As one of the major methods for the simulation of option pricing, Monte Carlo method assumes random fluctuations in the distribution of asset prices. Under certain uncertainties process, different evolution paths could be simulated so as to finally yield the expectation value of the asset price, which requires a lot of simulations to ensure the accuracy based on huge and expensive calculations. In order to solve the above computational problem, quantum Monte Carlo (QMC) has been established and applied in the relevant systems such as European call options. In this work, both MC and QM methods are adopted to simulate European call options. Based on the preparation of quantum states in QMC algorithm and the construction of quantum circuits by simulating a quantum hardware environment on a traditional computer, the amplitude estimation (AE) algorithm is found to play a secondary role in accelerating the pricing of European options. More importantly, the payoff function and the time required for the simulation in QMC method show some improvements than those in MC method.
An option refers to a contract that gives the holder the right to purchase or sell an asset at a fixed price (strike price) prior to a specific date (exercise window). In simple terms, The strike price is a fixed price, and the time range is also a single point in time. However, exotic variants can lead to multiple underlying assets, that is to say, the execution price can be a function with many execution windows. Option is not only a profitable tool but also the core of various hedging strategies [
Due to the randomness of the defined parameters, it is feasible but difficult to calculate the fair value [
As is known, quantum computers [
This article mainly focuses on the application of MC and QMC methods in European call options and further analysis for the certain examples. In the second section, the path based on the simulation with MC method is found similar to Brownian motion. The asset price on the exercise day is also introduced, with the basic principles of MC. And the relevant probability distribution on a quantum computer is also illustrated, with the basic principles of QMC. In the third section, the algorithm structure and some code examples of the MC and QMC methods are described, by giving a specific example of European call options. In the fourth section, the simulation results of MC and QMC algorithms are illustrated and discussed, and the advantages and disadvantages are analyzed and compared for both methods. Since the classic MC algorithm is relatively mature, this article stresses mainly on the AE and QMC algorithms.
The theoretical basis of Monte Carlo method is probability theory and mathematical statistics. The method is to assume that the asset price distribution is of random fluctuations. If the fluctuation process is known, different random paths can be simulated and yield an asset value. When this process is performed for several times, the obtained result could be regarded as an optimal asset value distribution, which can bring forward the desired asset price. The dominant advantage of MC method is that the error convergence rate does not depend on the dimensionality of the problem. However, accurate pricing demands MC method to perform millions of simulations with huge calculations [
Below we introduce the stock price volatility model, European option prices in efficient markets, with the Black-Scholes equation [
Set
In the above expression,
By using the Ito’s lemma [
Assuming that the stock price at the initial moment is
In a risk-neutral world,
Only European options is considered here. Since they can only be exercised on the exercise date (T), the strike price of the underlying asset can be set as
According to the principle of risk-neutral pricing (i.e., the expected rate of return of the underlying asset is the risk-free interest rate), application of MC simulation in option pricing needs a risk-neutral environment. The future cash flow is discounted by the risk-free interest rate, and It is only the transformation of probability space. Because it is not mentioned that investors are risk-neutral, so there is no need to assume investors’ risk preference. Therefore, on day
Simulate
The three building blocks are required to price options on a gate-based quantum computer. (1) Represents the probability distribution
Quantum Amplitude Estimation (QAE) is a fundamental quantum algorithm with the potential to achieve a quadratic speedup for many applications that are classically solved with MC simulation. Assumes a unitary operator
Here
For European option contracts, the random variables involved represent the possible value
Comparing
This shows that AE can allow us to calculate the price of the underlying asset on the exercise date of the option and create the state of
The quantum circuit may be illustrated in
The first step in option pricing is to build a circuit that takes the probability distribution implicit in the possible future asset prices and loads it into the quantum register. Its amplitude represents the corresponding probability and each ground state represents a possible value. Given an n qubit register, asset prices {
The Black-Scholes-Merton (BSM) model [
One can create the operator
Here we make algorithm analysis on the construction process of the two algorithms as an example. Suppose there is a stock option with a maturity of six months, the underlying asset stock price is 4.36 yuan, the option strike price is 5 yuan, the annualized risk-free interest rate and annualized volatility are 3% and 20%, respectively. Simulation is performed using MC method to find the price of call options and put options with loop iterations of 20,0000 times.
The MC method for option pricing is generally divided into the following steps:
Divide the time interval [0, T] before the expiration date into multiple equidistant Carry out I simulations, According to the first simulation, the expiration value (
The QMC method for option pricing is generally divided into the following steps:
Build uncertainty model
First of all, we need to prepare the quantum state.
The probability corresponding to the truncated discrete distribution is Building the payment function
For the spot price of the expiring
Approximation can be made for the linear part here. Apply
Based on the above process, an operator can be constructed to act as controlled Y-rotations.
Eventually, we focus on the measuring probability of Obtain the expectation value of a linear function f for a random variable X
One can obtain the expectation value of a linear function f for a random variable X with AE by creating the operator
Estimated value ($) | 0.070510 |
Time (s) | 10.375985 |
Exact value ($) | 0.0728 |
Estimated value ($) | 0.0892 |
Time (s) | 4.037455 |
From the above example of European call options, the path of the option can be simulated, and the discounted price and the time required for computer simulation are obtained with MC method. we can see that AE allows us to compute the undiscounted price of an option given a way to represent the option’s payoff as a quantum circuit and create the state. The above example can also be analyzed with QMC method, which would reduce the time required for the results with amplitude estimation (AE). The MC algorithm reveals that the expectation value of European options after six months is 0.070510, while that with QMC algorithm is 0.0892, which are comparable with each other. As for the simulation time, MC method takes the simulations of 200,000 times, much larger than different from QMC. Obviously, QMC method realizes considerable time reduction with respect to MC method by more than 60%. In fact, the number of random numbers that need to be processed will significantly increase with increasing the simulation time. So, the advantage of reduction of time required by using QMC method would be expected more and more important for such systems as European call options with increasing complicacy.
This article concerns mainly with the brief introduction for MC and QMC algorithms, especially their applications in the example of a specific European call option as well as comparisons for the two methods. Based on the simulations, the advantage of MC method in the error convergence rate independent of the dimensionality of the problem is verified. However, QMC algorithm shows significant advantage in the problems requiring relatively accurate simulation with a lot of calculations such as present option pricing. An option pricing and quantum circuit is proposed by a gate-based quantum computer, and the pricing of options depends on AE. Compared with MC method, QMC allows secondary acceleration. In view of the current resources of quantum computers available, the actual quantum hardware remains to be tested, and the algorithm is also necessary to be optimized according to the realistic situation. Therefore, it seems that QMC still has a lot of space for development.
This work was financially supported by the National Natural Science Foundation of China Granted No. 11764028. One of the authors (Jian-Guo Hu) would also like to thank Tian-tian Li for his helpful suggestions.