taylor's series application

say we wanna know f(x1), but we only know

• x1-x0 is small,
• f(x0),
• f'(x0), ie first derivative
• f''(x0), ie 2nd derivative
• higher order of derivatives, etc.

, what do we do?

using taylor's series, we can estimate by
f(x1) = f(x0) + (x1-x0)*f'(x0)/1! + (x1-x0)*f''(x0)/2! + ...

:D

a concrete example, say,

• with the current underlying price, x0,
• we calculate an option's value, f(x0),
• the associated delta, f'(x0),
• gamma, f''(x0)

if the underlying price moves a little bit from x0 to x1, how do we estimate the new option price, f(x1), without going through the option pricing model?