Implicit Solution Of Uncertain Volatility/Transaction Cost Option Pricing Models With Discretely Observed Barriers

. Option pricing models with uncertain volatility/transaction costs give rise to a nonlinear PDE. Previous work has focussed on explicit methods. However, pricing discretely observed barrier options requires a very small grid spacing near the barrier, and as a result, the maximumstable timestep for an explicit method is impractically small. A fully implicit method is developed for nonlinear option pricing models, and applied to arithmetic step options, where the option loses a fraction of its value for every day over the barrier. Keywords: Option pricing, nonlinear, implicit Running Title: Implicit uncertain volatility models Acknowledgment: This work was supported by the National Sciences and Engineering Research Council of Canada, and Communications and Information Technology Ontario (CITO), funded by the Province of Ontario. AMS Classification: 65N30 Last Revised: February 18, 1999 Department of Computer Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1, pafor..

A General Finite Element Approach For PDE Option Pricing Models

. This paper presents a general approach for solving two-factor (two-dimensional) option pricing problems. The finite element method provides greater flexibility over that of the finite difference schemes (or equivalently, lattice methods) which are often employed in finance. This paper will demonstrate how various two-dimensional pricing problems can all be solved using the same approach. The generality of the approach is in part due to the fact that changes caused by different model specifications are localized. Constraints on the solution are treated in a uniform manner using a penalty method. This uniform approach can readily accommodate constraints such as early-exercise opportunities and barriers. Keywords: Finite element, option pricing, local extremum diminishing Running Title: Finite element option pricing Acknowledgment: This work was supported by the National Sciences and Engineering Research Council of Canada, the Social Sciences and Humanities Research Council of Canada,..

Managing Capacity for Telecommunications Networks under Uncertainty

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

Penalty Methods For American Options With Stochastic Volatility

The American early exercise constraint can be viewed as transforming the two dimensional stochastic volatility option pricing PDE into a differential algebraic equation (DAE). Several methods are described for forcing the algebraic constraint by using a penalty source term in the discrete equations. The resulting nonlinear algebraic equations are solved using an approximate Newton iteration. The solution of the Jacobian is obtained using an incomplete LU (ILU) preconditioned PCG method. Some example computations are presented for option pricing problems based on a stochastic volatility model, including an exotic American chooser option written on a put and call with discrete double knockout barriers and discrete dividends

Discrete Asian Barrier Options

. 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 American-Asian 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..