# Oscillating errors in finite difference Black Scholes

I am writing an implementation of the explicit finite difference method to price a standard european call option, and comparing the results to the corresponding analytical value to gauge the error both in time and space/price. This is the error curve I get for fixed dt = 1.0e-5 and plotting as a function of ds:

Recalling that the finite difference for BS is of order ds^2, the behaviour of the curve seems appropriate, except for those strange oscillations. Has anyone seen anything like it? What could be the reason for such strange behaviour?

• How does the graph change if you change the size of $\delta t$? Do the dips move? What if you change the other parameters? What happens if you change the finite difference scheme, e.g. for a larger stencil/kernel? What is the shape of the f.d. result? What is the shape of the analytics solution you are comparing to? – Phil H Sep 18 '18 at 9:37
• These methods consider a discrete set of values for $S$. In my experience how this grid interacts with the exercise price K affects the accuracy (for example is one of the Ss exactly on top of K or not). I would speculate that you get a low error when as you vary $\Delta S$ you sometimes hit one of the favorable configurations of S's and K. I have seen this kind of periodic behavior before. – Alex C Sep 19 '18 at 3:56