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05 March 2023

**Introduction**

Derivatives have a huge impact on modern finance as the financial markets are benefited by them in several ways. Derivative contract is a financial instrument whose value is determined from the price of one or more underlying assets. Typically, it is issued between an issuer and the holder with a contract expiry date, which is valid until the contract expires. Derivative pricing measures the value of entering a derivative contract today, considering the uncertainty around the future values of underlying assets. The value at the expiration date is referred to as “payoff”.

In common, derivative contracts are priced by using Monte Carlo simulation. The Monte Carlo simulation is a probabilistic numerical model used to predict the outcome of an uncertain process, which requires a significant amount of computing power for predicting the probability of an outcome. Therefore, the computational requirement of Monte Carlo simulation is one of the most compelling arguments against itself. In addition, the independent samples produced by the Monte Carlo simulation will not always be the most effective for all kinds of problems. Quantum algorithms, such as Quantum Amplitude Estimation (QAE) and Improved Variational Quantum Optimization (IVQO), may solve them effectively by producing true independent random samples. Quantum machine learning models such as Quantum Boltzmann Machines (QBM) and quantum optimization models like Quantum Annealing (QA) can aid in derivative pricing while enhancing computing speed.

**Pricing of Financial Derivatives**

Derivatives are priced in a risk-free unique manner that eliminates the possibility of arbitrage (simultaneous buy and sell of the same asset in different markets in order to profit from tiny differences). In order to use Monte Carlo simulation, derivative pricing requires the following inputs: stock price, exercise price, volatility, expected term, risk free rate, and dividend yield. However, the volatility from unforeseen events, such as a stock market bubble and financial crises, is not considered. The Monte Carlo simulation model presumes that volatility will stay the same for the entire duration of the contract, which is untypical and unrealistic. In quantum computing, the Quantum Boltzmann Machines model has been shown to possess the best accuracy level in forecasting a stock market crash. Typically, a stock market crash occurs after a stock market bubble. Therefore, forecasting a stock market crash will aid in calculating the optimal derivative price. Quantum computer can also solve this problem in the quantum realm by speeding up the computation.

**Option Pricing**

Option is one type of derivative, which offers the holder a choice to buy or sell any of the underlying assets but not the obligation to do so. Option on various assets involves relatively expensive computation of possible asset price movements over time intervals known as pathways. Forecasting future asset price movements can aid in calculating the optimal option price. Therefore, identifying the least number of feasible pathways will help in calculating the price in a short period of time. The algorithm that uses Quantum Amplitude Estimation model on a quantum computer outperforms Monte Carlo simulation in terms of speed and accuracy by generating relevant samples for each pathway.

**Portfolios of options**

Multiple option contracts are entered simultaneously as opposed to single option in several popular trading and hedging strategies. An investor can create a more complex payoff rather than opting individual options by using these strategies. Without knowing which way, the volatility will push assets’ price, and an investor would want to profit from volatile assets with the same expiration date. A hybrid quantum-classical technique called Improved Variational Quantum Optimization can be used to determine the optimum trading value of an asset depending on the current market price.

A diverse portfolio of securities and the use of high probability options strategies are the best ways to be successful as an options trader. A diversified basket is a group of portfolios, each with one of the lowest risks. Finding a diverse basket is the difficult problem for a classical computer to solve. As a result, Quantum Annealing can aid in the simulation of a diverse basket.

**Future Forward**

The pricing of derivatives by using Monte Carlo simulation is an active area of research in capital markets. Despite its widespread use, it often necessitates a significant amount of computing power to provide improved option pricing, especially for complex options. It also has the limitation in accounting the bear markets, recessions, or other sorts of financial crises that may affect portfolios of options. By estimating the optimal trade value of the assets in the portfolio, quantum algorithms can help to improve option pricing accuracy. Meanwhile, quantum machine learning models handle this issue by accelerating computation power. In reaction to market situations and investor needs, quantum optimization models can give the finest diversified basket. This helps in precisely estimating the derivative's best price. Despite these facts, the asymptotic tendency of estimation error with respect to computation time prevents us from concluding that quantum algorithms outperform classical ones. Google has claimed a quantum computer breakthrough pertaining to the difficult problem of quantum error correction. As a result, it appears interesting that option pricing will be one of the first applications to gain from a real, practical quantum advantage, with the promise of increased quantum speedup.

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TATA Consultancy Services

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12 Aug 2021

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Chennai

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