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37
Quadratic Convergence For Valuing American Options Using A Penalty Method
 SIAM J. Sci. Comput
, 2002
"... . The convergence of a penalty method for solving the discrete regularized American option valuation problem is studied. Su#cient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution. These cond ..."
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Cited by 24 (4 self)
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. The convergence of a penalty method for solving the discrete regularized American option valuation problem is studied. Su#cient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution. These conditions also ensure that the solution of the penalty problem is an approximate solution to the discrete linear complementarity problem. The e#ciency and quality of solutions obtained using the implicit penalty method are compared with those produced with the commonly used technique of handling the American constraint explicitly. Convergence rates are studied as the timestep and mesh size tend to zero. It is observed that an implicit treatment of the American constraint does not converge quadratically (as the timestep is reduced) if constant timesteps are used. A timestep selector is suggested which restores quadratic convergence. Key words. American option, penalty iteration, linear complementarity AMS subject classifications. 65M12, 65M60, 91B28 Revised: May 18, 2001 1.
Managing Capacity for Telecommunications Networks under Uncertainty
, 2002
"... Bandwidth is set to become the next major commodity market. Many companies are rapidly moving into this arena. The existing telecommunications infrastructure in most of the world is adequate to deliver voice and text applications, but demand for broadband services such as streaming video and large f ..."
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Cited by 15 (4 self)
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Bandwidth is set to become the next major commodity market. Many companies are rapidly moving into this arena. The existing telecommunications infrastructure in most of the world is adequate to deliver voice and text applications, but demand for broadband services such as streaming video and large file transfer (e.g. movies) is accelerating. The explosion in Internet use has created huge demand for telecommunications capacity. Because of the high volatility present in demand for capacity, investment decision tools are highly desirable. In this paper, traditional financial methods are applied to the problem of investment decision timing. We study the underlying driving force of the bandwidth market, and then apply real options theory to the upgrade decision problem. We study how the uncertainty and growth rate of demand for capacity a#ect the decision to upgrade. In certain cases, our results contradict the anecdotal 50% upgrade rule. We notice that sometimes it is better to wait until the maximum transmission rate of the line is nearly reached before upgrading directly to the line with the highest known transmission rate (skipping the intermediate lines). Finally, we show that a small perturbation in the revenue function leads to earlier upgrades. To the best of our knowledge, this approach has not been used previously. Consequently, we believe that this methodology can o#er insights for the telecommunications industry. Keywords uncertain demand for capacity, real options, telecommunications I.
Discrete Asian Barrier Options
, 1998
"... . A partial differential equation method based on using auxiliary variables is described for pricing discretely monitored Asian options with or without barrier features. The barrier provisions can be applied to either the underlying asset or to the average. They may also be of either instantaneous o ..."
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Cited by 14 (4 self)
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. A partial differential equation method based on using auxiliary variables is described for pricing discretely monitored Asian options with or without barrier features. The barrier provisions can be applied to either the underlying asset or to the average. They may also be of either instantaneous or delayed effect (i.e. Parisian style). Numerical examples demonstrate that this method can be used for pricing floating strike, fixed strike, American, or European options. In addition, examples are provided which indicate that an upstream biased quadratic interpolation is superior to linear interpolation for handling the jump conditions at observation dates. Moreover, it is shown that defining the auxiliary variable as the average rather than the running sum is more rapidly convergent for AmericanAsian options. Keywords: Asian options, Barrier options, Parisian options, PDE option pricing Running Title: Discrete Asian Barrier Options Acknowledgment: This work was supported by the Nation...
A SemiLagrangian approach for American Asian options under jump diffusion
 SIAM Journal on Scientific Computing
, 2003
"... version 1.7 A semiLagrangian method is presented to price continuously observed fixed strike Asian options. At each timestep a set of one dimensional partial integral differential equations (PIDEs) is solved and the solution of each PIDE is updated using semiLagrangian timestepping. CrankNicolson ..."
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Cited by 14 (7 self)
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version 1.7 A semiLagrangian method is presented to price continuously observed fixed strike Asian options. At each timestep a set of one dimensional partial integral differential equations (PIDEs) is solved and the solution of each PIDE is updated using semiLagrangian timestepping. CrankNicolson and second order backward differencing timestepping schemes are studied. Monotonicity and stability results are derived. With low volatility values, it is observed that the nonsmoothness at the strike in the payoff affects the convergence rate; subquadratic convergence rate is observed.
A new PDE approach for pricing arithmetic average Asian options
, 2000
"... . In this paper, arithmetic average Asian options are studied. It is observed that the Asian option is a special case of the option on a traded account. The price of the Asian option is characterized by a simple onedimensional partial dierential equation which could be applied to both continuous an ..."
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Cited by 13 (1 self)
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. In this paper, arithmetic average Asian options are studied. It is observed that the Asian option is a special case of the option on a traded account. The price of the Asian option is characterized by a simple onedimensional partial dierential equation which could be applied to both continuous and discrete average Asian option. The article also provides numerical implementation of the pricing equation. The implementation is fast and accurate even for low volatility and/or short maturity cases. Key words: Asian options, Options on a traded account, Brownian motion, xed strike, oating strike. 1 Introduction Asian options are securities with payo which depends on the average of the underlying stock price over certain time interval. Since no general analytical solution for the price of the Asian option is known, a variety of techniques have been developed to analyze arithmetic average Asian options. A number of approximations that produce closed form expressions have appeared, se...
Competitive Monte Carlo methods for the Pricing of Asian Options
 Journal of Computational Finance
, 2000
"... We explain how a carefully chosen scheme can lead to competitive Monte Carlo algorithm for the computation of the price of Asian options. We give evidence of the eciency of these algorithms with a mathematical study of the rate of convergence and a numerical comparison with some existing methods. K ..."
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Cited by 13 (1 self)
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We explain how a carefully chosen scheme can lead to competitive Monte Carlo algorithm for the computation of the price of Asian options. We give evidence of the eciency of these algorithms with a mathematical study of the rate of convergence and a numerical comparison with some existing methods. Key Words: Asian option, Monte Carlo methods, Numerical methods, Diusion process. 1 Introduction Monte Carlo methods are known to be useful when the state dimension is large. This is widely true but we will give here an example of a small dimension problem coming from nance where a Monte Carlo (helped by a variance reduction technique) can be more ecient than other known methods. This example is based on the price of an Asian option (see subsection 2.1). This problem is known to be computationally hard and a lot of literature deals with this problem: using either analytic methods ([10], [9]), numerical methods based on the partial dierential equation associated ([4], [7], [12], [16]) or M...
On multigrid for linear complementarity problems with application to Americanâ€“style options
 ELECTRON. TRANS. NUMER. ANAL
, 2003
"... We discuss a nonlinear multigrid method for a linear complementarity problem. The convergence is improved by a recombination of iterants. The problem under consideration deals with option pricing from mathematical finance. Linear complementarity problems arise from socalled Americanstyle options. ..."
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Cited by 12 (1 self)
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We discuss a nonlinear multigrid method for a linear complementarity problem. The convergence is improved by a recombination of iterants. The problem under consideration deals with option pricing from mathematical finance. Linear complementarity problems arise from socalled Americanstyle options. A 2D convectiondiffusion type operator is discretized with the help of second order upwind discretizations. The properties of smoothers are analyzed with Fourier twogrid analysis. Numerical solutions obtained for the option pricing problem are compared with reference results.
2003. Optimal exercise policies and simulationbased valuation for AmericanAsian options. Operations Research 51: 52â€“66
 AUTHOR BIOGRAPHIES BARRY R. COBB
"... AmericanAsian options are averageprice options that allow early exercise. In this paper, we first derive structural properties of the optimal exercise policy for these call options in a general setting. In particular, we show that the optimal policy is a threshold policy: the option should be exer ..."
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Cited by 12 (6 self)
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AmericanAsian options are averageprice options that allow early exercise. In this paper, we first derive structural properties of the optimal exercise policy for these call options in a general setting. In particular, we show that the optimal policy is a threshold policy: the option should be exercised as soon as the average asset price reaches a characterized threshold, which can be written as a function of asset price at that time. After further characterizing the exercise boundary, we parameterize it, and then derive gradient estimators with respect to the parameters of the model. Implementing these estimators in an iterative gradientbased stochastic approximation algorithm, we approximate the optimal exercise boundary and consequently obtain an estimate for the price of the AmericanAsian option. Numerical experiments carried out indicate that the algorithm performs extremely well.
Shout Options: A Framework For Pricing Contracts Which Can Be Modified By The Investor
 J. Comp. Appl. Math
, 1999
"... A shout option may be broadly defined as a financial contract which can be modified by its holder according to specified rules. In a simple example, the holder could have the right to set the strike of an option equal to the current value of the underlying asset. In such a case, the holder effective ..."
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Cited by 12 (9 self)
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A shout option may be broadly defined as a financial contract which can be modified by its holder according to specified rules. In a simple example, the holder could have the right to set the strike of an option equal to the current value of the underlying asset. In such a case, the holder effectively has the right to select when to take ownership of an atthemoney option. More generally, the holder could have multiple rights along these lines, in some cases with a limit placed on the number of rights which may be exercised within a given time period (e.g. four times per year). The value of these types of contracts can be estimated by solving a system of interdependent linear complementarity problems. This paper describes a general framework for the valuation of complex types of shout options. Numerical issues related to interpolation and choice of timestepping method are considered in detail. Some illustrative examples are provided.
Quadratic Convergence Of A Penalty Method For Valuing American Options
 SIAM JOURNAL ON SCIENTIFIC COMPUTATION
, 2002
"... The convergence of a penalty method for solving the discrete regularized American option valuation problem is studied. Sufficient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution. These con ..."
Abstract

Cited by 10 (7 self)
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The convergence of a penalty method for solving the discrete regularized American option valuation problem is studied. Sufficient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution. These conditions also ensure that the solution of the penalty problem is an approximate solution to the discrete linear complementarity problem. The efficiency and quality of solutions obtained using the implicit penalty method are compared with those produced with the commonly used technique of handling the American constraint explicitly. Convergence rates are studied as the timestep and mesh size tend to zero. It is observed that an implicit treatment of the American constraint does not converge quadratically (as the timestep is reduced) if constant timesteps are used. A timestep selector is suggested which restores quadratic convergence.