Sensitivity Analysis Using Itô--Malliavin Calculus and Martingales, and Application to Stochastic Optimal Control

Gobet, Emmanuel; Munos, Rémi

doi:10.1137/s0363012902419059

Cited by 86 publications

(84 citation statements)

References 35 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%

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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%

“…Denote p ε := 4 1+ε ε . Then, for factor II 3 n we use (17) in connection with (16) and apply Minkowski's inequality, Burkholder-Davis-Gundy's inequality on the stochastic integrals and Hölder's Proof of Theorem 3.1. To prove that X x t is a strong solution, it only remains to prove that X x t is F t -measurable for every t ∈ [0, T ].…”

mentioning

confidence: 99%

Computing deltas without derivatives

Baños

Meyer‐Brandis²,

Proske

et al. 2017

Finance Stoch

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“…The implementation of procedures for all three methods have been outlined in [18] in the pure diffusion case. We shall not work on this issue here, since our main message is that one can implement precisely the same methods as in pure diffusion cases for jump-diffusions.…”

Section: Calculating the Greeks In Finite Dimensionmentioning

confidence: 99%

“…Moreover, by the explicit construction of a in the proof, we can assert that π v is the sum of an Itô integral and an integral with respect to Lebesgue measure, see for instance [18].…”

Section: Definition 1 Assume That H = Rmentioning

confidence: 99%

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Absolutely Continuous Laws of Jump-Diffusions in Finite and Infinite Dimensions with Applications to Mathematical Finance

Forster¹,

Lütkebohmert²,

Teichmann³

2009

SIAM J. Math. Anal.

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Abstract. In mathematical Finance calculating the Greeks by Malliavin weights has proved to be a numerically satisfactory procedure for finite-dimensional Itô-diffusions. The existence of Malliavin weights relies on absolute continuity of laws of the projected diffusion process and a sufficiently regular density. In this article we first prove results on absolute continuity for laws of projected jump-diffusion processes in finite and infinite dimensions, and a general result on the existence of Malliavin weights in finite dimension. In both cases we assume Hörmander conditions and hypotheses on the invertibility of the so-called linkage operators. The purpose of this article is to show that for the construction of numerical procedures for the calculation of the Greeks in fairly general jump-diffusion cases one can proceed as in a pure diffusion case. We also show how the given results apply to infinite dimensional questions in mathematical Finance. There we start from the Vasiček model, and add -by pertaining no arbitrage -a jump diffusion component. We prove that we can obtain in this case an interest rate model, where the law of any projection is absolutely continuous with respect to Lebesgue measure on R M .

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Sensitivity of the joint survival probability for reinsurance schemes

Roumelioti

Zazanis

Frangos

2013

Math Methods in App Sciences

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We model the joint risk process for an insurer and a reinsurer using a diffusion approximation and obtain expressions for the sensitivity of the joint survival probability with respect to parameters of the reinsurance scheme. The approach used leads, more generally, to explicit expressions for the sensitivity of functionals of diffusions in double-struckRm with constant coefficients, whose drift vector and covariance matrix are differentiable functions of a parameter, in a form suitable for efficient Monte Carlo simulation. The functionals examined depend on the values of the diffusion at a finite number of time epochs, and the sensitivities are calculated using the likelihood ratio method. An extension to dynamic reinsurance schemes is also briefly described, and sensitivity estimators are provided using the integration by parts formula of the Malliavin calculus. Copyright © 2013 John Wiley & Sons, Ltd.

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Sensitivity Analysis Using Itô--Malliavin Calculus and Martingales, and Application to Stochastic Optimal Control

Cited by 86 publications

References 35 publications

Computing deltas without derivatives

Computing deltas without derivatives

Absolutely Continuous Laws of Jump-Diffusions in Finite and Infinite Dimensions with Applications to Mathematical Finance

Sensitivity of the joint survival probability for reinsurance schemes

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