A LOGISTIC NONLINEAR BLACK-SCHOLES-MERTON PARTIAL DIFFERENTIAL EQUATION: EUROPEAN OPTION
DOI:
https://doi.org/10.29121/granthaalayah.v6.i6.2018.1393Keywords:
Non-Linear, Black Scholes, Brownian Motion, Logistic Brownian Motion, Illiquid MarketsAbstract [English]
Nonlinear Black-Scholes equations provide more accurate values by taking into account more realistic assumptions, such as transaction costs, illiquid markets, risks from an unprotected portfolio or large investor's preferences, which may have an impact on the stock price, the volatility, the drift and the option price itself. Most modern models are represented by nonlinear variations of the well-known Black-Scholes Equation. On the other hand, asset security prices may naturally not shoot up indefinitely (exponentially) leading to the use of Verhulst’s Logistic equation. The objective of this study was to derive a Logistic Nonlinear Black Scholes Merton Partial Differential equation by incorporating the Logistic geometric Brownian motion. The methodology involves, analysis of the geometric Brownian motion, review of logistic models, process and lemma, stochastic volatility models and the derivation of the linear and nonlinear Black-Scholes-Merton partial differential equation. Illiquid markets have also been analyzed alongside stochastic differential equations.
The result of this study may enhance reliable decision making based on a rational prediction of the future asset prices given that in reality the stock market may depict a nonlinear pattern.
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References
Akira, T. (1996). Mathematical Economics.2nd Edition; Cambridge university press.
Bachilier L. (1900). Th´eori´e de sp´eculation Annales scientific de L”E”cole norm sup 111 . DOI: https://doi.org/10.24033/asens.476
Bank P. and Baum D. (2002). Hedging and Portfolio Optimization in Illiquid Financial Markets.; Humboldt University Barlin.
Barles G. and Soner H.M. (1998). Option pricing with transaction costs and Nolinear Black- Scholes equation Finance and Stochastics Vol. 2.
Baum D. (2001). Realisierbarer Portifoliowert in illiquiden FinanzmA¨arkten; PhD thesis Department of Mathematics, Humboldt; UniversitA¨at Berlin.
Beaumont H.P. (2004). Financial Engineering Principles.A unified theory for Financial Product analysis and Valuation; John Wiley & Sons, Inc, Hoboken, New Jersey.
Black F. and Scholes M. (1973). The pricing of options and cor- porate liabilities; Journal for Political Economics, Vol. 81.
Buchanan J. R. (2006). An Undergraduate Introduction To Finan- cial Mathematics; World Scientific Publishing Co. Pte. Ltd.
Cox J.C.and Ross A. S. (1976). The Valuation of Options for alternative Stochastic Processes; Journal of financial Economics, Vol.3.
Doina C. and Jacques-Louis l. (2002). Nonlinear Partial Differ- ential Equations And Their Applications; Colle´ge de France Seminar Volume xiv; Elsevier Science B.V.
Ehrhardt M. (2008). Nonlinear Models in Mathematical Finance; Nova Science Publishers.
Frey R. (2000). Market Illiquidity as a source of Model Risk in Dynamic Hedging;Swiss Banking Institute, University of Zu¨rich ; Switzerland.
Frey R. and Pierre P. (2002). Risk Management of Derivatives under Illiquid Markets; Advances in Finance and Stochastics:“Essay in Honour of Dieter Sondermann”; Springer. DOI: https://doi.org/10.1007/978-3-662-04790-3_8
Frey R. and Stremme A. (1997). Market Volatility and Feedback effects from Dynamic Hedging; Mathematical Finance, Vol. 7.
Hodges S.D. and Neuberger A. (1989). Optimal replication of contigent claims under transaction costs; Review on future Markets, Vol. 8
Hull J. and White A. (1987). The pricing of options on assets with stochastic volatility Journal of finance, Vol. 42.
Hull C.J. (2000). Option futures and other derivatives 4th edition, prentice; Hall international.
Ito, K. (1944). Stochastic Integrals, Proceedings of the Imperial Academy of Tokyo, Vol. 20.
Jarrow R. (1992). Market manipulation, bubbles, corners and short squeezes; Financial and Qualitative analysis, Vol. 27.
Jarrow R. (1994). Derivatives Securities Markets, Market Manipu- lation and Option Pricing Theory;Journal of Financial and Quanti- tative analysis, Vol. 29.
Jarrow R. and Turnbull S. (1995). Pricing Derivatives on Finan- cial securities subject to Default Risk; Journal of finanance, Vol. 50.
Karuppiah, J. Los, C.A. (2005). Wavelet multiresolution analysis of high-frequency Asian FX rates; International Review of Financial Analysis, Vol. 14.
King, A. C., Billingham, J. and Otto, S. R. (2003). Differential Equations-Linear, Nonlinear, Ordinary, Partial; Cambridge Univer- sity Press. DOI: https://doi.org/10.1017/CBO9780511755293
Leland H. (1985). Option pricing and replication with transaction costs; The Journal of Finance, Vol. 40.
Leland H. and Gennote G. (1990). Market liquidity, hedging and
Crashes; American Economic Review, Vol. 80.
Liu H. and Yong J. (2005). Option Pricing With an Illiquid Un- derlying asseet Market;Journal of Economic Dynamic Control 29; Elsevier B.V.
Lo A. and MacKinlay C. (1997). The Econometrics of FinancialMarkets; Princeton University Press, Princeton New Jersey.
Lo A. and MacKinlay C. (1999). A Non-Random Walk Down WallStreet; Princeton University Press, Princeton New Jersey.
Lungu, E and B. Øksendal (1997). Optimal Harvesting from a Population in a Stochastic Crowded Environment; Mathematical Biosciences, Vol. 145.
Mandelbrot B.B. (1963). The Variation of Certain Speculative Prices; Journal of Business, Vol. 36.
Mandelbrot B.B. (1967). The Variation of Some Other Speculative Prices; Journal of Business, Vol. 40.
Martin, B. and Andrew R. (2000). An introduction to derivative pricing; Cambridge university press.
Merton R. (1973). The theory of rational option pricing; The Bell of Economics and management science, Vol. 4.
Muller E., Vijay M. and Frank M.B. (1990). New Product Diffus- sion Models in Marketing: Areview and Direction for reserch; Jour- nal of Marketing, Vol. 54.
Muhannad R. N. and Aurie´lie T. (2007). The Log-logistic Option Pricing Model; Graduate Student Research Paper; Lehigh Univer- sity.
Nyakinda J.O. (2007). Derivation of the Logistic Black Scholes- Merton Partial Differential Equation-A case of Stochastic Volatility; A Masters Thesis in Applied Mathematics, Maseno University.
Onyango S(2003) . Extracting stochastic process from market price data:A pattern recognition approach; PhD Thesis University of Hud- derfield.U.K.
Onyango, S. (2005). On the linear stochastic price adjustment of securities; The East African Journal of statistics, Jomo Kenyatta University press.
Ornstein L.S. and Uhlensbeck G. E (1930). On the theory of theBrownian Motion; The American Physical Society, Rev. 36.
Paul B, Robert L. D. and Glen R. H. (1996). Differential equa- tions; Brooks / Cole publishing company.
Platen E. and Schweizer M. (1998). On feedback effects from hedg- ing derivatives; Mathematical Finance, Vol. 8.
Polyanin, A. D. and Manzhirov A.V. (2007). Handbook of Math- ematics for Engineers and Scientists; Chapman & Hall. CRC.
Polyanin, A. D. and Manzhirov A.V. (2008). Handbook of ItegralMathematics; Chapman & Hall. CRC.
Ray R. with Dan P. (1992). Introduction to Differential equations; Jones and Barlett publisher; Boston.
Samuelson, P. A. (1941). The Stability of Equilibrium: Compara- tive Statics and dynamics; Econometrica, Vol. 9. [46] Samuelson, P. A. (1965). Proof that property anticipated prices fluc- tuate randomly, Industrial Management, Rev. VI.
Savit R. (1988). When random is not random:An introduction to chaos in market prices; The Journal of future markets, Vol. 8.
Savit R. (1989). Nonlinearities and chaotic effects in option prices; The Journal of future markets, Vol. 8.
Sergio M. F.& Fabozzi F. J. (2004). The Mathematics of Financial Modeling and Investment Management; John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada.
Sheldon M. R (1999). An introduction to mathematics of finance; University of Carlifonia Berkeley, Cambridge university press.
Sicar R. and Papanicolaou (1998). General Black Scholes Ac- counting for Increased Market Volatility from Hedging Strategies; Ap- plied mathematical finance, Vol. 5.
Van Djik D. and Franses P. (2000). Non-linear time series model in emperical finance; Cambridge University Press, Cambridge, United Kingdom. DOI: https://doi.org/10.1017/CBO9780511754067
Verhulst, P.F (1838). Notice sur la loi que la population suit dans son accroissement, Correspondence Mathmatique et physique, Vol.10.
Whalley E. A. (1998). Option pricing with transaction costs; A PhDthesis, University of Oxford.
Whalley E. A. and Wilmott P. (1997). An asymptotic analysis of an optimal hedging model for option pricing with transaction costs; Mathematical Finance, Vol. 7.
Wilmott P, Howlson S and Dewayne J. (1995). The Mathematics of Financial Derivatives; Pass Syndicate of The University of Cambridge. DOI: https://doi.org/10.1017/CBO9780511812545
Wilmott P. (1998). Derivatives-The Theory and Practice of Financial Engineering; John Wiley and Sons Ltd, Baffin Lane. Chichester, West Sussex 1019 IUD. England.
Wilmott P. (2006). Paul Wilmott on Quantitative Finance; John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England. (2005) found that…” OR “In a similar study, Jones and Smith (1999) found that…”.
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