Notes on Quantum Pricing & Arbitrage

Classical Monte Carlo simulations are foundational in financial modeling, used extensively to estimate expected outcomes for derivatives pricing and risk analysis. They work by repeatedly sampling random variables to approximate solutions to problems that lack closed-form analytical solutions. However, classical Monte Carlo methods suffer from significant limitations, primarily due to the “curse of dimensionality”. As the number of underlying assets or stochastic factors increases, the computational effort required grows exponentially. Tasks such as valuing complex instruments like American options, basket options, and performing nested risk assessments (like Value-at-Risk or VaR calculations) become highly inefficient or even infeasible using classical methods due to their intensive computational demands.

Similarly, traditional algorithms for currency arbitrage, such as Bellman-Ford, are employed to find profitable trading cycles by detecting negative cycles within graphs that represent currency exchanges. Yet, as these problems scale with more currencies or higher frequency data, classical algorithms become computationally expensive and slow, making them unsuitable for practical, real-time applications.

Fundamentals

Quantum computing represents information using quantum bits or “qubits,” which leverage quantum mechanical phenomena such as superposition and entanglement. Superposition allows qubits to simultaneously exist in multiple states, providing inherent parallelism that classical bits do not. Entanglement enables interconnected qubits to maintain correlated states regardless of their physical separation, enhancing computational power significantly. These quantum properties collectively permit quantum computers to effectively tackle computational problems far beyond practical reach of classical systems, particularly beneficial for financial problems that involve complex optimization and simulation tasks.

Quantum Monte Carlo Option Pricing (MCQP)

MCQP is a quantum method designed to simulate asset price paths concurrently, utilizing quantum parallelism. Unlike classical simulations, which sequentially compute each stochastic path independently, quantum methods allow simultaneous evolution of multiple asset paths. The fundamental quantum simulation follows the Black-Scholes stochastic evolution for asset pricing:

\[S_{t+1} = S_t \times \left(1 + \mu \Delta t + \sigma \Delta W_t \right)\]

where:

• \( S_t \) is the asset price at time \( t \),
• \( \mu \) is the drift rate (expected return),
• \( \sigma \) is the asset’s volatility,
• \( \Delta t \) is the time increment, and
• \( \Delta W_t \sim \mathcal{N}(0, \Delta t) \) is a normally distributed stochastic increment representing a Wiener process.

Quantum circuits generate these increments simultaneously using specialized quantum gates, ensuring the stochastic variables are appropriately distributed. Quantum operators then iteratively update quantum registers encoding these stochastic variables and corresponding asset prices, simulating the entire price evolution across discrete time intervals simultaneously.

The payoff for a European call option, calculated at maturity \( T \), is given by:

\[P(S_T) = \max(S_T - K, 0)\]

where $K$ represents the strike price. The Quantum Amplitude Estimation (QAE) algorithm efficiently computes the expected payoff encoded in the amplitude of an ancillary qubit, achieving a quadratic speedup over classical methods by improving error scaling from classical $O(1/\sqrt{N})$ to quantum $O(1/N)$.

Quantum Arbitrage Optimization via QUBO Formulation

Quantum arbitrage optimization methods reformulate currency arbitrage as Quadratic Unconstrained Binary Optimization (QUBO) problems. This approach maps the arbitrage maps the arbitrage problem onto a directed graph, where nodes represent currencies, and edges represent exchange rates. The overarching goal is to find optimal arbitrage cycles that maximize profits.

The mathematical formulation for arbitrage opportunities in a QUBO framework is:

\[C_{\text{tot}} = \sum_{i,j} -\log(r_{ij}) \cdot b_{ij} + m_p \left( \sum_i \sum_{j \ne j'} b_{i,j} \cdot b_{i,j'} + \sum_j \sum_{i \ne i'} b_{i,j} \cdot b_{i',j} \right)\]

where:

• $r_{ij}$ are the currency exchange rates,
• $b_{ij} \in {0, 1}$ are binary variables indicating whether a specific exchange from currency $i$ to currency $j$ is selected,
• $m_p$ is a penalty factor that enforces the cycle constraint by ensuring exactly one incoming and one outgoing edge per currency node.

Quantum Annealing, implemented in platforms like D-Wave quantum systems, efficiently solves QUBO problems by minimizing the energy states of a quantum system, enabling quick convergence to optimal or near-optimal arbitrage opportunities.

Quantum Approximate Optimization Algorithm (QAOA) Approach

Quantum Approximate Optimization Algorithm (QAOA) provides a quantum computing approach to solve combinatorial optimization problems through parameterized quantum circuits. QAOA encodes QUBO problems into quantum circuits characterized by parameters $\gamma$ and $\beta$. These parameters are iteratively optimized using classical algorithms like COBYLA to minimize the cost function:

\[\text{Cost Function} = \sum_{i,j} -\log(r_{ij}) b_{ij}\]

Despite theoretical promise, practical challenges persist due to current quantum hardware limitations, such as noise, qubit decoherence, limited connectivity, and gate errors, complicating effective optimization and solution accuracy.

The Heston Model

Quantum computational methods are also effective for complex financial models incorporating stochastic volatility, such as the Heston model. The Heston model describes the evolution of asset prices $S_t$ and volatility $\nu_t$ through correlated stochastic processes:

\[S_{t+1} = S_t + \mu \Delta t \times S_t + \sigma_t \Delta W_t \times S_t,\] \[\nu_{t+1} = \nu_t + \kappa(\theta - \nu_t)\Delta t + \xi \sqrt{\nu_t} \Delta W_t^\nu\]

where parameters $\kappa, \theta, \xi$ characterize volatility behavior, and $\Delta W_t, \Delta W_t^\nu$ represent correlated Brownian increments.

Quantum circuits effectively simulate these stochastic dynamics concurrently, providing linear scalability in complexity, and significantly improving computational efficiency by enabling parallel simulations.

Risk Analysis and Nested Monte Carlo

Quantum algorithms significantly enhance risk analysis by efficiently executing nested quantum Monte Carlo simulations, which are classically computationally demanding. Quantum parallel nested simulations utilize additional quantum registers dedicated to simulating nested paths and ancillary qubits for encoding complex risk metrics like Value-at-Risk (VaR). Through quantum amplitude estimation, nested simulations can quickly and accurately estimate risk metrics, drastically reducing computational overhead compared to classical methods. Quantum risk analysis techniques thereby offer significant practical advantages in real-time, complex risk management tasks.

Challenges

Despite significant potential, quantum computing methods in finance face critical practical challenges. Quantum systems currently struggle with errors arising from gate operations, qubit decoherence, hardware noise, and limitations in connectivity. Addressing these challenges involves focused research on:

1. Development and implementation of robust quantum error-correcting codes and noise mitigation strategies.
2. Optimization of quantum circuits to minimize depth and complexity.
3. Advances in quantum random number generation methods to ensure accuracy and reliability.
4. Exploration of hybrid quantum-classical computational methods that effectively combine the strengths of quantum speedups with classical computational reliability and flexibility.

References

This set of notes are based on the approachs outlined in Monte-Carlo Option Pricing in Quantum Parallel and Currency Arbitrage Optimization using Quantum Annealing, QAOA and Constraint Mapping.