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

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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.

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References

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

Capiński M., Kopp E. (2012), The Black–Scholes Model, Mastering Mathematical Finance, Cambridge University Press, Cambridge.
Google Scholar 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.
Google Scholar DOI: https://doi.org/10.1007/s00780-006-0020-6

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

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.
Google Scholar 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.
Google Scholar

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.
Google Scholar 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.
Google Scholar

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.
Google Scholar

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

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.
Google Scholar DOI: https://doi.org/10.1111/1467-9965.00080

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

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.
Google Scholar DOI: https://doi.org/10.1002/fut.2101

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

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

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.
Google Scholar 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.
Google Scholar

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

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

Rachev S.T., Ruschendorff L. (1994), Models for option process, “Theory of Probability Applications”, vol. 39, no. 1, pp. 120–152.
Google Scholar 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.
Google Scholar 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.
Google Scholar 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.
Google Scholar 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.
Google Scholar 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.
Google Scholar 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.
Google Scholar 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.
Google Scholar DOI: https://doi.org/10.1016/j.amc.2010.03.111

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2023-07-21

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How to Cite

Fraszka-Sobczyk, E. (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

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