Note

This page was generated from docs/tutorials/10_qgan_option_pricing.ipynb.

# 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.

```
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```

```
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

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

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```
# 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.

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```
# 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.

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```
# 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.

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```
# 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
```

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```
# 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,
)
```

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```
# 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})
)
```

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```
result = ae.estimate(problem)
```

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```
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]
```

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```
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 |

### This code is a part of Qiskit

© Copyright IBM 2017, 2023.

This code is licensed under the Apache License, Version 2.0. You may

obtain a copy of this license in the LICENSE.txt file in the root directory

of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.

Any modifications or derivative works of this code must retain this

copyright notice, and modified files need to carry a notice indicating

that they have been altered from the originals.

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