Finite difference: move forwards or backwards?

Question

In finite differences for the black scholes method, you move backwards in time, since of course you know the prices at time $t = T$, and then you iterate until you get to time $t = 0$.

However, why then in this code does the time move forwards? Here, cur_t is current time, and as you can see, he iterates and each time moves cut_r forwards by dt.

Entire code can be found here: https://www.quantstart.com/articles/C-Explicit-Euler-Finite-Difference-Method-for-Black-Scholes

Is this a mistake in the code?

Answer 1

They have written the equation to be solved as $$ -\frac{\partial C}{\partial t} + r S \frac{\partial C}{\partial S} + ... = 0 $$ instead of the more usual $$ \frac{\partial C}{\partial t} + r S \frac{\partial C}{\partial S} + ... = 0 $$ This means that in their setup $t$ represents the time to maturity, that is $t = T - \text{time}$. So they start from $t = 0$ where the option value is equal to its payoff, and they move forward in time to maturity until reaching $t = T$ which corresponds to $\text{time} = 0$ .