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Journal articles on the topic 'Discrete hedging'

Author: Grafiati

Published: 16 December 2023

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1

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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2

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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3

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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4

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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5

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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6

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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7

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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8

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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9

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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10

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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11

Machado-Santos, Carlos. "Portfolio insurance using traded options." Revista de Administração Contemporânea 5, no. 3 (December 2001): 187–214. http://dx.doi.org/10.1590/s1415-65552001000300010.

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Literature concerning the institutional use of options indicates that the main purpose of option trading is to provide investors with the opportunity to create return distributions previously unavailable, considering that options provide the means to manipulate portfolio returns. In such a context, this study intends to analyse the returns of insured portfolios generated by hedging strategies on underlying stock portfolios. Because dynamic hedging is too expensive, we have hedged the stock positions discretely, in a way that the positions were revised only when the daily hedge ratio has changed more than a specific amount. The results, provided by these hedging schemes, indicate that a small rise of the standard deviation seems to be largely compensated with the higher average returns. In fact, such strategies seem to be highly influenced by the price movements of underlying stocks, requiring more frequent (sparse) adjustments in periods of high (low) volatility. Thus, discrete hedging strategies seem more accurate and meaningful than the arbitrary regular intervals largely presented and discussed in literature.

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12

Hou, Songyan, Thomas Krabichler, and Marcus Wunsch. "Deep Partial Hedging." Journal of Risk and Financial Management 15, no. 5 (May 19, 2022): 223. http://dx.doi.org/10.3390/jrfm15050223.

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Using techniques from deep learning, we show that neural networks can be trained successfully to replicate the modified payoff functions that were first derived in the context of partial hedging by Föllmer and Leukert. Not only does this approach better accommodate the realistic setting of hedging in discrete time, it also allows for the inclusion of transaction costs as well as general market dynamics. It needs to be noted that, without further modifications, the approach works only if the risk aversion is beyond a certain level.

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13

MASTINSEK, MIKLAVZ. "ON ROBUSTNESS OF THE BLACK–SCHOLES PARTIAL DIFFERENTIAL EQUATION MODEL." International Journal of Theoretical and Applied Finance 19, no. 02 (March 2016): 1650013. http://dx.doi.org/10.1142/s0219024916500138.

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When the discretely adjusted option hedges are constructed by the continuous-time Black–Scholes delta, then the hedging errors appear. The first objective of the paper is to consider a discrete-time adjusted delta, such that the hedging error can be reduced. Consequently, a partial differential equation for option valuation associated with the problem is derived and solved. The second objective is to compare the obtained results with the results given by the Black–Scholes formula. The obtained option values may be higher than those given by the Black–Scholes formula, however, unless the option is near expiry, the difference is relatively small.

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14

Coleman, Thomas, Yuying Li, and Maria-Cristina Patron. "Discrete hedging under piecewise linear risk minimization." Journal of Risk 5, no. 3 (May 2003): 39–65. http://dx.doi.org/10.21314/jor.2003.079.

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15

Morozov, V. V., and A. I. Soloviev. "On optimal partial hedging in discrete markets." Optimization 62, no. 11 (November 2013): 1403–18. http://dx.doi.org/10.1080/02331934.2013.854784.

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16

Hussain, S., and M. Shashiashvili. "DISCRETE TIME HEDGING OF THE AMERICAN OPTION." Mathematical Finance 20, no. 4 (September 22, 2010): 647–70. http://dx.doi.org/10.1111/j.1467-9965.2010.00415.x.

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17

Schulmerich, Marco, and Siegfried Trautmann. "Local Expected Shortfall-Hedging in Discrete Time *." Review of Finance 7, no. 1 (April 1, 2003): 75–102. http://dx.doi.org/10.1023/a:1022506825795.

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18

Anthropelos, Michail, and Evmorfia Blontzou. "On Valuation and Investments of Pension Plans in Discrete Incomplete Markets." Risks 11, no. 6 (June 1, 2023): 103. http://dx.doi.org/10.3390/risks11060103.

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We study the valuation of a pension fund’s obligations in a discrete time and space incomplete market model. The market’s incompleteness stems from the non-replicability of the wage process that finances the pension plan through time. The contingent defined-benefit liability of the pension fund is a function of the wages, which can be seen as the payoff of a path-dependent derivative security. We apply the notion of the super-hedging value and propose its difference from the current pension’s fund capital as a measure of distance to liability hedging. The induced closed-form expressions of the values and the related investment strategies provide insightful comparative statistics. Furthermore, we use a utility-based optimization portfolio to point out that in cases of sufficient capital, the application of a subjective investment criterion may result in heavily different strategies than the super-hedging one. This means that the pension fund will be left with some liability risk, although it could have been fully hedged. Finally, we provide conditions under which the effect of a possible early exit leaves the super-hedging valuation unchanged.

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19

KOLKIEWICZ, ADAM W. "EFFICIENT HEDGING OF PATH–DEPENDENT OPTIONS." International Journal of Theoretical and Applied Finance 19, no. 05 (July 29, 2016): 1650032. http://dx.doi.org/10.1142/s0219024916500321.

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In this paper, we propose a novel method of hedging path-dependent options in a discrete-time setup. Assuming that prices are given by the Black–Scholes model, we first describe the residual risk when hedging a path-dependent option using only an European option. Then, for a fixed hedging interval, we find the hedging option that minimizes the shortfall risk, which we define as the expectation of the shortfall weighted by some loss function. We illustrate the method using Asian options, but the methodology is applicable to other path-dependent contacts.

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20

Thiha, Soe, Asaad Y. Shamseldin, and Bruce W. Melville. "Improving the Summer Power Generation of a Hydropower Reservoir Using the Modified Multi-Step Ahead Time-Varying Hedging Rule." Water Resources Management 36, no. 3 (January 26, 2022): 853–73. http://dx.doi.org/10.1007/s11269-021-03043-7.

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AbstractThis paper aims to improve summer power generation of the Yeywa Hydropower Reservoir in Myanmar using the modified multi-step ahead time-varying hedging (TVH) rule as a case study. The results of the TVH rules were compared with the standard operation policy (SOP) rule, the binary standard operation policy (BSOP) rule, the discrete hedging (DH) rule, the standard hedging (SH) rule, the one-point hedging (OPH) rule, and the two-point hedging (TPH) rule. The Multi-Objective Genetic Algorithm (MOGA) was utilized to drive the optimal Pareto fronts for the hedging rules. The results demonstrated that the TVH rules had higher performance than the other rules and showed improvements in power generation not only during the summer period but also over the entire period.

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21

Li, Meng, Xuefeng Wang, and Fangfang Sun. "Pricing of Proactive Hedging European Option with Dynamic Discrete Position Strategy." Discrete Dynamics in Nature and Society 2019 (April 1, 2019): 1–11. http://dx.doi.org/10.1155/2019/1070873.

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Proactive hedging European option is an exotic option for hedgers in the options market proposed recently by Wang et al. It extends the classical European option by requiring option holders to continuously trade in underlying assets according to a predesigned trading strategy, to proactively hedge part of the potential risk from underlying asset price changes. To generalize this option design for practical application, in this study, a proactive hedging option with discrete trading strategy is developed and its pricing formula is deducted assuming the underlying asset price follows Geometric Fractional Brownian Motion. Simulation studies show that proactive hedging option with discrete trading strategy still enjoys strong price advantage compared to the classical European option for majority of parameter space. The observed price advantage is stronger when the underlying asset has more volatility or when the asset price follows closer to Geometric Brownian Motion. Additionally, we found that a higher frequency trading strategy has stronger price advantage if there is no trading cost. The findings in this research strongly facilitate the practical application of the proactive hedging option, making this lower-cost trading tool more feasible.

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22

Josephy, Norman, Lucia Kimball, and Victoria Steblovskaya. "Optimal Hedging and Pricing of Equity-Linked Life Insurance Contracts in a Discrete-Time Incomplete Market." Journal of Probability and Statistics 2011 (2011): 1–23. http://dx.doi.org/10.1155/2011/850727.

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We present a method of optimal hedging and pricing of equity-linked life insurance products in an incomplete discrete-time financial market. A pure endowment life insurance contract with guarantee is used as an example. The financial market incompleteness is caused by the assumption that the underlying risky asset price ratios are distributed in a compact interval, generalizing the assumptions of multinomial incomplete market models. For a range of initial hedging capitals for the embedded financial option, we numerically solve an optimal hedging problem and determine a risk-return profile of each optimal non-self-financing hedging strategy. The fair price of the insurance contract is determined according to the insurer's risk-return preferences. Illustrative numerical results of testing our algorithm on hypothetical insurance contracts are documented. A discussion and a test of a hedging strategy recalibration technique for long-term contracts are presented.

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23

Fard, Farzad Alavi, Firmin Doko Tchatoka, and Sivagowry Sriananthakumar. "Maximum Entropy Evaluation of Asymptotic Hedging Error under a Generalised Jump-Diffusion Model." Journal of Risk and Financial Management 14, no. 3 (February 28, 2021): 97. http://dx.doi.org/10.3390/jrfm14030097.

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In this paper we propose a maximum entropy estimator for the asymptotic distribution of the hedging error for options. Perfect replication of financial derivatives is not possible, due to market incompleteness and discrete-time hedging. We derive the asymptotic hedging error for options under a generalised jump-diffusion model with kernel bias, which nests a number of very important processes in finance. We then obtain an estimation for the distribution of hedging error by maximising Shannon’s entropy subject to a set of moment constraints, which in turn yields the value-at-risk and expected shortfall of the hedging error. The significance of this approach lies in the fact that the maximum entropy estimator allows us to obtain a consistent estimate of the asymptotic distribution of hedging error, despite the non-normality of the underlying distribution of returns.

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24

Park, Minseok, Kyungsub Lee, and Geon Ho Choe. "Distribution of Discrete Time Delta-Hedging Error via a Recursive Relation." East Asian Journal on Applied Mathematics 6, no. 3 (July 20, 2016): 314–36. http://dx.doi.org/10.4208/eajam.010116.220516a.

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AbstractWe introduce a new method to compute the approximate distribution of the Delta-hedging error for a path-dependent option, and calculate its value over various strike prices via a recursive relation and numerical integration. Including geometric Brownian motion and Merton's jump diffusion model, we obtain the approximate distribution of the Delta-hedging error by differentiating its price with respect to the strike price. The distribution from Monte Carlo simulation is compared with that obtained by our method.

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25

Bhatia, Nikhil, Roshan Srivastav, and Kasthrirengan Srinivasan. "Season-Dependent Hedging Policies for Reservoir Operation—A Comparison Study." Water 10, no. 10 (September 22, 2018): 1311. http://dx.doi.org/10.3390/w10101311.

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During periods of significant water shortage or when drought is impending, it is customary to implement some kind of water supply reduction measures with a view to prevent the occurrence of severe shortages (vulnerability) in the near future. In the case of operation of a water supply reservoir, this reduction of water supply is affected by hedging schemes or hedging policies. This research work aims to compare the popular hedging policies: (i) linear two-point hedging; (ii) modified two-point hedging; and, (iii) discrete hedging based on time-varying and constant hedging parameters. A parameterization-simulation-optimization (PSO) framework is employed for the selection of the parameters of the compromising hedging policies. The multi-objective evolutionary search-based technique (Non-dominated Sorting based Genetic Algorithm-II) was used to identify the Pareto-optimal front of hedging policies that seek to obtain the trade-off between shortage ratio and vulnerability. The case example used for illustration is the Hemavathy reservoir in Karnataka, India. It is observed that the Pareto-optimal front that was obtained from time-varying hedging policies show significant improvement in reservoir performance when compared to constant hedging policies. The variation in the monthly parameters of the time-variant hedging policies shows a strong correlation with monthly inflows and available water.

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26

Josephy, Norman, Lucia Kimball, and Victoria Steblovskaya. "Alternative hedging in a discrete-time incomplete market." Journal of Risk 16, no. 1 (September 2013): 85–117. http://dx.doi.org/10.21314/jor.2013.268.

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27

Akyildirim, Erdnç, and Albert Altarovici. "Partial hedging and cash requirements in discrete time." Quantitative Finance 16, no. 6 (November 10, 2015): 929–45. http://dx.doi.org/10.1080/14697688.2015.1095347.

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28

Peeters, B., C. L. Dert, and A. Lucas. "Hedging Large Portfolios of Options in Discrete Time*." Applied Mathematical Finance 15, no. 3 (June 2008): 251–75. http://dx.doi.org/10.1080/13504860701718471.

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29

Bossaerts, Peter, and Pierre Hillion. "Local parametric analysis of hedging in discrete time." Journal of Econometrics 81, no. 1 (November 1997): 243–72. http://dx.doi.org/10.1016/s0304-4076(97)00046-8.

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30

Tse, Wai-Man, Eric C. Chang, Leong Kwan Li, and Henry M. K. Mok. "Pricing and Hedging of Discrete Dynamic Guaranteed Funds." Journal of Risk & Insurance 75, no. 1 (March 2008): 167–92. http://dx.doi.org/10.1111/j.1539-6975.2007.00253.x.

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31

Meirelles, Sofia Kusiak, and Marcelo Fernandes. "Estratégias de Imunização de Carteiras de Renda Fixa no Brasil." Brazilian Review of Finance 16, no. 2 (July 11, 2018): 179. http://dx.doi.org/10.12660/rbfin.v16n2.2018.69279.

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This paper aims to statistically compare the performance of two hedging strategies for Brazilian fixed income portfolios, with discrete rebalancing. The first hedging strategy matches duration, and hence it considers only small parallel changes in the yield curve. The alternative methodology ponders level, curvature and convexity shifts through a factor model. We first estimate the yield curve using the polynomial model of Nelson & Siegel (1987) and Diebold & Li (2006) and then immunize the fixed income portfolio using Litterman & Scheinkman’s (1991) hedging procedure. The alternative strategy for portfolio immunization outperforms duration matching in the empirical exercise we contemplate. Additionally, we show that rebalancing the hedging portfolio every month is more efficient than at other frequencies.

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32

Dahl, Mikkel. "A Discrete-Time Model for Reinvestment Risk in Bond Markets." ASTIN Bulletin 37, no. 02 (November 2007): 235–64. http://dx.doi.org/10.2143/ast.37.2.2024066.

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In this paper we propose a discrete-time model with fixed maximum time to maturity of traded bonds. At each trading time, a bond matures and a new bond is introduced in the market, such that the number of traded bonds is constant. The entry price of the newly issued bond depends on the prices of the bonds already traded and a stochastic term independent of the existing bond prices. Hence, we obtain a bond market model for the reinvestment risk, which is present in practice, when hedging long term contracts. In order to determine optimal hedging strategies we consider the criteria of super-replication and risk-minimization.

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33

Dahl, Mikkel. "A Discrete-Time Model for Reinvestment Risk in Bond Markets." ASTIN Bulletin 37, no. 2 (November 2007): 235–64. http://dx.doi.org/10.1017/s0515036100014859.

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In this paper we propose a discrete-time model with fixed maximum time to maturity of traded bonds. At each trading time, a bond matures and a new bond is introduced in the market, such that the number of traded bonds is constant. The entry price of the newly issued bond depends on the prices of the bonds already traded and a stochastic term independent of the existing bond prices. Hence, we obtain a bond market model for the reinvestment risk, which is present in practice, when hedging long term contracts. In order to determine optimal hedging strategies we consider the criteria of super-replication and risk-minimization.

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34

Saito, Taiga. "Hedging and pricing illiquid options with market impacts." International Journal of Financial Engineering 04, no. 02n03 (June 2017): 1750030. http://dx.doi.org/10.1142/s242478631750030x.

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In this paper, we consider hedging and pricing of illiquid options on an untradable underlying asset, where an alternative asset is used as a hedging instrument. Particularly, we consider the situation where the trade price of the hedging instrument is subject to market impacts caused by the hedger and the liquidity costs paid as a spread from the mid price. Pricing illiquid options, which often appears in trading of structured products, is a critical issue in practice because of its difficulties in hedging mainly due to untradability of the underlying asset as well as the liquidity costs and market impacts of the hedging instrument. First, by setting the problem under a discrete time model, where the optimal hedging strategy is defined by the local risk-minimization, we present algorithms to obtain the option price along with the hedging strategy by an asymptotic expansion. Moreover, we provide numerical examples. This model enables the estimation of the effect of both the market impacts and the liquidity costs on option prices, which is important in practice.

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35

Gugushvili, S. "Dynamic Programming and Mean-Variance Hedging in Discrete Time." gmj 10, no. 2 (June 2003): 237–46. http://dx.doi.org/10.1515/gmj.2003.237.

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Abstract We consider the mean-variance hedging problem in the discrete time setting. Using the dynamic programming approach we obtain recurrent equations for an optimal strategy. Additionally, some technical restrictions of the previous works are removed.

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36

Sepp, Artur. "When you hedge discretely: optimization of the Sharpe ratio for the Delta-hedging strategy under discrete hedging and transaction costs." Journal of Investment Strategies 3, no. 1 (December 2013): 19–59. http://dx.doi.org/10.21314/jois.2013.023.

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37

Alexandru Badescu, Joan del Castillo, and Juan-Pablo Ortega. "Hedging of Time Discrete Auto-Regressive Stochastic Volatility Options." Annals of Economics and Statistics, no. 123/124 (2016): 271. http://dx.doi.org/10.15609/annaeconstat2009.123-124.0271.

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38

Nian, Ke, Thomas F. Coleman, and Yuying Li. "Learning minimum variance discrete hedging directly from the market." Quantitative Finance 18, no. 7 (February 12, 2018): 1115–28. http://dx.doi.org/10.1080/14697688.2017.1413245.

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39

Dolinsky, Yan, and Yuri Kifer. "Hedging with risk for game options in discrete time." Stochastics 79, no. 1-2 (February 2007): 169–95. http://dx.doi.org/10.1080/17442500601097784.

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40

Schäl, Manfred. "Martingale Measures and Hedging for Discrete-Time Financial Markets." Mathematics of Operations Research 24, no. 2 (May 1999): 509–28. http://dx.doi.org/10.1287/moor.24.2.509.

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41

Trabelsi, Faouzi, and Abdelhamid Trad. "L 2 -discrete hedging in a continuous-time model." Applied Mathematical Finance 9, no. 3 (September 2002): 189–217. http://dx.doi.org/10.1080/1350486022000013672.

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42

Černý, Aleš. "Dynamic programming and mean‐variance hedging in discrete time." Applied Mathematical Finance 11, no. 1 (March 2004): 1–25. http://dx.doi.org/10.1080/1350486042000196164.

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43

Battauz, Anna. "Quadratic hedging for asset derivatives with discrete stochastic dividends." Insurance: Mathematics and Economics 32, no. 2 (April 2003): 229–43. http://dx.doi.org/10.1016/s0167-6687(02)00212-3.

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44

Gobet, Emmanuel, and Emmanuel Temam. "Discrete time hedging errors for options with irregular payoffs." Finance and Stochastics 5, no. 3 (July 2001): 357–67. http://dx.doi.org/10.1007/pl00013539.

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45

Angelini, Flavio, and Stefano Herzel. "Delta hedging in discrete time under stochastic interest rate." Journal of Computational and Applied Mathematics 259 (March 2014): 385–93. http://dx.doi.org/10.1016/j.cam.2013.06.022.

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46

Soloviev, A. I. "Partial Hedging of American Claims in a Discrete Market." Computational Mathematics and Modeling 25, no. 4 (September 16, 2014): 592–601. http://dx.doi.org/10.1007/s10598-014-9252-z.

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Shih, Jhih-Shyang, and Charles ReVelle. "Water supply operations during drought: A discrete hedging rule." European Journal of Operational Research 82, no. 1 (April 1995): 163–75. http://dx.doi.org/10.1016/0377-2217(93)e0237-r.

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Rémillard, Bruno, Alexandre Hocquard, Hugo Lamarre, and Nicolas Papageorgiou. "Option Pricing and Hedging for Discrete Time Regime-Switching Models." Modern Economy 08, no. 08 (2017): 1005–32. http://dx.doi.org/10.4236/me.2017.88070.

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Cheridito, Patrick, Michael Kupper, and Ludovic Tangpi. "Duality Formulas for Robust Pricing and Hedging in Discrete Time." SIAM Journal on Financial Mathematics 8, no. 1 (January 2017): 738–65. http://dx.doi.org/10.1137/16m1064088.

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Kifer, Yuri. "Hedging of game options in discrete markets with transaction costs." Stochastics 85, no. 4 (May 23, 2013): 667–81. http://dx.doi.org/10.1080/17442508.2013.795566.

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