184 research outputs found
Large liquidity expansion of super-hedging costs.
No full textWe consider a financial market with liquidity cost as in Cetin, Jarrow and Protter [3] where the supply function S"(s; ) depends on a parameter " 0 with S0(s; ) = s corresponding to the perfect liquid situation. Using the PDE characterization of Cetin, Soner and Touzi [6] of the super-hedging cost of an option written on such a stock, we provide a Taylor expansion of the super-hedging cost in powers of ". In particular, we explicitly compute the first term in the expansion for a European Call option and give bounds for the order of the expansion for a European Digital Option.Super-replication; liquidity; viscosity solutions; asymptotic expansions;
The dynamic programming equation for second order stochastic target problems
No full textMotivated by applications in mathematical finance [U. Cetin, H. M. Soner, and N. Touzi, "Options hedging for small investors under liquidity costs," Finance Stoch., to appear] we continue our study of second order backward stochastic equations. In this paper, we derive the dynamic programming equation for a certain class of problems which we call the second order stochastic target problems. In contrast with previous formulations of similar problems, we restrict control processes to be continuous. This new framework enables us to apply our results to a larger class of models. Also the resulting derivation is more transparent. The main technical tool is the geometric dynamic programming principle in this context, and it is proved by using the framework developed in [H. M. Soner and N. Touzi, J. Eur. Math. Soc. (JEMS), 8 (2002), pp. 201-236]
Large liquidity expansion of super-hedging costs
No full textInternational audienceWe consider a financial market with liquidity cost as in Çetin, Jarrow and Protter [2004], where the supply function S{\epsilon}(s,{\nu}) depends on a parameter {\epsilon}\geq0 with S0(s,{\nu})=s corresponding to the perfect liquid situation. Using the PDE characterization of Çetin, Soner and Touzi [2010], of the super-hedging cost of an option written on such a stock, we provide a Taylor expansion of the super-hedging cost in powers of {\epsilon}. In particular, we explicitly compute the first term in the expansion for a European Call option and give bounds for the order of the expansion for a European Digital Option
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Duality and convergence for binomial markets with friction
No full textWe prove limit theorems for the super-replication cost of European options in a binomial model with friction. Examples covered are markets with proportional transaction costs and illiquid markets. A dual representation for the super-replication cost in these models is obtained and used to prove the limit theorems. In particular, the existence of a liquidity premium for the continuous-time limit of the model proposed in Çetin et al. (Finance Stoch. 8:311-341, 2004) is proved. Hence, this paper extends the previous convergence result of Gökay and Soner (Math Finance 22:250-276, 2012) to the general non-Markovian case. Moreover, the special case of small transaction costs yields, in the continuous limit, the G-expectation of Peng as earlier proved by Kusuoka (Ann. Appl. Probab. 5:198-221, 1995). © 2012 Springer-Verlag
Merton Problem with Taxes: Characterization, computation and Approximation
No full textInternational audienceWe formulate a computationally tractable extension of the classical Merton optimal consumptioninvestment problem to include the capital gains taxes. This is the continuous-time version of the model introduced by Dammon, Spatt, and Zhang [Rev. Financ. Stud., 14 (2001), pp. 583-616]. In this model the tax basis is computed as the average cost of the stocks in the investor's portfolio. This average rule introduces only one additional state variable, namely the tax basis. Since the other tax rules such as the first in first out rule require the knowledge of all past transactions, the average model is computationally much easier. We emphasize the linear taxation rule, which allows for tax credits when capital gains losses are experienced. In this context wash sales are optimal, and we prove it rigorously. Our main contributions are a first order explicit approximation of the value function of the problem and a unique characterization by means of the corresponding dynamic programming equation. The latter characterization builds on technical results isolated in the accompanying paper [I. Ben Tahar, H. M. Soner, and N. Touzi, SIAM J. Control Optim., 46 (2007), pp. 1779-1801]. We also suggest a numerical computation technique based on a combination of finite differences and the Howard iteration algorithm. Finally, we provide some numerical results on the welfare consequences of taxes and the quality of the first order approximation
Super-replication under gamma constraints: An application to forward start options
No full textEsta tesis sirve como una introducción al problema y a los resultados de la super-replicación de derivados financieros con restricciones en gamma, tal como lo presentan Soner y Touzi. En el tercer capítulo se presenta una extensión leve e inmediata del resultado de Soner y Touzi para incluir el descuento con una tasa fija y determinista bajo un factor de descuento continuo. El objeto de estudio de esta tesis es la Opción Forward Start, específicamente con asignación lineal del precio de strike, para la cual se presentan la replicación de Black-Scholes y la super-replicación bajo una restricción sobre el gamma del portafolio y bajo un modelo subyacente de Movimiento Browniano Geométrico. Bajo la replicación de Black-Scholes, la estrategia de replicación resulta ser una estrategia estática antes de la fecha T' de asignación del precio de strike. Esta tesis muestra que la estrategia de super-replicación bajo la restricción gamma también resulta ser una estrategia estática antes de la fecha T' de asignación del precio de strike.Pregrad
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Vortex density models for Superconductivity and Superfluidity
No full textWe study some functionals that describe the density of vortex lines in superconductors subject to an applied magnetic field, and in Bose-Einstein condensates subject to rotational forcing, in quite general domains in 3 dimensions. These functionals are derived from more basic models via Gamma-convergence, here and in the companion paper (Baldo et al. in Arch Rat Mech Anal 205(3):699–752, 2012). In our main results, we use these functionals to obtain leading order descriptions of the first critical applied magnetic field (for superconductors) and forcing (for Bose-Einstein), above which ground states exhibit nontrivial vorticity, as well as a characterization of the vortex density in terms of a non local vector-valued generalization of the classical obstacle problem
Convertible Bond Pricing and Markets: An Analysis of the Spread between Theoretical and Trading Prices
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The Variance Gamma Option Pricing Model: An Empirical Evaluation and Comparison to Black-Scholes
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