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SIAM Journal on Financial Mathematics
Journal Prestige (SJR): 1.222  Citation Impact (citeScore): 1 Number of Followers: 3  Hybrid journal (It can contain Open Access articles)ISSN (Online) 1945-497X Published by Society for Industrial and Applied Mathematics  [17 journals]
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- Short Communication: Projection of Functionals and Fast Pricing of Exotic
Options-
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Authors: Valentin Tissot-Daguette
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page SC74-SC86, June 2022.
We investigate the approximation of path functionals. In particular, we advocate the use of the Karhunen--Loève expansion, the continuous analogue of principal component analysis, to extract relevant information from the image of a functional. Having an accurate estimate of functionals is of paramount importance in the context of exotic derivatives pricing, as presented in the practical applications. Specifically, we show how a simulation-based procedure, which we call the Karhunen--Loève Monte Carlo (KLMC) algorithm, allows fast and efficient computation of the price of path-dependent options. We also explore the path signature as an alternative tool to project both paths and functionals.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-06-27T07:00:00Z
DOI: 10.1137/21M1451439
Issue No: Vol. 13, No. 2 (2022)
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- Short Communication: On the Weak Convergence Rate in the Discretization of
Rough Volatility Models-
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Authors: Christian Bayer, Masaaki Fukasawa, Shonosuke Nakahara
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page SC66-SC73, June 2022.
We study the weak convergence rate in the discretization of rough volatility models. After showing a lower bound $2H$ under a general model, where $H$ is the Hurst index of the volatility process, we give a sharper bound $H + 1/2$ under a linear model.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-06-13T07:00:00Z
DOI: 10.1137/22M1482871
Issue No: Vol. 13, No. 2 (2022)
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- Short Communication: Super-Replication Prices with Multiple Priors in
Discrete Time-
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Authors: Romain Blanchard, Laurence Carassus
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page SC53-SC65, June 2022.
In the frictionless discrete time financial market of Bouchard and Nutz (2015), we propose a full characterization of the quasi-sure super-replication price, as the supremum of the mono-prior super-replication prices, through an extreme prior and through martingale measures.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-05-16T07:00:00Z
DOI: 10.1137/22M1470013
Issue No: Vol. 13, No. 2 (2022)
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- Short Communication: Chances for the Honest in Honest versus Insider
Trading-
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Authors: Mauricio Elizalde, Carlos Escudero
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page SC39-SC52, June 2022.
We study a Black--Scholes market with a finite time horizon and two investors: an honest and an insider trader. We analyze it with anticipating stochastic calculus in two steps. First, we recover the classical result on portfolio optimization that shows that the expected logarithmic utility of the insider is strictly greater than that of the honest trader. Then, we prove that whenever the market is viable, the honest trader can get a higher logarithmic utility, and therefore more wealth, than the insider with a strictly positive probability. Our proof relies on the analysis of a sort of forward integral variant of the Doléans--Dade exponential process. The main financial conclusion is that the logarithmic utility is perhaps too conservative for some insiders.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-04-28T07:00:00Z
DOI: 10.1137/21M1439547
Issue No: Vol. 13, No. 2 (2022)
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- Mean-Variance Portfolio Selection in Contagious Markets
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Authors: Yang Shen, Bin Zou
Pages: 391 - 425
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page 391-425, June 2022.
We consider a mean-variance portfolio selection problem in a financial market with contagion risk. The risky assets follow a jump-diffusion model, in which jumps are driven by a multivariate Hawkes process with mutual-excitation effect. The mutual-excitation feature of the Hawkes process captures the contagion risk in the sense that each price jump of an asset increases the likelihood of future jumps not only in the same asset but also in other assets. We apply the stochastic maximum principle, backward stochastic differential equation theory, and linear-quadratic control technique to solve the problem and obtain the efficient strategy and efficient frontier in semiclosed form, subject to a nonlocal partial differential equation. Numerical examples are provided to illustrate our results.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-04-07T07:00:00Z
DOI: 10.1137/20M1320560
Issue No: Vol. 13, No. 2 (2022)
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- American Options in the Volterra Heston Model
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Authors: Etienne Chevalier, Sergio Pulido, Elizabeth Zún͂iga
Pages: 426 - 458
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page 426-458, June 2022.
We price American options using kernel-based approximations of the Volterra Heston model. We choose these approximations because they allow simulation-based techniques for pricing. We prove the convergence of American option prices in the approximating sequence of models towards the prices in the Volterra Heston model. A crucial step in the proof is to exploit the affine structure of the model in order to establish explicit formulas and convergence results for the conditional Fourier--Laplace transform of the log price and an adjusted version of the forward variance. We illustrate with numerical examples our convergence result and the behavior of American option prices with respect to certain parameters of the model.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-04-27T07:00:00Z
DOI: 10.1137/21M140674X
Issue No: Vol. 13, No. 2 (2022)
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- Strong Convergence to the Mean Field Limit of a Finite Agent Equilibrium
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Authors: Masaaki Fujii, Akihiko Takahashi
Pages: 459 - 490
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page 459-490, June 2022.
We study an equilibrium-based continuous asset pricing problem for the securities market. In the previous work [M. Fujii and A. Takahashi (2022), SIAM J. Control Optim., 60, pp. 259--279], we have shown that a certain price process, which is given by the solution to a forward-backward stochastic differential equation of conditional McKean--Vlasov type, asymptotically clears the market in the large population limit. In the current work, under suitable conditions, we show the existence of a finite agent equilibrium and its strong convergence to the corresponding mean-field limit given in [M. Fujii and A. Takahashi (2022), SIAM J. Control Optim., 60, pp. 259--279]. As an important byproduct, we get the direct estimate on the difference of the equilibrium price between the two markets: the one consisting of heterogeneous agents of finite population size and the other of homogeneous agents of infinite population size.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-04-28T07:00:00Z
DOI: 10.1137/21M1441055
Issue No: Vol. 13, No. 2 (2022)
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- Exact Solutions and Approximations for Optimal Investment Strategies and
Indifference Prices-
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Authors: Michel Vellekoop, Marcellino Gaudenzi
Pages: 491 - 520
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page 491-520, June 2022.
In this paper a new approach is proposed to determine the optimal strategy for investment in risky assets by a risk averse investor. To generate approximations for such problems in continuous time, we define a sequence of models in discrete time with a finite state space and a restricted class of utility functions for which the exact optimal strategy can be found. We prove that the graphs of optimal policies form a connected subset of two bundles of parallel lines in the plane and that the optimization problem can be reduced to a sequence of simple binary decisions. This allows us to avoid the search over real numbers that is required for every possible value of the state when finite difference schemes for the Hamilton--Jacobi--Bellman equations in continuous time are used. A very efficient calculation scheme is defined which generates the exact solutions for our discrete time approximations, and we use known results from the theory of viscosity solutions to give conditions which guarantee that a sequence of such approximations for a given problem in continuous time converges to the correct limit. We show in a number of examples how the method can be used to find indifference prices in incomplete markets and that our approach can outperform alternative methods that are based on finite difference schemes.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-05-02T07:00:00Z
DOI: 10.1137/21M1393303
Issue No: Vol. 13, No. 2 (2022)
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- The Dispersion Bias
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Authors: Lisa R. Goldberg, Alex Papanicolaou, Alex Shkolnik
Pages: 521 - 550
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page 521-550, June 2022.
We identify and correct excess dispersion in the leading eigenvector of a sample covariance matrix when the number of variables vastly exceeds the number of observations. Our correction is data-driven, and it materially diminishes the substantial impact of estimation error on weights and risk forecasts of minimum variance portfolios. We quantify that impact with a novel metric, the optimization bias, which has a positive lower bound prior to correction and tends to zero almost surely after correction. Our analysis sheds light on aspects of how estimation error corrupts an estimated covariance matrix and is transmitted to portfolios via quadratic optimization.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-05-03T07:00:00Z
DOI: 10.1137/21M144058X
Issue No: Vol. 13, No. 2 (2022)
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- Optimal Signal-Adaptive Trading with Temporary and Transient Price Impact
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Authors: Eyal Neuman, Moritz Voß
Pages: 551 - 575
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page 551-575, June 2022.
We study optimal liquidation in the presence of linear temporary and transient price impact along with taking into account a general price predicting finite-variation signal. We formulate this problem as minimization of a cost-risk functional over a class of absolutely continuous and signal-adaptive strategies. The stochastic control problem is solved by following a probabilistic and convex analytic approach. We show that the optimal trading strategy is given by a system of four coupled forward-backward SDEs, which can be solved explicitly. Our results reveal how the induced transient price distortion provides together with the predictive signal an additional predictor about future price changes. As a consequence, the optimal signal-adaptive trading rate trades off exploiting the predictive signal against incurring the transient displacement of the execution price from its unaffected level. This answers an open question from [C. A. Lehalle and E. Neuman, Finance Stoch., 23 (2019), pp. 275--311] as we show how to derive the unique optimal signal-adaptive liquidation strategy when price impact is not only temporary but also transient.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-05-05T07:00:00Z
DOI: 10.1137/20M1375486
Issue No: Vol. 13, No. 2 (2022)
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- Functional Portfolio Optimization in Stochastic Portfolio Theory
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Authors: Steven Campbell, Ting-Kam Leonard Wong
Pages: 576 - 618
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page 576-618, June 2022.
In this paper we develop a concrete and fully implementable approach to the optimization of functionally generated portfolios in stochastic portfolio theory. The main idea is to optimize over a family of rank-based portfolios parameterized by an exponentially concave function on the unit interval. This choice can be motivated by the long term stability of the capital distribution observed in large equity markets and allows us to circumvent the curse of dimensionality. The resulting optimization problem, which is convex, allows for various regularizations and constraints to be imposed on the generating function. We prove an existence and uniqueness result for our optimization problem and provide a stability estimate in terms of a Wasserstein metric of the input measure. Then we formulate a discretization which can be implemented numerically using available software packages and analyze its approximation error. Finally, we present empirical examples using CRSP data from the U.S. stock market, including the performance of the portfolios allowing for dividends, defaults, and transaction costs.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-05-05T07:00:00Z
DOI: 10.1137/21M1417715
Issue No: Vol. 13, No. 2 (2022)
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- Performance Fees with Stochastic Benchmark
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Authors: Gu Wang
Pages: 619 - 652
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page 619-652, June 2022.
A hedge fund manager invests the fund in a constant investment opportunity, and receives performance fees when the fund reaches a new maximum relative to a stochastic benchmark, aiming to maximize the expected power utility from fees in the long run. The manager's optimal portfolio includes a Merton component with an effective risk aversion parameter shifted from his/her own risk aversion toward one, and an extra component which hedges the risks in the benchmark. The effective risk aversion and the hedging component depend on how the fund investment opportunity compares to the benchmark, which allows investors to regulate the manager's risk taking with a carefully chosen benchmark.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-05-19T07:00:00Z
DOI: 10.1137/21M1401826
Issue No: Vol. 13, No. 2 (2022)
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- Erratum: The Robust Superreplication Problem: A Dynamic Approach
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Authors: Laurence Carassus, Jan Obłój, Johannes Wiesel
Pages: 653 - 655
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page 653-655, June 2022.
The assertions of Proposition 3.7 in our paper “The robust superreplication problem: A dynamic approach” [L. Carassus, J. Obłój, and J. Wiesel, SIAM J. Financial Math., 10 (2019), pp. 907--941] may fail to hold without an additional assumption, which we detail in this erratum.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-06-28T07:00:00Z
DOI: 10.1137/21M1447040
Issue No: Vol. 13, No. 2 (2022)
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- Short Communication: An Axiomatization of $\Lambda$-Quantiles
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Authors: Fabio Bellini, Ilaria Peri
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 1, Page SC26-SC38, March 2022.
We give an axiomatic foundation to $\Lambda$-quantiles, a family of generalized quantiles introduced in [M. Frittelli, M. Maggis, and I. Peri, Math. Finance, 24 (2014), pp. 442--463] under the name Lambda Value at Risk. Under mild assumptions, we show that these functionals are characterized by a property that we call “locality,” which means that any change in the distribution of the probability mass that arises entirely above or below the value of the $\Lambda$-quantile does not modify its value. We make comparisons with a related axiomatization of the usual quantiles given by Chambers in [Math. Finance, 19 (2009), pp. 335--342], based on the stronger property of “ordinal covariance,” meaning that quantiles are covariant with respect to increasing transformations. Further, we present a systematic treatment of the properties of $\Lambda$-quantiles, refining some of the results of Frittelli, Maggis, and Peri and [M. Burzoni, I. Peri, and C. M. Ruffo, Quant. Finance, 17 (2017), pp. 1735--1743] and showing that in the case of a nonincreasing $\Lambda$ the properties of $\Lambda$-quantiles closely resemble those of the usual quantiles.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-03-17T07:00:00Z
DOI: 10.1137/21M1444278
Issue No: Vol. 13, No. 1 (2022)
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- Short Communication: Utility Indifference Pricing with High Risk Aversion
and Small Linear Price Impact-
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Authors: Yan Dolinsky, Shir Moshe
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 1, Page SC12-SC25, March 2022.
We consider the Bachelier model with linear price impact. Exponential utility indifference prices are studied for vanilla European options, and we compute their nontrivial scaling limit for a vanishing price impact which is inversely proportional to the risk aversion. Moreover, we find explicitly a family of portfolios which are asymptotically optimal.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-03-07T08:00:00Z
DOI: 10.1137/21M1456431
Issue No: Vol. 13, No. 1 (2022)
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- Short Communication: A Gaussian Kusuoka Approximation without Solving
Random ODEs-
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Authors: Toshihiro Yamada
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 1, Page SC1-SC11, January 2022.
This paper introduces an efficient implementation scheme for Kusuoka approximation for a stochastic differential equation (SDE) driven by Brownian motion. A second order weak approximation is shown for Lipschitz continuous test functions under the UFG condition on the vector fields defining the SDE. Contrary to other second order implementation schemes for the Kusuoka--Lyons--Victoir method, the scheme does not require any solver of random ordinary differential equations or tree-based method. A simple simulation method is proposed using polynomials of Brownian motion, and moreover, a semiclosed form Kusuoka approximation is introduced. The algorithm, sample code, and numerical examples are provided for high-dimensional models.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-01-10T08:00:00Z
DOI: 10.1137/21M1433915
Issue No: Vol. 13, No. 1 (2022)
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- Joint Modeling and Calibration of SPX and VIX by Optimal Transport
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Authors: Ivan Guo, Grégoire Loeper, Jan Obłój, Shiyi Wang
Pages: 1 - 31
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 1, Page 1-31, January 2022.
This paper addresses the joint calibration problem of SPX options and VIX options or futures. We show that the problem can be formulated as a semimartingale optimal transport problem under a finite number of discrete constraints, in the spirit of [Guo, Loeper, and Wang, Math. Finance, 32 (2021)]. We introduce a PDE formulation along with its dual counterpart. The solution, a calibrated diffusion process, can be represented via the solutions of Hamilton--Jacobi--Bellman equations arising from the dual formulation. The method is tested on both simulated data and market data. Numerical examples show that the model can be accurately calibrated to SPX options, VIX options, and VIX futures simultaneously.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-01-05T08:00:00Z
DOI: 10.1137/20M1375905
Issue No: Vol. 13, No. 1 (2022)
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- On Smile Properties of Volatility Derivatives: Understanding the VIX Skew
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Authors: Elisa Alòs, David García-Lorite, Aitor Muguruza Gonzalez
Pages: 32 - 69
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 1, Page 32-69, January 2022.
We develop a method to study the implied volatility of exotic underlyings, with special focus on volatility derivatives such as VIX options. Remarkably, our approach is flexible enough to be applied to any underlying, subject to mild technical conditions. Our method, built upon Malliavin calculus techniques, allows to transform any such underlying into the Black--Scholes model with a particular type of stochastic volatility. This, in turn, allows us to describe the properties of the at-the-money implied volatility (ATMI) in terms of the Malliavin derivatives of the transformed underlying process. Concretely, we study the short-time behavior of the ATMI level and skew. As an application, we describe the short-term behavior of the ATMI of VIX and realize variance options in terms of the Hurst parameter of the model, and most importantly, we describe the class of volatility processes that generate a positive skew for the VIX implied volatility. Several numerical examples are provided to support our theoretical results.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-01-10T08:00:00Z
DOI: 10.1137/19M1269981
Issue No: Vol. 13, No. 1 (2022)
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- Suffocating Fire Sales
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Authors: Nils Detering, Thilo Meyer-Brandis, Konstantinos Panagiotou, Daniel Ritter
Pages: 70 - 108
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 1, Page 70-108, January 2022.
Fire sales are among the major drivers of market instability in modern financial systems. Due to iterated distressed selling and the associated price impact, initial shocks to some institutions can be amplified dramatically through the network induced by portfolio overlaps. In this paper, we develop a mathematical framework that allows us to investigate central characteristics that drive or hinder the propagation of distress. We investigate single systems as well as ensembles of systems that are alike, where similarity is measured in terms of the empirical distribution of all defining properties of a system. This asymptotic approach ensures a great deal of robustness to statistical uncertainty and temporal fluctuations. A characterization of those systems that are resilient to small shocks emerges, and we provide criteria that regulators might exploit in order to assess the stability of a financial system. We illustrate the application of these criteria for some exemplary configurations in the context of capital requirements and test the applicability of our results for systems of moderate size by Monte Carlo simulations.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-01-27T08:00:00Z
DOI: 10.1137/20M1379800
Issue No: Vol. 13, No. 1 (2022)
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- Sub- and Supersolution Approach to Accuracy Analysis of Portfolio
Optimization Asymptotics in Multiscale Stochastic Factor Markets-
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Authors: Jean-Pierre Fouque, Ruimeng Hu, Ronnie Sircar
Pages: 109 - 128
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 1, Page 109-128, January 2022.
The problem of portfolio optimization when stochastic factors drive returns and volatilities has been studied in previous works by the authors. In particular, they proposed asymptotic approximations for value functions and optimal strategies in the regime where these factors are running on both slow and fast timescales. However, the rigorous justification of the accuracy of these approximations has been limited to power utilities and a single factor. In this paper, we provide an accurate analysis for cases with general utility functions and two timescale factors by constructing sub- and supersolutions to the fully nonlinear problem so that their difference is at the desired level of accuracy. This approach will be valuable in various related stochastic control problems.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-01-31T08:00:00Z
DOI: 10.1137/21M1428625
Issue No: Vol. 13, No. 1 (2022)
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- Reward Design in Risk-Taking Contests
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Authors: Marcel Nutz, Yuchong Zhang
Pages: 129 - 146
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 1, Page 129-146, March 2022.
Following the risk-taking model of Seel and Strack, $n$ players decide when to stop privately observed Brownian motions with drift and absorption at zero. They are then ranked according to their level of stopping and paid a rank-dependent reward. We study the problem of a principal who aims to induce a desirable equilibrium performance of the players by choosing how much reward is attributed to each rank. Specifically, we determine optimal reward schemes for principals interested in the average performance and the performance at a given rank. While the former can be related to reward inequality in the Lorenz sense, the latter can have a surprising shape.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-02-08T08:00:00Z
DOI: 10.1137/21M1397386
Issue No: Vol. 13, No. 1 (2022)
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- A High-Order Numerical Method for BSPDEs with Applications to Mathematical
Finance-
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Authors: Yunzhang Li
Pages: 147 - 178
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 1, Page 147-178, March 2022.
In this paper, we propose a local discontinuous Galerkin (LDG) method for backward stochastic partial differential equations (BSPDEs), which is a high-order numerical scheme. We prove the $L^2$-stability of the numerical scheme. For the superparabolic BSPDEs, the optimal error estimates are obtained for Cartesian meshes with $Q^k$ elements, and the suboptimal error estimates are derived for triangular meshes with $P^k$ elements. We also prove the suboptimal error estimates for the degenerate BSPDEs. Numerical examples in one and two space dimensions are given to display the performance of the LDG method. As an application in mathematical finance, the numerical scheme is applied to approximate the hedging price of a contingent claim and the corresponding optimal hedging strategy.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-02-14T08:00:00Z
DOI: 10.1137/20M1383252
Issue No: Vol. 13, No. 1 (2022)
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- Pricing Options under Rough Volatility with Backward SPDEs
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Authors: Christian Bayer, Jinniao Qiu, Yao Yao
Pages: 179 - 212
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 1, Page 179-212, March 2022.
In this paper, we study the option pricing problems for rough volatility models. As the framework is non-Markovian, the value function for a European option is not deterministic; rather, it is random and satisfies a backward stochastic partial differential equation (BSPDE). The existence and uniqueness of a weak solution is proved for general nonlinear BSPDEs with unbounded random leading coefficients whose connections with certain forward-backward stochastic differential equations are derived as well. These BSPDEs are then used to approximate American option prices. A deep leaning-based method is also investigated for the numerical approximations to such BSPDEs and associated non-Markovian pricing problems. Finally, the examples of rough Bergomi type are numerically computed for both European and American options.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-02-28T08:00:00Z
DOI: 10.1137/20M1357639
Issue No: Vol. 13, No. 1 (2022)
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- Robust Risk-Aware Reinforcement Learning
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Authors: Sebastian Jaimungal, Silvana M. Pesenti, Ye Sheng Wang, Hariom Tatsat
Pages: 213 - 226
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 1, Page 213-226, March 2022.
We present a reinforcement learning (RL) approach for robust optimization of risk-aware performance criteria. To allow agents to express a wide variety of risk-reward profiles, we assess the value of a policy using rank dependent expected utility (RDEU). RDEU allows agents to seek gains, while simultaneously protecting themselves against downside risk. To robustify optimal policies against model uncertainty, we assess a policy not by its distribution but rather by the worst possible distribution that lies within a Wasserstein ball around it. Thus, our problem formulation may be viewed as an actor/agent choosing a policy (the outer problem) and the adversary then acting to worsen the performance of that strategy (the inner problem). We develop explicit policy gradient formulae for the inner and outer problems and show their efficacy on three prototypical financial problems: robust portfolio allocation, benchmark optimization, and statistical arbitrage.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-03-01T08:00:00Z
DOI: 10.1137/21M144640X
Issue No: Vol. 13, No. 1 (2022)
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- No Arbitrage SVI
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Authors: Claude Martini, Arianna Mingone
Pages: 227 - 261
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 1, Page 227-261, March 2022.
We fully characterize the absence of butterfly arbitrage in the stochastic volatility inspired (SVI) formula for implied total variance proposed by Gatheral in 2004. The main ingredient is an intermediate characterization of the necessary condition for no arbitrage obtained for any model by Fukasawa in 2012 that the inverse functions of the $-d_1$ and $-d_2$ of the Black--Scholes formula, viewed as functions of the log-forward moneyness, should be increasing. A natural rescaling of the SVI parameters and a meticulous analysis of the Durrleman condition allow us then to obtain simple range conditions on the parameters. This leads to a straightforward implementation of a least-squares calibration algorithm on the no arbitrage domain, which yields an excellent fit on the market data we used for our tests, with the guarantee to yield smiles with no butterfly arbitrage.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-03-07T08:00:00Z
DOI: 10.1137/20M1351060
Issue No: Vol. 13, No. 1 (2022)
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- Optimal Cross-Border Electricity Trading
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Authors: Álvaro Cartea, Maria Flora, Tiziano Vargiolu, Georgi Slavov
Pages: 262 - 294
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 1, Page 262-294, March 2022.
We show that there exists a profitable cross-border trading strategy for an agent who trades electricity in the European electricity network. Data of the European markets are employed to show how electricity prices in all locations of the network are affected by the flow of power between any two locations that trade power between them. The optimal cross-border trading strategy is derived via the explicit solution of a nontrivial stochastic control problem in which prices at different locations are co-integrated and trading affects prices in all locations of the network.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-03-14T07:00:00Z
DOI: 10.1137/21M1398537
Issue No: Vol. 13, No. 1 (2022)
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- Tail Optimality and Preferences Consistency for Intertemporal Optimization
Problems-
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Authors: Elena Vigna
Pages: 295 - 320
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 1, Page 295-320, March 2022.
Given an intertemporal optimization problem over a time interval $[t_0,T]$ and a control plan associated to it, we introduce the four notions of local and global tail optimality of the control plan and local and global preferences consistency of the agent. While the notion of tail optimality of a control plan is not new, the main innovation of this paper is the definition of preferences consistency of an agent, which is a novel concept. We prove that, in the case of a linear time-consistent problem where dynamic programming can be applied, the optimal control plan is globally tail-optimal and the agent is globally preferences-consistent. Oppositely, in the case of a nonlinear problem that gives rise to time inconsistency, we find that global tail optimality and global preferences consistency do not coexist. We analyze three common ways to attack a time-inconsistent problem: (i) precommitment approach, (ii) dynamically optimal approach, and (iii) consistent planning approach. We find that none of the three approaches keeps simultaneously the desirable properties of global tail optimality and global preferences consistency: the existing approaches to time inconsistency are flawed in various ways. We also prove that if the performance criterion includes a convex function of expected final wealth and a globally tail-optimal plan exists, then the three approaches coincide and the problem is linear. The contribution of the paper is to disentangle the notion of time consistency into the two notions of tail optimality and preferences consistency. The analysis should shed light on the price to be paid in terms of tail optimality and preferences consistency with each of the three approaches currently available for time inconsistency.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-03-24T07:00:00Z
DOI: 10.1137/21M1435422
Issue No: Vol. 13, No. 1 (2022)
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- Optimal Investment and Consumption under a Habit-Formation Constraint
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Authors: Bahman Angoshtari, Erhan Bayraktar, Virginia R. Young
Pages: 321 - 352
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 1, Page 321-352, March 2022.
We formulate an infinite-horizon optimal investment and consumption problem, in which an individual forms a habit based on the exponentially weighted average of her past consumption rate, and in which she invests in a Black--Scholes market. The individual is constrained to consume at a rate higher than a certain proportion $\alpha$ of her consumption habit. Our habit-formation model allows for both addictive ($\alpha=1$) and nonaddictive ($0
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-03-29T07:00:00Z
DOI: 10.1137/21M1397891
Issue No: Vol. 13, No. 1 (2022)
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- Multi-asset Optimal Execution and Statistical Arbitrage Strategies under
Ornstein--Uhlenbeck Dynamics-
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Authors: Philippe Bergault, Fayçal Drissi, Olivier Guéant
Pages: 353 - 390
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 1, Page 353-390, March 2022.
In recent years, academics, regulators, and market practitioners have increasingly addressed liquidity issues. Among the numerous problems addressed, the optimal execution of large orders is probably the one that has attracted the most research works, mainly in the case of single-asset portfolios. In practice, however, optimal execution problems often involve large portfolios comprising numerous assets, and models should consequently account for risks at the portfolio level. In this paper, we address multi-asset optimal execution in a model where prices have multivariate Ornstein--Uhlenbeck dynamics and where the agent maximizes the expected (exponential) utility of her Profit and Loss (PnL). We use the tools of stochastic optimal control and simplify the initial multidimensional Hamilton--Jacobi--Bellman equation into a system of ordinary differential equations (ODEs) involving a matrix Riccati ODE for which classical existence theorems do not apply. By using a priori estimates obtained thanks to optimal control tools, we nevertheless prove an existence and uniqueness result for the latter ODE and then deduce a verification theorem that provides a rigorous solution to the execution problem. Using examples based on data from the foreign exchange and stock markets, we eventually illustrate our results and discuss their implications for both optimal execution and statistical arbitrage.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-03-31T07:00:00Z
DOI: 10.1137/21M1407756
Issue No: Vol. 13, No. 1 (2022)
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