Comparing an Intermediate-Time Option Grid with Black–Scholes Values

Summary

The document asks how to compare a numerical European call price from an explicit finite-difference scheme with the analytical Black–Scholes value at an intermediate time step. The model uses spatial grid points corresponding to underlying asset prices, and the goal is to calculate the error at every spatial point for one selected step.

The key issue raised is how to set the time input in Black–Scholes: the questioner knows the full option duration and the grid spacing, but is unsure how an intermediate step relates to the analytical valuation. The document itself provides no answer, computed errors, or numerical evidence, so it does not specify a complete comparison procedure. It identifies a useful distinction for such comparisons: Black–Scholes valuation depends on time remaining to maturity, which must be matched to the numerical scheme’s time convention at the selected step.

Key ideas

• The task compares an explicit finite-difference call price with its analytical Black–Scholes counterpart.
• The comparison is requested at one intermediate time step across all underlying-price grid points.
• The Black–Scholes valuation must use a time input consistent with the numerical grid’s time convention.
• The source poses the question but supplies no calculation or validation results.

Tags

# 63184


# Calculate error at all spatial indices for a given time step between BS equation and its numerical solution using explicit method












I am using the explicit finite backward difference scheme to discretize and calculate the price of an European call option in a discretization stencil.

My goal is to find the error at a given time step (e.g. at the 200th time step in a 360 time step model) evaluated at each spatial step (which represents the underlying asset price in this case) for the numerical solution, as compared to the analytical Black Scholes solution.

However, I don't understand how to supply the 200th time step parameter to the Black Scholes equation to calculate the option value at that time step for different asset prices. As far as my understanding goes, the BS equation only takes in the size of the time step, which is calculated by dividing the option duration into equal sizes. It then gives the option value at t=0.

How can I use the BS model to find the option value at, say, t = 200/360?

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