### Pricing European Call Options

#### 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:

$max\left\{{S}_{T}-K,0\right\}$

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:

$\mathbb{E}\left[max\left\{{S}_{T}-K,0\right\}\right]$

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

$\mathrm{\Delta }=\mathbb{P}\left[{S}_{T}\ge K\right]$

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:

#### 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 $\left[\text{low},\text{high}\right]$ 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:

$|0{⟩}_{n}↦|\psi {⟩}_{n}=\sum _{i=0}^{{2}^{n}-1}\sqrt{{p}_{i}}|i{⟩}_{n},$

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:

$\left\{0,\dots ,{2}^{n}-1\right\}\ni i↦\frac{\text{high}-\text{low}}{{2}^{n}-1}\ast i+\text{low}\in \left[\text{low},\text{high}\right].$

#### 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 ${\mathrm{sin}}^{2}\left(y+\pi /4\right)\approx y+1/2$ for small $|y|$. Thus, for a given approximation rescaling factor ${c}_{\text{approx}}\in \left[0,1\right]$ and $x\in \left[0,1\right]$ we consider

${\mathrm{sin}}^{2}\left(\pi /2\ast {c}_{\text{approx}}\ast \left(x-1/2\right)+\pi /4\right)\approx \pi /2\ast {c}_{\text{approx}}\ast \left(x-1/2\right)+1/2$

for small ${c}_{\text{approx}}$.

We can easily construct an operator that acts as

$|x⟩|0⟩↦|x⟩\left(\mathrm{cos}\left(a\ast x+b\right)|0⟩+\mathrm{sin}\left(a\ast x+b\right)|1⟩\right),$

using controlled Y-rotations.

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