Using Partial Derivatives to check B-S Equation holds and find constants

In summary, the speaker has used partial derivatives to simplify the Black Scholes equations and is now stuck on solving the remaining equations. They have found the solution to the first part by letting (a + 2bt + αt +r) = 0, but are unsure of how to complete the last part. They are seeking help and believe it is a straightforward process.
  • #1
robot1000
5
0
The question I'm trying to solve is part (ii) of the attached file

I've used partial derivatives to input back into the Black Scholes equations and after factorising it, I've got it down to:

(a + 2bt + αt +r) * (S².c.e^(at+bt²) = 0

I'm now stuck on what to do next, as there would need to be 2 equations in order to solve simultaneously which isn't the case above.

Help would appreciated
 

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  • #2
I think I got the solution to the first part by letting (a + 2bt + αt +r) = 0

Therefore a = -r and b = -α/2

However I'm not sure how to complete the last part, I'm sure it's quite straight forward, but I just can't see what to do.

Thanks
 

1. What are partial derivatives?

Partial derivatives are a mathematical concept used in multivariable calculus to measure the rate of change of a function with respect to one of its variables, while holding all other variables constant.

2. How are partial derivatives used to check if the Black-Scholes equation holds?

In the context of finance, the Black-Scholes equation is a mathematical model used to predict the price of a financial instrument over time. By taking partial derivatives of the equation with respect to the underlying asset price and time, we can verify if the equation holds and accurately predicts the price of the instrument.

3. Can partial derivatives be used to find the constants in the Black-Scholes equation?

Yes, by setting up a system of partial differential equations and solving for the constants, we can use partial derivatives to find the values of the parameters in the Black-Scholes equation.

4. Are there any limitations to using partial derivatives to check the Black-Scholes equation?

While partial derivatives are a powerful tool for analyzing mathematical models, they do have limitations. In the case of the Black-Scholes equation, it assumes certain ideal market conditions that may not always hold true in the real world.

5. How can partial derivatives be applied in other fields besides finance?

Partial derivatives are a fundamental concept in multivariable calculus and have applications in various fields such as physics, engineering, economics, and more. They are used to analyze and optimize complex systems with multiple variables.

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