Solution of option pricing equations using orthogonal polynomial expansion

Baustian, Falko; Filipová, Kateřina; Pospı́šil, J.

doi:10.21136/am.2021.0361-19

Cited by 4 publications

(4 citation statements)

References 43 publications

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“…11 (Appendix), §11.1- §11.2], pp. 48-50, in Takáč [44], and in the numerical simulations by Baustian et al [7]. Replacing a piecewise linear nonlinearity by a polynomial introduces a significant error into the algorithm in [25].…”

Section: Numerical Methods: Finite Differences/elements Compared To M...mentioning

confidence: 99%

Monotone methods in counterparty risk models with nonlinear Black–Scholes-type equations

Alziary

Takáč

2022

SeMA

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A nonlinear Black–Scholes-type equation is studied within counterparty risk models. The classical hypothesis on the uniform Lipschitz-continuity of the nonlinear reaction function allows for an equivalent transformation of the semilinear Black–Scholes equation into a standard parabolic problem with a monotone nonlinear reaction function and an inhomogeneous linear diffusion equation. This setting allows us to construct a scheme of monotone, increasing or decreasing, iterations that converge monotonically to the true solution. As typically any numerical solution of this problem uses most computational power for computing an approximate solution to the inhomogeneous linear diffusion equation, we discuss also this question and suggest several solution methods, including those based on Monte Carlo and finite differences/elements.

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Section: Numerical Methods: Finite Differences/elements Compared To M...mentioning

confidence: 99%

Monotone methods in counterparty risk models with nonlinear Black–Scholes-type equations

Alziary

Takáč

2022

SeMA

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show abstract

“…(4.7) that provides the required limit on the solution u(x, ξ, t) in Theorem 4. 4. This theorem has an important corollary applicable to a typical initial value problem in Mathematical Finance (see Corollary 4.5).…”

Section: Introductionmentioning

confidence: 91%

“…(4.5). This feature of Heston's model is used in the recent work by F. Baustian, K. Filipová, and J. Pospíšil [4] with an orthogonal polynomial expansion in the spatial domain H. Orthogonal polynomial expansions have been used recently also in D. Ackerer and D. Filipović [1] for numerical approximations. Earlier, the authors [3,Sect.…”

Section: Introductionmentioning

confidence: 99%

The Heston stochastic volatility model has a boundary trace at zero volatility

Alziary¹,

Takáč²

2020

Preprint

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We establish boundary regularity results in Hölder spaces for the degenerate parabolic problem obtained from the Heston stochastic volatility model in Mathematical Finance set up in the spatial domain (upper half-plane) H = R × (0, ∞) ⊂ R 2 . Starting with nonsmooth initial data u 0 ∈ H, we take advantage of smoothing properties of the parabolic semigroup e −tA : H → H, t ∈ R + , generated by the Heston model, to derive the smoothness of the solution u(t) = e −tA u 0 for all t > 0. The existence and uniqueness of a weak solution is obtained in a Hilbert space H = L 2 (H; w) with very weak growth restrictions at infinity and on the boundary ∂H = R × {0} ⊂ R 2 of the half-plane H. We investigate the influence of the boundary behavior of the initial data u 0 ∈ H on the boundary behavior of u(t) for t > 0.

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“…11 (Appendix), §11.1 - §11.2], pp. 48-50, in P. Takáč [38], and in the numerical simulations by F. Baustian, K. Filipová, and J. Pospíšil [7]. Replacing a piecewise linear nonlinearity by a polynomial introduces a significant error into the algorithm in [20].…”

Section: Then the Monotone Iterations Umentioning

confidence: 99%

Monotone methods in counterparty risk models with non-linear Black-Scholes-type equations

Alziary¹,

Takáč²

2022

Preprint

View full text Add to dashboard Cite

A nonlinear Black-Scholes-type equation is studied within counterparty risk models. The classical hypothesis on the uniform Lipschitz-continuity of the nonlinear reaction function allows for an equivalent transformation of the semilinear Black-Scholes equation into a standard parabolic problem with a monotone nonlinear reaction function and an inhomogeneous linear diffusion equation. This setting allows us to construct a scheme of monotone, increasing or decreasing, iterations that converge monotonically to the true solution. As typically any numerical solution of this problem uses most computational power for computing an approximate solution to the inhomogeneous linear diffusion equation, we discuss also this question and suggest several solution methods, including those based on Monte Carlo and finite differences/elements.

show abstract

Solution of option pricing equations using orthogonal polynomial expansion

Cited by 4 publications

References 43 publications

Monotone methods in counterparty risk models with nonlinear Black–Scholes-type equations

Monotone methods in counterparty risk models with nonlinear Black–Scholes-type equations

The Heston stochastic volatility model has a boundary trace at zero volatility

Monotone methods in counterparty risk models with non-linear Black-Scholes-type equations

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