Greeks Synopsis


$Delta$ - 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}   $   + * Only true for instantaneous rate of return. Note how $\Omega$ is similar to $\Delta$
  • $\Omega = \frac{\text{Risk premium for option}}{\text{Risk premium for Stock}} = \frac{(\gamma – r) }{(\alpha – r)}$   + $\gamma = \text{Expected return of option}$ and $\alpha = \text{Expected return of stock}$
  • $|\Omega| =\frac{ \sigma_{options} }{ \sigma_{stock}}$

$\Omega_{call} \geq 1$$\Omega_{put} \leq 0$

Greeks for a Portfolio

$Greek_{port} = \sum_{i=1}^{n} N_i . Greek_i$

$\Omega_{port} = \frac{\Delta_{port} S}{V_{port}} = \sum_{i=1}^{n} w_i . \Omega_i$

 



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