The American Put Option Valued Analytically

Geske, Robert; Johnson, Herbert Eugene

doi:10.1111/j.1540-6261.1984.tb04921.x

Cited by 504 publications

(210 citation statements)

References 16 publications

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“…This is the prescription of the RE we employ as opposed to the more usual post-processed form which requires two independently derived solutions for use in the extrapolation V 0 j ¼ 2V 0 j ðÁt=2Þ À V 0 j ðÁtÞ (e.g., Geske and Johnson 1984). The significant difference is that, in the former case, a properly second-order integration method is obtained in the sense that a second-order solution is available at all intermediate times.…”

Section: Methodsmentioning

confidence: 99%

On the acceleration of explicit finite difference methods for option pricing

O’Sullivan

2011

Quantitative Finance

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Implicit finite difference methods are conventionally preferred over their explicit counterparts for the numerical valuation of options. In large part the reason for this is a severe stability constraint known as the Courant-Friedrichs-Lewy (CFL) condition which limits the latter class's efficiency. Implicit methods, however, are difficult to implement for all but the most simple of pricing models, whereas explicit techniques are easily adapted to complex problems. For the first time in a financial context, we present an acceleration technique, applicable to explicit finite difference schemes describing diffusive processes with symmetric evolution operators, called Super-Time-Stepping. We show that this method can be implemented as part of a more general approach for non-symmetric operators. Formal stability is thereby deduced for the exemplar cases of European and American put options priced under the Black-Scholes equation. Furthermore, we introduce a novel approach to describing the efficiencies of finite difference schemes as semi-empirical power laws relating the minimal real time required to carry out the numerical integration to a solution with a specified accuracy. Tests are described in which the method is shown to significantly ameliorate the severity of the CFL constraint whilst retaining the simplicity of the underlying explicit method. Degrees of acceleration are achieved yielding comparable, or superior, efficiencies to a set of benchmark implicit schemes. We infer that the described method is a powerful tool, the explicit nature of which makes it ideally suited to the treatment of symmetric and non-symmetric diffusion operators describing complex financial instruments including multi-dimensional systems requiring representation on decomposed and/or adaptive meshes.Numerical methods for option pricing, Black-Scholes model, Computational finance, Equity options, American options, Exotic options,

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Section: Methodsmentioning

confidence: 99%

On the acceleration of explicit finite difference methods for option pricing

O’Sullivan

2011

Quantitative Finance

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“…Based on an early work of Geske [11], Roll [7] derived an analytical valuation formula for American call options on a stock that pays discrete dividends. This formula was further developed by Geske [12], Whaley [13] and Geske and Johnson [14]. These models attracted widely attention and were further extended later.…”

Section: Introductionmentioning

confidence: 99%

“…Shastri and Tandon [15,16] separately derived a closed-form pricing formula for American options on futures contracts and foreign exchange (FX) options on major currencies. Bunch and Johnson [17] presented a more efficient analytical approach for American puts based on the compound option approach of Geske [12] and Geske and Johnson [14]. Recently Gukhal [18] derived an analytical valuation formula for compound options when the underlying asset follows a jump-diffusion process.…”

Section: Introductionmentioning

confidence: 99%

Pricing convertible bonds based on a multi-stage compound-option model

Gong

Zhu

2006

Physica A: Statistical Mechanics and its Applications

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In this paper, we introduce the concept of multi-stage compound options to the valuation of convertible bonds. Rather than evaluating a nested high-dimensional integral that has arisen from the valuation of multistage compound options, we found that adopting the Finite Difference Method (FDM) to solve the BlackScholes equation for each stage actually resulted in a better numerical efficiency. By comparing our results with those obtained by solving the Black-Scholes equation directly, we can show that the new approach does provide an approximation approach for the valuation of convertible bonds and demonstrate that it offers a great potential for a further extension to CBs with more complex structures such as those with call and/or put provisions. Disciplines Physical Sciences and Mathematics Publication DetailsThis article was originally published as Gong, P, He, Z and Zhu, SP, Pricing convertible bonds based on a multi-stage compound option model, Physica A, 336, 2006, 449-462

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“…Cox and Rubinstein (1985) outline the principle of the multidimensional extension. Other numerical valuation techniques are presented in Geske and Johnson (1984), Barone-Adesi and Whaley (1987), Barone-Adesi and Elliott (1991).…”

Section: Relation To Other Workmentioning

confidence: 99%

Numerical Valuation of High Dimensional Multivariate European Securities

Barraquand¹

1995

Management Science

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an acknowledgement of the authors and individual contributors to the work; and all applicable portions of the copyright notice. Copying, reproducing, or republishing for any other purpose shall require a license with payment of fee to the Paris Research Laboratory. All rights reserved. ii AbstractWe consider the problem of pricing an American contingent claim whose payoff depends on several sources of uncertainty. Using classical assumptions from the Arbitrage Pricing Theory, the theoretical price can be computed as the maximum over all possible early exercise strategies of the discounted expected cash flows under the modified risk-neutral information process.Several efficient numerical techniques exist for pricing American securities depending on one or few (up to 3) risk sources. They are either lattice-based techniques or finite difference approximations of the Black-Scholes diffusion equation. However, these methods cannot be used for high-dimensional problems, since their memory requirement is exponential in the number of risk sources.In this paper, we present an efficient numerical technique that combines Monte Carlo simulation with a particular partitioning method of the underlying assets space, which we call Stratified State Aggregation (SSA). Using this technique we can compute accurate approximations of prices of American securities with an arbitrary number of underlying assets. Our numerical experiments show that the method is practical for pricing American claims depending on up to 400 risk sources. On all problems for which we could compare the method with known optimal solutions, the price computed through stratified state aggregation was indistinguishable from the optimal theoretical price. Several numerical examples are presented and discussed.iii Acknowledgements This research benefited from discussions with Bruno Langlois. We wish to thank Thierry Pudet for reviewing an earlier draft.iv

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The American Put Option Valued Analytically

Cited by 504 publications

References 16 publications

On the acceleration of explicit finite difference methods for option pricing

On the acceleration of explicit finite difference methods for option pricing

Pricing convertible bonds based on a multi-stage compound-option model

Numerical Valuation of High Dimensional Multivariate European Securities

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