black_76

module jetblack_options.european.black_76

Summary

Black (1976) Options on futures/forwards

Description

• The discounted futures price $F$,
• Strike price $K$,
• Risk-free rate $r$,
• Annual dividend yield $q$,
• Time to maturity $\tau =T-t$
• Volatility $\sigma$.

Most of the formula use one or both of the following terms.

$$d_1=\frac{\ln (F/K)+({\sigma }^2/2)T}{\sigma \sqrt{T}}$$ $$d_2=\frac{\ln (F/K)-({\sigma }^2/2)T}{\sigma \sqrt{T}}=d_1-\sigma \sqrt{T}$$ $$\phi (x)\&=\frac{1}{\sqrt{2\pi }}{e}^{-\frac{1}{2}{x}^2}$$ $$\Phi (x)\&=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^x{e}^{-\frac{1}{2}{y}^2}dy=1-\frac{1}{\sqrt{2\pi }}{\int }_x^{\infty }{e}^{-\frac{1}{2}{y}^2}dy$$

function jetblack_options.european.black_76.delta

Summary

The sensitivity of the option to a change in the asset price

Description

using Black 76.

For the call.

$$\frac{\partial C}{\partial S}=e^{-r\tau }\Phi (d_1)$$

For the put.

$$\frac{\partial P}{\partial S}=-e^{-r\tau }\Phi (-d_1)$$

jetblack_options.european.black_76.delta(

is_call: bool,

F: float,

K: float,

T: float,

r: float,

v: float

) -> float

Parameters

is_call: bool

True for a call, false for a put.

F: float

The current futures price.

K: float

The strike price.

T: float

The time to expiry in years.

r: float

The risk free rate.

v: float

The volatility.

Returns

float: The delta.

---

function jetblack_options.european.black_76.gamma

Summary

The second derivative to the change in asset price using Black 76.

Description

The gamma for both calls and puts.

$$\frac{{\partial }^{2}V}{\partial S^2}=e^{-r\tau }\frac{\phi (d_1)}{F\sigma \sqrt{\tau }}=K{e}^{-r\tau }\frac{\phi (d_2)}{F^2\sigma \sqrt{\tau }}$$

jetblack_options.european.black_76.gamma(

F: float,

K: float,

T: float,

r: float,

v: float

) -> float

Parameters

F: float

The current futures price.

K: float

The strike price.

T: float

The time to expiry in years.

r: float

The risk free rate.

v: float

The volatility.

Returns

float: The gamma.

---

function jetblack_options.european.black_76.ivol

Summary

Calculate the volatility of a Black 76 option that is implied by the price.

jetblack_options.european.black_76.ivol(

is_call: bool,

F: float,

K: float,

T: float,

r: float,

p: float,

*,

max_iterations: int, Optional,

epsilon: float, Optional

) -> float

Parameters

is_call: bool

True for a call, false for a put.

F: float

The current asset price.

K: float

The option strike price

T: float

The time to maturity of the option in years.

r: float

The risk free rate.

p: float

The option price.

max_iterations: int, Optional

The maximum number of iterations before a price is returned. Defaults to 20.

epsilon: float, Optional (optional)

The largest acceptable error. Defaults to 1e-8.

Returns

float: The implied volatility.

---

function jetblack_options.european.black_76.make_numeric_greeks

Summary

Make a class to generate greeks numerically using finite difference methods.

jetblack_options.european.black_76.make_numeric_greeks(

is_call: bool

) -> NumericGreeks

Parameters

is_call: bool

If true the options is a call; otherwise it is a put.

Returns

NumericGreeks: A class which can generate Greeks using finite difference methods.

---

function jetblack_options.european.black_76.price

Summary

Fair value of a futures/forward using Black 76.

Description

For a call:

$$C=e^{-r\tau }[F\Phi (d_1)-K\Phi (d_2)]$$

For a put:

$$P=e^{-r\tau }[K\Phi (-d_2)-F\Phi (-d_1)]$$

jetblack_options.european.black_76.price(

is_call: bool,

F: float,

K: float,

T: float,

r: float,

v: float

) -> float

Parameters

is_call: bool

True for a call, false for a put.

F: float

The price of the future.

K: float

The strike price.

T: float

The time to expiry in years.

r: float

The risk free rate.

v: float

The asset volatility.

Returns

float: The option price.

---

function jetblack_options.european.black_76.rho

Summary

The sensitivity of the option price to a change in the risk free rate

Description

using Black 76.

For a call:

$$\frac{\partial C}{\partial r}=-\tau e^{-r\tau }[F\Phi (d_1)-K\Phi (d_2)]$$

For a put:

$$\frac{\partial P}{\partial r}=-\tau e^{-r\tau }[K\Phi (-d_2)-F\Phi (-d_1)]$$

jetblack_options.european.black_76.rho(

is_call: bool,

F: float,

K: float,

T: float,

r: float,

v: float

) -> float

Parameters

is_call: bool

True for a call, false for a put.

F: float

The price of the future.

K: float

The strike price.

T: float

The time to expiry in years.

r: float

The risk free rate.

v: float

The asset volatility.

Returns

float: The rho.

---

function jetblack_options.european.black_76.theta

Summary

The change in the value of the option with respect to time to expiry

Description

using Black 76.

For the call.

$$\frac{\partial C}{\partial T}-\frac{F{e}^{-r\tau }\phi (d_1)\sigma }{2\sqrt{\tau }}-rK{e}^{-r\tau }\Phi (d_2)+rF{e}^{-r\tau }\Phi (d_1)$$

For the put.

$$\frac{\partial P}{\partial T}-\frac{F{e}^{-r\tau }\phi (d_1)\sigma }{2\sqrt{\tau }}+rK{e}^{-r\tau }\Phi (-d_2)-rF{e}^{-r\tau }\Phi (-d_1)$$

jetblack_options.european.black_76.theta(

is_call: bool,

F: float,

K: float,

T: float,

r: float,

v: float

) -> float

Parameters

is_call: bool

True for a call, false for a put.

F: float

The current futures price.

K: float

The strike price.

T: float

The time to expiry in years.

r: float

The risk free rate.

v: float

The volatility.

Returns

float: The theta.

---

function jetblack_options.european.black_76.vanna

Summary

The sensitivity of the option value to the underlying

Description

asset price and the volatility.

For both calls and puts.

$$\frac{{\partial }^{2}V}{\partial F\partial \sigma }=-e^{-r\tau }\phi (d_1)\frac{d_2}{\sigma }=\frac{𝒱}{F}[1-\frac{d_1}{\sigma \sqrt{\tau }}]$$

jetblack_options.european.black_76.vanna(

F: float,

K: float,

T: float,

r: float,

v: float

) -> float

Parameters

F: float

The price of the future.

K: float

The strike price.

T: float

The time to expiry in years.

r: float

The risk free rate.

v: float

The asset volatility.

Returns

float: The vanna.

---

function jetblack_options.european.black_76.vega

Summary

The sensitivity of the options price or a change in the asset volatility

Description

using Black 76.

For both calls and puts.

$$\frac{\partial V}{\partial \sigma }=F{e}^{-r\tau }\phi (d_1)\sqrt{\tau }=K{e}^{-r\tau }\phi (d_2)\sqrt{\tau }$$

jetblack_options.european.black_76.vega(

F: float,

K: float,

T: float,

r: float,

v: float

) -> float

Parameters

F: float

The current futures price.

K: float

The strike price.

T: float

The time to expiry in years.

r: float

The risk free rate.

v: float

The volatility.

Returns

float: The vega.

---

function jetblack_options.european.black_76.vomma

Summary

The second order sensitivity to volatility.

Description

For both puts and calls.

$$\frac{{\partial }^{2}V}{\partial {\sigma }^2}=F{e}^{-r\tau }\phi (d_1)\sqrt{\tau }\frac{d_1{d}_2}{\sigma }=𝒱\frac{d_1{d}_2}{\sigma }$$

jetblack_options.european.black_76.vomma(

F: float,

K: float,

T: float,

r: float,

v: float

) -> float

Parameters

F: float

The price of the future.

K: float

The strike price.

T: float

The time to expiry in years.

r: float

The risk free rate.

v: float

The asset volatility.

Returns

float: The vomma

---