УЧЕТ НЕДОСТАТОЧНОЙ ЛИКВИДНОСТИ И ТРАНЗАКЦИОННЫХ ИЗДЕРЖЕК ПРИ ДЕЛЬТА-ХЕДЖИРОВАНИИ

М. М. Дышаев; В. Е. Федоров

doi:10.52575/2687-0959-2021-53-2-132–143

Авторы

М. М. Дышаев Челябинский государственный университет http://orcid.org/0000-0003-4265-1752
В. Е. Федоров Челябинский государственный университет http://orcid.org/0000-0002-0787-3272

DOI:

https://doi.org/10.52575/2687-0959-2021-53-2-132–143

Ключевые слова:

хеджирование, ликвидность, транзакционные издержки, книга лимитных ордеров, нелинейные уравнения типа Блэка - Шоулса, численное решение

Аннотация

Статья посвящена исследованию оптимального временного интервала хеджирования при недостаточной ликвидности и наличии транзакционных издержек. Получены нелинейные уравнения типа Блэка – Шоулса для случая, когда функция затрат неликвидности является линейной и квадратичной. Для определения транзакционных издержек используется модель методологии ценообразования с поправкой на риск (risk-adjusted pricing methodology (RAPM) model). Практическое применение продемонстрировано для опционной комбинации «long butterfly».

Скачивания

Данные скачивания пока недоступны.

Библиографические ссылки

Agliardi R., GenCay R. 2014. Hedging through a limit order book with varying liquidity. The Journal of Derivatives, 22(2):32--49. DOI:10.3905/jod.2014.22.2.032.

Alfonsi A., Fruth A., Schied A. 2010. Optimal execution strategies in limit order books with general shape functions. Quantitative Finance, 10(2):143--157. DOI:10.1080/14697680802595700.

Almgren R., Chriss N. 2001. Optimal execution of portfolio transactions. Journal of Risk, 3(2):5--39. DOI:10.21314/JOR.2001.041.

Almgren R., Li T.M. 2016. Option hedging with smooth market impact. Mark. Microstructure Liq., 02(01):1650002. DOI:10.1142/s2382626616500027.

Ankudinova J., Ehrhardt M. 2008. On the numerical solution of nonlinear Black-Scholes equations. Computers & Mathematics with Applications, 56(3):799--812. DOI:10.1016/j.camwa.2008.02.005.

Avellaneda M., Paras A. 1994. Dynamic hedging portfolios for derivative securities in the presence of large transaction costs. Applied Mathematical Finance, 1(2):165--194. DOI:10.1080/13504869400000010.

Bank P., Soner H.M., Vos, M. 2017. Hedging with temporary price impact. Mathematics and Financial Economics, 11(2):215--239. DOI:10.1007/s11579-016-0178-4.

Barles G., Soner H.M. 1998. Option pricing with transaction costs and a nonlinear Black-Scholes equation. Finance and Stochastics, 2(4):369--397. DOI:10.1007/s007800050046.

Bertsimas D., Lo A.W. 1998. Optimal control of execution costs. Journal of Financial Markets, 1(1):1--50.DOI:10.1016/S1386-4181(97)00012-8.

Biais B., Glosten L., Spatt C. 2005. Market microstructure: A survey of microfoundations empirical results policy implications. Journal of Financial Markets, 8(2):217--264. DOI:10.1016/j.finmar.2004.11.001.

Black F., Scholes M. 1973. The pricing of options and corporate liabilities. Journal of Political Economy, 81(3):637--654. DOI:10.1086/260062.

Bouchard B., Loeper G., Zou Y. 2017. Hedging of covered options with linear market impact and gamma

constraint. SIAM Journal on Control and Optimization, 55(5):3319--3348. DOI:10.1137/15M1054109.

Cai J., Fukasawa M., Rosenbaum M., Tankov P. 2016. Optimal discretization of hedging strategies with directional views. SIAM Journal on Financial Mathematics, 7(1):34--69. DOI:10.1137/151004306.

Cartea A., Gan L., Jaimungal S. 2019. Hedge and speculate: replicating option payoffs with limit and market orders. SIAM Journal on Financial Mathematics, 10(3):790--814. DOI:10.1137/18M1192706.

Cartea A., Jaimungal S. 2015. Optimal execution with limit and market orders. Quantitative Finance, 15(8):1279--1291. DOI:10.1080/14697688.2015.1032543.

Company R., Navarro E., Pintos J.R., Ponsoda E. 2008. Numerical solution of linear and nonlinear Black-Scholes option pricing equations. Computers & Mathematics with Applications, 56(3):813--821. DOI:10.1016/j.camwa.2008.02.010.

Davis M. H.A., Panas V.G., Zariphopoulou T. 1993. European option pricing with transaction costs. SIAM Journal on Control and Optimization, 31(2):470--493. DOI:10.1137/0331022.

Dyshaev M.M. 2020. On measuring the cost of liquidity in the limit order book. Chelyabinsk Physical and Mathematical Journal, 5(1):96--104. DOI:10.24411/2500-0101-2020-15107.

Dyshaev M.M., Fedorov V.E. 2019. Comparing of some sensitivities (Greeks) for nonlinear models of option pricing with market illiquidity. Mathematical notes of NEFU, 26(2):94--108. DOI:10.25587/SVFU.2019.102.31514.

Dyshaev M.M., Fedorov V.E. 2020. The optimal rehedging interval for the options portfolio within the RAPM, taking into account transaction costs and liquidity costs. The Bulletin of Irkutsk State University. Series: Mathematics, 31:3--17. DOI:10.26516/1997-7670.2020.31.3.

Frey R., Stremme A. 1997. Market volatility and feedback effects from dynamic hedging. Mathematical Finance, 7(4):351--374. DOI:10.1111/1467-9965.00036.

Gould M.D., Porter M.A., Williams S., McDonald M., Fenn D.J., Howison S.D. 2013. Limit order books. Quantitative Finance, 13(11):1709--1742. DOI:10.1080/14697688.2013.803148.

Gueant O. 2016. The financial mathematics of market liquidity: from optimal execution to market making. Chapman and Hall/CRC.DOI:10.1201/b21350.

Gueant O., Pu J. 2017. Option pricing and hedging with execution costs and market impact. Mathematical Finance, 27(3):803--831. DOI:10.1111/mafi.12102.

Harris L., Hasbrouck J. 1996. Market vs. limit orders: the SuperDOT evidence on order submission strategy. Journal of Financial and Quantitative Analysis, 31(2):213--231. DOI:10.2307/2331180.

Heider P. 2010. Numerical methods for non-linear Black--- Scholes equations. Applied Mathematical Finance, 17:59--81. DOI:10.1080/13504860903075670.

Jandacka M., Sevcovic, D. 2005. On the risk-adjusted pricing-methodology-based valuation of vanilla options and explanation of the volatility smile. Journal of Applied Mathematics, 2005(3):235--258. DOI:10.1155/JAM.2005.235.

Kabanov Y.M., Safarian M.M. 1997. On Leland's strategy of option pricing with transactions costs. Finance and Stochastics, 1(3):239--250. DOI:10.1007/s007800050023.

Lehoczky J., Schervish M. 2018. Overview and history of statistics for equity markets. Annual Review of Statistics and Its Application, 5:265--288. DOI:10.1146/annurev-statistics-031017-100518.

Leland H.E. 1985. Option pricing and replication with transactions costs. The Journal of Finance, 40(5):1283--1301. DOI:10.1111/j.1540-6261.1985.tb02383.x.

Malo P., Pennanen T. 2012. Reduced form modeling of limit order markets. Quantitative Finance, 12(7):1025--1036. DOI:10.1080/14697688.2011.589402.

McAleer M., Medeiros M.C. 2008. Realized volatility: A review. Econometric Reviews, 27(1-3):10--45. DOI:10.1080/07474930701853509.

Moscow Exchange. Search by contracts 2020. Accessed Nov 2020. URL:https://www.moex.com/en/derivatives/contracts.aspx?p=act.

Obizhaeva A.A., Wang J. 2013. Optimal trading strategy and supply/demand dynamics. Journal of Financial Markets, 16(1):1--32. DOI:10.1016/j.finmar.2012.09.001.

Ranaldo A. 2004. Order aggressiveness in limit order book markets. Journal of Financial Markets, 7(1):53--74. DOI:10.1016/S1386-4181(02)00069-1.

Rogers L. C.G., Singh S. 2010. The cost of illiquidity and its effects on hedging. Mathematical Finance, 20(4):597--615. DOI:10.1111/j.1467-9965.2010.00413.x.

Schonbucher P.J., Wilmott P. 2000. The feedback effect of hedging in illiquid markets. SIAM J. Appl. Math., 61(1):232--272. DOI:10.1137/S0036139996308534.

Sepp A. 2013. When you hedge discretely: optimization of sharpe ratio for delta-hedging strategy under discrete hedging and transaction costs. Journal of Investment Strategies, 3(1):19--59. DOI:10.2139/ssrn.1865998.

Sircar R.K., Papanicolaou G. 1998. General Black-Scholes models accounting for increased market volatility from hedging strategies. Appl. Math. Finance, 5(1):45--82. DOI:10.1080/135048698334727.

Tarasov V.E. 2020. Fractional econophysics: market price dynamics with memory effects. Physica A: Statistical Mechanics and its Applications, p.124865. DOI:10.1016/j.physa.2020.124865.

Whalley A.E., Wilmott P. 1997. An asymptotic analysis of an optimal hedging model for option pricing with transaction costs. Mathematical Finance, 7(3):307--324. DOI:10.1111/1467-9965.00034.

Zakamouline V.I. 2006. Efficient analytic approximation of the optimal hedging strategy for a European call option with transaction costs. Quantitative Finance, 6(5):435--445. DOI:10.1080/14697680600724809.