$Delta$ - Delta
$\Delta = \frac{\text{Change in option price}}{\text{Change in Stock price}} = \frac{\partial V}{\partial S}$
$\Delta _C = e^{-\delta T}N(d_1)$ | $\Delta _P= -e^{-\delta T}N(-d_1)$
$ 0 \leq \Delta _C \leq 1$ | $ -1 \leq \Delta _P \leq 0$
$\Delta_C – \Delta_P = e^{-\delta T} $
$ \Delta$ INCREASES as $S$ INCREASES
$Gamma$ - Gamma
$\Gamma = \frac{Change in \Delta}{\text{Change in Stock price}} = \frac{\partial \Delta}{\partial S} = \frac{\partial ^2V}{\partial S^2}$
$\Gamma_C \geq 0$ | $\Gamma_P \geq 0$
$\Gamma_C = \Gamma_P = \frac{F^P(S)}{(S_0^2)v}*\frac{e^{-\frac{d_{1}^2}{2}}}{\sqrt{2\pi}}$
$Theta$ - Theta
$\Theta = \text{Change in option price as time advances} = \frac{\partial V}{\partial t}$
$\Theta$ is usually negative.
$ u$ - Vega
$\nu = \frac{\text{Change in option price}}{\text{Change in volatility}} = \frac{\partial V}{\partial \sigma} => \nu $ is positive
$Vega_C \geq 0$ | $Vega_P \geq 0 $
$Vega_C = Vega_P$
As $\sigma$ INCREASES $V$ INCREASES
$ ho$ - Rho
$\rho = \frac{\text{Change in option price}}{\text{Change in risk free rate}} = \frac{\partial V}{\partial r} $
$\rho_C \geq 0$ | $\rho_P \leq 0$
$Psi$ - Psi
$\Psi = \frac{\text{Change in option price}}{\text{Change in dividend yield}} = \frac{\partial V}{\partial \delta} $
$\Psi_C \leq 0$ | $\Psi_P \geq 0 $
Elasticity ($Omega $)
- $\Omega = \frac{\text{% change in option price}}{\text{% change in stock price}}= \frac{\Delta S}{V} $ [btw] * Only true for instantaneous rate of return. Note how $\Omega$ is similar to $\Delta$[/btw]
- $\Omega = \frac{\text{Risk premium for option}}{\text{Risk premium for Stock}} = \frac{(\gamma – r) }{(\alpha – r)}$ [btw]$\gamma = \text{Expected return of option}$ and $\alpha = \text{Expected return of stock}$[/btw]
- $|\Omega| =\frac{ \sigma_{options} }{ \sigma_{stock}}$
$\Omega_{call} \geq 1$ | $\Omega_{put} \leq 0$
Greeks for a Portfolio
$Greek_{port} = \sum_{i=1}^{n} N_i . Greek_i$
$\Omega_{port} = \frac{\Delta_{port} S}{V_{port}} = \sum_{i=1}^{n} w_i . \Omega_i$