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This notebook covers the implementation of the Integration Method for the rainbow option presented in [1]. In finance, a crucial aspect of asset pricing pertains to derivatives. Derivatives are contracts whose value is contingent upon another source, known as the underlying. The pricing of options—a specific derivative instrument—involves determining the fair market value (discounted payoff) of contracts that afford their holders the right, though not the obligation, to buy (call) or sell (put) one or more underlying assets at a predefined strike price by a specified future expiration date (maturity date). This process relies on mathematical models, considering variables such as current asset prices, time to expiration, volatility, and interest rates.

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Data Definitions

The problem inputs:
  • NUM_QUBITS: the number of qubits representing an underlying asset
  • NUM_ASSETS: the number of underlying assets
  • K: the strike price
  • S0: the arrays of underlying asset prices
  • dt: the number of days to the maturity date
  • COV: the covariance matrix that correlate to the underlyings
  • MU_LOG_RET: the array containing the mean of the log return of each underlying
import numpy as np
import scipy

NUM_QUBITS = 2
NUM_ASSETS = 2

K = 190
S0 = [193.97, 189.12]
dt = 250

COV = np.array([[0.000335, 0.000257], [0.000257, 0.000418]])
MU_LOG_RET = np.array([0.00050963, 0.00062552])


MU = MU_LOG_RET * dt
CHOLESKY = np.linalg.cholesky(COV) * np.sqrt(dt)
SCALING_FACTOR = 1 / CHOLESKY[0, 0]


from classiq import *

EPSILON = 0.05
ALPHA = 0.1

​
Gaussian State Preparation

Encode the probability distribution of a discrete multivariate random variable WW taking values in {w0,..,wN−1}\{w_0, .., w_{N-1}\} describing the asset prices at the maturity date. The number of discretized values, denoted as NN, depends on the precision of the state preparation module and is consequently connected to the number of qubits (n=n=NUM_QUBITS) according to the formula N=2nN=2^n: ∑i=0N−1p(wi)∣wi⟩\sum_{i=0}^{N-1} \sqrt{p(w_i)}\left|w_i\right\rangle 
def gaussian_discretization(num_qubits, mu=0, sigma=1, stds_around_mean_to_include=3):
    lower = mu - stds_around_mean_to_include * sigma
    upper = mu + stds_around_mean_to_include * sigma
    num_of_bins = 2**num_qubits
    sample_points = np.linspace(lower, upper, num_of_bins + 1)

    def single_gaussian(x: np.ndarray, _mu: float, _sigma: float) -> np.ndarray:
        cdf = scipy.stats.norm.cdf(x, loc=_mu, scale=_sigma)
        return cdf[1:] - cdf[0:-1]

    non_normalized_pmf = (single_gaussian(sample_points, mu, sigma),)
    real_probs = non_normalized_pmf / np.sum(non_normalized_pmf)
    return sample_points[:-1], real_probs[0].tolist()


grid_points, probabilities = gaussian_discretization(NUM_QUBITS)

STEP_X = grid_points[1] - grid_points[0]
MIN_X = grid_points[0]

​

​
Sanity Check

To avoid meaningless results, the process must stop if the strike price KK is greater than the maximum value reacheable by the assets during the simulation. In this case, the payoff is 00, so there is no need to simulate: 
from IPython.display import Markdown

if K >= max(S0 * np.exp(np.dot(CHOLESKY, [grid_points[-1]] * 2) + MU)):
    display(
        Markdown(
            "<font color='red'> K always greater than the maximum asset values.

Stop the run, the payoff is 0</font>"
        )
    )

​
Maximum Computation

​

​
Precision Utils

FRAC_PLACES = 2


def round_factor(a):
    precision_factor = 2**FRAC_PLACES
    return round(a * precision_factor) / precision_factor


def floor_factor(a):
    precision_factor = 2**FRAC_PLACES
    return np.floor(a * precision_factor) / precision_factor

​

​
Affine and Maximum Arithmetic Definitions

from functools import reduce

from classiq.qmod.symbolic import max as qmax

a = STEP_X / SCALING_FACTOR
b = np.log(S0[0]) + MU[0] + MIN_X * CHOLESKY[0].sum()


def get_affine_formula(assets, i):
    return reduce(
        lambda x, y: x + y,
        [
            assets[j] * round_factor(SCALING_FACTOR * CHOLESKY[i, j])
            for j in range(NUM_ASSETS)
            if CHOLESKY[i, j]
        ],
    )


c = (
    SCALING_FACTOR
    * (
        np.log(S0[1])
        + MU[1]
        - (np.log(S0[0]) + MU[0])
        + MIN_X * sum(CHOLESKY[1] 

- CHOLESKY[0])
    )
    / (STEP_X)
)

c = round_factor(c)


def calculate_max_reg_type():
    x1 = QNum(size=NUM_QUBITS)
    x2 = QNum(size=NUM_QUBITS)
    expr = qmax(get_affine_formula([x1, x2], 0), get_affine_formula([x1, x2], 1) + c)
    size_in_bits, sign, fraction_digits = get_expression_numeric_attributes(
        [x1, x2], expr
    )
    return size_in_bits, fraction_digits


MAX_NUM_QUBITS = calculate_max_reg_type()[0]
MAX_FRAC_PLACES = calculate_max_reg_type()[1]


@qperm
def affine_max(x1: Const[QNum], x2: Const[QNum], res: Output[QNum]):
    res |= qmax(get_affine_formula([x1, x2], 0), get_affine_formula([x1, x2], 1) + c)

​
Integration Method

The comparator collects the probabilities g(r)g(r) of ∣r⟩|r\rangle state until ∣r⟩|r\rangle register is lower than ∣x⟩|x\rangle: ∑r=02R−1g(r)∣x⟩∣r⟩∣r≤x⟩=∣x⟩⊗[∑r=0xg(r)∣r⟩∣1⟩+∑r=x2R−1g(r)∣r⟩∣0⟩]\begin{equation} \begin{split} &\sum_{r=0}^{2^R-1}{\sqrt{g(r)}}|x\rangle|r\rangle|r\leq x\rangle \\ = &|x\rangle \otimes \left[ \sum_{r=0}^{x}{\sqrt{g(r)}} |r\rangle |1\rangle + \sum_{r=x}^{2^R-1}{\sqrt{g(r)}} |r\rangle |0\rangle \right] \end{split} \end{equation} Collecting the probability to have r≤xr\leq x, define the function: h~(x)=∑r=0xg(r)\begin{equation} \tilde{h}(x)=\sum_{r=0}^{x}g(r) \end{equation} Evaluating the probability to get a ∣1⟩|1\rangle results in ∑x=02R−1h~(x)\sum_{x = 0}^{2^R-1}{\tilde{h}(x)}. To obtain a given function h~\tilde{h}, choose a proper function g(r)g(r). The g(r)g(r) for r=0r=0 value must therefore be: g(0)={~h}(0)g(0) = \tilde\{h\}(0) and for all the other rr: g(r)=h~(r)−h~(r−1)g(r) = \tilde{h}(r)-\tilde{h}(r-1) 
@qfunc
def integrator(x: Const[QNum], ref: QNum, res: QBit) -> None:
    exp_rate = (1 / (2**x.fraction_digits)) * a
    prepare_exponential_state(-exp_rate, ref)
    res ^= x >= ref


from classiq.qmod.symbolic import asin, exp, sqrt


def get_strike_price_theta_integration(x: QNum):
    exp_rate = (1 / (2**x.fraction_digits)) * a
    B = (exp((2**x.size) * exp_rate) - 1) / exp(exp_rate)
    A = 1 / exp(exp_rate)
    C = S0[0] * exp((MU[0] + MIN_X * CHOLESKY[0].sum()))
    return 2 * asin(sqrt((K - (C * A)) / (C * B)))


# this is not a qfunc, just a utility function
def is_geq_strike_price(
    x: Const[QNum],
) -> None:
    a = STEP_X / SCALING_FACTOR
    b = np.log(S0[0]) + MU[0] + MIN_X * CHOLESKY[0].sum()
    COMP_VALUE = (np.log(K) - b) / a
    return x > floor_factor(COMP_VALUE)


@qfunc
def integration_payoff(max_reg: Const[QNum], integrator_reg: QNum, ind_reg: QBit):
    control(
        is_geq_strike_price(max_reg),
        lambda: integrator(max_reg, integrator_reg, ind_reg),
        lambda: RY(get_strike_price_theta_integration(max_reg), ind_reg),
    )


class EstimationVars(QStruct):
    x1: QNum[NUM_QUBITS]
    x2: QNum[NUM_QUBITS]
    integrator: QNum[MAX_NUM_QUBITS, False, MAX_FRAC_PLACES]


@qfunc
def rainbow_integration(qvars: EstimationVars, ind: QBit) -> None:
    inplace_prepare_state(probabilities, 0, qvars.x1)
    inplace_prepare_state(probabilities, 0, qvars.x2)

    max_out = QNum()
    affine_max(qvars.x1, qvars.x2, max_out),
    integration_payoff(max_out, qvars.integrator, ind)


@qfunc
def main(qvars: Output[EstimationVars], ind: Output[QBit]) -> None:
    allocate(qvars)
    allocate(ind)

    rainbow_integration(qvars, ind)


MAX_WIDTH_1 = 23
qmod_1 = create_model(main, constraints=Constraints(max_width=MAX_WIDTH_1))
print("Starting synthesis")
qprog_1 = synthesize(qmod_1)
show(qprog_1)
Output:

Starting synthesis
  Quantum program link: https://platform.classiq.io/circuit/3560ShwR9pHLDmG2n4tA0ra7QDu

​
Iterative Quantum Amplitude Estimation (IQAE) Algorithm

from classiq.applications.iqae.iqae import IQAE

MAX_WIDTH_2 = 25

iqae = IQAE(
    state_prep_op=rainbow_integration,
    problem_vars_size=NUM_QUBITS * NUM_ASSETS + MAX_NUM_QUBITS,
    constraints=Constraints(max_width=MAX_WIDTH_2),
    preferences=Preferences(optimization_level=1),
)


qmod_2 = iqae.get_model()

print("Starting synthesis")
qprog_2 = iqae.get_qprog()
show(qprog_2)
print("Starting execution")
result = iqae.run(EPSILON, ALPHA)
print("raw iqae results:", result.estimation, result.confidence_interval)
Output:

Starting synthesis
  Quantum program link: https://platform.classiq.io/circuit/3560YutokLfva7nGAyKraA8CECv
  Starting execution
  raw iqae results: 0.05270338444908511 [0.04886018991028241, 0.05654657898788781]

​
Post-process

Add a term to the post-processing function: E[max⁡(ea(x+1)−1ea(xmax+1)−1c+1ea,Ke−b′)]eb′−K=E[max⁡(ea(x+1)−1ea(xmax+1)−1,Ke−b′c−e−ac)]ceb′+eb′e−a−K\begin{equation} \begin{split} \mathbb{E} \left[\max\left(\frac{e^{a(x+1)} - 1}{e^{a(x_{max} +1)}-1}c + \frac{1}{e^a} , Ke^{-b'}\right)\right] e^{b'} - K \\ =\mathbb{E} \left[\max\left(\frac{e^{a(x+1)} - 1}{e^{a(x_{max} +1)}-1}, \frac{Ke^{-b'}}{c} - \frac{e^{-a}}{c}\right)\right]ce^{b'} + e^{b'}e^{-a} - K \end{split} \end{equation} 
exp_rate = (1 / (2**MAX_FRAC_PLACES)) * a
B = (np.exp((2**MAX_NUM_QUBITS) * exp_rate) - 1) / np.exp(exp_rate)
A = 1 / np.exp(exp_rate)
C = S0[0] * np.exp((MU[0] + MIN_X * CHOLESKY[0].sum()))


def parse_result_integration(result):
    option_value = (result.estimation * (C * B)) + (C * A) 

- K
    confidence_interval = (np.array(result.confidence_interval) * (C * B)) + (C * A) 

- K
    return (option_value, confidence_interval)

​
Run Method

parsed_result, conf_interval = parse_result_integration(result)
print(
    f"raw iqae results: {result.estimation} with confidence interval {result.confidence_interval}"
)
print(
    f"option estimated value: {parsed_result} with confidence interval {conf_interval}"
)
Output:
raw iqae results: 0.05270338444908511 with confidence interval [0.04886018991028241, 0.05654657898788781]
  option estimated value: 29.49446224830868 with confidence interval [19.5384534 39.4504711]

​

​
Assertions

expected_payoff = 23.0238
ALPHA_ASSERTION = 1e-5
measured_confidence = conf_interval[1] - conf_interval[0]
confidence_scale_by_alpha = np.sqrt(
    np.log(ALPHA / ALPHA_ASSERTION)
)  # based on e^2=(1/2N)*log(2T/alpha) from "Iterative Quantum Amplitude Estimation" since our alpha is low, we want to check within a bigger confidence interval
assert (
    np.abs(parsed_result - expected_payoff)
    <= 0.5 * measured_confidence * confidence_scale_by_alpha
), f"Payoff result is out of the {ALPHA_ASSERTION*100}% confidence interval: |{parsed_result} - {expected_payoff}| > {0.5*measured_confidence * confidence_scale_by_alpha}"

​
References

[1] Francesca Cibrario et al., Quantum Amplitude Loading for Rainbow Options Pricing. Preprint.