Skip to content
All library documents

Connecting Probability Distributions to Differential Equations in Derivatives

Article Quant Q&A · Author: David Addison
Read the original

Summary

The document explores whether probability distributions can be derived as solutions to differential equations, motivated by the Black–Scholes–Merton model. It questions whether dynamic hedging is best understood as a modeling assumption that leads to an equation with a known distribution, and whether similar transformations could help analyze other derivative products, including path-dependent options.

As examples, it points to inverse-chi-squared and normal-inverse-gamma distributions and asks why differential-equation forms once shown for these distributions were removed from reference entries. The document does not supply derivations or establish whether the examples are correct. It is a research question seeking a comprehensive source that maps known distributions to differential equations, rather than a developed method or an evaluation of a trading strategy.

Key ideas

  • The Black–Scholes–Merton model motivates asking how differential equations relate to probability distributions.
  • The author proposes that such relationships could aid analysis of derivatives, including path-dependent options.
  • Inverse-chi-squared and normal-inverse-gamma distributions are cited as examples needing further investigation.
  • The document raises, but does not answer, why some distribution-related equations were removed from reference entries.

Tags

OptionsDerivatives pricingStatistics
Full text
# Probability distributions as solutions to differential equations


# Probability distributions as solutions to differential equations












As far as what I can tell, the popularity of the Black-Scholes-Merton model partly stems from the fact that it formulates the value of a derivative in a differential form in which the solution has a known distribution. From this perspective, the dynamic hedging argument was simply a modeling assumption which allowed its authors to pose the problem in a way which already had a known solution. Anyway, I was hoping to make use of such transformations to identify solutions to other similarly posed problems. However, I have not been able to find a comprehensive reference on the topic of transformations of differential equations to probability distributions.

My intuition is that such a resource might be helpful in researching solutions to other types derivatives, such as path dependent options.

For example, the current Wiki entries on the Inverse-chi-squared distribution and Normal-inverse-gamma distribution lack the differential equations to which these distributions are solutions. Moreover, historical Wiki entries actually contained such solutions, such as this redacted entry on Inverse-chi-squared distribution:

$\left\{2x^{2}f_{}'(x)+f_{}(x)(-\nu +\nu x+2x)=0,f_{}(1)={\frac {(2e)^{{-\nu /2}}v^{{\nu /2}}}{\Gamma \left({\frac {\nu }{2}}\right)}\sim\frac{1}{\chi^2[\nu]}}\right\}$

Were these solutions redacted because they were wrong? If not, why?

Moreover, does comprehensive resource exist which connects known distributions as solutions to various differential equations?

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.