Call option

From WilmottWiki

A call option is a contract that gives the holder the right, but not the obligation, to buy the underlying asset in the future for a specified amount, the strike. If the option can only be exercised on a specified date the option is called European, if it can be exercised any time up to a specified date it is called American, and it there are specified dates on, or periods during, which it can be exercised then this is a Bermudan option.

1 Value
2 Delta
3 Gamma
4 Theta
5 Speed
6 Charm
7 Colour
8 Vega
9 Rho (r)
10 Rho (D)
11 Vanna
12 Vomma or Volga

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Value

The value of a European call option in the Black-Scholes world is

$Se^{-D(T-t)}N\left(d_1\right)-Ke^{-r(T-t)}N\left(d_2\right)$

where the asset price is $S$ , time is $t$ , strike $K$ , expiration $T$ , asset volatility $\sigma$ , dividend yield $D$ and risk-free interest rate $r$ and

$d_1=\frac{\ln(S/K)+\left(r-D+\frac{1}{2}\sigma^2\right)(T-t)}{\sigma \sqrt{T-t}}$ , $d_2=d_1-\sigma \sqrt{T-t}$ and $N(x)=\frac{1}{\sqrt{2 \pi}}\int_{-\infty}^x e^{-\frac{1}{2}s^2}ds$ .

All formulae below are for European exercise.

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Delta

The delta formula is $e^{-D(T-t)}N(d_{1})$ .

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Gamma

The gamma formula is $\frac{e^{-D(T-t)}N^{\prime }(d_{1})}{\sigma S\sqrt{T-t}}$ .

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Theta

The theta formula is $-\frac{\sigma Se^{-D(T-t)}N^{\prime }(d_{1})}{2\sqrt{T-t}}+DSN(d_{1})e^{-D(T-t)}-rKe^{-r(T-t)}N(d_{2})$ .

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Speed

The speed formula is $-\frac{e^{-D(T-t)}N^{\prime }(d_{1})}{\sigma^2 S^2(T-t)}\times\left( d_1+\sigma \sqrt{T-t} \right)$ .

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Charm

The charm formula is $D e^{-D(T-t)}N(d_1)+ e^{-D(T-t)}N^{\prime}(d_1)\times \left( \frac{d_2}{2(T-t)} - \frac{r-D}{\sigma \sqrt{T-t}} \right)$ .

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Colour

The colour formula is $\frac{e^{-D(T-t)}N^{\prime}(d_1)}{\sigma S \sqrt{T-t}} \times\left( D+\frac{1-d_1 d_2}{2(T-t)}-\frac{d_1(r-D)}{\sigma \sqrt{T-t}} \right)$ .