# Tag Info

• 2,432
Accepted

### Errors on Finite Differences + Implicit Scheme + Black & Scholes

The PDE is defined for $x \in ]-\infty, +\infty[$ but the finite difference scheme requires a truncated domain $[x_{\min}, x_{\max}]$, and the choice of $x_{\min}$ and $x_{\max}$ will affect the ...
• 5,672

### Why does it make sense that $S$ and $e^{rt}$ are solutions to the Black-Scholes PDE?

These are well known trivial solutions to the Black-Scholes PDE. The first one is just the price of the underlying stock and the second is interest bearing money in a bank. These are trivially true ...
• 27.5k
Accepted

### Why does it make sense that $S$ and $e^{rt}$ are solutions to the Black-Scholes PDE?

Under the standard assumptions, generally speaking, any contract that depends on the current values of t and S, and which are paid for at the start satisfy this PDE. In the financial context, the ...

### Alternative derivation of the Black Scholes formula

The option pricing formula must satisfy the PDE you have derived for all values of $K$. The only way this can be the case is if the two parts that you separate are both equal to zero. Suppose the ...
• 51
Accepted

• 1,906
Accepted

### Cash deposit in replicating portfolio for BS equation unnecessary?

The key point here is that the portfolio must be self-financing, namely the initial option premium $V_0$ should be enough to allow you to hedge it throughout its life. If not, the option price $V_0$ ...
• 8,119
Accepted

• 16k

### Nonlinear Black-Scholes model Vs linear Black-Scholes

In general you have don't have an exact solution for the non linear equation - so you have to use numerical methods. Have you seen this Non linear option pricing book. It covers the topics that you ...
Accepted

### Implicit finite difference method always guarantees positive and stable price of derivative?

For the original PDE, the positivity can be deduced from the maximum principle for a parabolic operator. There is also a discrete version of the maximum principle for the finite difference parabolic ...
• 2,806
Accepted

### Boundary Conditions for Call Option in Black Scholes Model

Note that: $$C(t,S) =S-K{\rm e}^{-r(T-t)}$$ as $S\rightarrow \infty$, for all $t$. Basically because one can easily accept $$P(t,S) =0$$ as $S\rightarrow \infty$, for all $t$, and one still ...
• 5,043

• 5,043

### What are the parallels between the Black-Scholes equation and the heat equation?

For heat equation, it describes how heat diffuses (usually measured by temperature) through the length of the material and over time. For Black Scholes, it describes how the value of the option ...
• 131
Judging from the oscillations near $S=0$, it looks like the payoff function is causing these problems. Your payoff should go towards -1 as $S$ goes towards zero, but your computer might just ...