# Analytical solution for a modified Black-Scholes equation

Recently, a modified Black-Scholes equation was proposed (Zheng), namely

$$\sigma \left( S,t \right) =\sigma\,{S}^{k/2}$$

and with the European put option

Using Maple I am obtaining the following analytical solution in terms of the associated Laguerre polynomials

Such solution can be used with Maple to compute the price for many instances of the European put option. For example when $k=3$ the solution is (please do right click on the image to enlarge it)

and taking the first three terms in the series we have (please do right click on the image to enlarge it)

A numerical example. Please consider the following values for the parameters $$S=100,K=95,T=90/365,r=4/100, \sigma=0.5;$$

using the standard Black-Scholes formula, the price of the put option is $6.9082$; and using my formula with $300$ terms in the series, the price of the put option is $70.51873101$. Assuming that the standard Black-Scholes formula underestimates the price of the put option and my formula overestimates the price of the put option, it is possible to fix the price of the put option near to the simple average between the two results, namely $38.5$.

My questions are:

1. I claim that such solution is new. Do you agree?

2. I claim that such solution could have important applications in computational finance. Do you agree?

• No , you just approximated option price by Laguerre polynomials.Using this method is very popular. – user16891 Aug 17 '15 at 6:29
• @Farahvartish,. Please let me know a paper where the solution in my post was presented. – Juan Ospina Aug 17 '15 at 13:00
• There is no paper because Numerical solution different from close form solution. you just solve a improper integral by numerical solution. – user16891 Aug 17 '15 at 13:29
• @ Farahvartish, excuseme but I think that you do not understand the question in my post. I am not using numerical methods, I am solving analytically a PDE which is a modified Black-Scholes equation. The analytical solution is a series in associated Laguerre polynomials. It is not an nuerical approximation. – Juan Ospina Aug 17 '15 at 13:36
• In Quantitative finance analytical solution should be like Black-Scholes formula or Heston formula or Bates formula – user16891 Aug 17 '15 at 14:01

The term of art in our industry for this type of option pricing formula is a series solution. As Farahvartish indicates in the comments, a series solution is not considered to be an "analytical solution" due to the reliance on a converging infinite sum for actual numeric output.(*)

Series solutions have been employed at least since the 1990s, when they were used along with the reflection principle to estimate prices of options with knock-out features.

More specific to your case, this paper also gives a series solution for option prices with volatility dependent on asset level. It is more general than your formula above, though it uses Hermite polynomials rather than Laguerre. In it, Xiu allows for $\sigma(S)$ to be any function whose reciprocal is Lebesgue integrable.

(Note: I have not checked your result or the Xiu paper for correctness)

(*) I might add that the attitude about series solutions versus analytical solutions is logically inconsistent in practice, since the cumulative distribution function $N(\cdot)$ of the standard gaussian is numerically obtained from a series expansion in Legendre polynomials.

• The Xiu paper is using a method of expansion using Hermite polynomials but such polynomials do not arise directly from the solution of the corresponing PDE. Xiu is using a method similar to the orthogonal colocation method. In contrast I am using a purely analytical method where the associated Laguerre polynomials arise as the exact solutions of the considered PDE. My solution does not appear in the Xiu paper. I am almost sure that my solution is correct, new and it could be important in computational finance. – Juan Ospina Aug 17 '15 at 20:45
• Your solution is fine. But the correct word for it (in the mathematicians world) is Numerical solution (using infinite series for approximation), not Analytical solution (finite number of calculation). – dns Sep 8 '15 at 9:24
• @dns. My solution is an exact solution it is not an approximation. Please take a look about the method of solution at lanl.arxiv.org/ftp/arxiv/papers/1508/1508.03841.pdf – Juan Ospina Sep 9 '15 at 1:16
• Your solution only can get the true value if n=∞ (infinite). However if n<∞ less than infinite then the value you get is just approximation to the real "true value". Just like binomial option pricing but with infinitely growing number of branch (infinite series). – dns Sep 22 '15 at 15:45