P. JAILLET, D. LAMBERTON, AND B. LAPEYRE, Variational inequalities and the pricing of American options, Acta Appl. Math., 21 (1990), pp. 263--289. 24  Home/Search   Document Details and Download   Summary   Related Articles   Check

....The infimum is attained by taking M = M # . 2.2. 3 The variational inequality formulation Formulating the problem as a variational inequality invites implications from the large number of results that have been developed in this field, for example the work of Glowinski et al. 8] Jaillet et al. [11] applied this approach to the analysis of American option pricing. One must first define an elliptic operator giving the di#usion of the process. This is given by tr## # ( ##) # #x r (8) where r is the riskfree rate and # : t, T ] is the function satisfying V rv = ....

Patrick Jaillet, Damien Lamberton, and Bernard Lapeyre. Variational inequalities and the pricing of American options. Acta Applicandae Mathematicae, 21:263--289, 1990.

....manifests itself as a free boundary in the PDE. The problem becomes to find the solution v(x, t)tothe following variational inequality # #t (v #) 0 for (x, t) 4) Again some regularity conditions are required for the problem to possess a unique solution (see Jaillet et al. [7]) The most popular numerical method to solve the problem in up to two dimensions is formulated by adapting a finite di#erence method using projected SOR (PSOR) so that the extra constraint is satisfied at each time step. The discretised system can be treated as a linear complementarity problem. ....

.... Cottle, Pang and Stone [4] The convergence of PSOR for real symmetric positive definite ML is proved by Cryer [5] and the convergence of the overall computed solution at t =0 (allowing for numerical errors at each time step) is proved for certain classes of payo# functions # by Jaillet et al. [7]. The European and American pricing problems in one dimension are treated in detail in this paradigm by Wilmott, Dewynne and Howison [#5] for example. 3 An Irregular Grid Method The method we propose basically follows the second of the two paradigms mentioned in Section 2, but we also make use ....

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Jaillet, P., Lamberton, D., Lapeyre, B.: Variational inequalities and the pricing of American options. Acta Applicandae Mathematicae 21 (1990) 263---289

....the front xing method as a reference solution for studying the convergence properties of our penalty schemes. In addition to penalty, singularity separating and front xing methods for solving option problems several schemes have been proposed. Among these are the Brennan and Schwartz algorithm [3, 9], the projected SOR scheme [15] the binomial method [8] and Monte Carlo simulation techniques [7, 13, 14] The outline of the paper is as follows. The next section contains the Black Scholes model for American put problems. In Section 3 we de ne the Front Fixing Method and the associated explicit ....

P. Jaillet, D. Lamberton, and B. Lapeyre. Variational-inequalities and the pricing of american options. ACTA Applicandae Mathematicae, 21(3):263-289, 1990.

....(4.1) that leverages the use of more e ective solvers that are available today. The results presented here reply on the connection between the optimal stopping problem and the variational inequalities as established by Bensoussan and Lions [3] and recently by Jaillet, Lamberton and Lapeyre [16] in their study of American options. Those results are also extensively The Valuation of American Passport Options Current Draft: July 10, 1999 11 used by Wilmott, Dewynne and Howison [26] in pricing various option securities with early exercise features, and recently in Huang and Pang [14] ....

Jaillet, P., Lamberton, D., and Lapeyre, B. Variational inequalities and the pricing of American options. Acta Applicandae Mathematicae 21 (1990), 263-289.

....were strictly less than zero. Therefore we obtain the third condition: #Lv# # #v , ##=0: #6# The inequalities #4#, #5#, and #6# subject to v#T;s#=##s# form a parabolic obstacle problem. We refer to Friedman #1988# for the existence and the uniqueness of the solution to such problems. Jaillet, Lamberton, and Lapeyre #1990# showed that the solution of the parabolic variational inequalities #4#, #5#, and #6# has a continuous gradient at the free boundary #i.e. a smooth #t#, and Van Moerbeke #1976# showed that the optimal stopping 4 boundary is continuously di#erentiable. Thus Ito s formula #3# is valid at least in ....

Jaillet, P., D. Lamberton, and B. Lapeyre #1990#, Variational inequalities and the pricing of American options, Acta Appl. Math., 21., pp 263-289.

....strictly less than zero. Therefore we obtain the third condition: Lv) Delta (v Gamma OE) 0 : 6) The inequalities (4) 5) and (6) subject to v(T; s) OE(s) form a parabolic obstacle problem. We refer to Friedman (1988) for the existence and the uniqueness of the solution to such problems. Jaillet, Lamberton, and Lapeyre (1990) showed that the solution of the parabolic variational inequalities (4) 5) and (6) has a continuous gradient at the free boundary (i.e. a smooth fit) and Van Moerbeke (1976) showed that the optimal stopping 4 boundary is continuously differentiable. Thus Ito s formula (3) is valid at least ....

Jaillet, P., D. Lamberton, and B. Lapeyre (1990), Variational inequalities and the pricing of American options, Acta Appl. Math., 21., pp 263-289.

....the front xing method as a reference solution for studying the convergence properties of our penalty schemes. In addition to penalty, singularity separating and front xing methods for solving option problems several schemes have been proposed. Among these are the Brennan and Schwartz algorithm [3, 9], the projected SOR scheme [15] the binomial method [8] and Monte Carlo simulation techniques [7, 13, 14] The outline of the paper is as follows. The next section contains the Black Scholes model for American put problems. In Section 3 we de ne the Front Fixing Method and the associated explicit ....

P. Jaillet, D. Lamberton, and B. Lapeyre. Variational-inequalities and the pricing of american options. ACTA Applicandae Mathematicae, 21(3):263-289, 1990.

....such) for pricing an American put option is by Brennan and Schwartz [2] In spite of the pioneering e orts of these authors This work was based on research supported by the National Science Research Foundation under grant CCR9624018. 1 and others, including Jaillet, Lamberton, and Lapeyre [18] and Dempster and Hutton [10, 11] it is our belief that the linear complementarity approach to pricing American options is very much at its infancy and its full potential has yet to be realized. This paper aims to further explore the linear complementarity problem (LCP) and its extensions as a ....

P. Jaillet, D. Lamberton, and B. Lapeyre, \Variational inequalities and the pricing of American options", Acta Applicandae Mathematicae 21 (1990) 263-289.

....NEWTON METHOD FOR AMERICAN OPTION PRICING THOMAS F. COLEMAN#, YUYING LI#, AND ARUN VERMA# December 13, 1999 Abstract. The variational inequality formulation provides a mechanism to determine both the option value and the early exercise curve implicitly [17]. Standard finite difference approximation typically leads to linear complementarity problems with tridiagonal coefficient matrices. The second order upwind finite difference formulation gives rise to finite dimensional linear complementarity problems with nontridiagonal matrices, whereas the ....

....e.g. calculating VaR. Brennan and Schwartz [3] introduce a simple procedure using the standard implicit finite difference method for the classical Black Scholes partial differential operator. Convergence of the Brenan and Schwartz method is established by Jaillet, Lamberton and Lapeyre in [17]. It is also shown, in [17] that determination of the early exercise curve can be made implicit with a variational inequality formulation in a generalized Black Scholes framework. The variational inequality formulation for the American option in a jump diffusion model is analyzed in [21] ....

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P. JAILLET, D. LAMBERTON, AND B. LAPEYRE, Variational inequalities and pricing of American options, Acta Applicanda Mathematicae, 21 (1990), pp. 263--289.

....curve. For example, closed form solutions in terms of exogenous variables and the optimal exercise boundary are presented by Carr, Jarrow, and Myneni [9] Jacka [11] and Kim [13] The method of variational inequalities was employed by Bensoussan and Lions [4] and Jaillet, Lamberton, and Lapeyre [12] for a study in the solution uniqueness and existence. In the numerical methods, implicit finite difference methods of Brennan and Schwartz [6, 7] and Brenner, Courtadon, Subrahmanyam [8] and quadratic approximation techniques of Barone Adesi 1991 Mathematics Subject Classification. 90A09, 65K10, ....

P. Jaillet, D. Lamberton, and B. Lapeyre, Variational inequalities and the pricing of American options, Acta Appl. Math., 21 (1990), 263--289.

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Jaillet, P., D. Lamberton, and B. Lapeyre, 1989, Variational Inequalities and the Pricing of American Options, CERMA-ENPC working paper.

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P. JAILLET, D. LAMBERTON, AND B. LAPEYRE, Variational inequalities and the pricing of American options, Acta Appl. Math., 21 (1990), pp. 263--289. 24

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P. Jaillet, D. Lamberton, and B. Lapeyre, Variational inequalities and the pricing of American options, Acta Applicandae Mathematicae, 21 (1990), pp. 263-- 289.

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P. Jaillet, D. Lamberton, B. Lapeyre, Variational Inequalities and the Pricing of American Options Acta Applicandae Mathematicae 21 (1990) 263-289.

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P. Jaillet, D. Lamberton and B. Lapeyre, (1990), Variational inequalities and the pricing of American options. Acta Applicandae Mathematicae, Vol. 21, pp. 263--289.

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4th Edition. #13# Jaillet, P., Lamberton, D. and Lapeyre, B. #1990#. Variational Inequalities and the Pricing of American Options. Acta Appl. Math.

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P. Jaillet, D. Lamberton, and B. Lapeyre, Variational inequalities and the pricing of American options, Acta Appl. Math., 21 (1990), 263--289.

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