# 04 Greeks and Elasticity #### $Delta$ - 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

Financial MathematicsDerivative Markets

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