Discrepancies between theoretical option pricing models and actual market
prices create arbitrage opportunities in financial markets. Despite being
widely used in option pricing, the famous Black-Scholes model estimates option
values based on the strict assumption of no arbitrage. In addition, its
assumptions of constant volatility and log-normal asset price distribution may
not fully capture real-world market dynamics, resulting in mispricing and
potential arbitrage opportunities. The Information-based model is adopted as an
alternative to address this, allowing for stochastic volatility, non-specific
asset price distributions, and variable transaction costs. This study extends
the IBM by developing a pricing equation incorporating weak arbitrage
possibilities using the weaker form of no-arbitrage termed as the Zero
Curvature condition. The equation incorporates an adjusted risk-free rate,
influenced by an arbitrage measure and option derivatives. Empirical findings
based on the iShares S&P 100 ETF American call options dataset demonstrate
that capturing weak arbitrage improves theoretical option price estimates,
reducing discrepancies and potential arbitrage opportunities. Further research
can focus on validating and enhancing the Information-based model using
alternative financial assets data.
References
[1]
Harrison, J.M. and Kreps, D.M. (1979) Martingales and Arbitrage in Multiperiod Securities Markets. Journal of Economic Theory, 20, 381-408. https://doi.org/10.1016/0022-0531(79)90043-7
[2]
Dalang, R.C., Morton, A. and Willinger, W. (1990) Equivalent Martingale Measures and No-Arbitrage in Stochastic Securities Market Models. Stochastics: An International Journal of Probability and Stochastic Processes, 29, 185-201. https://doi.org/10.1080/17442509008833613
[3]
Schachermayer, W. (1993) A Counterexample to Several Problems in the Theory of Asset Pricing. Mathematical Finance, 3, 217-229. https://doi.org/10.1111/j.1467-9965.1993.tb00089.x
[4]
Harrison, J.M. and Pliska, S.R. (1981) Martingales and Stochastic Integrals in the Theory of Continuous Trading. Stochastic Processes and Their Applications, 11, 215-260. https://doi.org/10.1016/0304-4149(81)90026-0
[5]
Delbaen, F. (1992) Representing Martingale Measures When Asset Prices Are Continuous and Bounded. Mathematical Finance, 2, 107-130. https://doi.org/10.1111/j.1467-9965.1992.tb00041.x
[6]
Kusuoka, S. (1993) A Remark on Arbitrage and Martingale Measure. Publications of the Research Institute for Mathematical Sciences, 29, 833-840. https://doi.org/10.2977/prims/1195166576
[7]
Delbaen, F. and Schachermayer, W. (1994) A General Version of the Fundamental Theorem of Asset Pricing. Mathematische Annalen, 300, 463-520. https://doi.org/10.1007/BF01450498
[8]
Ball, C.A. and Torous, W.N. (1983) Bond Price Dynamics and Options. Journal of Financial and Quantitative Analysis, 18, 517-531. https://doi.org/10.2307/2330945
[9]
Hodges, S. (1989) Optimal Replication of Contingent Claims under Transaction Costs. Review Futures Market, 8, 222-239.
[10]
Hull, J. and White, A. (1990) Valuing Derivative Securities Using the Explicit Finite Difference Method. Journal of Financial and Quantitative Analysis, 25, 87-100. https://doi.org/10.2307/2330889
[11]
Geman, H. and Yor, M. (1993) Bessel Processes, Asian Options, and Perpetuities. Mathematical Finance, 3, 349-375. https://doi.org/10.1111/j.1467-9965.1993.tb00092.x
[12]
Corrado, C.J. and Su, T. (1996) S&P 500 Index Option Tests of Jarrow and Rudd’s Approximate Option Valuation Formula. Journal of Futures Markets: Futures, Options, and Other Derivative Products, 16, 611-629. https://doi.org/10.1002/(SICI)1096-9934(199609)16:6<611::AID-FUT1>3.0.CO;2-I
[13]
Haug, E.G. and Taleb, N.N. (2011) Option Traders Use (Very) Sophisticated Heuristics, Never the Black-Scholes-Merton Formula. Journal of Economic Behavior & Organization, 77, 97-106. https://doi.org/10.1016/j.jebo.2010.09.013
[14]
Chakrabarti, B. and Santra, A. (2017) Comparison of Black Scholes and Heston Models for Pricing Index Options. Social Science Research Network, 10, 1-14. https://doi.org/10.2139/ssrn.2943608
[15]
Liu, G.F. and Xu, W.J. (2017) Application of Heston’s Model to the Chinese Stock Market. Emerging Markets Finance and Trade, 53, 1749-1763. https://doi.org/10.1080/1540496X.2016.1219849
[16]
Wattanatorn, W. and Sombultawee, K. (2021) The Stochastic Volatility Option Pricing Model: Evidence from a Highly Volatile Market. The Journal of Asian Finance, Economics and Business, 8, 685-695.
[17]
Zhang, Y.M. (2021) Dynamic Optimal Mean-Variance Investment with Mispricing in the Family of 4/2 Stochastic Volatility Models. Mathematics, 9, 2293-2298. https://doi.org/10.3390/math9182293
[18]
Fullwood, J., James, J. and Marsh, I.W. (2021) Volatility and the Cross-Section of Returns on FX Options. Journal of Financial Economics, 141, 1262-1284. https://doi.org/10.1016/j.jfineco.2021.04.030
[19]
Alfeus, M., He, X.-J. and Zhu, S.-P. (2022) An Empirical Analysis of Option Pricing with Short Sell Bans. International Journal of Theoretical and Applied Finance, 25, 2250012-2250014. https://doi.org/10.1142/S0219024922500121
[20]
Sood, S., Jain, T., Batra, N. and Taneja, H.C. (2023) Black-Scholes Option Pricing Using Machine Learning. Proceedings of International Conference on Data Science and Applications: ICDSA 2022, Volume 1, 481-493. https://doi.org/10.1007/978-981-19-6631-6_34
[21]
Adamchuk, A.N. and Esipov, S.E. (1997) Collectively Fluctuating Assets in the Presence of Arbitrage Opportunities, and Option Pricing. Physics-Uspekhi, 40, 1239-1240. https://doi.org/10.1070/PU1997v040n12ABEH000319
[22]
Ilinski, K. (1999) How to Account for Virtual Arbitrage in the Standard Derivative Pricing.
[23]
Ilinski, K. and Stepanenko, A. (1999) Derivative Pricing with Virtual Arbitrage.
[24]
Otto, M. (2000) Stochastic Relaxational Dynamics Applied to Finance: Towards Nonequilibrium Option Pricing Theory. The European Physical Journal B-Condensed Matter and Complex Systems, 14, 383-394. https://doi.org/10.1007/s100510050143
[25]
Otto, M., et al. (2000) Towards Non-Equilibrium Option Pricing Theory. International Journal of Theoretical and Applied Finance, 3, 565-566. https://doi.org/10.1142/S0219024900000607
[26]
Contreras, M., Montalva, R., Pellicer, R. and Villena, M. (2010) Dynamic Option Pricing with Endogenous Stochastic Arbitrage. Physica A: Statistical Mechanics and Its Applications, 389, 3552-3564. https://doi.org/10.1016/j.physa.2010.04.019
[27]
Fedotov, S. and Panayides, S. (2005) Stochastic Arbitrage Return and Its Implication for Option Pricing. Physica A: Statistical Mechanics and Its Applications, 345, 207-217. https://doi.org/10.1016/S0378-4371(04)00989-6
[28]
Cassese, G. (2005) A Note on Asset Bubbles in Continuous-Time. International Journal of Theoretical and Applied Finance, 8, 523-536. https://doi.org/10.1142/S0219024905003074
[29]
Christensen, M.M. and Larsen, K. (2007) No Arbitrage and the Growth Optimal Portfolio. Stochastic Analysis and Applications, 25, 255-280. https://doi.org/10.1080/07362990600870488
[30]
Kardaras, C. (2012) Market Viability via Absence of Arbitrage of the First Kind. Finance and Stochastics, 16, 651-667. https://doi.org/10.1007/s00780-012-0172-5
[31]
Ruf, J. (2013) Hedging under Arbitrage. Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics, 23, 297-317. https://doi.org/10.1111/j.1467-9965.2011.00502.x
[32]
Karatzas, I. and Kardaras, C. (2007) The Numeraire Portfolio in Semimartingale Financial Models. Finance and Stochastics, 11, 447-493. https://doi.org/10.1007/s00780-007-0047-3
[33]
Takaoka, K. (2010) On the Condition of No Unbounded Profit with Bounded Risk. Graduate School of Commerce and Management, Hitotsubashi University, Kunitachi, Tokyo.
[34]
Fontana, C. (2015) Weak and Strong No-Arbitrage Conditions for Continuous Financial Markets. International Journal of Theoretical and Applied Finance, 18, Article ID: 1550005. https://doi.org/10.1142/S0219024915500053
[35]
Vazquez, S.E. and Farinelli, S. (2009) Gauge Invariance, Geometry and Arbitrage.
[36]
Farinelli, S. (2015) Geometric Arbitrage Theory and Market Dynamics. Social Science Research Network, 1113292(92).
[37]
Farinelli, S. and Takada, H. (2022) The Black-Scholes Equation in the Presence of Arbitrage. Quantitative Finance, 22, 2155-2170. https://doi.org/10.1080/14697688.2022.2117075
[38]
Brody, D.C., Hughston, L.P. and Macrina, A. (2008) Information-Based Asset Pricing. International Journal of Theoretical and Applied Finance, 11, 107-142. https://doi.org/10.1142/S0219024908004749
[39]
Odin, M., Aduda, J.A. and Omari, C.O. (2022) Pricing Bermudan Option with Variable Transaction Costs under the Information-Based Model. Open Journal of Statistics, 12, 549-562. https://doi.org/10.4236/ojs.2022.125033
[40]
Bjork, T. (2009) Arbitrage Theory in Continuous Times. Oxford University Press, Oxford.
[41]
Macrina, A. (2022) An Information-Based Framework for Asset Pricing: X-Factor Theory and Its Applications. In: Brody, D.C., Hughston, L.P. and Macrina, A., Eds., Financial Informatics: An Information-Based Approach to Asset Pricing, Springer, New York, 231-257.
[42]
Odin, M., Aduda, J.A. and Omari, C.O. (2023) Numerical Approximation of Information-Based Model Equation for Bermudan Option with Variable Transaction Costs. Journal of Mathematical Finance, 13, 89-111. https://doi.org/10.4236/jmf.2023.131006
