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Relevant bibliographies by topics / Discrete hedging

Academic literature on the topic 'Discrete hedging'

Author: Grafiati

Published: 16 December 2023

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Journal articles on the topic "Discrete hedging"

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Hussain, Sultan, Salman Zeb, Muhammad Saleem, and Nasir Rehman. "Hedging error estimate of the american put option problem in jump-diffusion processes." Filomat 32, no. 8 (2018): 2813–24. http://dx.doi.org/10.2298/fil1808813h.

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We consider discrete time hedging error of the American put option in case of brusque fluctuations in the price of assets. Since continuous time hedging is not possible in practice so we consider discrete time hedging process. We show that if the proportions of jump sizes in the asset price are identically distributed independent random variables having finite moments then the value process of the discrete time hedging uniformly approximates the value process of the corresponding continuous-time hedging in the sense of L1 and L2-norms under the real world probability measure.

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Rémillard, Bruno, and Sylvain Rubenthaler. "Optimal hedging in discrete time." Quantitative Finance 13, no. 6 (June 2013): 819–25. http://dx.doi.org/10.1080/14697688.2012.745012.

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Brealey, R. A., and E. C. Kaplanis. "Discrete exchange rate hedging strategies." Journal of Banking & Finance 19, no. 5 (August 1995): 765–84. http://dx.doi.org/10.1016/0378-4266(95)00089-y.

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BRODÉN, MATS, and PETER TANKOV. "TRACKING ERRORS FROM DISCRETE HEDGING IN EXPONENTIAL LÉVY MODELS." International Journal of Theoretical and Applied Finance 14, no. 06 (September 2011): 803–37. http://dx.doi.org/10.1142/s0219024911006760.

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We analyze the errors arising from discrete readjustment of the hedging portfolio when hedging options in exponential Lévy models, and establish the rate at which the expected squared error goes to zero when the readjustment frequency increases. We compare the quadratic hedging strategy with the common market practice of delta hedging, and show that for discontinuous option pay-offs the latter strategy may suffer from very large discretization errors. For options with discontinuous pay-offs, the convergence rate depends on the underlying Lévy process, and we give an explicit relation between the rate and the Blumenthal-Getoor index of the process.

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Dariane, Alireza B., Mohammad M. Sabokdast, Farzane Karami, Roza Asadi, Kumaraswamy Ponnambalam, and Seyed Jamshid Mousavi. "Integrated Operation of Multi-Reservoir and Many-Objective System Using Fuzzified Hedging Rule and Strength Pareto Evolutionary Optimization Algorithm (SPEA2)." Water 13, no. 15 (July 21, 2021): 1995. http://dx.doi.org/10.3390/w13151995.

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In this paper, a many-objective optimization algorithm was developed using SPEA2 for a system of four reservoirs in the Karun basin, including hydropower, municipal and industrial, agricultural, and environmental objectives. For this purpose, using 53 years of available data, hedging rules were developed in two modes: with and without applying fuzzy logic. SPEA2 was used to optimize hedging coefficients using the first 43 years of data and the last 10 years of data were used to test the optimized rule curves. The results were compared with those of non-hedging methods, including the standard operating procedures (SOP) and water evaluation and planning (WEAP) model. The results indicate that the combination of fuzzy logic and hedging rules in a many-objectives system is more efficient than the discrete hedging rule alone. For instance, the reliability of the hydropower requirement in the fuzzified discrete hedging method in a drought scenario was found to be 0.68, which is substantially higher than the 0.52 from the discrete hedging method. Moreover, reduction of the maximum monthly shortage is another advantage of this rule. Fuzzy logic reduced 118 million cubic meters (MCM) of deficit in the Karun-3 reservoir alone. Moreover, as expected, the non-hedging SOP and WEAP model produced higher reliabilities, lower average storages, and less water losses through spills.

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Hamdi, Haykel, and Jihed Majdoub. "Risk-sharing finance governance: Islamic vs conventional indexes option pricing." Managerial Finance 44, no. 5 (May 14, 2018): 540–50. http://dx.doi.org/10.1108/mf-05-2017-0199.

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Purpose Risk governance has an important influence on the hedging performances in option pricing and portfolio hedging in both discrete and dynamic case for both conventional and Islamic indexes. The paper aims to discuss these issues. Design/methodology/approach This paper explores option pricing and portfolio hedging in a discrete and dynamic case with transaction costs. Monte Carlo simulations are applied to both conventional and Islamic indexes in US and UK markets. Simulations show that conventional and Islamic assets do not exhibit the same price and portfolio hedging strategy governance. Findings The authors conclude that Islamic assets show different option price and hedging strategy compared to their conventional counterpart. Originality/value The research question of this paper aims at filling the gap in the empirical literature by exploring option price and hedging structure for both conventional and Islamic indexes in US and UK stock markets.

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Schweizer, Martin. "Variance-Optimal Hedging in Discrete Time." Mathematics of Operations Research 20, no. 1 (February 1995): 1–32. http://dx.doi.org/10.1287/moor.20.1.1.

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Ku, Hyejin, Kiseop Lee, and Huaiping Zhu. "Discrete time hedging with liquidity risk." Finance Research Letters 9, no. 3 (September 2012): 135–43. http://dx.doi.org/10.1016/j.frl.2012.02.002.

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Park, Sang-Hyeon, and Kiseop Lee. "Hedging with Liquidity Risk under CEV Diffusion." Risks 8, no. 2 (June 5, 2020): 62. http://dx.doi.org/10.3390/risks8020062.

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We study a discrete time hedging and pricing problem in a market with the liquidity risk. We consider a discrete version of the constant elasticity of variance (CEV) model by applying Leland’s discrete time replication scheme. The pricing equation becomes a nonlinear partial differential equation, and we solve it by a multi scale perturbation method. A numerical example is provided.

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Baule, Rainer, and Philip Rosenthal. "Time-Discrete Hedging of Down-and-Out Puts with Overnight Trading Gaps." Journal of Risk and Financial Management 15, no. 1 (January 11, 2022): 29. http://dx.doi.org/10.3390/jrfm15010029.

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Hedging down-and-out puts (and up-and-out calls), where the maximum payoff is reached just before a barrier is hit that would render the claim worthless afterwards, is challenging. All hedging methods potentially lead to large errors when the underlying is already close to the barrier and the hedge portfolio can only be adjusted in discrete time intervals. In this paper, we analyze this hedging situation, especially the case of overnight trading gaps. We show how a position in a short-term vanilla call option can be used for efficient hedging. Using a mean-variance hedging approach, we calculate optimal hedge ratios for both the underlying and call options as hedge instruments. We derive semi-analytical formulas for optimal hedge ratios in a Black–Scholes setting for continuous trading (as a benchmark) and in the case of trading gaps. For more complex models, we show in a numerical study that the semi-analytical formulas can be used as a sufficient approximation, even when stochastic volatility and jumps are present.

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More sources

Dissertations / Theses on the topic "Discrete hedging"

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Brodén, Mats. "On the convergence of discrete time hedging schemes /." Lund : Centre for Mathematical Sciences, Mathematical Statistics, Faculty of Engineering, Lund University, 2008. http://www.maths.lth.se.

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Xu, Kun. "Static hedging of barrier options in discrete and continuous time." Thesis, Uppsala University, Department of Mathematics, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-120232.

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Ng, Desmond Siew Wai. "Nonlinear Pricing in Discrete-time under Default and Optimal Collateral." Thesis, The University of Sydney, 2018. http://hdl.handle.net/2123/19637.

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This thesis addresses a re-examination of the classical no-arbitrage pricing theory of mathematical finance through Backward Stochastic Difference Equations (BSdEs), their extensions and connections to Nonlinear Evaluations and Generalized Game Contingent Claims (GGCCs). A theory is developed in discrete-time encompassing the nonlinear features introduced into the pricing and hedging problems stemming from three salient features prevalent in modern day derivatives markets; nonlinear differential funding, default and collateralization. Their implications upon the arbitrage-free nature of market models and the nonlinear pricing of contingent claims is examined. A common arbitrage-free framework encompassing all three features, including an endogenous method for the determination of optimal collateral is presented in discrete-time.

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Lazier, Iuri. "Hedge de opção utilizando estratégias dinâmicas multiperiódicas autofinanciáveis em tempo discreto em mercado incompleto." Universidade de São Paulo, 2009. http://www.teses.usp.br/teses/disponiveis/12/12139/tde-11092009-103057/.

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Este trabalho analisa três estratégias de hedge de opção, buscando identificar a importância da escolha da estratégia para a obtenção de um bom desempenho do hedge. O conceito de hedge é analisado de forma retrospectiva e uma teoria geral de hedge é apresentada. Em seguida são descritos alguns estudos comparativos de desempenho de estratégias de hedge de opção e suas metodologias de implementação. Para esta análise comparativa são selecionadas três estratégias de hedge de opção de compra do tipo européia: a primeira utiliza o modelo Black-Scholes-Merton de precificação de opções, a segunda utiliza uma solução de programação dinâmica para hedge dinâmico multiperiódico e a terceira utiliza um modelo GARCH para precificação de opções. As estratégias são comentadas e comparadas do ponto de vista de suas premissas teóricas e por meio de testes comparativos de desempenho. O desempenho das estratégias é comparado sob uma perspectiva dinâmicamente ajustada, multiperiódica e autofinanciável. Os dados para comparação de desempenho são gerados por simulação e o desempenho é avaliado pelos erros absolutos médios e erros quadráticos médios, resultantes na carteira de hedge. São feitas ainda considerações a respeito de alternativas de estimação e suas implicações no desempenho das estratégias.
This work analyzes three option hedging strategies, to identify the importance of choosing a strategy in order to achieve a good hedging performance. A retrospective analysis of the concept of hedging is conducted and a general hedging theory is presented. Following, some comparative papers of hedging performance and their implementation methodologies are described. For the present comparative analysis, three hedging strategies for European options have been selected: the first one based on the Black-Scholes-Merton model for option pricing, the second one based on a dynamic programming solution for dynamic multiperiod hedging and the third one based on a GARCH model for option pricing. The strategies are compared under their theoric premisses and through comparative performance testes. The performances of the strategies are compared under a dynamically adjusted multiperiodic and self-financing perspective. Data for performance comparison are generated by simulation and performance is evaluated by mean absolute errors and mean squared errors resulting on the hedging portfolio. An analysis is also done regarding estimation approaches and their implications over the performance of the strategies.

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Menes, Matheus Dorival Leonardo Bombonato. "Versão discreta do modelo de elasticidade constante da variância." Universidade de São Paulo, 2012. http://www.teses.usp.br/teses/disponiveis/55/55134/tde-16042013-151325/.

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Neste trabalho propomos um modelo de mercado através de uma discretização aleatória do movimento browniano proposta por Leão & Ohashi (2010). Com este modelo, dada uma função payoff, vamos desenvolver uma estratégia de hedging e uma metodologia para precificação de opções
In this work we propose a market model using a discretization scheme of the random Brownian motion proposed by Leão & Ohashi (2010). With this model, for any given payoff function, we develop a hedging strategy and a methodology to option pricing

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De, Vita Renato. "Arbitragem estatística com opções utilizando modelo de volatilidade incerta e hedging estático." reponame:Repositório Institucional do FGV, 2014. http://hdl.handle.net/10438/11992.

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Com o objetivo de identificar oportunidades de arbitragem estatística no mercado de opções brasileiro, este trabalho utiliza o modelo de volatilidade incerta e o conceito de Hedging Estático, no apreçamento de um portfólio composto por diversas opções. São também incluídos os custos de transação relacionados a estruturação de um portfólio livre de risco, obtendo assim um modelo que pode ser facilmente implementado através da utilização do método de diferenças finitas explicito. Na aplicação do modelo ao mercado de opções sobre a ação preferencial da Petrobrás (PETR4), foi estabelecido um critério para estabelecer a frequência do ajuste do delta hedge do portfólio livre de risco de maneira a não incorrer em custos de transação elevados. Foi escolhido o período entre 19/05/08 e 20/01/14 para analisar o desempenho do modelo, selecionando-se em cada data de cálculo um conjunto de 11 opções de diferentes strikes e mesmo vencimento para compor o portfólio. O modelo apresentou um bom desempenho quando desconsiderados os custos de transação na apuração do resultado da estratégia de arbitragem, obtendo na maior parte dos casos resultados positivos. No entanto, ao incorporar os custos de transação, o desempenho obtido pelo modelo foi ruim, uma vez que na maior parte dos casos o resultado apresentado foi negativo.
For the purpose to identify statistical arbitrage opportunities in the Brazilian options market, this paper uses the model of uncertain volatility and the concept of Static Hedging, in the pricing of a portfolio comprised of several options. Also were included transaction costs related to structure a risk free portfolio, thus obtaining a model that can be easily implemented by using the explicit finite difference method. In applying the model for the Petrobras preferred (PETR4) options, were established a criterion to define the frequency of delta hedges adjustment of the risk-free portfolio in order to do not incur in high transaction costs. The period between 19/05/08 and 01/20/14 was chosen to analyze the performance of the model, selecting at each calculation date a set of 11 options of different strikes and the same maturity to compose the portfolio. The model performed well when ignored transaction costs in determining the outcome of the arbitration strategy, obtaining in most cases positive results. However, once incorporated transaction costs, the performance obtained by the model was bad, since in most cases the result presented was negative.

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Cai, Jiatu. "Méthodes asymptotiques en contrôle stochastique et applications à la finance." Sorbonne Paris Cité, 2016. http://www.theses.fr/2016USPCC338.

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Dans cette thèse, nous étudions plusieurs problèmes de mathématiques financières liés à la présence d’imperfections sur les marchés. Notre approche principale pour leur résolution est l’utilisation d’un cadre asymptotique pertinent dans lequel nous parvenons à obtenir des solutions approchées explicites pour les problèmes de contrôle associés. Dans la première partie de cette thèse, nous nous intéressons à l’évaluation et la couverture des options européennes. Nous considérons tout d’abord la problématique de l’optimisation des dates de rebalancement d’une couverture à temps discret en présence d’une tendance dans la dynamique du sous-jacent. Nous montrons que dans cette situation, il est possible de générer un rendement positif tout en couvrant l’option et nous décrivons une stratégie de rebalancement asymptotiquement optimale pour un critère de type moyenne-variance. Ensuite, nous proposons un cadre asymptotique pour la gestion des options européennes en présence de coûts de transaction proportionnels. En s’inspirant des travaux de Leland, nous développons une méthode alternative de construction de portefeuilles de réplication permettant de minimiser les erreurs de couverture. La seconde partie de ce manuscrit est dédiée à la question du suivi d’une cible stochastique. L’objectif de l’agent est de rester proche de cette cible tout en minimisant le coût de suivi. Dans une asymptotique de coûts petits, nous démontrons l’existence d’une borne inférieure pour la fonction valeur associée à ce problème d’optimisation. Cette borne est interprétée en terme du contrôle ergodique du mouvement brownien. Nous fournissons également de nombreux exemples pour lesquels la borne inférieure est explicite et atteinte par une stratégie que nous décrivons. Dans la dernière partie de cette thèse, nous considérons le problème de consommation et investissement en présence de taxes sur le rendement des capitaux. Nous obtenons tout d’abord un développement asymptotique de la fonction valeur associée que nous interprétons de manière probabiliste. Puis, dans le cas d’un marché avec changements de régime et pour un investisseur dont l’utilité est du type Epstein-Zin, nous résolvons explicitement le problème en décrivant une stratégie de consommation-investissement optimale. Enfin, nous étudions l’impact joint de coûts de transaction et de taxes sur le rendement des capitaux. Nous établissons dans ce cadre un système d’équations avec termes correcteurs permettant d’unifier les résultats de [ST13] et[CD13]
In this thesis, we study several mathematical finance problems related to the presence of market imperfections. Our main approach for solving them is to establish a relevant asymptotic framework in which explicit approximate solutions can be obtained for the associated control problems. In the first part of this thesis, we are interested in the pricing and hedging of European options. We first consider the question of determining the optimal rebalancing dates for a replicating portfolio in the presence of a drift in the underlying dynamics. We show that in this situation, it is possible to generate positive returns while hedging the option and describe a rebalancing strategy which is asymptotically optimal for a mean-variance type criterion. Then we propose an asymptotic framework for options risk management under proportional transaction costs. Inspired by Leland’s approach, we develop an alternative way to build hedging portfolios enabling us to minimize hedging errors. The second part of this manuscript is devoted to the issue of tracking a stochastic target. The agent aims at staying close to the target while minimizing tracking efforts. In a small costs asymptotics, we establish a lower bound for the value function associated to this optimization problem. This bound is interpreted in term of ergodic control of Brownian motion. We also provide numerous examples for which the lower bound is explicit and attained by a strategy that we describe. In the last part of this thesis, we focus on the problem of consumption-investment with capital gains taxes. We first obtain an asymptotic expansion for the associated value function that we interpret in a probabilistic way. Then, in the case of a market with regime-switching and for an investor with recursive utility of Epstein-Zin type, we solve the problem explicitly by providing a closed-form consumption-investment strategy. Finally, we study the joint impact of transaction costs and capital gains taxes. We provide a system of corrector equations which enables us to unify the results in [ST13] and [CD13]

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Chou, Ting-Hsuan, and 周庭萱. "Optimal Variance Hedging in Discrete Time." Thesis, 2015. http://ndltd.ncl.edu.tw/handle/3q52pw.

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Chiou, Shang-Chan, and 邱上展. "The Pricing and Hedging of Discrete Reset Option." Thesis, 2000. http://ndltd.ncl.edu.tw/handle/21874996374717783062.

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碩士
國立臺灣大學
財務金融學研究所
88
Abstract : In recent years, many kinds of reset option were issued with the character of resetting the strike price when the price of underlying asset touches the barrier. Reset Options can be classified according to whether the price of underlying assets has been continuous observed or not. Reset Options with continuous observation reset the strike price when the stock price hits the barrier while discrete reset options reset only when the stock price touches the barrier at some specific observation time point, such as one time a week, or two times a month. This thesis focus on the discrete observation type. This thesis proves the following results: 1. With the character of “discreteness”, Discrete Reset Option will be cheaper than reset options with continuous observation of stock price. Moreover, its cost increases little when compared with the one whose reset property just covers a partial period of option’s life. Thus, these advantageous properties let Discrete Reset Option more easily be promoted in the financial market. This thesis generalizes In-Out Parity to prove that Discrete Reset Option can be treated as a portfolio of discrete barrier options. M-steps recursion method [AitSahlia and Lai (1997)] and generalized In-Out Parity are mainly used to price discrete reset option. 2. When dynamic hedging is applied to a reset option, two problems are encountered: delta jump and negative delta. Unfortunately, put-call symmetry (PCS), the powerful static hedging method used in reset options or barrier options of continuous type, does not work when applying to discrete reset option. Under the dilemma, this thesis designs a “multi-barriers reset option”. It can be proved to be free of hedging problems without losing its function of protecting downside risk. 3. American type discrete barrier option can be priced with modified trinomial tree. Still, it has hedging problems mentioned above. By the same concept as introduced in the European type, “multi-barriers” can be applied to solve this problems.

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Huang, Chun-chang, and 黃俊彰. "Optimal Hedging Strategy with Partial Information in Discrete Financial Model." Thesis, 2007. http://ndltd.ncl.edu.tw/handle/54266453015473003298.

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碩士
國立高雄大學
統計學研究所
95
From the perspectives of the investors, we invest in some assets in discrete time and thus, we usually observe the discrete information from the market. In this paper we give the investors three main results for our discrete financial model. On the other hand, from the perspectives of the financial institutions, we face the problem about how to reduce the risk of the investment with consumptions. Then we have similar conclusion with Follmer and Sondermann (1986).

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Books on the topic "Discrete hedging"

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Traugott, John. Hedging Options in Discrete Time. Independently Published, 2018.

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Book chapters on the topic "Discrete hedging"

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Fukasawa, Masaaki. "Asymptotically Efficient Discrete Hedging." In Stochastic Analysis with Financial Applications, 331–46. Basel: Springer Basel, 2011. http://dx.doi.org/10.1007/978-3-0348-0097-6_19.

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Privault, Nicolas. "Pricing and Hedging in Discrete Time." In Introduction to Stochastic Finance with Market Examples, 65–112. 2nd ed. Boca Raton: Chapman and Hall/CRC, 2022. http://dx.doi.org/10.1201/9781003298670-3.

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Huang, Shih-Feng, and Meihui Guo. "Dynamic Programming and Hedging Strategies in Discrete Time." In Handbook of Computational Finance, 605–31. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-17254-0_22.

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Rémillard, Bruno, Alexandre Hocquard, Hugues Langlois, and Nicolas Papageorgiou. "Optimal Hedging of American Options in Discrete Time." In Springer Proceedings in Mathematics, 145–70. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-25746-9_5.

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Kolkiewicz, Adam W. "Optimal Static Hedging of Non-tradable Risks with Discrete Distributions." In Springer Proceedings in Mathematics & Statistics, 531–41. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-99719-3_48.

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Duong, Thanh, Quyen Ho, An Tran, and Minh Tran. "Optimal Discrete Hedging in Garman-Kohlhagen Model with Liquidity Risk." In Advances in Intelligent Systems and Computing, 377–88. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-18167-7_33.

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Di Masi, G. B., E. Platen, and W. J. Runggaldier. "Hedging of Options under Discrete Observation on Assets with Stochastic Volatility." In Seminar on Stochastic Analysis, Random Fields and Applications, 359–64. Basel: Birkhäuser Basel, 1995. http://dx.doi.org/10.1007/978-3-0348-7026-9_25.

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Černý, Aleš. "Discrete-Time Quadratic Hedging of Barrier Options in Exponential Lévy Model." In Springer Proceedings in Mathematics & Statistics, 257–75. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-45875-5_12.

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El-Férik, Sami, and Roland P. Malhamé. "Optimizing the transient behavior of hedging control policies in manufacturing systems." In 11th International Conference on Analysis and Optimization of Systems Discrete Event Systems, 565–71. Berlin, Heidelberg: Springer Berlin Heidelberg, 1994. http://dx.doi.org/10.1007/bfb0033588.

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Viens, Frederi. "A Didactic Introduction to Risk Management via Hedging in Discrete and Continuous Time." In Statistical Methods and Applications in Insurance and Finance, 3–37. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-30417-5_1.

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Conference papers on the topic "Discrete hedging"

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Bhat, Sanjay, VijaySekhar Chellaboina, Anil Bhatia, Sandeep Prasad, and M. Uday Kumar. "Discrete-time, minimum-variance hedging of European contingent claims." In 2009 Joint 48th IEEE Conference on Decision and Control (CDC) and 28th Chinese Control Conference (CCC). IEEE, 2009. http://dx.doi.org/10.1109/cdc.2009.5399522.

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Subramanian, Easwar, and Sanjay P. Bhat. "Discrete-Time Quadratic-Optimal Hedging Strategies for European Contingent Claims." In 2015 IEEE Symposium Series on Computational Intelligence (SSCI). IEEE, 2015. http://dx.doi.org/10.1109/ssci.2015.249.

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Subramanian, Easwar, Vijaysekhar Chellaboina, and Arihant Jain. "Explicit Solutions of Discrete-Time Hedging Strategies for Multi-Asset Options." In 2016 International Conference on Industrial Engineering, Management Science and Application (ICIMSA). IEEE, 2016. http://dx.doi.org/10.1109/icimsa.2016.7504008.

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Subramanian, Easwar, Vijaysekhar Chellaboina, and Arihant Jain. "Performance Evaluation of Discrete-Time Hedging Strategies for European Contingent Claims." In 2016 International Conference on Industrial Engineering, Management Science and Application (ICIMSA). IEEE, 2016. http://dx.doi.org/10.1109/icimsa.2016.7504026.

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Caccia, Massimo, and Bruno Rémillard. "Option Pricing and Hedging for Discrete Time Autoregressive Hidden Markov Model." In Innovations in Insurance, Risk- and Asset Management. WORLD SCIENTIFIC, 2018. http://dx.doi.org/10.1142/9789813272569_0012.

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Costa, O. L. V., A. C. Maiali, and A. de C. Pinto. "Mean-variance hedging strategies in discrete time and continuous state space." In COMPUTATIONAL FINANCE 2006. Southampton, UK: WIT Press, 2006. http://dx.doi.org/10.2495/cf060111.

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Uday Kumar, M., Vijaysekhar Chellaboina, Sanjay Bhat, Sandeep Prasad, and Anil Bhatia. "Discrete-time optimal hedging for multi-asset path-dependent European contingent claims." In 2009 Joint 48th IEEE Conference on Decision and Control (CDC) and 28th Chinese Control Conference (CCC). IEEE, 2009. http://dx.doi.org/10.1109/cdc.2009.5399932.

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Subramanian, Easwar, and Vijaysekhar Chellaboina. "Explicit solutions of discrete-time quadratic optimal hedging strategies for European contingent claims." In 2014 IEEE Conference on Computational Intelligence for Financial Engineering & Economics (CIFEr). IEEE, 2014. http://dx.doi.org/10.1109/cifer.2014.6924108.

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