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AIMS Mathematics

2021, Volume 6, Issue 7: 7199-7214. doi: 10.3934/math.2021422
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Research article

Valuation of bid and ask prices for European options under mixed fractional Brownian motion

  • 1.
    School of Business, Nanjing Normal University, Nanjing 210023, China
  • 2.
    School of Mathematics, South China University of Technology, Guangzhou 510640, China
  • Received: 09 January 2021 Accepted: 26 April 2021 Published: 28 April 2021

  • MSC : 91G20, 91-10, 62P05

  • In this paper, we study the valuation of the bid and ask prices for European options under the mixed fractional Brownian motion with Hurst index $ H > 3/4 $, which is able to capture the long range dependence of the underlying asset returns in real markets. As we know, the classical option pricing theories are usually built on the law of one price, while ignoring the impact of market liquidity on bid-ask spreads. The theory of conic finance replaces the law of one price by the law of two prices, allowing for market participants sell to the market at the bid price and buy from the market at the higher ask price. Within the framework of conic finance, we then derive the explicit formulas for the bid and ask prices of European call and put options by using WANG-transform as a distortion function. Moreover, numerical experiment is performed to illustrate the effects of the Hurst index and market liquidity level on bid and ask prices.

    Citation: Zhe Li, Xiao-Tian Wang. Valuation of bid and ask prices for European options under mixed fractional Brownian motion[J]. AIMS Mathematics, 2021, 6(7): 7199-7214. doi: 10.3934/math.2021422

    Related Papers:

  • In this paper, we study the valuation of the bid and ask prices for European options under the mixed fractional Brownian motion with Hurst index $ H > 3/4 $, which is able to capture the long range dependence of the underlying asset returns in real markets. As we know, the classical option pricing theories are usually built on the law of one price, while ignoring the impact of market liquidity on bid-ask spreads. The theory of conic finance replaces the law of one price by the law of two prices, allowing for market participants sell to the market at the bid price and buy from the market at the higher ask price. Within the framework of conic finance, we then derive the explicit formulas for the bid and ask prices of European call and put options by using WANG-transform as a distortion function. Moreover, numerical experiment is performed to illustrate the effects of the Hurst index and market liquidity level on bid and ask prices.


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    [1] H. Albrecher, F. Guillaume, W. Schoutens, Implied liquidity: model sensitivity, J. Empir. Financ., 23 (2013), 48–67. doi: 10.1016/j.jempfin.2013.05.003
    [2] F. Black, M. Scholes, The pricing of option and corporate liabilities, J. Polit. Econ., 81 (1973), 637–654. doi: 10.1086/260062
    [3] J. Barkoulas, A. Barilla, W. Wells, Long-memory exchange rate dynamics in the euro era, Chaos Solitons Fractals, 86 (2016), 92–100. doi: 10.1016/j.chaos.2016.02.007
    [4] L. Ballestra, G. Pacelli, D. Radi, A very efficient approach for pricing barrier options on an underlying described by the mixed fractional Brownian motion, Chaos Solitons Fractals, 87 (2016), 240–248. doi: 10.1016/j.chaos.2016.04.008
    [5] C. Bender, T. Sottinen, E. Valkeila, Pricing by hedging and no-arbitrage beyond semimartingales, Financ. Stoch., 12 (2008), 441–468. doi: 10.1007/s00780-008-0074-8
    [6] P. Cheridito, Mixed Fractional Brownian Motion, Bernoulli, 7 (2001), 913–934. doi: 10.2307/3318626
    [7] P. Cheridito, Arbitrage in fractional Brownian motion models, Financ. Stoch., 7 (2003), 533–553. doi: 10.1007/s007800300101
    [8] A. Cherny, D. Madan, New measures of performance evaluation, Rev. Financ. Stud., 22 (2009), 2571–2606. doi: 10.1093/rfs/hhn081
    [9] J. Corcuera, F. Guillaume, D. Madan, W. Schoutens, Implied liquidity: towards stochastic liquidity modelling and liquidity trading, Int. J. Portf. Anal. Manage., 1 (2012), 80–91.
    [10] C. El-Nouty, The fractional mixed fractional Brownian motion, Stat. Probab. Lett., 625 (2003), 111–120.
    [11] F. Guillaume, The LIX: A model-independent liquidity index, J. Bank Financ., 58 (2015), 214–231. doi: 10.1016/j.jbankfin.2015.04.015
    [12] F. Guillaume, G. Junike, P. Leoni, W. Schoutens, Implied liquidity risk premia in option markets, Ann. Finance, 15 (2019), 233–246. doi: 10.1007/s10436-018-0339-y
    [13] G. Junike, A. Arratia, A. Cabaña, W. Schoutens, American and exotic options in a market with frictions, Eur. J. Financ., 26 (2020), 179–199. doi: 10.1080/1351847X.2019.1599407
    [14] A. Lo, Long-Term Memory in Stock Market Prices, Econometrica, 59 (1991), 1279–1313. doi: 10.2307/2938368
    [15] M. Leippold, S. Schärer, Discrete-time option pricing with stochastic liquidity, J. Bank Financ., 75 (2017), 1–16. doi: 10.1016/j.jbankfin.2016.11.014
    [16] Z. Li, W. Zhang, Y. Zhang, Z. Yi, An analytical approximation approach for pricing European options in a two-price economy, N. Am. Econ. Financ., 50 (2019), 100986. doi: 10.1016/j.najef.2019.100986
    [17] D. Madan, Conserving capital by adjusting deltas for gamma in the presence of skewness, J. Risk Financial Manag., 3 (2010), 1–25. doi: 10.3390/jrfm3010001
    [18] Y. Mishura, Stochastic Calculus for Fractional Brownian Motions and Related Processes, Berlin: Springer Press, 2008.
    [19] D. Madan, A. Cherny, Markets as a counterparty: an introduction to conic finance, Int. J. Theor. Appl. Finance, 13 (2010), 1149–1177. doi: 10.1142/S0219024910006157
    [20] D. Madan, W. Schoutens, Conic Option Pricing, J. Deriv., 25 (2017), 10–36. doi: 10.3905/jod.2017.25.1.010
    [21] D. Madan, W. Schoutens, Applied Conic Finance, Cambridge: Cambridge University Press, 2016.
    [22] B. L. S. Prakasa Rao, Pricing geometric Asian power options under mixed fractional Brownian motion environment, Phys. A, 446 (2016), 92–99. doi: 10.1016/j.physa.2015.11.013
    [23] L. Rogers, Arbitrage with Fractional Brownian Motion, Math. Financ., 7 (1997), 95–105. doi: 10.1111/1467-9965.00025
    [24] L. Sun, Pricing currency options in the mixed fractional Brownian motion, Phys. A, 392 (2013), 3441–3458. doi: 10.1016/j.physa.2013.03.055
    [25] S. Sadique, P. Silvapulle, Long-term memory in stock market returns: international evidence, Int. J. Financ. Econ., 6 (2001), 59–67. doi: 10.1002/ijfe.143
    [26] M. Sonono, H. Mashele, Estimation of bid-ask prices for options on LIBOR based instruments, Financ. Res. Lett., 19 (2016), 33–41. doi: 10.1016/j.frl.2016.05.013
    [27] A. Sensoy, B. Tabak, Time-varying long term memory in the European Union stock markets, Phys. A, 436 (2015), 147–158. doi: 10.1016/j.physa.2015.05.034
    [28] A. Sensoy, B. Tabak, Dynamic efficiency of stock markets and exchange rates, Int. Rev. Financ. Anal., 47 (2016), 353–371. doi: 10.1016/j.irfa.2016.06.001
    [29] W. Xiao, W. Zhang, X. Zhang, Pricing model for equity warrants in a mixed fractional Brownian environment and its algorithm, Phys. A, 391 (2012), 6418–6431. doi: 10.1016/j.physa.2012.07.041
    [30] M. Zili, On the mixed fractional Brownian motion, J. Appl. Math. Stoch. Anal., 2006 (2006), 3245.
    [31] W. Zhang, Z. Li, Y. Liu, Analytical pricing of geometric Asian power options on an underlying driven by a mixed fractional Brownian motion, Phys. A, 490 (2018), 402–418. doi: 10.1016/j.physa.2017.08.070
    [32] W. Zhang, Z. Li, Y. Liu, Y. Zhang, Pricing European Option Under Fuzzy Mixed Fractional Brownian Motion Model with Jumps, Comput. Econ., (2020), 1–33.
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