option price change

Question

I am trying to match change in European Call option price to greeks using the calculator here

e.g. for S=95, K=100, r=0, V=25, t=5 and dividend=0, I get

Theoretical Price   0.046
Delta   0.041
Gamma   0.032
Vega    0.01
Theta   -0.024
Rho 0.001

Now I move 1 day forward and change S by 1,

So now for S=96, K=100, r=0, V=25, t=4 and dividend=0, I get

Theoretical Price   0.065
Delta   0.061
Gamma   0.048
Vega    0.012
Theta   -0.038
Rho 0.001

However, If i use

$dc = \Delta ds + 0.5 \Gamma ds^2 + \theta dt$

I get

dc = 0.041 * 1 + 0.5 * 0.032 * 1^2 + (-0.024) *1
   = 0.033 

compared to (0.065-0.046) = 0.019 fromt he numbers above.

Am I missing something?

Answer 1

Your equation:

$$dC = \Delta S + 0.5\Gamma (\Delta S)^2 + \theta \Delta t$$

is actually an approximation of the option price changes (more precisely a "delta-gamma-theta" approximation) which is relevant only for sufficiently small underlying price movements. It basically captures first and second-order moves in the stock price along with first order move in the time-to-maturity. See Taylor Series for more details on this.

If you want a better approximation of what will be your call option price the next day given an underlying price move, you should either:

1. capture higher-order variations of your option price by introducing third, fourth, ..., n-th order greeks (e.g. Speed, equal to $d\Gamma/dS = d^3C/dS^3$) in your approximation equation;

2. reduce your underlying price move: for instance, with a new underlying price of 95.1, you get a new option value equal to 0.027 so 0.046 - 0.027 = 0.019.

   The delta-gamma-theta approximation yields to 0.019 as well: (0.041 * 0.1) + (0.5 * 0.032 * 0.1 * 0.1) - 0.024*1 = 0.019.