A BSDE with Delayed Generator Approach to Pricing under Counterparty Risk and Collateralization

Cordoni, Francesco; Persio, Luca Di

doi:10.1155/2016/1059303

Cited by 12 publications

(13 citation statements)

References 36 publications

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“…That theorem is very useful to understand the deep relation between generators of stochastic processes and their Nelson's derivatives. We would like to underline that such relation can be used to better study a wide range of models as those arise in the mathematical finance scenario, particularly in the case of delayed forward-backward S(P)DEs, see, e.g., [5] Theorem 3. Let X ∈ Λ 1 which writes dX(t) = b(t, X(t))dt+σ(t, X(t))dW (t).…”

Section: The Nelson's Derivatives For Good Diffusion Processesmentioning

confidence: 99%

On the Relation Between the Nelson Stochastic Derivative and the Feynman-Kac Theorem

Guida¹

2017

Int. J. of Pure and Appl. Math.

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This paper proposes a different point of view to look at the Nelson's derivatives. In particular we relate the Nelson's forward derivative of a function of a diffusion, with certain properties, to the partial differential equation coming from the Feynman-Kac's theorem. We will see that whenever the Nelson forward derivative of functions of a diffusion process, that satisfies a certain stochastic differential equations, is computed at the initial point of the process and whenever we put it equal to zero, we obtain a partial differential equation. The latter is the same equation coming from the Feynman-Kac's theorem. Finally we will see how we could reduce the stochastic derivatives to the forward Nelson's derivative of a certain process by supposing the existence of a new complex processes related to the backward filtration.

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Section: The Nelson's Derivatives For Good Diffusion Processesmentioning

confidence: 99%

On the Relation Between the Nelson Stochastic Derivative and the Feynman-Kac Theorem

Guida¹

2017

Int. J. of Pure and Appl. Math.

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“…basically as an application of the Fubini Theorem, in the second equality, and by the very definition of the BLT density given in eq. (19). Hence, as an intermediate first application, we use the above pricing formula for our Accumulator, and we compare three different pricing techniques for the Accumulator defined by (C − 2P ), where C and P are the Call option price and the Put option price, namely: BSD, the straight BS evaluation, i.e.…”

Section: The Proposal For the Ee Evaluationmentioning

confidence: 99%

“…A smaller execution time and a better EE appraisal accuracy, makes our method a competitive tool, suggesting to extend the local time approach to more general derivatives, such as barrier options or Asian options. Next step consists in comparing our results with those derived in [18,19].…”

Section: Conclusion and Further Researchmentioning

confidence: 99%

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Estimating the Counterparty Risk Exposure by Using the Brownian Motion Local Time

Bonollo

Persio

Mammi

et al. 2017

International Journal of Applied Mathematics and Computer Science

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In recent years, the counterparty credit risk measure, namely the default risk in over-the-counter (OTC) derivatives contracts, has received great attention by banking regulators, specifically within the frameworks of Basel II and Basel III. More explicitly, to obtain the related risk figures, one is first obliged to compute intermediate output functionals related to the mark-to-market position at a given time no exceeding a positive and finite time horizon. The latter implies an enormous amount of computational effort is needed, with related highly time consuming procedures to be carried out, turning out into significant costs. To overcome the latter issue, we propose a smart exploitation of the properties of the (local) time spent by the Brownian motion close to a given value.

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“…In fact, recursive utility corresponds to the solution of a particular BSDE associated with a generator which does not depend on z-component, see below for more details. Moreover, the BSDEs theory has been extensively used in stochastic control, see, e.g., [3] and references therein, as they appear as adjoint equations in the stochastic version of Pontryagin maximum principle, and in Mathematical Finance, since any pricing problem by replication can be written in terms of linear BSDEs, or non-linear BSDEs when portfolios constraints are taken into account, see [19,11,12,13,14].…”

Section: Introductionmentioning

confidence: 99%

A Rbsdes Approach for the Numerical Evaluation of the American Option Pricing Problem

Franco¹

2016

Int. J. of Pure and Appl. Math.

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The problem of pricing American type options is a typical example of a non linear problem characterized by the absence of closed expressions for its evaluation. Therefore, during recent years, such an issue has been approached , both deterministically and randomly, from the algorithmic point of view, trying to derive suitable numerical approximations. In this paper, starting from the aforementioned solutions, we review some computational, stochastic inspired, methods, mainly based on the the existing link between the above recalled pricing task, and the theory of Reflected Backward Stochastic Differential Equations (BSDEs).In particular we show how suitable numerical schemes can be developed within the SBDEs framework by mean of quantization techniques as well as considering Monte Carlo methods. AMS Subject Classification: 60H15, 60H35, 90-08, 91G20, 91G80, 91B25

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A BSDE with Delayed Generator Approach to Pricing under Counterparty Risk and Collateralization

Cited by 12 publications

References 36 publications

On the Relation Between the Nelson Stochastic Derivative and the Feynman-Kac Theorem

On the Relation Between the Nelson Stochastic Derivative and the Feynman-Kac Theorem

Estimating the Counterparty Risk Exposure by Using the Brownian Motion Local Time

A Rbsdes Approach for the Numerical Evaluation of the American Option Pricing Problem

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