참고

이 페이지는 “tutorials/finance/04_유로퓸 _option_pricing.ipynb” 로부터 생성되었다.

IBM 퀀텀 랩 에서 대화식으로 실행하시오.

가격 책정 유럽 통화 풋 옵션

소개

Suppose a European put option with strike price \(K\) and an underlying asset whose spot price at maturity \(S_T\) follows a given random distribution. The corresponding payoff function is defined as:

\[\max\{K - S_T, 0\}\]

다음에서, 진폭 추정에 기초한 양자 알고리즘은 옵션에 대하여 예상되는 페이오프, 즉 할인 전에 적정 가격을 추정하는데 사용된다.

\[\mathbb{E}\left[ \max\{K - S_T, 0\} \right]\]

뿐만 아니라 대응하는 :math:”Delta”, 즉, 현물 가격에 대한 옵션 가격의 미분은 다음과 같이 정의된다:

\[\Delta = -\mathbb{P}\left[S_T \leq K\right]\]

The approximation of the objective function and a general introduction to option pricing and risk analysis on quantum computers are given in the following papers:

• 양자 위험 분석. [Woerner, Egger. 2018.]

• 양자컴퓨터를 이용한 옵션 가격 책정법 (Stamatopoulos et al. 2019.)

import matplotlib.pyplot as plt
%matplotlib inline
import numpy as np

from qiskit import Aer
from qiskit.aqua.algorithms import IterativeAmplitudeEstimation
from qiskit.circuit.library import LogNormalDistribution, LinearAmplitudeFunction

불확실성 모델

We construct a circuit factory to load a log-normal random distribution into a quantum state. The distribution is truncated to a given interval \([\text{low}, \text{high}]\) and discretized using \(2^n\) grid points, where \(n\) denotes the number of qubits used. The unitary operator corresponding to the circuit factory implements the following:

\[\big|0\rangle_{n} \mapsto \big|\psi\rangle_{n} = \sum_{i=0}^{2^n-1} \sqrt{p_i}\big|i\rangle_{n},\]

where \(p_i\) denote the probabilities corresponding to the truncated and discretized distribution and where \(i\) is mapped to the right interval using the affine map:

\[\{0, \ldot, 2 ^ n-1\} \ni \mapsto \frac {\text{high} - \text{low}} {2 ^ n-1} * i+ \text{low} \in [\text{low}, \text{high}].\]

# number of qubits to represent the uncertainty
num_uncertainty_qubits = 3

# parameters for considered random distribution
S = 2.0 # initial spot price
vol = 0.4 # volatility of 40%
r = 0.05 # annual interest rate of 4%
T = 40 / 365 # 40 days to maturity

# resulting parameters for log-normal distribution
mu = ((r - 0.5 * vol**2) * T + np.log(S))
sigma = vol * np.sqrt(T)
mean = np.exp(mu + sigma**2/2)
variance = (np.exp(sigma**2) - 1) * np.exp(2*mu + sigma**2)
stddev = np.sqrt(variance)

# lowest and highest value considered for the spot price; in between, an equidistant discretization is considered.
low  = np.maximum(0, mean - 3*stddev)
high = mean + 3*stddev

# construct A operator for QAE for the payoff function by
# composing the uncertainty model and the objective
uncertainty_model = LogNormalDistribution(num_uncertainty_qubits, mu=mu, sigma=sigma**2, bounds=(low, high))

# plot probability distribution
x = uncertainty_model.values
y = uncertainty_model.probabilities
plt.bar(x, y, width=0.2)
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.show()

페이오프 함수

만기시 현물 가격의 합 \((S_T^1 + S_T^2)\) 보다 적고 선형 적으로 증가하는 한 지불 함수는 0이다. 만약 \((S_T^1 + S_T^2) \geq K\) 라면, 구현에서는 먼저 가중치 합산 연산자를 사용하여 현물 가격의 합계를 ancilla 레지스터로 계산 한 다음 비교기를 사용하여 ancilla 큐 비트를 \(\big|0\rangle\)big|1rangle`로 뒤집는다.이 ancilla는 페이오프함수의 선형 부분을 제어하는 ​​데 사용된다.

The linear part itself is then approximated as follows. We exploit the fact that \(\sin^2(y + \pi/4) \approx y + 1/2\) for small \(|y|\). Thus, for a given approximation rescaling scaling factor \(c_\text{approx} \in [0, 1]\) and \(x \in [0, 1]\) we consider

\[\sin^2( \pi/2 * c_\text{approx} * ( x - 1/2 ) + \pi/4) \approx \pi/2 * c_\text{approx} * ( x - 1/2 ) + 1/2\]

for small \(c_\text{approx}\).

다음과 같이 작동하는 연산자를 쉽게 구성할 수 있습니다.

\[\big|x\rangle \big|0\rangle \mapsto \big|x\rangle \left( \cos(a*x+b) \big|0\rangle + \sin(a*x+b) \big|1\rangle \right),\]

제어된 Y 회전을 사용하였다.

Eventually, we are interested in the probability of measuring \(\big|1\rangle\) in the last qubit, which corresponds to \(\sin^2(a*x+b)\). Together with the approximation above, this allows to approximate the values of interest. The smaller we choose \(c_\text{approx}\), the better the approximation. However, since we are then estimating a property scaled by \(c_\text{approx}\), the number of evaluation qubits \(m\) needs to be adjusted accordingly.

For more details on the approximation, we refer to: Quantum Risk Analysis. Woerner, Egger. 2018.

# set the strike price (should be within the low and the high value of the uncertainty)
strike_price = 2.126

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

# setup piecewise linear objective fcuntion
breakpoints = [low, strike_price]
slopes = [-1, 0]
offsets = [strike_price - low, 0]
f_min = 0
f_max = strike_price - low
european_put_objective = LinearAmplitudeFunction(
    num_uncertainty_qubits,
    slopes,
    offsets,
    domain=(low, high),
    image=(f_min, f_max),
    breakpoints=breakpoints,
    rescaling_factor=rescaling_factor,
)

# construct A operator for QAE for the payoff function by
# composing the uncertainty model and the objective
european_put = european_put_objective.compose(uncertainty_model, front=True)

# plot exact payoff function (evaluated on the grid of the uncertainty model)
x = uncertainty_model.values
y = np.maximum(0, strike_price - x)
plt.plot(x, y, '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()

# evaluate exact expected value (normalized to the [0, 1] interval)
exact_value = np.dot(uncertainty_model.probabilities, y)
exact_delta = -sum(uncertainty_model.probabilities[x <= strike_price])
print('exact expected value:\t%.4f' % exact_value)
print('exact delta value:   \t%.4f' % exact_delta)

exact expected value:   0.1709
exact delta value:      -0.8193

기대수익 (Expected Payoff) 평가

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

# construct amplitude estimation
ae = IterativeAmplitudeEstimation(epsilon=epsilon, alpha=alpha,
                                  state_preparation=european_put,
                                  objective_qubits=[num_uncertainty_qubits],
                                  post_processing=european_put_objective.post_processing)

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

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

Exact value:            0.1709
Estimated value:        0.1801
Confidence interval:    [0.1742, 0.1860]

델타 (Delta) 평가

The Delta is a bit simpler to evaluate than the expected payoff. Similarly to the expected payoff, we use a comparator circuit and an ancilla qubit to identify the cases where \(S_T \leq K\). However, since we are only interested in the (negative) probability of this condition being true, we can directly use this ancilla qubit as the objective qubit in amplitude estimation without any further approximation.

# setup piecewise linear objective fcuntion
breakpoints = [low, strike_price]
slopes = [0, 0]
offsets = [1, 0]
f_min = 0
f_max = 1

european_put_delta_objective = LinearAmplitudeFunction(
    num_uncertainty_qubits,
    slopes,
    offsets,
    domain=(low, high),
    image=(f_min, f_max),
    breakpoints=breakpoints
)

# construct circuit factory for payoff function
european_put_delta = european_put_delta_objective.compose(uncertainty_model, front=True)

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

# construct amplitude estimation
ae_delta = IterativeAmplitudeEstimation(epsilon=epsilon, alpha=alpha,
                                        state_preparation=european_put_delta,
                                        objective_qubits=[num_uncertainty_qubits])

result_delta = ae_delta.run(quantum_instance=Aer.get_backend('qasm_simulator'), shots=100)

conf_int = -np.array(result_delta['confidence_interval'])[::-1]
print('Exact delta:    \t%.4f' % exact_delta)
print('Esimated value: \t%.4f' % -result_delta['estimation'])
print('Confidence interval: \t[%.4f, %.4f]' % tuple(conf_int))

Exact delta:            -0.8193
Esimated value:         -0.8180
Confidence interval:    [-0.8244, -0.8117]

import qiskit.tools.jupyter
%qiskit_version_table
%qiskit_copyright

Version Information

Qiskit Software	Version
Qiskit	None
Terra	0.16.0.dev0+28d8c6a
Aer	0.6.1
Ignis	0.5.0.dev0+470d8cc
Aqua	0.8.0.dev0+ce81016
IBM Q Provider	0.8.0
System information
Python	3.7.7 (default, May 6 2020, 04:59:01) [Clang 4.0.1 (tags/RELEASE_401/final)]
OS	Darwin
CPUs	2
Memory (Gb)	16.0
Fri Oct 16 10:45:35 2020 CEST

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