Derivative-free Greeks for the Barndorff-Nielsen and Shephard stochastic volatility model

Benth, Fred Espen; Groth, Martin; Wallin, Olli

doi:10.1080/17442501003629554

Cited by 7 publications

(6 citation statements)

References 24 publications

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“…The first paper to address this application was [15], which has consecutively triggered an active research interest in this topic, see e.g. [14], [4], [1], [17], [21]. See also [7], [11], [18] and references therein for a related approach based on functional Itô calculus.…”

Section: Introductionmentioning

confidence: 99%

“…We remark that the probabilistic representation (4) of the space derivative of the solution to the associated Kolmogorov equation is also referred to as Bismuth-Elworthy-Li type formula in the literature due to [13], [6]. The strength of (4) is that the Delta is expressed again as an expectation of the pay-off multiplied by the so-called Malliavin weight T 0 a(t) σ −1 (X x t ) Z t dB t . Computing the Delta by Monte-Carlo via this reformulation then guarantees a convergence rate that is independent of the regularity of the pay-off function Φ and the dimensionality.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Computing deltas without derivatives

Baños

Meyer‐Brandis²,

Proske

et al. 2017

Finance Stoch

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A well-known application of Malliavin calculus in Mathematical Finance is the probabilistic representation of option price sensitivities, the socalled Greeks, as expectation functionals that do not involve the derivative of the pay-off function. This allows for numerically tractable computation of the Greeks even for discontinuous pay-off functions. However, while the pay-off function is allowed to be irregular, the coefficients of the underlying diffusion are required to be smooth in the existing literature, which for example excludes already simple regime switching diffusion models. The aim of this article is to generalise this application of Malliavin calculus to Itô diffusions with irregular drift coefficients, whereat we here focus on the computation of the Delta, which is the option price sensitivity with respect to the initial value of the underlying. To this purpose we first show existence, Malliavin differentiability, and (Sobolev) differentiability in the initial condition of strong solutions of Itô diffusions with drift coefficients that can be decomposed into the sum of a bounded but merely measurable and a Lipschitz part. Furthermore, we give explicit expressions for the corresponding Malliavin and Sobolev derivatives in terms of the local time of the diffusion, respectively. We then turn to the main objective of this article and analyse the existence and probabilistic representation of the corresponding Deltas for European and path-dependent options. We conclude with a small simulation study of several regime-switching examples.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Computing deltas without derivatives

Baños

Meyer‐Brandis²,

Proske

et al. 2017

Finance Stoch

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show abstract

“…In the following, we study the robustness of the BN-S model and the associated option price. The computation of the delta is studied in Benth, Groth, and Wallin [3]. As e λεs ≤ e aT , then by dominated convergence theorem, we can take the limit inside the integral in (4.7) and we have the almost sure convergence of the process Y ε to the process Y when ε goes to 0.…”

Section: ∂Xmentioning

confidence: 99%

“…As the market is incomplete, We consider a structure preserving class of equivalent martingale measures introduced by Nicolato and Venardos [27] and we prove the convergence of the option price after a change of measure in this class. For the computation of the delta of options written in such models, we refer to Benth, Groth, and Wallin [3].…”

Section: Introductionmentioning

confidence: 99%

Computation of the Delta in Multidimensional Jump-Diffusion Setting with Applications to Stochastic Volatility Models

Khedher¹

2012

Stochastic Analysis and Applications

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We study the robustness of option prices to model variation in a multidimensional jump-diffusion framework. In particular we consider price dynamics in which small variations are modeled either by a Poisson random measure with infinite activity or by a Brownian motion. We consider both European and Exotic option and we study their deltas using two approaches: the Malliavin method and the Fourier method. We prove robustness of the deltas to model variation. We apply these results to the study of stochastic volatility models for the underlying and the corresponding options.

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“…0:n ) and return to the start of 1. In order to select the M n , a conditionally optimal density is [26]: (5) M n (x n |x 0:n−1 ) = π n (x n |x 0:n−1 ).…”

Section: Formulationmentioning

confidence: 99%

Sequential Monte Carlo Methods for Option Pricing

Jasra

Moral

2011

Stochastic Analysis and Applications

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International audienceIn the following paper we provide a review and development of sequential Monte Carlo (SMC) methods for option pricing. SMC are a class of Monte Carlo-based algorithms, that are designed to approximate expectations w.r.t a sequence of related probability measures. These approaches have been used, successfully, for a wide class of applications in engineering, statistics, physics and operations research. SMC methods are highly suited to many option pricing problems and sensitivity/Greek calculations due to the nature of the sequential simulation. However, it is seldom the case that such ideas are explicitly used in the option pricing literature. This article provides an up-to date review of SMC methods, which are appropriate for option pricing. In addition, it is illustrated how a number of existing approaches for option pricing can be enhanced via SMC. Specifically, when pricing the arithmetic Asian option w.r.t a complex stochastic volatility model, it is shown that SMC methods provide additional strategies to improve estimation

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Derivative-free Greeks for the Barndorff-Nielsen and Shephard stochastic volatility model

Cited by 7 publications

References 24 publications

Computing deltas without derivatives

Computing deltas without derivatives

Computation of the Delta in Multidimensional Jump-Diffusion Setting with Applications to Stochastic Volatility Models

Sequential Monte Carlo Methods for Option Pricing

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