Olkhov, Victor (2020): Classical Option Pricing and Some Steps Further.
There is a more recent version of this item available. 

PDF
MPRA_paper_99918.pdf Download (150kB)  Preview 
Abstract
This paper modifies single assumption in the base of classical option pricing model and derives further extensions for the BlackScholesMerton equation. We regard the price as the ratio of the cost and the volume of market transaction and apply classical assumptions on stochastic Brownian motion not to the price but to the cost and the volume. This simple replacement leads to 2dimensional BSMlike equation with two constant volatilities. We argue that decisions on the cost and the volume of market transactions are made under agents expectations. Random perturbations of expectations impact the market transactions and through them induce stochastic behavior of the underlying price. We derive BSMlike equation driven by Brownian motion of agents expectations. Agents expectations can be based on option trading data. We show how such expectations can lead to nonlinear BSMlike equations. Further we show that the Heston stochastic volatility option pricing model can be applied to our approximations and as example derive 3dimensional BSMlike equation that describes option pricing with stochastic cost volatility and constant volume volatility. Diversity of BSMlike equations with 2 – 5 or more dimensions emphasizes complexity of option pricing problem. Such variety states the problem of reasonable balance between the accuracy of asset and option price description and the complexity of the equations under consideration. We hope that some of BSMlike equations derived in this paper may be useful for further development of assets and option market modeling.
Item Type:  MPRA Paper 

Original Title:  Classical Option Pricing and Some Steps Further 
English Title:  Classical Option Pricing and Some Steps Further 
Language:  English 
Keywords:  Option Pricing; BlackScholesMerton Equations; Stochastic Volatility; Market Transactions; Expectations; Nonlinear equations 
Subjects:  G  Financial Economics > G1  General Financial Markets G  Financial Economics > G1  General Financial Markets > G12  Asset Pricing ; Trading Volume ; Bond Interest Rates G  Financial Economics > G1  General Financial Markets > G17  Financial Forecasting and Simulation 
Item ID:  99918 
Depositing User:  Victor Olkhov 
Date Deposited:  29 Apr 2020 07:26 
Last Modified:  29 Apr 2020 07:26 
References:  Ball, C.A., Roma, A., (1994). Stochastic Volatility Option Pricing, The Journal of Financial and Quantitative Analysis, 29 (4), 589607 Bates, D.S., (1996). Testing Option Pricing Models, G.S. Maddale and C.R. Rao, (ed), Handbook of Statistics: Statistical Methods in Finance, 14, 567611 Bensaid, B., Lesne, J., Pages, H., and Scheinkman, J., (1992). Derivative asset pricing with transaction costs. Math. Finance, 2, 63–86 Black, F., Scholes, M., (1973). The Pricing of Options and Corporate Liabilities, The Journal of Political Economy, 81, 63765 Black, F., Derman, E., Toy, W., (1990). A OneFactor Model of Interest Rates and Its Application to Treasury Bond Options, Financial Analysts Journal; 46 (1), 3339 Blume, L. E., and Easley, D., (1984). Rational Expectations Equilibrium: An Alternative Approach. Journal of Economic Theory 34, 116–29 Borland, L., (2004). A Theory of NonGaussian Option Pricing, arXiv:condmat/0205078 v3 BrittenJones, M., Neuberger, A., (2000). Option Prices, Implied Price Processes, and Stochastic Volatility, The Journal of Finance, 55 (2), 839866 Broadie, M., Detemple, J., (1997). The Valuation Of American Options On Multiple Assets, Mathematical Finance, 7 (3), 241–286 Carmona, R., and Durrleman, V., (2006). Generalizing the BlackScholes formula to multivariate contingent claims. Journal of Computational Finance, 9(2), 43 Choi, J., (2018). Sum Of All BlackScholesMerton Models: An Efficient Pricing Method For Spread, Basket, And Asian Options, arXiv:1805.03172v1 Cohen, S.N. Tegner, M., (2018). European Option Pricing with Stochastic Volatility models under Parameter, Uncertainty, arXiv:1807.03882v1 Cox, J.C., Ross, S.A., (1976). The Valuation Of Options For Alternative Stochastic Processes, Journal of Financial Economics, 3, 145166 Engle, R., Figlewski, S., (2014). Modeling the Dynamics of Correlations among Implied Volatilities. Review of Finance 19(3), 9911018 Figlewski, S., (1998). Derivatives Risks, Old and New, BrookingsWharton Papers on Financial Services, I (1), 159217 Frey, R., (2008). Pricing and Hedging of Credit Derivatives via Nonlinear Filtering, Dep. Math., Univ. Leipzig Frey,R., Polte, U., (2011). Nonlinear Black–Scholes Equations In Finance: Associated Control Problems And Properties Of Solutions, Siam J. Control Optim., 49 (1), 185–204 Greenwood, R., Shleifer, A., (2014). Expectations of Returns and Expected Returns, The Review of Financial Studies 27, 714–46 Hansen, L.P., Sargent, T.J., (1979). Formulating and Estimating Dynamic Rational Expectations Models, Cambridge, NBER Heston, S.L., (1993). A ClosedForm Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, The Review of Financial Studies, 6 (2), 327343 Hull, J., White, A., (1987). The Pricing Of Options On Assets With Stochastic Volatilities, The Journal Of Finance, 42 (2), 281300 Hull, J.C., (2009). Options, Futures and other Derivatives, 7th.ed. Englewood Cliffs, NJ: PrenticeHall Hull J. C., White, A., (2001), The General HullWhite Model and Super Calibration, Financial Analysts Journal, 57 (6), 3443 Keynes, J.M., (1936). The General Theory of Employment, Interest and Money. London: Macmillan Cambridge Univ. Press Kleinert, H., Korbel, J., (2016). Option Pricing Beyond BlackScholes Based on DoubleFractional Diffusion, arXiv:1503.05655v2 Kydland, F., Prescott, E.C., (1980). A Competitive Theory of Fluctuations and the Feasibility and Desirability of Stabilization Policy. In Rational Expectations and Economic Policy. Edited by S. Fisher. Cambridge, NBER, pp. 169–98 Li, M., Deng, S., Zhou, J., (2010). Multiasset Spread Option Pricing and Hedging, Quantitative Finance, 10 (3), 305–324 Loeper, G., (2018). Option Pricing With Linear Market Impact and Nonlinear Black–Scholes Equations, The Annals of Applied Probability, 28 (5), 2664–2726 Lucas, R.E., (1972). Expectations and the Neutrality of Money. Journal of Economic Theory 4, 103–24 Manski, C., (2017). Survey Measurement of Probabilistic Macroeconomic Expectations: Progress and Promise. Cambridge, NBER Merton, R., (1973). Theory of Rational Option Pricing, The Bell Journal of Economic and management Sci, 4, 141183 Merton, R. C., (1997). Applications of optionpricing theory: twenty years later, Nobel Lecture Muth, J.F., (1961). Rational Expectations and the Theory of Price Movements. Econometrica, 29, 315–35 Olkhov, V., (2016a). On Economic space Notion, International Review of Financial Analysis, 47, 372381, DOI10.1016/j.irfa.2016.01.001 Olkhov, V., (2016b). Finance, Risk and Economic space, ACRN Oxford Journal of Finance and Risk Perspectives, Special Issue of Finance Risk and Accounting Perspectives, 5, 209221 Olkhov, V., (2016c). On Hidden Problems of Option Pricing, https://ssrn.com/abstract=2788108 Olkhov, V., (2019). Financial Variables, Market Transactions, and Expectations as Functions of Risk. Int. J. Financial Stud., 7, 66; 127. doi:10.3390/ijfs7040066 Poon, S.H., (2005). A Practical Guide to Forecasting Financial Market Volatility, J.Wiley & Sons Ltd, The Atrium, England Rapuch, G., Roncalli, T., (2004). Some remarks on twoasset options pricing and stochastic dependence of asset prices. Journal of Computational Finance 7(4), 2333 Saikat, N., (1996). Pricing and hedging index options under stochastic volatility: an empirical examination, WP 969, Federal Reserve Bank of Atlanta, Atlanta, GA Sargent, T.J., Wallace, N., (1976). Rational Expectations And The Theory Of Economic Policy. Journal of Monetary Economics, 2, 169183 Scholes, M.S., (1997). Derivatives in Dynamic Environment, Nobel Lecture Sircar, R., Papanicolaou, G., (1998). General BlackScholes models accounting for increased market volatility from hedging strategies. Appl. Math. Finance, 5,45–82 Shiryaev, A.N., (1999). Essentials of Stochastic Finance: Facts, Models, Theory. World Sci 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/99918 
Available Versions of this Item
 Classical Option Pricing and Some Steps Further. (deposited 29 Apr 2020 07:26) [Currently Displayed]