This is an outdated version published on 2023-07-21. Read the most recent version.

Limiting Cases of the Black-Scholes Type Asymptotics of Call Option Pricing in the Generalised CRR Model

Authors

Emilia Fraszka-Sobczyk University of Lodz, Faculty of Economics and Sociology, Department of Theory and Analysis of Economic Systems, Lodz, Poland https://orcid.org/0000-0001-9736-4406

DOI:

https://doi.org/10.18778/0208-6018.363.01

Keywords:

Cox‑Ross‑Rubinstein model (CRR model), binomial model, Black‑Scholes formula, option pricing

Abstract

The article concerns the generalised Cox‑Ross‑Rubinstein (CRR) option pricing model with new formulas for changes in upper and lower stock prices. The formula for option pricing in this model, which is the Black‑Scholes type formula, and its asymptotics are presented. The aim of the paper is to analyse limiting cases of the obtained asymptotics using probability theory and later data from the Warsaw Stock Exchange. Empirical analyses of option pricing in the generalised CRR model confirm the calculated limits.

Downloads

Download data is not yet available.

References

Black F., Scholes M. (1973), The pricing of options and corporate liabilities, “Journal of Political Economy”, vol. 81, pp. 637–654. DOI: https://doi.org/10.1086/260062

Capiński M., Kopp E. (2012), The Black–Scholes Model, Mastering Mathematical Finance, Cambridge University Press, Cambridge. DOI: https://doi.org/10.1017/CBO9781139026130

Chang L.B., Palmer K. (2007), Smooth convergence in the binomial model, “Finance and Stochastics”, vol. 11, no. 1, pp. 91–105. DOI: https://doi.org/10.1007/s00780-006-0020-6

Cox J.C., Rubinstein M. (1985), Options Markets, Prentice-Hall, New Jersey.

Cox J.C., Ross S.A., Rubinstein M. (1979), Option Pricing. A Simplified Approach, “Journal of Financial Economics”, vol. 7, no. 3, pp. 229–263. DOI: https://doi.org/10.1016/0304-405X(79)90015-1

Dana R.A., Jeanblanc M. (2007), Financial Markets in Continuous Time, Springer-Verlag, Berlin.

Diener F., Diener M. (2004), Asymptotics of the price oscillations of a European call option, “Journal of Mathematical Finance”, vol. 14, no. 2, pp. 271–293. DOI: https://doi.org/10.1111/j.0960-1627.2004.00192.x

Elliot R.J., Kopp P.E. (2005), Mathematics of Financial Markets, Springer-Verlag, New York.

Fraszka-Sobczyk E. (2014), On some generalization of the Cox-Ross-Rubinstein model and its asymptotics of Black-Scholes type, “Bulletin de la Société des Sciences et des Lettres de Łódź”, vol. LXIV, no. 1, pp. 25–34.

Fraszka-Sobczyk E. (2020), Wycena europejskich opcji kupna w modelach rynku z czasem dyskretnym. Uogólnienia formuły Blacka-Scholesa, Wydawnictwo Uniwersytetu Łódzkiego, Łódź.

Heston S., Zhou G. (2000), On the rate of convergence of discrete-time contingent claims, “Journal of Mathematical Finance”, vol. 10, no. 1, pp. 53–75. DOI: https://doi.org/10.1111/1467-9965.00080

Hull J. (1998), Kontrakty terminowe i opcje, Wydawnictwo WIG-Press, Warszawa.

Jabbour G., Kramin M., Young S. (2001), Two-state option pricing. Binomial model revisited, “Journal of Futures Markets”, vol. 21, no. 11, pp. 987–1001. DOI: https://doi.org/10.1002/fut.2101

Jakubowski J. (2006), Modelowanie rynków finansowych, Wydawnictwo SCRIPT, Warszawa.

Jakubowski J., Palczewski A., Rutkowski M., Stettner Ł. (2006), Matematyka finansowa. Instrumenty pochodne, Wydawnictwa Naukowo-Techniczne, Warszawa.

Joshi M. (2010), Achieving higher order convergence for the process of European options in binomial trees, “Mathematical Finance”, vol. 20, no. 1, pp. 89–103. DOI: https://doi.org/10.1111/j.1467-9965.2009.00390.x

Karandikar R.L., Rachev S.T. (1995), A generalized binomial model and option pricing formulae for subordinated stock-price processes, “Probability and Mathematical Statistics”, vol. 15, pp. 427–447.

Leisen D., Reimer M. (2006), Binomial models for option valuation – examining and improving convergence, “Applied Mathematical Finance”, vol. 3, no. 4, pp. 319–346. DOI: https://doi.org/10.1080/13504869600000015

Musiela M., Rutkowski M. (2008), Martingale Methods in Financial Modelling, Springer-Verlag, Berlin.

Rachev S.T., Ruschendorff L. (1994), Models for option process, “Theory of Probability Applications”, vol. 39, no. 1, pp. 120–152. DOI: https://doi.org/10.1137/1139005

Ratibenyakool Y., Neammanee K. (2019), Rate of convergence of binomial formula for option pricing, “Communications in Statistics – Theory and Methods”, vol. 3, no. 4, pp. 3537–3556. DOI: https://doi.org/10.1080/03610926.2019.1590600

Rendleman R., Bartter B. (1979), Two-State option pricing, “The Journal of Finance”, vol. 34, no. 4, pp. 1092–1110. DOI: https://doi.org/10.1111/j.1540-6261.1979.tb00058.x

Rubinstein M. (2000), On the relation between binomial and trinomial option pricing models, “The Journal of Derivatives”, vol. 8, no. 2, pp. 47–50. DOI: https://doi.org/10.3905/jod.2000.319149

Shreve S.E. (2004), Stochastic Calculus for Finance I . The Binomial Asset Pricing Model, Springer-Verlag, New York. DOI: https://doi.org/10.1007/978-0-387-22527-2

Stettner Ł. (1997), Option pricing in the CRR model with proportional transaction costs. A cone transformation approach, “Applicationes Mathematicae”, vol. 24, no. 4, pp. 475–514. DOI: https://doi.org/10.4064/am-24-4-475-514

Walsh J.B. (2003), The rate of convergence of the binomial tree scheme, “The Journal of Finance and Stochastics”, vol. 7, no. 3, pp. 337–361. DOI: https://doi.org/10.1007/s007800200094

Xiao X . (2010), Improving speed of convergence for the prices of European options in binomial trees with even numbers of steps, “Applied Mathematics and Computation”, vol. 216, no. 1, pp. 2659–2670. DOI: https://doi.org/10.1016/j.amc.2010.03.111

Downloads

Published

2023-07-21

Versions

Issue

Vol. 2 No. 363 (2023)

Section

Articles

License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Fraszka-Sobczyk, Emilia. 2023. “Limiting Cases of the Black-Scholes Type Asymptotics of Call Option Pricing in the Generalised CRR Model”. Acta Universitatis Lodziensis. Folia Oeconomica 2 (363): 1-24. https://doi.org/10.18778/0208-6018.363.01.