このページは docs/tutorials/03_european_call_option_pricing.ipynb から生成されました。

# ヨーロピアン・コール・オプションの価格推定#

## はじめに#

$\max\{S_T - K, 0\}$

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

$\Delta = \mathbb{P}\left[S_T \geq K\right]$

[1]:

import matplotlib.pyplot as plt

%matplotlib inline
import numpy as np

from qiskit import QuantumCircuit
from qiskit_algorithms import IterativeAmplitudeEstimation, EstimationProblem
from qiskit.circuit.library import LinearAmplitudeFunction
from qiskit_aer.primitives import Sampler
from qiskit_finance.circuit.library import LogNormalDistribution


## 不確実性モデル#

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

ここで $$p_i$$ は切り捨てられ離散化された分布を与える確率を表し、$$i$$ はアフィン写像で得られた正しい区間です。

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

# 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)
)

[3]:

# 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()


## ペイオフ関数#

ペイオフ関数は、満期時のスポット価格 $$S_T$$ が権利行使価格 $$K$$ よりも小さい間はゼロであり、その後、線形に増加します。実装はコンパレーターを使用し、もし $$S_T \geq K$$ ならば補助量子ビットを $$\big|0\rangle$$ から $$\big|1\rangle$$ にフリップし、この補助量子ビットがペイオフ関数の線形部分を制御します。

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

ここで、$$c_\text{approx}$$ は十分小さいものとします。

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

[4]:

# set the strike price (should be within the low and the high value of the uncertainty)
strike_price = 1.896

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

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

# construct A operator for QAE for the payoff function by
# composing the uncertainty model and the objective
num_qubits = european_call_objective.num_qubits
european_call = QuantumCircuit(num_qubits)
european_call.append(uncertainty_model, range(num_uncertainty_qubits))
european_call.append(european_call_objective, range(num_qubits))

# draw the circuit
european_call.draw()

[4]:

     ┌───────┐┌────┐
q_0: ┤0      ├┤0   ├
│       ││    │
q_1: ┤1 P(X) ├┤1   ├
│       ││    │
q_2: ┤2      ├┤2   ├
└───────┘│    │
q_3: ─────────┤3 F ├
│    │
q_4: ─────────┤4   ├
│    │
q_5: ─────────┤5   ├
│    │
q_6: ─────────┤6   ├
└────┘
[5]:

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

[6]:

# 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:   0.1623
exact delta value:      0.8098


## 期待ペイオフの評価#

[7]:

european_call.draw()

[7]:

     ┌───────┐┌────┐
q_0: ┤0      ├┤0   ├
│       ││    │
q_1: ┤1 P(X) ├┤1   ├
│       ││    │
q_2: ┤2      ├┤2   ├
└───────┘│    │
q_3: ─────────┤3 F ├
│    │
q_4: ─────────┤4   ├
│    │
q_5: ─────────┤5   ├
│    │
q_6: ─────────┤6   ├
└────┘
[8]:

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

problem = EstimationProblem(
state_preparation=european_call,
objective_qubits=[3],
post_processing=european_call_objective.post_processing,
)
# construct amplitude estimation
ae = IterativeAmplitudeEstimation(
epsilon_target=epsilon, alpha=alpha, sampler=Sampler(run_options={"shots": 100, "seed": 75})
)

[9]:

result = ae.estimate(problem)

[10]:

conf_int = np.array(result.confidence_interval_processed)
print("Exact value:        \t%.4f" % exact_value)
print("Estimated value:    \t%.4f" % (result.estimation_processed))
print("Confidence interval:\t[%.4f, %.4f]" % tuple(conf_int))

Exact value:            0.1623
Estimated value:        0.1687
Confidence interval:    [0.1637, 0.1737]


Instead of constructing these circuits manually, the Qiskit Finance module offers the EuropeanCallPricing circuit, which already implements this functionality as a building block.

[11]:

from qiskit_finance.applications.estimation import EuropeanCallPricing

european_call_pricing = EuropeanCallPricing(
num_state_qubits=num_uncertainty_qubits,
strike_price=strike_price,
rescaling_factor=c_approx,
bounds=(low, high),
uncertainty_model=uncertainty_model,
)

[12]:

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

problem = european_call_pricing.to_estimation_problem()
# construct amplitude estimation
ae = IterativeAmplitudeEstimation(
epsilon_target=epsilon, alpha=alpha, sampler=Sampler(run_options={"shots": 100, "seed": 75})
)
result = ae.estimate(problem)

conf_int = np.array(result.confidence_interval_processed)
print("Exact value:        \t%.4f" % exact_value)
print("Estimated value:    \t%.4f" % (european_call_pricing.interpret(result)))
print("Confidence interval:\t[%.4f, %.4f]" % tuple(conf_int))

Exact value:            0.1623
Estimated value:        0.1687
Confidence interval:    [0.1637, 0.1737]


## デルタの評価#

デルタは期待ペイオフよりも少し評価が簡単です。 期待ペイオフと同様に、コンパレータ回路と補助量子ビットを使用して、 $$S_T > K$$ の場合を特定します。しかしながら、この条件が真であるときの確率のみに興味があるので、振幅推定において他に何も仮定せずにこの補助量子ビットを直接目的量子ビットとして使うことができます。

[13]:

from qiskit_finance.applications.estimation import EuropeanCallDelta

european_call_delta = EuropeanCallDelta(
num_state_qubits=num_uncertainty_qubits,
strike_price=strike_price,
bounds=(low, high),
uncertainty_model=uncertainty_model,
)

[14]:

european_call_delta._objective.decompose().draw()

[14]:

         ┌──────┐
state_0: ┤0     ├
│      │
state_1: ┤1     ├
│      │
state_2: ┤2     ├
│  cmp │
state_3: ┤3     ├
│      │
work_0: ┤4     ├
│      │
work_1: ┤5     ├
└──────┘
[15]:

european_call_delta_circ = QuantumCircuit(european_call_delta._objective.num_qubits)
european_call_delta_circ.append(uncertainty_model, range(num_uncertainty_qubits))
european_call_delta_circ.append(
european_call_delta._objective, range(european_call_delta._objective.num_qubits)
)

european_call_delta_circ.draw()

[15]:

     ┌───────┐┌──────┐
q_0: ┤0      ├┤0     ├
│       ││      │
q_1: ┤1 P(X) ├┤1     ├
│       ││      │
q_2: ┤2      ├┤2     ├
└───────┘│  ECD │
q_3: ─────────┤3     ├
│      │
q_4: ─────────┤4     ├
│      │
q_5: ─────────┤5     ├
└──────┘
[16]:

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

problem = european_call_delta.to_estimation_problem()

# construct amplitude estimation
ae_delta = IterativeAmplitudeEstimation(
epsilon_target=epsilon, alpha=alpha, sampler=Sampler(run_options={"shots": 100, "seed": 75})
)

[17]:

result_delta = ae_delta.estimate(problem)

[18]:

conf_int = np.array(result_delta.confidence_interval_processed)
print("Exact delta:    \t%.4f" % exact_delta)
print("Estimated value: \t%.4f" % european_call_delta.interpret(result_delta))
print("Confidence interval: \t[%.4f, %.4f]" % tuple(conf_int))

Exact delta:            0.8098
Estimated value:        0.8091
Confidence interval:    [0.8034, 0.8148]

[19]:

import qiskit.tools.jupyter

%qiskit_version_table


### Version Information

SoftwareVersion
qiskitNone
qiskit-terra0.45.0.dev0+c626be7
qiskit_finance0.4.0
qiskit_aer0.12.0
qiskit_algorithms0.2.0
qiskit_ibm_provider0.6.1
qiskit_optimization0.6.0
System information
Python version3.9.7
Python compilerGCC 7.5.0
Python builddefault, Sep 16 2021 13:09:58
OSLinux
CPUs2
Memory (Gb)5.778430938720703
Fri Aug 18 16:00:58 2023 EDT

### This code is a part of Qiskit

[ ]: