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Tangent Lévy Market Models
"... 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 timeinhomogeneous Lévy measure (an alternative to the implied volatility surface), ..."
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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 timeinhomogeneous Lévy measure (an alternative to the implied volatility surface), and we set this static codebook 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 codebook (dynamic Lévy density) coincide with the conditional expectations of the respective payoffs if and only if certain restrictions on the dynamics of the codebook are satisfied (including a drift condition à 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.
Approximate solutions to second order parabolic equations I: Analytic estimates
, 2009
"... We establish a new type of local asymptotic formula for the Green's function G t (x, y) of a uniformly parabolic linear operator ∂ t − L with nonconstant coefficients using dilations and Taylor expansions at a point z = z(x, y), for a function z with bounded derivatives such that z(x, x) = x ..."
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We establish a new type of local asymptotic formula for the Green's function G t (x, y) of a uniformly parabolic linear operator ∂ t − L with nonconstant coefficients using dilations and Taylor expansions at a point z = z(x, y), for a function z with bounded derivatives such that z(x, x) = x ∈ R N . Our method is based on dilation at z, Dyson and Taylor series expansions. We use the BakerCampbellHausdorff commutator formula to explicitly compute the terms in the Dyson series. Our procedure leads to an explicit, elementary, algorithmic construction of approximate solutions to parabolic equations which are accurate to arbitrary prescribed order in the shorttime limit. We establish mapping properties and precise error estimates in the exponentially weighted, L p type Sobolev spaces W s,p a (R N ) that appear in practice.
TANGENT MODELS AS A MATHEMATICAL FRAMEWORK FOR DYNAMIC CALIBRATION
"... 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 abse ..."
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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 timetomaturity 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. 1.
PARABOLIC EQUATIONS I: ANALYTIC ESTIMATES RADU CONSTANTINESCU, NICOLA COSTANZINO,
, 2009
"... Abstract. We establish a new type of local asymptotic formula for the Green’s function of a parabolic operator with nonconstant coefficients. Our procedure leads to a construction of approximate solutions to parabolic equations which are accurate to arbitrary prescribed order in time. Contents ..."
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Abstract. We establish a new type of local asymptotic formula for the Green’s function of a parabolic operator with nonconstant coefficients. Our procedure leads to a construction of approximate solutions to parabolic equations which are accurate to arbitrary prescribed order in time. Contents