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doi: 10.3934/dcdss.2022063
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## Expected vs. real transaction costs in European option pricing

 1 Università degli Studi di Bari "Aldo Moro", Department of Economics, Management and Business Law, Largo Abbazia Santa Scolastica 53, Bari, I-70124, Italy 2 Università degli Studi di Roma "La Sapienza", Department of Methods and Models for Economics, Territory and Finance, Via del Castro Laurenziano 9, Roma, I-00185, Italy

* Corresponding author: Michele Bufalo

In loving memory of Rosa Maria Mininni

Received  August 2021 Revised  January 2022 Early access March 2022

As an application and extension of some previous results contained in [1], we face up the problem of the option pricing in presence of transaction costs and hence in the framework of incomplete markets. The model proposed herein passes through defining properly the expected transaction costs, opposite to the real transaction costs in trading. The analysis is carried out both in the discrete and the continuous case and leads to suitable modifications of Cox-Ross-Rubinstein and Black-Scholes formulas. An application to a specific case referred to real market data at the end of the paper seems to validate our approach.

Citation: Antonio Attalienti, Michele Bufalo. Expected vs. real transaction costs in European option pricing. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022063
##### References:
 [1] A. Attalienti and M. Bufalo, Option pricing formulas under a change of numèraire, Opuscula Mathematica, 40 (2020), 451-473.  doi: 10.7494/OpMath.2020.40.4.451. [2] BIS, Is the Unthinkable Becoming Routine?, Technical Report, Bank for International Settlements, (2015). [3] F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81, (1973), 637–654. doi: 10.1086/260062. [4] M. Bufalo, R. M. Mininni and S. Romanelli, A semigroup approach to generalized Black-Scholes type equation in incomplete markets, Journal of Mathematical Analysis and Applications, 447, (2019), 1195–1223. doi: 10.1016/j.jmaa.2019.05.008. [5] J. C. Cox, S. A. Ross and M. Rubinstein, Option Pricing: A Simplified Approach, Journal of Financial Economics, 7, (1979), 229–263. doi: 10.1016/0304-405X(79)90015-1. [6] M. H. Davis, V. Panas and T. Zariphopoulou, European option pricing with transaction costs, SIAM Journal of Control and Optimization, 31, (1993), 470–493. doi: 10.1137/0331022. [7] K. C. Engelen, The unthinkable as the new normal, The International Economy, 29, (2015). [8] R. Engle, R. Ferstenberg and J. Russell, Measuring and modeling execution cost and risk, The Journal of Portfolio Management, 38 (2006), 14-28.  doi: 10.2139/ssrn.1211162. [9] H. Geman, N. El Karoui and J. C. Rochet, Changes of numéraire, change of probability measure and option pricing, Journal of Applied Probability, 32 (1995), 443-458.  doi: 10.2307/3215299. [10] J. A. Goldstein, R. M. Mininni and S. Romanelli, A new explicit formula for the solution of the Black-Merton-Scholes equation, Infinite Dimensional Stochastic Analysis, QP–PQ: Quantum Probab. White Noise Anal., World Sci. Publ., Hackensack, NJ, 22 (2008), 226-235.  doi: 10.1142/9789812779557_0013. [11] P. Guasoni, E. Lépinette and M. Rásonyi, The fundamental theorem of asset pricing under transaction costs, Finance and Stochastics, 16 (2012), 741-777.  doi: 10.1007/s00780-012-0185-0. [12] P. Guasoni, M. Rásonyi and W. Shachermayer, Consistent price systems and face-lifting pricing under transaction costs, The Annals of Applied Probability, 18, (2008), 491-520. doi: 10.1214/07-AAP461. [13] J. Kallsen and J. Muhle-Karbe, Option pricing and hedging with small transaction costs, Mathematical Finance, 25 (2015), 702-723.  doi: 10.1111/mafi.12035. [14] M. A. Kociński, On transaction costs in stock trading, Quantitative Methods in Economics, 18 (2017), 58-67. [15] H. E. Leland, Option pricing and replication with transaction costs, The Journal of Finance, 40 (1985), 1283-1301. [16] R. C. Merton, Theory of rational option pricing, Journal of Economy and Management Sciences, 4 (1973), 141-183.  doi: 10.2307/3003143. [17] M. Musiela and T. Zariphopoulou, An example of indifference price under exponential preferences, Finance and Stochastics, 8 (2014), 229-239.  doi: 10.1007/s00780-003-0112-5. [18] L. T. Nielsen, Understanding $N(d_1)$ and $N(d_2)$: Risk-adjusted probabilities in the Black-Scholes model, Revue Finance, 14 (1993), 95-106. [19] W. Shachermayer, Asymptotic Theory of Transaction Costs, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2017. doi: 10.4171/173. [20] S. E. Shreve, Stochastic Calculus for Finance I: The Binomial Asset Pricing Model, Springer-Verlag, New York, 2004. [21] S. E. Shreve, Stochastic Calculus for Finance. II. Continuous-Time Models, Springer-Verlag, New York, 2004. [22] A. E. Whalley and P. Wilmott, An asymptotic analysis of an optimal hedging model for option pricing with transaction costs, Mathematical Finance, 7 (1997), 307-324.  doi: 10.1111/1467-9965.00034.

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##### References:
 [1] A. Attalienti and M. Bufalo, Option pricing formulas under a change of numèraire, Opuscula Mathematica, 40 (2020), 451-473.  doi: 10.7494/OpMath.2020.40.4.451. [2] BIS, Is the Unthinkable Becoming Routine?, Technical Report, Bank for International Settlements, (2015). [3] F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81, (1973), 637–654. doi: 10.1086/260062. [4] M. Bufalo, R. M. Mininni and S. Romanelli, A semigroup approach to generalized Black-Scholes type equation in incomplete markets, Journal of Mathematical Analysis and Applications, 447, (2019), 1195–1223. doi: 10.1016/j.jmaa.2019.05.008. [5] J. C. Cox, S. A. Ross and M. Rubinstein, Option Pricing: A Simplified Approach, Journal of Financial Economics, 7, (1979), 229–263. doi: 10.1016/0304-405X(79)90015-1. [6] M. H. Davis, V. Panas and T. Zariphopoulou, European option pricing with transaction costs, SIAM Journal of Control and Optimization, 31, (1993), 470–493. doi: 10.1137/0331022. [7] K. C. Engelen, The unthinkable as the new normal, The International Economy, 29, (2015). [8] R. Engle, R. Ferstenberg and J. Russell, Measuring and modeling execution cost and risk, The Journal of Portfolio Management, 38 (2006), 14-28.  doi: 10.2139/ssrn.1211162. [9] H. Geman, N. El Karoui and J. C. Rochet, Changes of numéraire, change of probability measure and option pricing, Journal of Applied Probability, 32 (1995), 443-458.  doi: 10.2307/3215299. [10] J. A. Goldstein, R. M. Mininni and S. Romanelli, A new explicit formula for the solution of the Black-Merton-Scholes equation, Infinite Dimensional Stochastic Analysis, QP–PQ: Quantum Probab. White Noise Anal., World Sci. Publ., Hackensack, NJ, 22 (2008), 226-235.  doi: 10.1142/9789812779557_0013. [11] P. Guasoni, E. Lépinette and M. Rásonyi, The fundamental theorem of asset pricing under transaction costs, Finance and Stochastics, 16 (2012), 741-777.  doi: 10.1007/s00780-012-0185-0. [12] P. Guasoni, M. Rásonyi and W. Shachermayer, Consistent price systems and face-lifting pricing under transaction costs, The Annals of Applied Probability, 18, (2008), 491-520. doi: 10.1214/07-AAP461. [13] J. Kallsen and J. Muhle-Karbe, Option pricing and hedging with small transaction costs, Mathematical Finance, 25 (2015), 702-723.  doi: 10.1111/mafi.12035. [14] M. A. Kociński, On transaction costs in stock trading, Quantitative Methods in Economics, 18 (2017), 58-67. [15] H. E. Leland, Option pricing and replication with transaction costs, The Journal of Finance, 40 (1985), 1283-1301. [16] R. C. Merton, Theory of rational option pricing, Journal of Economy and Management Sciences, 4 (1973), 141-183.  doi: 10.2307/3003143. [17] M. Musiela and T. Zariphopoulou, An example of indifference price under exponential preferences, Finance and Stochastics, 8 (2014), 229-239.  doi: 10.1007/s00780-003-0112-5. [18] L. T. Nielsen, Understanding $N(d_1)$ and $N(d_2)$: Risk-adjusted probabilities in the Black-Scholes model, Revue Finance, 14 (1993), 95-106. [19] W. Shachermayer, Asymptotic Theory of Transaction Costs, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2017. doi: 10.4171/173. [20] S. E. Shreve, Stochastic Calculus for Finance I: The Binomial Asset Pricing Model, Springer-Verlag, New York, 2004. [21] S. E. Shreve, Stochastic Calculus for Finance. II. Continuous-Time Models, Springer-Verlag, New York, 2004. [22] A. E. Whalley and P. Wilmott, An asymptotic analysis of an optimal hedging model for option pricing with transaction costs, Mathematical Finance, 7 (1997), 307-324.  doi: 10.1111/1467-9965.00034.
Expected transaction cost coefficient $c_{\beta}$ for different risk-free rate $r$ and volatility $\sigma$
Ask and Bid Unicredit daily stock prices observed from 8 October 2018 to 4 October 2019
Real transaction cost coefficient $c_{\gamma}$ versus the expected transaction cost coefficient $c_{\beta}$
True call/put prices versus simulated call/put prices (with $c_{\gamma}$ and $c_{\beta}$, respectively)
Ask/Bid prices (€) of call and put options for different strike $X$ and volatility $\sigma$, with maturity $T = 1$ month, risk-free rate $r = -0.47\%$, dividend $\delta = 0.27$€ and (initial) underlying asset prices $S^{ask}_0 = 10.05$€ and $S^{bid}_0 = 10.26$, €
 Bid call prices Ask call prices Strike Bid put prices Ask put prices Volatility 2.571 3.071 7.4 0.0001 0.15 0.9500 2.376 2.876 7.6 0.0001 0.15 0.9330 2.185 2.685 7.8 0.0001 0.15 0.8670 2.002 2.502 8 0.0001 0.15 0.8050 1.865 2.165 8.2 0.0001 0.15 0.7450 1.6775 1.9775 8.4 0.044 0.0855 0.6850 1.582 1.7315 8.6 0.061 0.105 0.6260 1.4055 1.5525 8.8 0.09 0.1215 0.5740 1.2325 1.3805 9 0.1155 0.1505 0.5500 0.85 0.9745 9.5 0.2125 0.257 0.5280 0.553 0.604 10 0.379 0.43 0.5080 0.307 0.358 10.5 0.5275 0.808 0.4890 0.151 0.1945 11 0.935 1.063 0.4480 0.066 0.099 11.5 1.3355 1.4865 0.4150 0.0006 0.1465 12 1.7165 2.0165 0.3660 0.0001 0.15 12.5 2.085 2.585 0.3780 0.0001 0.15 13 2.591 3.091 0.3950 0.0001 0.15 13.5 3.0905 3.5905 0.4160 0.0001 0.15 14 3.5525 4.0525 0.4410 0.0001 0.15 14.5 4.09 4.59 0.4700 0.0001 0.15 15 4.558 5.058 0.5020 0.0001 0.15 15.5 5.00 6.00 0.5380 0.0001 0.15 16 5.288 6.288 0.5730
 Bid call prices Ask call prices Strike Bid put prices Ask put prices Volatility 2.571 3.071 7.4 0.0001 0.15 0.9500 2.376 2.876 7.6 0.0001 0.15 0.9330 2.185 2.685 7.8 0.0001 0.15 0.8670 2.002 2.502 8 0.0001 0.15 0.8050 1.865 2.165 8.2 0.0001 0.15 0.7450 1.6775 1.9775 8.4 0.044 0.0855 0.6850 1.582 1.7315 8.6 0.061 0.105 0.6260 1.4055 1.5525 8.8 0.09 0.1215 0.5740 1.2325 1.3805 9 0.1155 0.1505 0.5500 0.85 0.9745 9.5 0.2125 0.257 0.5280 0.553 0.604 10 0.379 0.43 0.5080 0.307 0.358 10.5 0.5275 0.808 0.4890 0.151 0.1945 11 0.935 1.063 0.4480 0.066 0.099 11.5 1.3355 1.4865 0.4150 0.0006 0.1465 12 1.7165 2.0165 0.3660 0.0001 0.15 12.5 2.085 2.585 0.3780 0.0001 0.15 13 2.591 3.091 0.3950 0.0001 0.15 13.5 3.0905 3.5905 0.4160 0.0001 0.15 14 3.5525 4.0525 0.4410 0.0001 0.15 14.5 4.09 4.59 0.4700 0.0001 0.15 15 4.558 5.058 0.5020 0.0001 0.15 15.5 5.00 6.00 0.5380 0.0001 0.15 16 5.288 6.288 0.5730
RMSE between the true ask/bid call and put prices and simulated values
 our model Leland model ask call price 12.52% 18.47% bid call price 10.38% 13.40% ask put price 20.46% 22.10% bid put price 19.85% 22.23%
 our model Leland model ask call price 12.52% 18.47% bid call price 10.38% 13.40% ask put price 20.46% 22.10% bid put price 19.85% 22.23%
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