Calibration

Carmona¹,

Nadtochiy²

2012

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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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“…The following tangent Lévy model was proposed in [2]. Its analysis and implementation on real market data is being carried out in [10].…”

Section: Example Of Dynamic Tangent Lévy Modelmentioning

confidence: 99%

Tangent Models as a Mathematical Framework for Dynamic Calibration

Carmona

¹

,

Nadtochiy

²

2011

Int. J. Theor. Appl. Finan.

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7

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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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Tangent Lévy market models

Cited by 19 publications

References 36 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 Models as a Mathematical Framework for Dynamic Calibration

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