Home Problem Definition Pricing Parameters Evaluate Expected Payoff

Pricing European Call Options

1. Problem Definition

2. Pricing Parameters

3. Evaluate Expected Payoff


Suppose a European call option with strike price K and an underlying asset whose spot price at maturity ST 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 Δ, 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 [low,high] and discretized using 2n grid points, where n denotes the number of qubits used. The unitary operator corresponding to the circuit factory implements the following:


where pi 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 ST 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 STK, 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 sin2(y+π/4)y+1/2 for small |y|. Thus, for a given approximation rescaling factor capprox[0,1] and x[0,1] we consider


for small capprox.

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 sin2(ax+b). Together with the approximation above, this allows to approximate the values of interest. The smaller we choose capprox, the better the approximation. However, since we are then estimating a property scaled by capprox, 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.