#### $Delta$ - 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

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

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

#### Exam FM/2

**Derivative Markets**