The Black–Scholes equation in stochastic volatility models

Ekström, Erik; Tysk, Johan

doi:10.1016/j.jmaa.2010.04.014

Cited by 49 publications

(44 citation statements)

References 22 publications

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“…For put options, we extend u for s > s max as u(s, v, τ ) = 0. On the boundary v = 0, the PIDEs (2.2) and (2.5) themselves can be posed as a boundary condition; see [16] for a discussion on this boundary and its treatment. …”

Section: Governing Equationsmentioning

confidence: 99%

An IMEX-Scheme for Pricing Options under Stochastic Volatility Models with Jumps

Salmi¹,

Toivanen²,

Sydow³

2014

SIAM J. Sci. Comput.

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Abstract. Partial integro-differential equation (PIDE) formulations are often preferable for pricing options under models with stochastic volatility and jumps, especially for American-style option contracts. We consider the pricing of options under such models, namely the Bates model and the so-called stochastic volatility with contemporaneous jumps (SVCJ) model. The nonlocality of the jump terms in these models leads to matrices with full matrix blocks. Standard discretization methods are not viable directly since they would require the inversion of such a matrix. Instead, we adopt a two-step implicit-explicit (IMEX) time discretization scheme, the IMEX-CNAB scheme, where the jump term is treated explicitly using the second-order Adams-Bashforth (AB) method, while the rest is treated implicitly using the Crank-Nicolson (CN) method. The resulting linear systems can then be solved directly by employing LU decomposition. Alternatively, the systems can be iterated under a scalable algebraic multigrid (AMG) method. For pricing American options, LU decomposition is employed with an operator splitting method for the early exercise constraint or, alternatively, a projected AMG method can be used to solve the resulting linear complementarity problems. We price European and American options in numerical experiments, which demonstrate the good efficiency of the proposed methods. It is well known that fitting empirically observed option prices into the Black-Scholes model typically implies a volatility distribution with a smile-like shape. This volatility smile becomes more pronounced near the maturity date. The usual modifications of the BlackScholes model to explain such implied volatility patterns include models with jumps and/or stochastic volatility. The underlying asset price can be completely modeled by an infinite activity model such as the Carr-Geman-Madan-Yor (CGMY) model [8]. Over long time intervals the component behaving like the geometric Brownian motion becomes the dominant part in the model. For options with long maturities, the stochastic volatility models, for example the Heston model [24], are often regarded as more appropriate. For options with short maturities, however, jumps become increasingly important as a purely geometric Brownian motion driven process would require extremely high levels of volatility to explain the pronounced volatility smile pattern. Well-known jump-diffusion models in the literature include the Merton [34]

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Section: Governing Equationsmentioning

confidence: 99%

An IMEX-Scheme for Pricing Options under Stochastic Volatility Models with Jumps

Salmi¹,

Toivanen²,

Sydow³

2014

SIAM J. Sci. Comput.

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“…In [13], Ekström and Tysk extend their results in [14] to the case of two-dimensional stochastic volatility models for option prices, where the variance process satisfies the assumptions of [14, Hypothesis 2.1].…”

Section: Uniqueness Of Solutions To Parabolic Obstacle Problemsmentioning

confidence: 90%

Stochastic representation of solutions to degenerate elliptic and parabolic boundary value and obstacle problems with Dirichlet boundary conditions

Feehan¹,

Pop²

2014

Trans. Amer. Math. Soc.

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Abstract. We prove existence and uniqueness of stochastic representations for solutions to elliptic and parabolic boundary value and obstacle problems associated with a degenerate Markov diffusion process. In particular, our article focuses on the Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance and a paradigm for a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this process with killing, called the elliptic Heston operator, is a second-order, degenerate, elliptic partial differential operator whose coefficients have linear growth in the spatial variables and where the degeneracy in the operator symbol is proportional to the distance to the boundary of the half-plane. In mathematical finance, solutions to terminal/boundary value or obstacle problems for the parabolic Heston operator correspond to value functions for American-style options on the underlying asset.

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“…In this and other works (e.g. Ekström and Tysk [15]), Tysk addresses the issue of absence of boundary conditions by adding appropriate ones. As mentioned by the authors, these boundary conditions are redundant if the state vector X does not reach the boundary.…”

Section: The Bates Model With Downward Jumps In the Volatilitymentioning

confidence: 99%

Singular risk-neutral valuation equations

2011

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Many risk-neutral pricing problems proposed in the finance literature do not admit closed-form expressions and have to be dealt with by solving the corresponding partial integro-differential equation. Often, these PIDEs have singular diffusion matrices and coefficients that are not Lipschitz-continuous up to the boundary. In addition, in general, boundary conditions are not specified. In this paper, we prove existence and uniqueness of (continuous) viscosity solutions for linear PIDEs with all the above features, under a Lyapunov-type condition. Our results apply to European and Asian option pricing, in jump-diffusion stochastic volatility and path-dependent volatility models. We verify our Lyapunov-type condition in several examples, including the arithmetic Asian option in the Heston model.

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The Black–Scholes equation in stochastic volatility models

Cited by 49 publications

References 22 publications

An IMEX-Scheme for Pricing Options under Stochastic Volatility Models with Jumps

An IMEX-Scheme for Pricing Options under Stochastic Volatility Models with Jumps

Stochastic representation of solutions to degenerate elliptic and parabolic boundary value and obstacle problems with Dirichlet boundary conditions

Singular risk-neutral valuation equations

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