QQuantLib.finance

QQuantLib.finance.classical_finance

This module contains several auxiliary functions used in quantitative classical finances

Authors: Alberto Pedro Manzano Herrero & Gonzalo Ferro Costas

tnbs.BTC_02_AE.QQuantLib.finance.classical_finance.bs_call_price(s_0: float, risk_free_rate: float, volatility: float, maturity: float, strike: float, **kwargs)

Computes the price for a European call option.

Notes

The formula is:

\[C(S, T) = S\Phi(d_1)-Ke^{-rT}\Phi(d_2)\]

Parameters

• s_0 (float) – current price of the underlying

• risk_free_rate (float) – risk free rate

• volatility (float) – the volatility

• maturity (float) – the maturity

• strike (float) – the strike

Returns

price – price of the European call option

Return type

float

tnbs.BTC_02_AE.QQuantLib.finance.classical_finance.bs_density(s_t: float, s_0: float, risk_free_rate: float, volatility: float, maturity: float, **kwargs)

Evaluates the Black-Scholes density function at s_t for a given set of parameters. The formula is:

Notes

\[\dfrac{1}{S_T\sigma\sqrt{2\pi T}}\exp\left(-\dfrac{ \ \left(\log(S_T)-\mu\right)}{2\sigma^2T}\right)\]

where \(\mu = (risk\_free\_rate-0.5\sigma)T+\log(S_0)\).

Parameters

• s_t (float) – point where we do the evaluation

• s_0 (float) – current price

• risk_free_rate (float) – risk free rate

• volatility (float) – the volatility

• maturity (float) – the maturity

Returns

density – value of the Black-Scholes density function in s_t

Return type

float

tnbs.BTC_02_AE.QQuantLib.finance.classical_finance.bs_digital_call_price(s_0: float, risk_free_rate: float, volatility: float, maturity: float, strike: float, coupon: float, **kwargs)

Computes the price for a digital(binary) call option.

Notes

The formula is:

\[DC(S, T) = e^{-rT}Coupon N(d_2)\]

Parameters

• s_0 (float) – current price of the underlying

• risk_free_rate (float) – risk free rate

• volatility (float) – the volatility

• maturity (float) – the maturity

• strike (float) – the strike

• coupon (float) – the amount received in case that the underlying pring exceeds the strike

Returns

price – price of the digital call option

Return type

float

tnbs.BTC_02_AE.QQuantLib.finance.classical_finance.bs_digital_put_price(s_0: float, risk_free_rate: float, volatility: float, maturity: float, strike: float, coupon: float, **kwargs)

Computes the price for a digital (binary) put option.

Notes

The formula is:

\[DC(S, T) = e^{-rT}Coupon N(-d_2)\]

Parameters

• s_0 (float) – current price of the underlying

• risk_free_rate (float) – risk free rate

• volatility (float) – the volatility

• maturity (float) – the maturity

• strike (float) – the strike

• coupon (float) – the amount received in case that the underlying pring exceeds the strike

Returns

price – price of the digital call option

Return type

float

tnbs.BTC_02_AE.QQuantLib.finance.classical_finance.bs_em_samples(s_0: float, risk_free_rate: float, volatility: float, maturity: float, number_samples: int, time_steps: int, **kwargs)

Computes samples from the approximated solution of the Black-Scholes SDE using the Euler-Maruyama discrimination.

Notes

The formula is:

\[S_{t+\Delta t} = S_t+rS_tdt+\sigma S_t N(0, 1)\sqrt{dt}\]

Parameters

• s_0 (float) – current price of the underlying

• risk_free_rate (float) – risk free rate

• volatility (float) – the volatility

• maturity (float) – the maturity

• strike (float) – the strike

• number_samples (int) – number of samples

• steps (time) – number of time steps

Returns

s_t – array of samples from the SDE.

Return type

numpy array of floats

tnbs.BTC_02_AE.QQuantLib.finance.classical_finance.bs_exact_samples(s_0: float, risk_free_rate: float, volatility: float, maturity: float, number_samples: int, **kwargs)

Computes samples from the exact solution of the Black-Scholes SDE.

Notes

The formula is:

\[S_T = S_0e^{\sigma*W_t+(risk_free_rate-\sigma^2/2)t}\]

Parameters

• s_0 (float) – current price of the underlying

• risk_free_rate (float) – risk free rate

• volatility (float) – the volatility

• maturity (float) – the maturity

• strike (float) – the strike

• number_samples (int) – number of samples

Returns

s_t – array of samples from the SDE.

Return type

numpy array of floats

tnbs.BTC_02_AE.QQuantLib.finance.classical_finance.bs_forward_price(s_0: float, risk_free_rate: float, maturity: float, strike: float, **kwargs)

Computes the price for a forward contract. Note that it doesn’t assume that we are at the begining of the lifetime of the contract

Notes

The formula is:

\[V(S, T) = S-Ke^{-r(T-t)}\]

Parameters

• s_0 (float) – current price of the underlying

• risk_free_rate (float) – risk free rate

• maturity (float) – time to expiration of the contract

• strike (float) – the strike

Returns

price – price of the forward

Return type

float

tnbs.BTC_02_AE.QQuantLib.finance.classical_finance.bs_probability(s_t: numpy.array, s_0: float, risk_free_rate: float, volatility: float, maturity: float, **kwargs)

Computes a discrete probability distribution from the Black-Scholes density function for a given set of parameters. This is done by evaluating the Black-Scholes density function in s_t and the normlising this result.

Parameters

• s_t (numpy array) – points where we define the discrete probability distribution

• s_0 (float) – current price

• risk_free_rate (float) – risk free rate

• volatility (float) – the volatility

• maturity (float) – the maturity

Returns

distribution – discrete probability distribution from Black-Scholes density

Return type

numpy array

tnbs.BTC_02_AE.QQuantLib.finance.classical_finance.bs_put_price(s_0: float, risk_free_rate: float, volatility: float, maturity: float, strike: float, **kwargs)

Computes the price for a European put option.

Notes

The formula is:

\[C(S, T) = Ke^{-rT}\Phi(-d_2)-S\Phi(-d_1)\]

Parameters

• s_0 (float) – current price of the underlying

• risk_free_rate (float) – risk free rate

• volatility (float) – the volatility

• maturity (float) – the maturity

• strike (float) – the strike

Returns

price – price of the European put option

Return type

float

tnbs.BTC_02_AE.QQuantLib.finance.classical_finance.bs_sde_solution(x_: numpy.array, s_0: float, risk_free_rate: float, volatility: float, maturity: float, **kwargs)

For a certain parametrization \(x\) it returns a value of the underlying \(S_T(x)\) and the probability density of that value of the underlying.

Notes

The formulas are:

\[S_T = S_0e^{\sigma x+(risk_free_rate-\sigma^2/2)t} p(S_T(x)) = N(x;mean = 0, variance = T)\]

Parameters

• x_ (numpy array) – parametrization

• s_0 (float) – current price of the underlying

• risk_free_rate (float) – risk free rate

• volatility (float) – the volatility

• maturity (float) – the maturity

Returns

• s_t (numpy array) – value of the underlying corresponding to parameter x

• probability_density (numpy array) – probability density corresponding to s_t

tnbs.BTC_02_AE.QQuantLib.finance.classical_finance.bs_tree(s_0: float, risk_free_rate: float, volatility: float, maturity: float, number_samples: int, time_steps: int, discretization: int, bounds: float, **kwargs)

Computes the probabilities of all possible pahts from the approximated solution of the Black-Scholes SDE using the Euler-Maruyama discretization for a given discretization of the Brownian motion.

Parameters

• s_0 (float) – current price of the underlying

• risk_free_rate (float) – risk free rate

• volatility (float) – the volatility

• maturity (float) – the maturity

• strike (float) – the strike

• number_samples (int) – number of samples

• steps (time) – number of time steps

• discretization (float) – number of points to build the discrete version of the gaussian density

• bounds (float) – bounds of the Gaussian density

Returns

s_t – array of samples from the SDE.

Return type

numpy array of floats

tnbs.BTC_02_AE.QQuantLib.finance.classical_finance.call_payoff(s_t: float, strike: float, **kwargs)

Computes the payoff of a European call option.

Notes

\[C(S_T, K) = \left(S_T-K, 0\right)^+\]

Parameters

• s_t (float) – price

• strike (float) – the strike

Returns

payoff – the payoff

Return type

float

tnbs.BTC_02_AE.QQuantLib.finance.classical_finance.digital_call_payoff(s_t: float, strike: float, coupon: float = 1.0, **kwargs)

Computes the payoff for a digital(binary) call option. The formula is:

Notes

\[DC(S_T, K, Coupon) = Coupon\mathbb{1}_{\{S_T-K\}}\]

Parameters

• s_t (float) – current price of the underlying

• strike (float) – the strike

• coupon (float) – the amount received in case that the underlying pring exceeds the strike

Returns

payoff – payoff of the european digital call option

Return type

float

tnbs.BTC_02_AE.QQuantLib.finance.classical_finance.digital_put_payoff(s_t: float, strike: float, coupon: float = 1.0, **kwargs)

Computes the payoff for a digital(binary) put option.

Notes

The formula is:

\[DC(S_T, K, Coupon) = Coupon\mathbb{1}_{\{K-S_T\}}\]

Parameters

• s_t (float) – current price of the underlying

• strike (float) – the strike

• coupon (float) – the amount received in case that the underlying pring exceeds the strike

Returns

payoff – payoff of the european digital call option

Return type

float

tnbs.BTC_02_AE.QQuantLib.finance.classical_finance.futures_payoff(s_t: float, strike: float, **kwargs)

Computes the payoff of a futures contract.

Notes

\[F(S_T, K) = \left(S_T-K, 0\right)\]

Parameters

• s_t (float) – price

• strike (float) – the strike

Returns

payoff – the payoff

Return type

float

tnbs.BTC_02_AE.QQuantLib.finance.classical_finance.geometric_sum(base: float, exponent: int, coeficient: float = 1.0, **kwargs)

Computes a geometric sum

Notes

\[\begin{split}s = a+a*base+a*base^2+a*base^3+...+a*base^{exponent} = \\ = \sum_{k=0}^{exponent}a*base^k = \ a\big(\frac{1-base^{exponent+1}}{1-base}\big)\end{split}\]

Parameters

• base (float) – base of the geometric sum

• exponent (int) – maximum exponent for the geometric sum

• coeficient (float) – coefficient for the geometric sum

Returns

return – result of the geometric sum

Return type

float

tnbs.BTC_02_AE.QQuantLib.finance.classical_finance.put_payoff(s_t: float, strike: float, **kwargs)

Computes the payoff of a European put option.

Notes

\[P(S_T, K) = \left(K-S_T, 0\right)^+\]

Parameters

• s_t (float) – price

• strike (float) – the strike

Returns

payoff – the payoff

Return type

float

QQuantLib.finance.payoff_class

Class definition for the PayOff

Authors: Alberto Pedro Manzano Herrero & Gonzalo Ferro

class tnbs.BTC_02_AE.QQuantLib.finance.payoff_class.PayOff(**kwargs)

Bases: object

Class for selecting derivative options and configuring them.

Parameters

kwargs (dictionary) –

Dictionary for selecting and configuring the derivative option. ç Implemented keys:

• strike: float

  strike of the option.

• coupon: float

  only valid for Digital Options

• pay_offstring

  type of pay_off function to load

get_pay_off()

Select of a PayOff

QQuantLib.finance.probability_class

Definition for DensityProbability Class.

Authors: Alberto Pedro Manzano Herrero & Gonzalo Ferro

class tnbs.BTC_02_AE.QQuantLib.finance.probability_class.DensityProbability(probability_type: str, **kwargs)

Bases: object

Class for selecting pay off functions algorithm

Parameters

• probability_type (string) – type of probability density function to load

• kwargs (dictionary) –

  Dictionary for configuring the asset and the probability used for simulating its behaviour.Implemented keys:

  s_0float

  initial value of the asset

  risk_free_ratefloat

  risk free ratio

  maturityfloat

  time where the probability wants to be calculated.

  volatiliyfloat

  volatility of the asset.

static get_density(probability_type, **kwargs)

Create the probability function

Parameters

• probability_type (string) – type of probability density function to load

• kwargs (dictionary) –

  with necessary information for configuring the probability density

  s_0float

  initial value of the asset

  risk_free_ratefloat

  risk free ratio

  maturityfloat

  time where the probability wants to be calculated

  volatiliyfloat

  volatility of the asset

static get_density_prob(probability_type, **kwargs)

Configures a probability density

Parameters

• probability_type (string) – type of probability density function to load

• kwargs (dictionary) –

  with necessary information for configuring the probability density.

  s_0float

  initial value of the asset

  risk_free_ratefloat

  risk free ratio

  maturityfloat

  time where the probability wants to be calculated

  volatiliyfloat

  volatility of the asset

QQuantLib.finance.quantum_integration

This module contains a function for solving integrals using Amplitude Estimation techniques. The function uses the Encoding class for loading data into the quantum circuit by creating an oracle and the AE class for getting the amplitude of a selected state using AE techniques. The function deals with all the normalisations needed for transform this amplitude into the desired integral.

Authors: Alberto Pedro Manzano Herrero & Gonzalo Ferro

tnbs.BTC_02_AE.QQuantLib.finance.quantum_integration.q_solve_integral(**kwargs)

Function for solving an integral using quantum amplitude estimation techniques.

Parameters

kwars (dictionary) –

Dictionary for configuring the integration to compute and the AE technique used for getting the solution.

array_functionnumpy array

numpy array with the desired function for encoding into the Quantum Circuit.

encodingint

Selecting the encode protocol

array_probabilitynumpy array

numpy array with the desired probability for encoding into the Quantum Circuit. It can be None (uniform distribution will be used)

ae_typestring

string with the desired AE algorithm: MLAE, CQPEAE, IQPEAE, IQAE, RQAE

Note

Other kwargs input dictionary keys will be related with the encoding of the integral into the quantum circuit (see QQuantLib.DL.encoding_protocols) and for the configuration of the AE algorithm used (see QQuantLib.AE.ae_class)

Returns

• ae_estimation (pandas DataFrame) – DataFrame with the desired integral computation.

• solver_ae (objet based on the AE class)