Natural Science

Analytical Solution to the Black-Scholes Equation

Masters Degree

The study involves the analytical methods to the Black-Scholes partial differential equation (PDE) for option pricing. The various methods examined have certain limitations and drawbacks that affect the accuracy of results obtained at certain points in the life of the option. Two numerical techniques are adopted to improve on the accuracy – the Method of Lines, and the Method of Separation of Variables, were applied to a modified version of the PDE (inhomogeneous form).


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In an expedition for a robust model for option pricing, the 1997 Nobel-prize winners, Fischer Black, Myron Scholes and Merton formulated a partial differential equation (PDE).

The aim of the project is to price the European call options, using analytical methods to Black-Scholes, and comparing the results gotten from the methods of lines, and the method of separation of variables.

The Black-Scholes model and the Cox, Ross and Rubinstein Binomial models are both based on the same theoretical foundations and assumptions (such as the geometric Brownian motion theory of stock price behaviour and risk-neutral valuation). To whatever degree, there are some important differences between the two: the Cox binomial option pricing model uses an iterative procedure, whereby permitting for the detailed description of modes, or points in time, during the time span between the valuation data and the options expiration data.

The Cox model diminishes likelihoods of changes in prices, eliminates the chance for arbitrage, presumes a perfectly efficient market and bridges the period or duration of the option. Further down this propagation, it has the proficiency of presenting a mathematical valuation of the option at every situation in the stated time.

However, it takes a risk- neutral approach to valuation, meanwhile the assumption here is that the underlying security prices can only either increase or decrease with time until the option expires worthless.

Certain exceptional rewards anticipated to be as a result of its straightforward and trouble free iterative structure attach the model and that is one of its advantages. In illustration, it is useful however, for valuing securities vig, American options which gives the owner freedom of exercising the option contract through the lifespan until expiration date. This is possible since it makes provisions for a stream of valuations for a security at every mode within time span. On the contrary, it is not so with the European options which can be exercised only at expiration.

Comparing with its contending rival – the Black-Scholes model, we can readily deduce or ascertain that the model is easier in assessment and for this reason is comfortably uncomplicated in numerical computation and implementation using spreadsheets.

The Black- Scholes model is used to calculate a theoretical call price (ignoring dividends paid during the life of the option) using the five determinants of an option’s price: stock price, strike price, volatility, time to expiration, and short-term (risk free) interest rate.

Apart from the fact that the Black-Scholes option- pricing theory provided a new way to value stock options, it also commenced a revolution on how hedge funds and other market participants think about and value financial assets.



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Title Page


Table of contents


1.1 Introduction

1.2 Options (the main types)

1.3 Black-Scholes formula and parameters

1.4 Derivation of black-Scholes equation

1.5 Expected return and the black-Scholes formula



2.1 Black-Scholes Model

2.2 Analytical solutions by black-Scholes (Black f 1989)

2.3 The pseudo spectral (Ps) Chebyshev method of solution to black-Scholes equation (Bunnin F.O., Ren Y and Darlington, 1999)

2.3.1 Application of the pseudo spectral method to the black-Scholes equation

2.4 Put-Call parity




3.1 Introduction

3.2 Method of Lines (MOL)

3.3 Application of MOL to the Black-Scholes Model

3.4 Method of Separation of Variables

3.5 Application of Method of Separation of Variables to Black-Scholes Model

3.5.1 Results

3.5.2 Results Using Black-Scholes’ Analytical and Pseudospectral        Chebyshev’s Methods of Solution

3.5.3 Results using Method of Lines

3.5.4 Results Using Method of Separation of Variables



4.1 Findings

4.2 Conclusion

4.3 Recommendation






Table 1: Coefficients of option values at various modes and corresponding values of Bi

Table 2: Option prices using Black-Scholes analytical formula (calculator) and the Pseudo spectral Chebyshev methods

Table 3: Option Prices Using Method of Lines

Table 4: Option prices using the method of separation of variables


Figure 1: Graph of option values using Black-Scholes formula

Figure 2: Graph of option values using Ps formula

Figure 3: Graph of option values using method of lines

Figure 4: Graph of option values using the method of separation of valuables          


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