Black-Scholes PDE and finding the general solution

In summary, the speaker is seeking help with finding a solution for a PDE using a specific form and given equations. They are also asking if anyone is familiar with transforming the given equation to a standard PDE heat equation.
  • #1
meghibbert17
3
0
Hello, I have the PDE

[itex]\frac{-∂v}{∂τ}[/itex]+[itex]\frac{1}{2}[/itex]σ[itex]^{2}[/itex]ε[itex]^{2}[/itex][itex]\frac{∂^{2}v}{∂ε^{2}}[/itex]+([itex]\frac{1}{T}[/itex]+(r-D)ε)[itex]\frac{∂v}{∂ε}[/itex]=0

and firstly I need to seek a solution of the form v=α[itex]_{1}[/itex](τ)ε + α[itex]_{0}[/itex](τ) and then determine the general solution for α[itex]_{1}[/itex](τ) and α[itex]_{0}[/itex](τ).

I am given that ε=[itex]\frac{I}{TS}[/itex] - [itex]\frac{X}{S}[/itex] and that τ=T-t.

Can anybody help me with this problem?

Thankyou
 
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  • #2
meghibbert17 said:
Hello, I have the PDE

[itex]\frac{-∂v}{∂τ}[/itex]+[itex]\frac{1}{2}[/itex]σ[itex]^{2}[/itex]ε[itex]^{2}[/itex][itex]\frac{∂^{2}v}{∂ε^{2}}[/itex]+([itex]\frac{1}{T}[/itex]+(r-D)ε)[itex]\frac{∂v}{∂ε}[/itex]=0

and firstly I need to seek a solution of the form v=α[itex]_{1}[/itex](τ)ε + α[itex]_{0}[/itex](τ) and then determine the general solution for α[itex]_{1}[/itex](τ) and α[itex]_{0}[/itex](τ).

I am given that ε=[itex]\frac{I}{TS}[/itex] - [itex]\frac{X}{S}[/itex] and that τ=T-t.

Can anybody help me with this problem?

Thankyou

Hey meghibbert17 and welcome to the forums.

Are you familiar with the idea for transforming the B.S. to a standard PDE heat equation?
 

1. What is the Black-Scholes PDE and why is it important?

The Black-Scholes Partial Differential Equation (PDE) is a mathematical model used to determine the price of a financial option. It is important because it provides a way to calculate the fair price of an option and is widely used in the financial industry.

2. How does the Black-Scholes PDE work?

The Black-Scholes PDE is based on the assumption that the price of an asset follows a geometric Brownian motion. It takes into account factors such as the current price of the underlying asset, the strike price, time to expiration, and volatility to calculate the fair price of an option.

3. What is the general solution to the Black-Scholes PDE?

The general solution to the Black-Scholes PDE is known as the Black-Scholes formula. It provides a way to calculate the fair price of an option at any given time, taking into account the current market conditions and the specific parameters of the option.

4. How accurate is the Black-Scholes PDE?

The Black-Scholes PDE is a widely accepted model for option pricing, but it does have its limitations. It assumes a constant volatility and a continuous trading environment, which may not always accurately reflect real market conditions. Therefore, it is important to regularly reassess and adjust the inputs to ensure accurate pricing.

5. Can the Black-Scholes PDE be used for all types of options?

The Black-Scholes PDE was originally developed for European-style options, which can only be exercised at expiration. However, it has been adapted for other types of options, such as American-style options, by incorporating additional factors. It is important to use the correct version of the Black-Scholes PDE for the type of option being priced.

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