Why do we use Taylor series approximations - Black-Scholes related

Hey guys. I was reading some side literature that was talking about Taylor series approximations to update the options value when some of the factors change.

For example, the dW = delta of option value depends on the partial derivatives wrt underlying asset, gamma, vega, etc.

Then we plug in the “delta” input values to get the “delta” change in the option value. Why do we need to do this? Why can’t we just plug in the new updated input values into the complete formula to get the “actual” values instead of bothering with Taylor series approximations? i.e. if we have the result of (2, 3, 4), instead of “updating” with (0.1, 0.1, -0.1), why don’t we just recompute the whole option value with (2.1, 3.1, 3.9)?

I understand that in the past, due to low computational power, this may have been a factor… But we are in 2013 now. Computing power is cheap…


the practical use of the Taylor approximation is for delta hedging rather than for recomputing the option value

Thanks. I am not familiar with delta hedging… Could you please elaborate on what you meant a bit more? Why is it impractical to “calculate the whole thing” with delta hedging?


Not to sound impertinent, but you should really look up delta hedging on Wikipedia or Investopedia before asking for elaboration. Once you understand the concept, the answer should be pretty obvious.

Not 100% sure what you’re asking with your original question, but for the Taylor series, the higher order terms in the expansion are bounded (by constant times x-squared, where x is between -1 and +1) and the limit taken in the B-S formula ensures convergence to what you get as a result. That is kind of far removed from computing “deltas” though.

It kind of baffles me that you’re reading “side literature” on Black-Scholes, yet you don’t know what delta hedging is, which is like Options 101.

But that’s only a first-order Taylor series approximation. In reality, it’s Options 101.01010101 . . .