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Option Pricing with qGANs

Introduction

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.

import matplotlib.pyplot as plt
import numpy as np

from qiskit.circuit import ParameterVector
from qiskit.circuit.library import TwoLocal
from qiskit.quantum_info import Statevector

from qiskit.algorithms import IterativeAmplitudeEstimation, EstimationProblem
from qiskit_aer.primitives import Sampler
from qiskit_finance.applications.estimation import EuropeanCallPricing
from qiskit_finance.circuit.library import NormalDistribution

/tmp/ipykernel_4570/1301515233.py:8: DeprecationWarning: ``qiskit.algorithms`` has been migrated to an independent package: https://github.com/qiskit-community/qiskit-algorithms. The ``qiskit.algorithms`` import path is deprecated as of qiskit-terra 0.25.0 and will be removed no earlier than 3 months after the release date. Please run ``pip install qiskit_algorithms`` and use ``import qiskit_algorithms`` instead.
  from qiskit.algorithms import IterativeAmplitudeEstimation, EstimationProblem

Uncertainty Model

The Black-Scholes model assumes that the spot price at maturity \(S_T\) for a European call option is log-normally distributed. Thus, we can train a qGAN on samples from a log-normal distribution and use the result as an uncertainty model underlying the option. In the following, we construct a quantum circuit that loads the uncertainty model. The circuit output reads

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

where the probabilities \(p_{\theta}^{j}\), for \(j\in \left\{0, \ldots, {2^n-1} \right\}\), represent a model of the target distribution.

# Set upper and lower data values
bounds = np.array([0.0, 7.0])
# 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=1.0, sigma=1.0, 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)

Evaluate Expected Payoff

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

# 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

Plot the probability distribution

Next, we plot the trained probability distribution and, for reasons of comparison, also the target probability distribution.

# 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()

Evaluate Expected Payoff

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.

# 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:  1.0633
Analytically calculated expected payoff w.r.t. the trained distribution: 0.9805

# construct circuit for payoff function
european_call_pricing = EuropeanCallPricing(
    num_qubits,
    strike_price=strike_price,
    rescaling_factor=c_approx,
    bounds=bounds,
    uncertainty_model=uncertainty_model,
)

# 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})
)

result = ae.estimate(problem)

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

Exact value:            0.9805
Estimated value:        1.0232
Confidence interval:    [0.9908, 1.0555]

import qiskit.tools.jupyter

%qiskit_version_table
%qiskit_copyright

Version Information

Software	Version
qiskit	None
qiskit-terra	0.25.1
qiskit_aer	0.12.2
qiskit_optimization	0.5.0
qiskit_finance	0.3.4
System information
Python version	3.8.17
Python compiler	GCC 11.3.0
Python build	default, Jun 7 2023 12:29:56
OS	Linux
CPUs	2
Memory (Gb)	6.7694854736328125
Thu Aug 31 21:48:40 2023 UTC

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