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Mathematics
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Application of Stochastic Analysis in Mathematical Finance
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Application of Stochastic Analysis in Mathematical Finance

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Financial Mathematics".

Deadline for manuscript submissions: closed (30 June 2021) | Viewed by 17459

Special Issue Editors

Prof. Dr. Elisa Alòs

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Guest Editor
Department d’Economia i Empresa and Barcelona Graduate School of Economics, University of Pompeu Fabra, Barcelona, Spain
Interests: mathematical finance; stochastic modeling; fractional Brownian motion
Prof. Dr. Jorge A. León

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Guest Editor
Control Automático, CINVESTAV-IPN, Mexico City, Mexico

Special Issue Information

Dear Colleagues,

This Special Issue is devoted to applications of stochastic analysis to the area of quantitative finance. The increasing complexity of markets needs the tools of stochastic analysis to be implemented to address problems associated with quantitative finance as, for example, hedging, option pricing, portfolio optimization, and study of volatilities, among others. Indeed, we cannot think about addressing or understanding problems of modern quantitative finance without using tools of stochastic analysis such as the Feynman–Kac formula, Girsanov’s theorem, Itô’s lemma, derivatives in the Malliavin calculus sense, the results used for the study of local volatilities, etc.

This Special Issue will contain 5 invited survey papers written by prestigious experts in quantitative finance, as well as research papers on applications of stochastic analysis to quantitative finance.

Prof. Dr. Elisa Alòs
Prof. Dr. Jorge A. León
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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Keywords

  • Applications of stochastic analysis
  • Modeling
  • Numerical methods
  • Option pricing
  • Portfolio optimization
  • Stochastic volatility models
  • Study of volatility

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Published Papers (5 papers)

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Research

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20 pages, 356 KiB  
Article
SWIFT Calibration of the Heston Model
by Eudald Romo and Luis Ortiz-Gracia
Mathematics 2021, 9(5), 529; https://doi.org/10.3390/math9050529 - 3 Mar 2021
Cited by 3 | Viewed by 2017
Abstract
In the present work, the SWIFT method for pricing European options is extended to Heston model calibration. The computation of the option price gradient is simplified thanks to the knowledge of the characteristic function in closed form. The proposed calibration machinery appears to [...] Read more.
In the present work, the SWIFT method for pricing European options is extended to Heston model calibration. The computation of the option price gradient is simplified thanks to the knowledge of the characteristic function in closed form. The proposed calibration machinery appears to be extremely fast, in particular for a single expiry and multiple strikes, outperforming the state-of-the-art method we compare it with. Further, the a priori knowledge of SWIFT parameters makes a reliable and practical implementation of the presented calibration method possible. A wide range of stress, speed and convergence numerical experiments is carried out, with deep in-the-money, at-the-money and deep out-of-the-money options for very short and very long maturities. Full article
(This article belongs to the Special Issue Application of Stochastic Analysis in Mathematical Finance)
21 pages, 564 KiB  
Article
Markovian Approximation of the Rough Bergomi Model for Monte Carlo Option Pricing
by Qinwen Zhu, Grégoire Loeper, Wen Chen and Nicolas Langrené
Mathematics 2021, 9(5), 528; https://doi.org/10.3390/math9050528 - 3 Mar 2021
Cited by 4 | Viewed by 3234
Abstract
The recently developed rough Bergomi (rBergomi) model is a rough fractional stochastic volatility (RFSV) model which can generate a more realistic term structure of at-the-money volatility skews compared with other RFSV models. However, its non-Markovianity brings mathematical and computational challenges for model calibration [...] Read more.
The recently developed rough Bergomi (rBergomi) model is a rough fractional stochastic volatility (RFSV) model which can generate a more realistic term structure of at-the-money volatility skews compared with other RFSV models. However, its non-Markovianity brings mathematical and computational challenges for model calibration and simulation. To overcome these difficulties, we show that the rBergomi model can be well-approximated by the forward-variance Bergomi model with wisely chosen weights and mean-reversion speed parameters (aBergomi), which has the Markovian property. We establish an explicit bound on the L2-error between the respective kernels of these two models, which is explicitly controlled by the number of terms in the aBergomi model. We establish and describe the affine structure of the rBergomi model, and show the convergence of the affine structure of the aBergomi model to the one of the rBergomi model. We demonstrate the efficiency and accuracy of our method by implementing a classical Markovian Monte Carlo simulation scheme for the aBergomi model, which we compare to the hybrid scheme of the rBergomi model. Full article
(This article belongs to the Special Issue Application of Stochastic Analysis in Mathematical Finance)
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32 pages, 682 KiB  
Article
Between Scylla and Charybdis: The Bermudan Swaptions Pricing Odyssey
by Dariusz Gatarek and Juliusz Jabłecki
Mathematics 2021, 9(2), 112; https://doi.org/10.3390/math9020112 - 6 Jan 2021
Cited by 2 | Viewed by 5303
Abstract
Bermudan swaptions are options on interest rate swaps which can be exercised on one or more dates before the final maturity of the swap. Because the exercise boundary between the continuation area and stopping area is inherently complex and multi-dimensional for interest rate [...] Read more.
Bermudan swaptions are options on interest rate swaps which can be exercised on one or more dates before the final maturity of the swap. Because the exercise boundary between the continuation area and stopping area is inherently complex and multi-dimensional for interest rate products, there is an inherent “tug of war” between the pursuit of calibration and pricing precision, tractability, and implementation efficiency. After reviewing the main ideas and implementation techniques underlying both single- and multi-factor models, we offer our own approach based on dimension reduction via Markovian projection. Specifically, on the theoretical side, we provide a reinterpretation and extension of the classic result due to Gyöngy which covers non-probabilistic, discounted, distributions relevant in option pricing. Thus, we show that for purposes of swaption pricing, a potentially complex and multidimensional process for the underlying swap rate can be collapsed to a one-dimensional one. The empirical contribution of the paper consists in demonstrating that even though we only match the marginal distributions of the two processes, Bermudan swaptions prices calculated using such an approach appear well-behaved and closely aligned to counterparts from more sophisticated models. Full article
(This article belongs to the Special Issue Application of Stochastic Analysis in Mathematical Finance)
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16 pages, 305 KiB  
Article
Examining the Feasibility of the Sturm–Liouville Theory for Ross Recovery
by Shinmi Ahn and Hyungbin Park
Mathematics 2020, 8(4), 550; https://doi.org/10.3390/math8040550 - 9 Apr 2020
Viewed by 1936
Abstract
Recent studies have suggested that it is feasible to recover a physical measure from a risk-neutral measure. Given a market state variable modeled as a Markov process, the key concept is to extract a unique positive eigenfunction of the generator of the Markov [...] Read more.
Recent studies have suggested that it is feasible to recover a physical measure from a risk-neutral measure. Given a market state variable modeled as a Markov process, the key concept is to extract a unique positive eigenfunction of the generator of the Markov process. In this work, the feasibility of this recovery theory is examined. We prove that, under a restrictive integrability condition, recovery is feasible if and only if both endpoints of the state variable are limit-point. Several examples with explicit positive eigenfunctions are considered. However, in general, a physical measure cannot be recovered from a risk-neutral measure. We provide a financial and mathematical rationale for such recovery failure. Full article
(This article belongs to the Special Issue Application of Stochastic Analysis in Mathematical Finance)

Review

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22 pages, 629 KiB  
Review
An Intuitive Introduction to Fractional and Rough Volatilities
by Elisa Alòs and Jorge A. León
Mathematics 2021, 9(9), 994; https://doi.org/10.3390/math9090994 - 28 Apr 2021
Cited by 4 | Viewed by 3792
Abstract
Here, we review some results of fractional volatility models, where the volatility is driven by fractional Brownian motion (fBm). In these models, the future average volatility is not a process adapted to the underlying filtration, and fBm is not a semimartingale in general. [...] Read more.
Here, we review some results of fractional volatility models, where the volatility is driven by fractional Brownian motion (fBm). In these models, the future average volatility is not a process adapted to the underlying filtration, and fBm is not a semimartingale in general. So, we cannot use the classical Itô’s calculus to explain how the memory properties of fBm allow us to describe some empirical findings of the implied volatility surface through Hull and White type formulas. Thus, Malliavin calculus provides a natural approach to deal with the implied volatility without assuming any particular structure of the volatility. The aim of this paper is to provides the basic tools of Malliavin calculus for the study of fractional volatility models. That is, we explain how the long and short memory of fBm improves the description of the implied volatility. In particular, we consider in detail a model that combines the long and short memory properties of fBm as an example of the approach introduced in this paper. The theoretical results are tested with numerical experiments. Full article
(This article belongs to the Special Issue Application of Stochastic Analysis in Mathematical Finance)
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