Black-Scholes Equation

Black-Scholes Equation

Introduction

Differential equation shave a quantity of applications in various fields of science such as engineering, physics, biology, pharmacokinetics (Liet. al., 2014). Yet, there are only few of their applications in economics or finance. Particularly, well-known models involving differential equations are only economic growth model and Black-Scholes equation. The latter one will be discussed in the paper. In 1977,  Myron Scholes together with Fischer Black were awarded a Nobel Prize in economics for the formulation of stock options formula through «new method of determining the value of derivatives» (Jarrow, 1999).

So, Black-Scholes model deals with one of the most important issues in quantitative finance – pricing of options (Mamon and Rodrigo, 2005). This model has significant implications – both theoretical and practical – since finance plays a great role in economies around the world (Bohner and Zheng, 2008).

 

Main body

1. Background information and underlying assumptions

In practice, Black-Scholes model of option pricing was applied to various «commodities and payoff structures» (Salaet. al., 2004). Black-Scholes model is widely used for American options as well as for European options. Therefore, the model has wide variety of applications. Before considering Black-Scholes model, there is a number of assumptions that should be made. Fischer Black calls them «ideal conditions» of the market (Black and Scholes, 2014). These assumptions are important to emphasize because it is well-known that stock markets are often volatile compared to other parts of the economy.

There are five underlying assumptions:

1. First assumption that should be made is information about values of short-term rates is available and short-term interest rates are constant (Black and Scholes, 2014).
2. Secondly, stock pays no dividend (Black and Scholes, 2014).
3. Thirdly, transaction costs that occur while buying or selling securities are eliminated (Black and Scholes, 2014).
4. Fourthly, it is achievable to borrow fraction of price of stock one wants in order to hold the stock (Black and Scholes, 2014).
5. The last assumption states that short selling of a security in necessary situations is allowed in the market (Black and Scholes, 2014).

Thanks to these assumptions, option price will be the function of time period and stock price only. In the following paragraphs option price will be reduced to the function of stock price only for simplicity.

Generally, option values increase when stock prices rise. Positive relationship between option value and stock price may be easily seen from the following graph:(Black and Scholes, 2014):

 

As it can be seen from the figure, graphs representing the relationship between option price and stock price at different time periods () lie below 45-degree line, which shows that option prices are more volatile than the stock prices (Black and Scholes, 2014). The volatility of option prices lead to the following statement: if price of the stock increases by a certain amount, greater percentage change will be generated in option prices.

The graph illustrated above shows what the paper seeks to explain through Black-Scholes model.

 

 

2. Transformation into an Ordinary Differential Equation.

Black-Scholes equation is given by the following expression:

,

where C(s, t) = price of option, s = price of the stock, t = period of time, r = interest rate (Companyet. al., 2007).

Firstly, it is useful to transform this partial differential equation (PDE) into an ordinary differential equation (ODE) by proposing the following solution: C(s, t) = C(s)*.

Given that

  and  ,

and substituting into the PDE we get:

.

The next step is to rearrange the equation to get second order ODE:

.

The latter expression can be reduced to the following equation:

since

 

3. Euler equation.

To get rid of the coefficient of the first term lets divide everything by

 

This equation reminds the Euler equation:

L[y] =

with real constants (Boyce and DiPrima, 2009). In our case, and , which are positive constants.

Euler equation has the solution of the form

 

in case of distinct real roots, and characteristic equation of the form:

(Boyce and DiPrima, 2009).

 

4. Solution of the Black-Scholes equation.

By the assumption given, and r are positive real numbers because r is an interest rate and is volatility of the stock as noted earlier in the paper.

Now, solution in the form of C(s) = can be proposed and applied to the Black-Scholes equation.

The following derivations will be useful in solving our problem:

C(s) =

 

 

Substituting the derivations back into the earlier equation we get:

.

The next step is to take out of bracket and derive characteristic equation introduced earlier:

 

 

To find the roots of characteristic equation, let us find discriminant:

by assumption.

So, the two distinct roots of characteristic equation will be:

 

Therefore, the solution of our problem can be written as:

 

The solution represents option value as a function of stock prices. By the assumption must be positive constants because of positive relationship between option price and the stock price introduced earlier.

 

 

Figure 2. Fluctuations in stock prices from 2000 to 2009

 

The given figure above shows fluctuations in stock prices from 2000 to 2009 time period (Wikipedia, 2014). The Black-Scholes presented in the paper is useful to explain, predict and estimate option prices based on stock prices in the financial world. Black-Scholes model gives more accurate estimates of option prices than other earlier developed models because it takes into account such factors influencing the stock prices as transaction costs, riskiness of assets, illiquid markets (Ankudinova and Ehrhardt, 2008).Therefore, the model is used to estimate European call options, which consolidates its role in applied economics (Barad, 2014).

 

 

 

 

 

 

 

 

 

 

Conclusion

To conclude, Black-Scholes model is highly appreciated in quantitave finance because of its accurate and useful estimation of stock prices. Black-Scholes equation represents derivation of option pricing though taking into account such factors as time period (t), risk-free interest rate (r) and volatility of stock prices() (Sheraz and Preda, 2014).Derived solution for the option value is closely related to corporate liabilities, therefore, the formula derived may be used to securities, including common stock and bond(Black and Scholes, 2014). This feature of Black-Scholes model illustrates its flexibility and efficiency of being applied to different contexts in the financial world.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

References

Ankudinova J. and Ehrhardt M. (2008). On the numerical solution of nonlinear Black–Scholes equations. Computers and Mathematics with Applications, 56, 799–812. doi:10.1016/j.camwa.2008.02.005

Barad G. (2014). Differential Geometry techniques in the Black-Scholes option pricing; theoretical results and approximations. Procedia Economics and Finance, 8, 48 – 52. doi:10.1016/S2212-5671(14)00061-6

Black F. and Scholes M. (2014).The Pricing of Options and Corporate Liabilities. Journal of     Political Economy, 81 (3), 637-654. Retrieved from http://www.jstor.org/stable/1831029

Bohner M. and Zheng Y. (2009). On analytical solutions of the Black-Scholes equation. Applied Mathematical Letters, 22, 309-313. doi:10.1016/j.aml.2008.04.002

Boyce W. and Di Prima R. (2009). Elementary Differential Equations. US: John Wiley & Sons Inc.

Company R., Jodar L., Rubio G. and Villanueva R. (2007). Explicit solution of Black–Scholes option pricing mathematical models with an impulsive payoff function. Mathematical and Computer Modelling, 45, 80-92. doi:10.1016/j.mcm.2006.04.006

Jarrow R. (1999). In Honor of the Nobel Laureates Robert C. Merton and Myron S. Scholes: A Partial Differential Equation That Changed the World. The Journal of Economic Perspectives,  13(4), 229-248. Retrieved from http://www.jstor.org/stable/2647021

Li X., Zhu Q. and O’Regan D. (2014). pth Moment exponential stability of impulsive stochastic functional differential equations and application to control problems of NNs. Journal of the Franklin Institute, 351, 4435–4456.  doi:org/10.1016/j.jfranklin.2014.04.008

Mammon R.and Rodrigo M. (2005). An alternative approach to solving the Black–Scholes equation with time-varying parameters. Applied Mathematical Letters, 19, 398-402. doi:10.1016/j.aml.2005.06.012

Sala R., Jodar L., Sevilla-Peris P. and Cortes C. (2005).A new direct method for solving the Black-Scholes equation. Applied Mathematics Letters, 18, 29-32.doi: 10.1016/j.aml.2002.12.016

Sheraz M. and Preda V. (2014). Implied volatility in Black-Scholes model with GARCH volatility. Procedia Economics and Finance,8, 658 – 663. doi:10.1016/S2212-5671(14)00141-5

Wikipedia. Figure «Stock price simulation». Retrievedfromhttp://commons.wikimedia.org/wiki/File%3AStockpricesimulation.jpg