TD - Using the Black-Scholes equation for a continuous-time value of an option

Using the Black-Scholes Equation for a Continuous-Time Value of an Option

Problem Overview
This study material walks through solving the Black-Scholes equation for the continuous-time value of an option. The Black-Scholes equation is fundamental in financial mathematics, particularly for pricing European options. We aim to demonstrate transformations of this equation into simpler forms, leading up to a solution.

Key Topics Covered:

• Black-Scholes Equation: The starting point of the problem is the Black-Scholes equation:

• Discounted Option Value: The equation is first transformed by introducing the discounted option value 

• Change of Variables: A further change of variables to backwards time

  Diffusion Equation: With a final change of variables (to be specified), the equation can be converted into the diffusion equation:​

• Solution to Diffusion Equation: By substitution, we demonstrate that the solution to the diffusion equation is:

  
  Solution to Black-Scholes Equation: Finally, the solution to the original Black-Scholes equation can be written down using the transformations performed throughout the problem.

Applications:

• Option Pricing: This result can be used to derive the price of a European option, given boundary conditions at the maturity date $T$.

This material is essential for students and professionals dealing with quantitative finance, derivatives pricing, and stochastic processes.

Why Choose This Material?

• Step-by-step walkthrough of transforming and solving the Black-Scholes equation.
• Clear explanation of variable transformations and their physical meaning.
• Practical application to real-world option pricing problems.