Background

Suppose a European call 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:

In the following, a quantum algorithm based on amplitude estimation is used to estimate the expected payoff, i.e., the fair price before discounting, for the option:

as well as the corresponding $\mathrm{\Delta }$, i.e., the derivative of the option price with respect to the spot price, defined as:

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:

• Quantum Risk Analysis. Woerner, Egger. 2018.

• Option Pricing using Quantum Computers. Stamatopoulos et al. 2019.

Uncertainty Model

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:

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:

Payoff Function

The payoff function equals zero as long as the spot price at maturity $S_T$ is less than the strike price $K$ and then increases linearly. The implementation uses a comparator, that flips an ancilla qubit from $|0⟩$ to $|1⟩$ if $S_T\ge K$, and this ancilla is used to control the linear part of the payoff function.

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 factor $c_{\text{approx}}\in [0,1]$ and $x\in [0,1]$ we consider

for small $c_{\text{approx}}$.

We can easily construct an operator that acts as

using controlled Y-rotations.

Eventually, we are interested in the probability of measuring $|1⟩$ in the last qubit, which corresponds to ${\sin }^2(a\ast x+b)$. 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.