black_scholes_merton

module jetblack_options.european.black_scholes_merton

Summary

Black-Scholes-Merton options pricing formulae using dividend yield.

Description

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

where:

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

function jetblack_options.european.black_scholes_merton.break_even_probability

jetblack_options.european.black_scholes_merton.break_even_probability(

is_call: bool,

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

is_call: bool

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.carry

Summary

Sensitivity to the cost of carry.

jetblack_options.european.black_scholes_merton.carry(

is_call: bool,

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

is_call: bool

True for a call, false for a put.

S: float

The asset price.

K: float

The strike price.

T: float

The time to expiry in years.

r: float

The risk free rate.

q: float

The dividend yield.

v: float

The asset volatility.

Returns

float: The carry.

---

function jetblack_options.european.black_scholes_merton.charm

jetblack_options.european.black_scholes_merton.charm(

is_call: bool,

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

is_call: bool

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.ddelta_dvol_dvol

jetblack_options.european.black_scholes_merton.ddelta_dvol_dvol(

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.delta

Summary

The sensitivity of the open to a change in the asset price.

Description

Call $\Delta$ $e^{-q\tau }\Phi (d_1)$

Put $\Delta$ $-e^{-q\tau }\Phi (-d_1)$

jetblack_options.european.black_scholes_merton.delta(

is_call: bool,

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

is_call: bool

True for a call, false for a put.

S: 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.

q: float

The dividend yield.

v: float

The volatility of the asset.

Returns

float: the delta.

---

function jetblack_options.european.black_scholes_merton.delta_from_in_the_money_prob

jetblack_options.european.black_scholes_merton.delta_from_in_the_money_prob(

is_call: bool,

T: float,

r: float,

q: float,

v: float,

in_the_money_prob: float

) -> float

Parameters

is_call: bool

T: float

r: float

q: float

v: float

in_the_money_prob: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.delta_mirror_call_put_strike

jetblack_options.european.black_scholes_merton.delta_mirror_call_put_strike(

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.delta_mirror_strike

jetblack_options.european.black_scholes_merton.delta_mirror_strike(

S: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.dgamma_dspot

jetblack_options.european.black_scholes_merton.dgamma_dspot(

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.dgamma_dtime

jetblack_options.european.black_scholes_merton.dgamma_dtime(

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.dgamma_dvol

jetblack_options.european.black_scholes_merton.dgamma_dvol(

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.dgammap_dspot

jetblack_options.european.black_scholes_merton.dgammap_dspot(

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.dgammap_dtime

jetblack_options.european.black_scholes_merton.dgammap_dtime(

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.dgammap_dvol

jetblack_options.european.black_scholes_merton.dgammap_dvol(

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.dvega_dtime

jetblack_options.european.black_scholes_merton.dvega_dtime(

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.dvegap_dvol

jetblack_options.european.black_scholes_merton.dvegap_dvol(

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.dvomma_dvol

jetblack_options.european.black_scholes_merton.dvomma_dvol(

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.dzeta_dtime

jetblack_options.european.black_scholes_merton.dzeta_dtime(

is_call: bool,

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

is_call: bool

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.dzeta_dvol

jetblack_options.european.black_scholes_merton.dzeta_dvol(

is_call: bool,

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

is_call: bool

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.elasticity

Summary

The option elasticity.

jetblack_options.european.black_scholes_merton.elasticity(

is_call: bool,

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

is_call: bool

True for a call, false for a put.

S: float

The asset price.

K: float

The strike price.

T: float

The time to expiry in years.

r: float

The risk free rate.

q: float

The dividend yield.

v: float

The asset volatility.

Returns

float: The elasticity.

---

function jetblack_options.european.black_scholes_merton.forward_delta

jetblack_options.european.black_scholes_merton.forward_delta(

is_call: bool,

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

is_call: bool

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.futures_rho

jetblack_options.european.black_scholes_merton.futures_rho(

is_call: bool,

S: float,

K: float,

T: float,

r: float,

v: float

) -> float

Parameters

is_call: bool

S: float

K: float

T: float

r: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.gamma

Summary

The second derivative to the change in the asset price.

Description

Γ$$e−qτφ(d1)Sστ=Ke−rτφ(d2)S2στ

jetblack_options.european.black_scholes_merton.gamma(

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: 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.

q: float

The dividend yield.

v: float

The volatility of the asset.

Returns

float: The gamma.

---

function jetblack_options.european.black_scholes_merton.gamma_from_delta

jetblack_options.european.black_scholes_merton.gamma_from_delta(

S: float,

T: float,

q: float,

v: float,

delta_: float

) -> float

Parameters

S: float

T: float

q: float

v: float

delta_: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.gammap

jetblack_options.european.black_scholes_merton.gammap(

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.gammap_from_delta

jetblack_options.european.black_scholes_merton.gammap_from_delta(

S: float,

T: float,

q: float,

v: float,

delta_: float

) -> float

Parameters

S: float

T: float

q: float

v: float

delta_: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.in_the_money_prob_from_delta

jetblack_options.european.black_scholes_merton.in_the_money_prob_from_delta(

is_call: bool,

T: float,

r: float,

q: float,

v: float,

delta_: float

) -> float

Parameters

is_call: bool

T: float

r: float

q: float

v: float

delta_: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.in_the_money_probability

jetblack_options.european.black_scholes_merton.in_the_money_probability(

is_call: bool,

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

is_call: bool

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.ivol

Summary

Calculate the volatility of an option that is implied by the price.

jetblack_options.european.black_scholes_merton.ivol(

is_call: bool,

S: float,

K: float,

T: float,

r: float,

q: float,

p: float,

*,

max_iterations: int, Optional,

epsilon: float, Optional

) -> float

Parameters

is_call: bool

True for a call, false for a put.

S: 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.

q: float

The dividend yield.

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_scholes_merton.make_numeric_greeks

Summary

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

jetblack_options.european.black_scholes_merton.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_scholes_merton.max_ddelta_dvol_asset

jetblack_options.european.black_scholes_merton.max_ddelta_dvol_asset(

is_lower: bool,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

is_lower: bool

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.max_ddelta_dvol_strike

jetblack_options.european.black_scholes_merton.max_ddelta_dvol_strike(

is_lower: bool,

S: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

is_lower: bool

S: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.max_gamma_at_S

jetblack_options.european.black_scholes_merton.max_gamma_at_S(

x: float,

r: float,

q: float,

T: float,

v: float

) -> float

Parameters

x: float

r: float

q: float

T: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.max_gamma_vega_at_X

jetblack_options.european.black_scholes_merton.max_gamma_vega_at_X(

S: float,

r: float,

q: float,

T: float,

v: float

) -> float

Parameters

S: float

r: float

q: float

T: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.max_vega_at_S

jetblack_options.european.black_scholes_merton.max_vega_at_S(

K: float,

r: float,

q: float,

T: float,

v: float

) -> float

Parameters

K: float

r: float

q: float

T: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.phi

jetblack_options.european.black_scholes_merton.phi(

is_call: bool,

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

is_call: bool

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.price

Summary

The fair value of a European option, using Black-Scholes-Merton.

Description

Call price: $S{e}^{-q\tau }\Phi (d_1)-e^{-r\tau }K\Phi (d_2)$

Put price: $e^{-r\tau }K\Phi (-d_2)-S{e}^{-q\tau }\Phi (-d_1)$

jetblack_options.european.black_scholes_merton.price(

is_call: bool,

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

is_call: bool

True for a call, false for a put.

S: 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.

q: float

The dividend yield.

v: float

The volatility of the asset.

Returns

float: The price of the options.

---

function jetblack_options.european.black_scholes_merton.probability_mirror_strike

jetblack_options.european.black_scholes_merton.probability_mirror_strike(

S: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.profit_loss_std

jetblack_options.european.black_scholes_merton.profit_loss_std(

TypeFlag: Literal['a', 'p'],

is_call: bool,

S: float,

K: float,

T: float,

r: float,

q: float,

v: float,

NHedges: int

) -> float

Parameters

TypeFlag: Literal['a', 'p']

is_call: bool

S: float

K: float

T: float

r: float

q: float

v: float

NHedges: int

Returns

float:

---

function jetblack_options.european.black_scholes_merton.rho

Summary

The sensitivity of the option price to the risk free rate.

Description

Call $\rho$ is $K\tau e^{-r\tau }\Phi (d_2)$

Put $\rho$ is $-K\tau e^{-r\tau }\Phi (-d_2)$

Useful for all options except futures options which should use futures_rho.

jetblack_options.european.black_scholes_merton.rho(

is_call: bool,

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

is_call: bool

True for a call, false for a put.

S: float

The asset price.

K: float

The strike price.

T: float

The time to expiry in years.

r: float

The risk free rate.

q: float

The dividend yield.

v: float

The asset volatility.

Returns

float: The rho.

---

function jetblack_options.european.black_scholes_merton.risk_neutral_density

jetblack_options.european.black_scholes_merton.risk_neutral_density(

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.rnd_from_in_the_money_prob

jetblack_options.european.black_scholes_merton.rnd_from_in_the_money_prob(

K: float,

T: float,

r: float,

v: float,

in_the_money_prob: float

) -> float

Parameters

K: float

T: float

r: float

v: float

in_the_money_prob: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.saddle_gamma

jetblack_options.european.black_scholes_merton.saddle_gamma(

K: float,

q: float,

v: float

) -> float

Parameters

K: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.strike_delta

jetblack_options.european.black_scholes_merton.strike_delta(

is_call: bool,

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

is_call: bool

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.strike_from_delta

jetblack_options.european.black_scholes_merton.strike_from_delta(

is_call: bool,

S: float,

T: float,

r: float,

q: float,

v: float,

delta_: float

) -> float

Parameters

is_call: bool

S: float

T: float

r: float

q: float

v: float

delta_: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.strike_from_in_the_money_prob

jetblack_options.european.black_scholes_merton.strike_from_in_the_money_prob(

is_call: bool,

S: float,

v: float,

T: float,

r: float,

q: float,

in_the_money_prob: float

) -> float

Parameters

is_call: bool

S: float

v: float

T: float

r: float

q: float

in_the_money_prob: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.theta

Summary

The theta or time decay of the value of the option.

Description

CallΘ$$−e−qτSφ(d1)σ2τ−rKe−rτΦ(d2)+qSe−qτΦ(d1)

jetblack_options.european.black_scholes_merton.theta(

is_call: bool,

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

is_call: bool

True for a call, false for a put.

S: float

The asset price.

K: float

The strike price.

T: float

The time to expiry in years.

r: float

The risk free rate.

q: float

The dividend yield.

v: float

The asset volatility.

Returns

float: The theta.

---

function jetblack_options.european.black_scholes_merton.theta_driftless

jetblack_options.european.black_scholes_merton.theta_driftless(

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.vanna

Summary

The sensitivity to the spot price and volatility.

Description

−e−qτφ(d1)d2σ=𝒱S[1−d1στ]

jetblack_options.european.black_scholes_merton.vanna(

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: float

The asset price.

K: float

The strike price.

T: float

The time to expiry in years.

r: float

The risk free rate.

q: float

The dividend yield.

v: float

The asset volatility.

Returns

float: The vanna.

---

function jetblack_options.european.black_scholes_merton.variance_delta

jetblack_options.european.black_scholes_merton.variance_delta(

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.variance_ultima

jetblack_options.european.black_scholes_merton.variance_ultima(

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.variance_vega

jetblack_options.european.black_scholes_merton.variance_vega(

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.variance_vomma

jetblack_options.european.black_scholes_merton.variance_vomma(

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.vega

Summary

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

Description

𝒱$is$Se−qτφ(d1)τ=Ke−rτφ(d2)τ

jetblack_options.european.black_scholes_merton.vega(

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: 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.

q: float

The dividend yield.

v: float

The volatility of the asset.

Returns

float: The vega

---

function jetblack_options.european.black_scholes_merton.vega_from_delta

jetblack_options.european.black_scholes_merton.vega_from_delta(

S: float,

T: float,

q: float,

delta_: float

) -> float

Parameters

S: float

T: float

q: float

delta_: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.vega_leverage

jetblack_options.european.black_scholes_merton.vega_leverage(

is_call: bool,

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

is_call: bool

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.vegap

jetblack_options.european.black_scholes_merton.vegap(

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.vegap_from_delta

jetblack_options.european.black_scholes_merton.vegap_from_delta(

S: float,

T: float,

q: float,

v: float,

delta_: float

) -> float

Parameters

S: float

T: float

q: float

v: float

delta_: float

Returns

float:

---

function jetblack_options.european.black_scholes_merton.vomma

jetblack_options.european.black_scholes_merton.vomma(

S: float,

K: float,

T: float,

r: float,

q: float,

v: float

) -> float

Parameters

S: float

K: float

T: float

r: float

q: float

v: float

Returns

float:

---