Series Solutions for Options with State-Dependent Volatility
Summary
The document discusses a proposed solution to a modified Black–Scholes equation for a European put when volatility depends on the underlying asset price. The proposed expression uses associated Laguerre polynomials and is evaluated as a series. A numerical example compares its output with the standard Black–Scholes price, but the large discrepancy is not resolved: the author speculates that the true price lies between the two and suggests their average without providing validation.
The answer clarifies that such formulas are generally called series solutions and points to earlier work on option pricing with state-dependent volatility, including a more general approach using Hermite polynomials. It cautions that neither the proposed result nor the cited paper was checked for correctness. The discussion is useful for distinguishing a series representation from a closed-form expression and for recognizing that a numerical comparison alone does not establish accuracy, novelty, or practical value.
Key ideas
- A volatility function that depends on the underlying price can lead to option pricing equations beyond the standard Black–Scholes setup.
- The proposed put price is represented by an infinite series involving associated Laguerre polynomials.
- The example reports a large difference from the standard Black–Scholes price, but supplies no independent validation of either estimate.
- Series methods for options with asset-dependent volatility predate the proposed expression and have been studied using other polynomial families.
- A numerical midpoint between two estimates is not supported as a pricing method by the evidence presented.
Tags
Full text
# Analytical solution for a modified Black-Scholes equation
# Analytical solution for a modified Black-Scholes equation
Recently, a modified Black-Scholes equation was proposed (Zheng), namely
Please consider the case when
$$\sigma \left( S,t \right) =\sigma\,{S}^{k/2}$$
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:
- I claim that such solution is new. Do you agree?
- I claim that such solution could have important applications in computational finance. Do you agree?
## Answer by Brian B (score 2)
https://quant.stackexchange.com/a/19359
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.