M.Sc. Project Report Submitted by Navjot Parmar and Kuldip Singh Patel 2010MAS7142 2010MAS7114

under the supervision of Dr. Mani Mehra

Department of Mathematics Indian Institute of Technology Delhi April, 2012

Certiﬁcate

This is to certify that the dissertation entitled Comparison between numerical and analytical solution of Black Scholes partial diﬀerential equation which is being submitted by Navjot Parmar and Kuldip Singh Patel, for the award of the degree of Master of Science in Mathematics, to the Indian Institute of Technology, Delhi. It is a record of bonaﬁde work carried out by them under my sustained guidance and supervision.

Dr. Mani Mehra Department of Mathematics I.I.T. Delhi

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Acknowledgement

We would like to take this opportunity to express our sincere gratitude to Dr. Mani Mehra for her invaluable help, constant guidance, and moral support. We are highly obliged to her for constantly encouraging us by giving her critics on our work and sparing time to guide us. We are grateful to her for signiﬁcantly increasing our insight on the subject, by sharing her vast knowledge with us. Without her motivation and useful suggestions, this project would not have been possible.

Date : April 27, 2012

Navjot Parmar (2010MAS7114) Kuldip Singh Patel (2010MAS7142)

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Summary

In our project report we studied Black Scholes partial diﬀerential equation(PDE) and evaluated its analytical solution. Then we applied three well known ﬁnite diﬀerence methods such as Explicit Method, Implicit Method and Crank-Nicolson method for solving it numerically. We plot comparison and error graphs using MATLAB. Then we applied wavelet methods to the Black Scholes PDE to obtain its solution and then compare it with the analytical solution.

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List of Figures

3.1 4.1 db4 wavelet function . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Price of call options calculated with Black Scholes equation for X = 20, r = 0.10, δ = 0, σ = 0.60 . . . . . . . . . . . . . . . . . . . . . 36 4.2 Price of put options calculated with Black Scholes equation for X = 20, r = 0.10, δ = 0, σ = 0.60 . . . . . . . . . . . . . . . . . . . . . 36 4.3 Price of call options calculated with Black Scholes equation for X = 50, r = 0.10, δ = 0, σ = 0.40 . . . . . . . . . . . . . . . . . . . . . 37 4.4 Price of put options calculated with Black Scholes equation for X = 50, r = 0.10, δ = 0, σ = 0.40 . . . . . . . . . . . . . . . . . . . . . 37 4.5 Price of put options calculated with Explicit method for X = 50, r = 0.10, δ = 0, σ = 0.40 . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.6 Price of put options calculated with Crank-Nicolson method for X = 50, r = 0.10, δ = 0, σ = 0.40 . . . . . . . . . . . . . . . . . . . . . 38 4.7 Price of put options calculated with Implicit method for X = 50, r = 0.10, δ = 0, σ = 0.40 . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.8 4.9 5.1 5.2 5.3 Adaptive grid with terminal and initial conditions . . . . . . . . . . . 39 Crank-Nicolson stencil . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ∆t = ∆t = ∆t = 5 , 1200 5 , 2400 5 , 1200

N = 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 N = 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 N = 100, σ = 0.2 . . . . . . . . . . . . . . . . . . . . . . 45 5

5.4 5.5 5.6 5.7

∆t = ∆t =

5 , 1200 5 , 1200

N = 100, σ = 0.6 . . . . . . . . . . . . . . . . . . . . . . 45 N = 100, σ = 0.6 . . . . . . . . . . . . . . . . . . . . . . 46

Error analysis of Black Scholes equation by Crank-Nicolson Method . 46 Error analysis of Analytic solution by Wavelets Method . . . . . . . . 47

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Contents

1 Preliminaries 1.1 1.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9

Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.1 1.2.2...