qGANs 을 이용한 옵션 가격 책정 (Option Pricing)¶

소개¶

In this notebook, we discuss how a Quantum Machine Learning Algorithm, namely a quantum Generative Adversarial Network (qGAN), can facilitate the pricing of a European call option. More specifically, a qGAN can be trained such that a quantum circuit models the spot price of an asset underlying a European call option. The resulting model can then be integrated into a Quantum Amplitude Estimation based algorithm to evaluate the expected payoff - see European Call Option Pricing. For further details on learning and loading random distributions by training a qGAN please refer to Quantum Generative Adversarial Networks for Learning and Loading Random Distributions. Zoufal, Lucchi, Woerner. 2019.

[1]:

import matplotlib.pyplot as plt
import numpy as np

from qiskit import Aer, QuantumRegister, QuantumCircuit
from qiskit.circuit import ParameterVector
from qiskit.circuit.library import TwoLocal, NormalDistribution
from qiskit.quantum_info import Statevector

from qiskit.aqua import aqua_globals, QuantumInstance
from qiskit.aqua.algorithms import IterativeAmplitudeEstimation
from qiskit.finance.applications import EuropeanCallExpectedValue

불확실성 모델¶

블랙 스콜스 모델은 유럽 호출 옵션에 대한 성숙도 :math:S_T의 현물 가격이 정상적으로 분산되어 있다고 가정합니다. 따라서, 로그 정규 분포의 샘플에 대해 qGAN을 훈련하고 결과를 옵션에 기반한 불확실성 모델로 사용할 수 있다. 다음에, 불확실성 모델을 로딩하는 양자 회로를 구성한다. 회로 출력 판독

\[\big| g_{\theta}\rangle = \sum_{j=0}^{2^n-1}\sqrt{p_{\theta}^{j}} \big| j \rangle ,\]

여기서, :math:` jin left{0, ldot, {2 ^ n-1} r] 에 대한 확률 :math:”p_ {theta} ^{j}” 는 대상 분포의 모델을 나타냅니다.

[2]:

# Set upper and lower data values
bounds = np.array([0.,7.])
# Set number of qubits used in the uncertainty model
num_qubits = 3

# Load the trained circuit parameters
g_params = [0.29399714, 0.38853322, 0.9557694, 0.07245791, 6.02626428, 0.13537225]

# Set an initial state for the generator circuit
init_dist = NormalDistribution(num_qubits, mu=1., sigma=1., bounds=bounds)

# construct the variational form
var_form = TwoLocal(num_qubits, 'ry', 'cz', entanglement='circular', reps=1)

# keep a list of the parameters so we can associate them to the list of numerical values
# (otherwise we need a dictionary)
theta = var_form.ordered_parameters

# compose the generator circuit, this is the circuit loading the uncertainty model
g_circuit = init_dist.compose(var_form)

기대수익 (Expected Payoff) 평가¶

이제 훈련된 불확실성 모델을 사용하여 퀀텀 진폭 추정으로 옵션의 페이오프 함수의 기대값을 평가할 수 있다.

[3]:

# set the strike price (should be within the low and the high value of the uncertainty)
strike_price = 2

# set the approximation scaling for the payoff function
c_approx = 0.25

# construct circuit for payoff function
european_call_objective = EuropeanCallExpectedValue(
    num_qubits,
    strike_price=strike_price,
    rescaling_factor=c_approx,
    bounds=bounds
)

확률 분포를 플롯합니다.¶

다음으로, 우리는 훈련된 확률 분포를 구성하고, 비교의 이유로, 또한 타겟 확률 분포를 구성한다.

[4]:

# Evaluate trained probability distribution
values = [bounds[0] + (bounds[1] - bounds[0]) * x / (2 ** num_qubits - 1) for x in range(2**num_qubits)]
uncertainty_model = g_circuit.assign_parameters(dict(zip(theta, g_params)))
amplitudes = Statevector.from_instruction(uncertainty_model).data

x = np.array(values)
y = np.abs(amplitudes) ** 2

# Sample from target probability distribution
N = 100000
log_normal = np.random.lognormal(mean=1, sigma=1, size=N)
log_normal = np.round(log_normal)
log_normal = log_normal[log_normal <= 7]
log_normal_samples = []
for i in range(8):
    log_normal_samples += [np.sum(log_normal==i)]
log_normal_samples = np.array(log_normal_samples / sum(log_normal_samples))

# Plot distributions
plt.bar(x, y, width=0.2, label='trained distribution', color='royalblue')
plt.xticks(x, size=15, rotation=90)
plt.yticks(size=15)
plt.grid()
plt.xlabel('Spot Price at Maturity $S_T$ (\$)', size=15)
plt.ylabel('Probability ($\%$)', size=15)
plt.plot(log_normal_samples,'-o', color ='deepskyblue', label='target distribution', linewidth=4, markersize=12)
plt.legend(loc='best')
plt.show()

../../_images/tutorials_finance_10_qgan_option_pricing_7_0.png

기대수익 (Expected Payoff) 평가¶

Now, the trained uncertainty model can be used to evaluate the expectation value of the option’s payoff function analytically and with Quantum Amplitude Estimation.

[5]:

# Evaluate payoff for different distributions
payoff = np.array([0,0,0,1,2,3,4,5])
ep = np.dot(log_normal_samples, payoff)
print("Analytically calculated expected payoff w.r.t. the target distribution:  %.4f" % ep)
ep_trained = np.dot(y, payoff)
print("Analytically calculated expected payoff w.r.t. the trained distribution: %.4f" % ep_trained)

# Plot exact payoff function (evaluated on the grid of the trained uncertainty model)
x = np.array(values)
y_strike = np.maximum(0, x - strike_price)
plt.plot(x, y_strike, 'ro-')
plt.grid()
plt.title('Payoff Function', size=15)
plt.xlabel('Spot Price', size=15)
plt.ylabel('Payoff', size=15)
plt.xticks(x, size=15, rotation=90)
plt.yticks(size=15)
plt.show()

Analytically calculated expected payoff w.r.t. the target distribution:  1.0591
Analytically calculated expected payoff w.r.t. the trained distribution: 0.9805

../../_images/tutorials_finance_10_qgan_option_pricing_9_1.png

[6]:

# construct A operator for QAE
european_call = european_call_objective.compose(uncertainty_model, front=True)

[7]:

# set target precision and confidence level
epsilon = 0.01
alpha = 0.05

# construct amplitude estimation
ae = IterativeAmplitudeEstimation(epsilon=epsilon, alpha=alpha,
                                  state_preparation=european_call,
                                  objective_qubits=[num_qubits],
                                  post_processing=european_call_objective.post_processing)

[8]:

result = ae.run(quantum_instance=Aer.get_backend('qasm_simulator'), shots=100)

[9]:

conf_int = np.array(result['confidence_interval'])
print('Exact value:        \t%.4f' % ep_trained)
print('Estimated value:    \t%.4f' % (result['estimation']))
print('Confidence interval:\t[%.4f, %.4f]' % tuple(conf_int))

Exact value:            0.9805
Estimated value:        1.0070
Confidence interval:    [0.9806, 1.0334]

[10]:

import qiskit.tools.jupyter
%qiskit_version_table
%qiskit_copyright

Version Information

Qiskit Software	Version
Qiskit	0.23.3
Terra	0.16.2
Aer	0.7.3
Ignis	0.5.1
Aqua	0.8.1
IBM Q Provider	0.11.1
System information
Python	3.8.5 (default, Sep 4 2020, 07:30:14) [GCC 7.3.0]
OS	Linux
CPUs	2
Memory (Gb)	3.7363624572753906
Thu Jan 28 21:29:04 2021 IST

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