The Greeks

The Greeks

The Greeks are a simple shorthand way of describing the way that the value of an option (or other derivative) varies as one some other underlying variable varies

You will probably be familiar with the Greeks in the context of the Black-Scholes model but the Greeks also exist for Gram-Charlier and the Heston SV model and they are defined in exactly the same way

The Greeks can be worked out analytically through calculus (some of which becomes very complicated) or more easily they can be calculated (approximated) by the finite difference method. This method although strictly an approximation can be performed to an arbitrary degree of accuracy and so is perfectly adequate for the vast majority of purposes

Black Scholes Greeks

The standard Greeks for the Black Scholes model are as follows:

Delta: $\Delta = \frac{\partial C}{\partial S} = \Theta(d_1)$

Theta: $\theta = \frac{\partial C}{\partial t} = - \frac{S \phi(d_1) \sigma}{2 \sqrt{T-t}}-r K e^{-r(T-t)} \Theta(d_2)$

Gamma: $\Gamma = \frac{\partial C}{\partial \Delta} = \frac{\phi(d_1)}{S \sigma \sqrt{T-t}}$

Rho: $\rho = \frac{\partial C}{\partial r} = K(T-t) e^{-r(T-t)} \Theta(d_2)$

Vega: $\nu = \frac{\partial C}{\partial \sigma} = S\phi(d_1) \sqrt{T-t}$

Formulas given are for call options

This spreadsheet allows you to experiment with the Greeks and different parameters

Interim Result: pre-lemma

For an option $S_t \phi(d_1) = K e^{-r(T-t)} \phi(d_2)$

where: $d_1=\frac{ln\left(\frac{S_t}{K}\right) + \left(r+\frac{\sigma^2}{2}\right)(T-t)}{\sigma \sqrt{T-t}}$

and $d_2=d_1 - \sigma \sqrt{T-t}$

Proof

$\phi(x)=\frac{1}{\sqrt{2 \pi}} e^{\frac{-x^2}{2}}$

$\therefore \frac{\phi(d_1)}{\phi(d_2)}=e^{\frac{1}{2}\left(d_2^2-d_1^2\right)}$

$\therefore \frac{\phi(d_1)}{\phi(d_2)}=e^{\frac{1}{2}\left(d_2-d_1\right)\left(d_2+d_1\right)}$

$\therefore \frac{\phi(d_1)}{\phi(d_2)}=e^{-\frac{1}{2} \sigma \sqrt{T-t} \left(2 d_1 - \sigma \sqrt{T-t}\right)}$

$\therefore \frac{\phi(d_1)}{\phi(d_2)}=e^{-\frac{1}{2} \sigma \sqrt{T-t} \left( \frac{2 ln \frac{S_t}{K} + (2r + \sigma^2)(T-t)}{\sigma \sqrt{T-t}}-\sigma \sqrt{T-t}\right)}$

$\therefore \frac{\phi(d_1)}{\phi(d_2)}=e^{-ln \frac{S_t}{K} -\frac{1}{2} (2r+\sigma^2)(T-t) + \frac{1}{2} \sigma^2(T-t)}$

$\therefore \frac{\phi(d_1)}{\phi(d_2)}=e^{-ln \frac{S_t}{K} -r(T-t)}$

$\therefore \frac{\phi(d_1)}{\phi(d_2)}=\frac{K}{S_t} e^{-r(T-t)}$

$\therefore S_t \phi(d_1) = Ke^{-r(T-t)}\phi(d_2)$

Delta $\left(\Delta \right) = \Theta(d_1)$

Proof

We wish to calculate the rate of change of a call option with the underlying share price

That is we want $\frac{\partial C}{\partial S}$

where $C=S_t \Theta(d_1) - K e^{-r(T-t)} \Theta(d_2)$

$d_1=\frac{ln\left(\frac{S_t}{K}\right) + \left(r+\frac{\sigma^2}{2}\right)(T-t)}{\sigma \sqrt{T-t}}$

$d_2=d_1 - \sigma \sqrt{T-t}$

We start with a straightforward differentiation

$\Delta=\frac{\partial C_t}{S_t} = \Theta(d_1)+S_t \phi(d_1) \frac{\partial d_1}{\partial S_t} - K e ^{-r(T-t)} \phi(d_2) \frac{\partial d_2}{\partial S_t}$

$\Delta=\frac{\partial C_t}{S_t} = \Theta(d_1)+S_t \phi(d_1) \left[ \frac{\partial d_1}{\partial S_t} - \frac{\partial d_2}{\partial S_t}\right]$

$\Delta=\frac{\partial C_t}{S_t} = \Theta(d_1)+S_t \phi(d_1) \frac{\partial (d_1 - d_2)}{\partial S_t}$

$\Delta=\frac{\partial C_t}{S_t} = \Theta(d_1)$

Gamma $\left(\Gamma \right) = \frac{\phi(d_1)}{S_t \sigma \sqrt{T-t}}$

Proof

We wish to calculate the rate of change of a the Delta of a call option with the underlying share price

That is we want $\Gamma = \frac{\partial \Delta}{\partial S_t} = \frac{\partial}{\partial S_t} \Theta(d_1)$

$\Gamma = \phi(d_1) \times \frac{\partial}{\partial S_t} \left(\frac{ln\left(\frac{S_t}{K}\right) + \left(r+\frac{\sigma^2}{2}\right)(T-t)}{\sigma \sqrt{T-t}} \right)$

$\Gamma = \phi(d_1) \times \frac{1}{S_t} \frac{1}{\sigma \sqrt{T-t}}$

$\Gamma = \frac{\phi(d_1)}{S_t \sigma \sqrt{T-t}}$

Theta $\left(\theta \right) = \frac{-S_t \phi(d_2) \sigma}{2 \sqrt{T-t}} - K r e^{-r(T-t)} \Theta(d_2)$

Proof

We wish to calculate the rate of change of a call option with respect to time

That is we want $\theta = \frac{\partial C_t}{\partial t}$

$\theta = S_t \phi(d_1).\frac{\partial d_1}{\partial t} - K r e^{-r(T-t)} \Theta(d_2) - K e^{-r(T-t)} \phi(d_2) \frac{\partial d_2}{\partial t}$

and

$d_1-d_2 = \sigma \sqrt{T-t}$

$\therefore \frac{\partial(d_1 - d_2)}{\partial t} = \sigma \frac{1}{2} (T-t)^{\frac{-1}{2}}.(-1)$

$\therefore \frac{\partial(d_1 - d_2)}{\partial t} = \frac{-\sigma}{2 \sqrt{T-t}}$

$\theta = S_t \phi(d_1) \left[\frac{\partial d_1}{\partial t} - \frac{\partial d_2}{\partial t} \right] - K r e^{-r(T-t)} \Theta(d_2)$

$\theta = \frac{-S_t \phi(d_1) \sigma}{2 \sqrt{T-t}} - K r e^{-r(T-t)} \Theta(d_2)$

Vega $\left(\nu \right) = S_t \phi(d_1) \sqrt{T-t}$

Proof

We wish to calculate the rate of change of a call option with the volatility of the underlying share

That is we want $\frac{\partial C}{\partial \sigma}$

We start with a straightforward differentiation

$\nu=\frac{\partial C_t}{\partial \sigma} = S_t \phi(d_1) \frac{\partial d_1}{\partial \sigma} - K e^{-r(T-t)} \phi(d_2) \frac{\partial d_2}{\partial \sigma}$

and

$d_1-d_2 = \sigma \sqrt{T-t}$

$\therefore \frac{\partial(d_1 - d_2)}{\partial \sigma} = \sqrt{T-t}$

$\nu=S_t \phi(d_1) \frac{\partial(d_1 - d_2)}{\partial \sigma}$

$\nu=S_t \phi(d_1) \sqrt{T-t}$

Rho $\left(\rho \right) = K(T-t) e^{-r(T-t)} \Theta(d_2)$

Proof

We wish to calculate the rate of change of a call option with the risk free rate of interest

That is we want $\frac{\partial C}{\partial r}$

We start with a straightforward differentiation

$\rho=S_t \phi(d_1) \frac{\partial d_1}{\partial r} + K(T-t) e^{-r(T-t)} \Theta(d_2) - K e^{-r(T-t)} \phi(d_2) \frac{\partial d_2}{\partial r}$

$\rho=S_t \phi(d_1) \frac{\partial (d_1-d_2)}{\partial r} + K(T-t) e^{-r(T-t)} \Theta(d_2)$

$\rho=S_t \phi(d_1) \times 0 + K(T-t) e^{-r(T-t)} \Theta(d_2)$

$\rho=K(T-t) e^{-r(T-t)} \Theta(d_2)$

The Gram-Charlier Greeks

We do not attempt to derive the Gram-Charlier Greeks as the algebra is very complicated and does not add to any fuller understanding of the subject

The formulas for a call option are as follows:

$\Delta=\Theta(d)-\frac{\gamma_{1T}}{3!}\phi(d)(1-d^2+3d\sigma_T - 2\sigma^2_T) + \frac{\gamma_{2T}}{4!} \phi(d)[3d(1-\sigma^2_T)+4d^2\sigma_T-d^3-4\sigma_T+3\sigma_T^2]$

$\Gamma=\frac{\phi(d)}{S \sigma_T} - \frac{\gamma_{1T}}{3!} \frac{\phi(d)}{S \sigma_T} [d^3 + d(2 \sigma^2_T-3)+3\sigma_T(1-d^2)]+\frac{\gamma_{2T}}{4!} \frac{\phi(d)}{s \sigma_T} [d^4 + 3(1-d^2)(1-2 \sigma^2_T)+4 d \sigma_T(3-d^2)-3d(\sigma^2_T+d)]$

$V=S \phi(d) \sqrt{T} + \frac{\gamma_{1T}}{3!} S \phi(d) \sqrt{T} [3 \sigma \sqrt{T}(1+d^2)-d(d^2+2 \sigma^2 T)] + \frac{\gamma_{2T}}{4!} S \phi(d) \sqrt{T} [(1-d^2+3 d \sigma \sqrt{T} - 3 \sigma^2 T)(\sigma d d' - 1)+2 d d' \sigma + 3 \sigma^2 T]$

where $d=\frac{ln\left(\frac{S}{K}\right)+ rT +\frac{\sigma^2_T}{2}}{\sigma_T}$ and $d'=\sqrt{T} - \frac{d}{\sigma}$

$\rho = KTe^{-rT}\Theta(d-\sigma_T) - \frac{\gamma_{1T}}{3!} ST \phi(d)[-d^2+2d \sigma_T+1] + \frac{\gamma_{2T}}{4!} S T \phi(d)[-d^3+3(d \sigma_T+1)(d-\sigma_T)]$

$\Theta=rKe^{-rT} \Theta(d - \sigma \sqrt{T}) - S \phi(d) \frac{\sigma}{2 \sqrt{T}}-\frac{\gamma_1}{3!} S \sigma \phi(d) \left[(d-2\sigma\sqrt{T})dd' + \left(\frac{\sigma}{\sqrt{T}}-d'\right)dd'+\frac{1}{2T^{3/2}}(1-d^2)-3\sigma d' + \frac{3 \sigma^2}{2\sqrt{T}}\right]$

where $d'=\frac{r+\sigma^2/2}{\sigma\sqrt{T}} - \frac{d}{2T}$

The Heston SV Greeks

Happily having defined a number of interim values such as $P$ and $f$ for the Heston SV model the definitions of the Greeks are 'relatively' straightforward

Using the same definitions as in the original model of the price of a call option the Greeks are as follows

$\Delta = P_1$

$\Gamma = \frac{1}{\pi S} \displaystyle\int_0^{\infty}Re\left[e^{-i \phi ln K}f_1 \right] d\phi$

$\rho = K e^{-r(T-t)}(T-t)P_2$

$V=S_t \frac{\partial P_1}{\partial \sigma} - Ke^{-r(T-t)} \frac{\partial P_2}{\partial \sigma}$

where $\frac{\partial P_j}{\partial \sigma} = \frac{1}{\pi} \displaystyle\int_0^{\infty} Re\left[\frac{e^{-i \phi ln K} f_j B_j}{i \phi} \right] d \phi$

$\Theta = -S \frac{\partial P_1}{\partial \tau} + K e^{-r \tau} \left(\frac{\partial P_2}{\partial \tau} - r P_2 \right)$

Numerical Greeks by Finite Differences