How Underlying Dynamics Enter Black-Scholes PDE Discretization
Summary
The document addresses why the probability distribution of the underlying seems absent when the Black-Scholes partial differential equation is discretized on a stock-price grid. The question contrasts a uniform spatial grid, which appears to encode only a price range, with the fact that changing the underlying process should change an option’s value.
The response explains that a numerical scheme propagates option values backward through time using weighted contributions from neighboring stock-price nodes. Those weights depend on the process parameters, including the risk-free rate and volatility, as well as the spatial and time step sizes. Thus, the grid specifies where values are calculated, while the process influences how those values are combined at each step. The discussion sketches a recursive approximation but does not give a full derivation, boundary treatment, stability conditions, or a worked numerical example; its explanation applies at a conceptual level to a chosen discretization.
Key ideas
- A uniform stock-price grid sets the locations where the PDE solution is approximated.
- The recursion combines values at neighboring stock nodes using weights determined by the underlying process and grid spacing.
- Risk-free rate and volatility affect the numerical solution through those weights.
- The exchange gives a conceptual outline rather than a complete discretization recipe or stability analysis.
Tags
Full text
# Black Scholes PDE discretization
# Black Scholes PDE discretization
We can solve the Black Scholes PDE by numerical methods like Euler \begin{equation} \frac{\partial V}{\partial t} + rS\frac{\partial V}{\partial S}+\frac{1}{2} \sigma^2 S^2\frac{\partial^2 V}{\partial S^2}-rV=0 \end{equation} In order to do that, we need to discretize the stock space. A simple way is to set a range of potential stock prices and then uniformly divide it.
My confusion is that, in this process, the distribution of the underlying stock price, i.e., $dS_t=rS_tdt+\sigma dW$, seems to only factor in determining the range of the space grid. So the probability of getting to each space grid does not affect the PDE solution. But on the other hand, if the stock price follows another process even if the distribution has a similar range the option price should certainly change. What am I missing here?
## Answer by Kermittfrog (score 1, accepted)
https://quant.stackexchange.com/a/69727
Assume a sufficiently 'wide and dense' mesh of sampling points $(S_i,t_j), i=0..N, j=0..M$, $S_i=S_{low}+i\Delta S$, $t_j=j\Delta t$. Given the payoff encoded in $v_{i,M}$ and some boundary conditions for $j=M$, sarting from $j=M$ backwards, an explicit recursive numerical approximation scheme to the value equation then works along the lines of
$$ v_{i,j}=\alpha v_{i-1,j+1}+\beta v_{i,j+1}+\gamma v_{i+1,j+1} $$
where $\alpha,\beta,\gamma$ are some weights defined by the parameters of the underlying process, i.e. $r,\sigma$, and your choice discretization step sizes $\Delta S,\Delta t$. As you have already 'fixed' some method and (sensible) discretization, the parameters $\alpha,\beta,\gamma$ are solely driven by the underlying process - and that's what will influence the option price.
Did that help?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.