### Pricing European Call Options

1. Problem Definition

2. Pricing Parameters

3. Evaluate Expected Payoff

#### Background

Suppose a European call option with strike price and an underlying asset whose spot price at maturity 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 , 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:

#### 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 and discretized using grid points, where denotes the number of qubits used. The unitary operator corresponding to the circuit factory implements the following:

where denote the probabilities corresponding to the truncated and discretized distribution and where 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 is less than the strike price and then increases linearly. The implementation uses a comparator, that flips an ancilla qubit from to if , 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 for small . Thus, for a given approximation rescaling factor and we consider

for small .

We can easily construct an operator that acts as

using controlled Y-rotations.

Eventually, we are interested in the probability of measuring in the last qubit, which corresponds to . Together with the approximation above, this allows to approximate the values of interest. The smaller we choose , the better the approximation. However, since we are then estimating a property scaled by , the number of evaluation qubits needs to be adjusted accordingly.

For more details on the approximation, we refer to: Quantum Risk Analysis. Woerner, Egger. 2018.