Fast numerical valuation of American, exotic and complex options

Dempster, M. A. H.; Hutton, J.P.

doi:10.1080/135048697334809

Cited by 29 publications

(11 citation statements)

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“…Note that implicit finite difference methods have occasionally been used for multidimensional problems in finance (Brennan and Schwartz, 1980;Wilmott et al, 1993;Dempster and Hutton, 1997), but it has recently been demonstrated in that the FVM has several advantages over finite difference schemes in this context.…”

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confidence: 99%

Negative coefficients in two-factor option pricing models

Zvan¹,

Vetzal²

2003

JCF

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The importance of positive coefficients in numerical schemes is frequently emphasized in the finance literature. This topic is explored in detail in this paper, in the particular context of two factor models. First, several two factor lattice type methods are derived using a finite difference/finite element methodology. Some of these methods have negative coefficients, but are nevertheless stable and consistent. Second, we outline the conditions under which finite volume/element methods applied to two factor option pricing partial differential equations give rise to discretizations with positive coefficients. Numerical experiments indicate that constructing a mesh which satisfies positive coefficient conditions may not only be unnecessary, but in some cases even detrimental. As well, it is shown that schemes with negative coefficients due to the discretization of the diffusion term satisfy approximate local maximum and minimum conditions as the mesh spacing approaches zero. This finding is of significance since, for arbitrary diffusion tensors, it may not be possible to construct a positive coefficient discretization for a given set of nodes.

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confidence: 99%

Negative coefficients in two-factor option pricing models

Zvan¹,

Vetzal²

2003

JCF

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“…After that convergence becomes fast linear. Discretized nonlinear complementarity problems generated from the upstream weighting with the Van Leer nonlinear flux limiter have additional degeneracy: the Jacobian of the constraintsḠ(x, y) = 0 can be singular when V down = V up , as can be seen from the definition of G given in (10). We observe that, though the proposed method takes slightly more iterations when G(x, y) is nonlinear, convergence is still rapid.…”

Section: G(x Y)mentioning

confidence: 77%

“…In addition, linear complementarity problems with non-tridiagonal coefficient matrices can arise in different asset pricing models, e.g., a jump diffusion model [21]. Computational investigation of American option pricing using the discretized linear complementarity has been made in [11,10,12,15]. In [20], the projected SOR approach has been considered.…”

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confidence: 99%

A Newton method for American option pricing

Coleman¹,

Li²,

Verma³

2002

JCF

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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 upstream weighting finite difference approach with the van Leer flux limiter for the convection term [19,22] yields nonlinear complementarity problems.We propose a Newton type interior-point method for solving discretized complementarity/variational inequality problems that arise in the American option valuation. We illustrate that the proposed method on average solves a discretized problem in 2 ∼ 5 iterations to an appropriate accuracy. More importantly, the average number of iterations required does not seem to depend on the number of discretization points in the spatial dimension; the average number of iterations actually decreases as the time discretization becomes finer.The arbitrage condition for the fair value of the American option requires that the delta hedge factor be continuous. We investigate continuity of the delta factor approximation using the complementarity approach, the binomial method, and the explicit payoff method. We illustrate that, while the (implicit finite difference) complementarity approach yields continuous delta hedge factors, both the binomial method and the explicit payoff method (with the implicit finite difference) yield discontinuous delta approximations. Hence the early exercise curve computed from the binomial method and the explicit payoff method can be inaccurate. In addition, it is demonstrated that the delta factor computed using the Crank-Nicolson method with complementarity approach oscillates around the early exercise curve. Introduction. An American option gives the holder of the option the right, but not the obligation, to buy or sell the underlying for a fixed exercise price any time before expiry. Valuation of the American option is an optimal stopping (or a free boundary) problem; it is significantly more complex than the European option pricing. Computing the fair value of an American option requires determination of the early exercise curve. This early exercise curve divides the domain of the underlying asset price and time into a continuation region C and a stopping region S; it is optimal to exercise in S while continually hold the contract in C. This early exercise curve is characterized by two boundary conditions. First, the option value is equal to the payoff at the early exercise curve. Second, the delta hedge factor is continuous at the early exercise curve.American option valuation has been an active research area; many methods have been proposed to approximate American option values. Reviews of these methods can be found in Broadie and Detemple [4,5]. A method frequently used in practice is the binomial method propose...

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“…The application of our method to gammas (second-order sensitivities to interest rate shocks) are given in Sec. 11. In Sec.…”

Section: Introductionmentioning

confidence: 99%

Risk Sensitivities of Bermuda Swaptions

Piterbarg

2004

Int. J. Theor. Appl. Finan.

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A new approach to the problem of computing risk sensitivities of Bermuda swaptions in a lattice, or PDE, framework is presented. The algorithms developed perform the task much faster and more accurately that the traditional approach in which the Greeks are computed numerically by shocking the appropriate inputs and revaluing the instrument. The time needed to execute the tradition scheme grows linearly with the number of Greeks required, whereas our approach computes any number of Greeks for a Bermuda swaption in nearly constant time. The new method explores symmetries in the structure of Bermuda swaptions to derive recursive relations between different Greeks, and is essentially model-independent. These recursive relations allow us to represent risk sensitivities in a number of ways, in particular as integrals over the "survival" density. The survival density is obtained as a solution to a forward Kolmogorov equation. This representation is the basis for practical applications of our approach. Market Model; Cheyette model. Int. J. Theor. Appl. Finan. 2004.07:465-509. Downloaded from www.worldscientific.com by UNIVERSITY OF CALIFORNIA @ SAN DIEGO on 02/03/15. For personal use only.466 V. V. Piterbargare called deltas. Typically, they are calculated and presented in the following way. A time line is split into time intervals (buckets) of common length, for example three months. Forward interest rates that correspond to all buckets are shocked in turn. A shocked interest rate curve is fed into the model, and the Bermuda swaption is revalued. The sensitivity of the Bermuda swaption value to this shock (often referred to as a "bucketed" delta) is then computed by subtracting a base value of the Bermuda swaption from its shocked value, and normalizing the difference by the size of the bump.Second-order derivatives to the changes in rates are also common, and usually called bucketed gammas.Bermuda swaptions are sensitive to changes in interest rate volatility as well. This sensitivity is also usually bucketed, in the sense that the Bermuda swaption sensitivity to various interest rate volatilities is computed. Bermuda swaptions are related to European swaptions. In most markets, European swaptions are actively traded and their implied volatilities are readily observable. For that reason, interest rate volatility sensitivities, called vegas, of Bermuda swaptions are computed with respect to shocks to European swaption volatilities.The Greeks calculation scheme, as presented above, requires a new value of a Bermuda swaption to be obtained for each bucketed Greek (sometimes even two new valuations are required if two-sided numerical derivatives are used). Thus, computing bucketed deltas to individual shocks of all 3-month tenor rates for a 40 year Bermuda swaption will require about 120 calls to the valuation model. Each valuation requires solving a PDE equation or a rollback on a lattice/tree. Adding gammas and vegas increases the number of valuations further. Multiplied by the size of a typical portfolio of Bermuda swap...

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Fast numerical valuation of American, exotic and complex options

Cited by 29 publications

References 5 publications

Negative coefficients in two-factor option pricing models

Negative coefficients in two-factor option pricing models

A Newton method for American option pricing

Risk Sensitivities of Bermuda Swaptions

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