Financial Engineering
Financial engineering (FE) is the discipline that applies mathematical and statistical methods and econometric tools to the theoretical analysis, design, valuation, and risk management of financial instruments and financial markets, with the main objective of modeling the prices and risks of financial instruments and developing efficient investment and risk management strategies.
The areas in which financial engineering is applied are as follows
- Design and pricing of derivatives: Derivatives are derivative forms of financial instruments and include options, futures, and swaps. Financial engineering involves the design of derivative pricing models (e.g., Black-Scholes models) and derivative strategies.
- Portfolio Optimization: Financial engineering develops portfolio theory to optimize the trade-off between risk and return. This allows investors to construct efficient portfolios and maximize expected returns while minimizing risk.
- Risk Management: Financial engineering provides tools to assess the risk of financial instruments and portfolios and to develop hedging strategies. This is the case, for example, Value at Risk (VaR) and stress testing are widely used risk management techniques.
- Financial Modeling: In financial engineering, mathematical models and statistical methods are developed to model the behavior of financial markets and instruments. This makes it possible to analyze market trends, price fluctuations, and return projections.
- Financial Data Analysis: Financial engineering uses statistical and machine learning techniques to analyze financial data to identify market trends and patterns. This can then be used to make investment decisions and build predictive models.
Financial engineering is a discipline widely used by financial institutions, investment firms, insurance companies, hedge funds, stock exchanges, etc. For individual investors, the theories and tools of financial engineering provide tools that enable efficient investment and risk management.
Black Scholes Model
The Black-Scholes model is a valuation model for option prices in financial engineering, which was published by Fisher Black and Merton Scholes in 1973 and is used to calculate the prices of equity options and other financial derivatives. The model has been used to calculate the prices of equity options and other financial derivatives, and has also been applied to pricing and forecasting the volatility of derivative instruments.
The Black-Scholes model is based on several assumptions. These assumptions include the efficiency of the market, random walk described in “Overview of Random Walks, Algorithms, and Examples of Implementations” in stock prices, and the lognormal distribution of the asset’s rate of return. The model uses factors such as the current price of the stock, strike price, time to maturity, interest rate, and volatility to calculate the option price.
The Black-Scholes model has been used as a common tool for calculating option prices, but it may not perfectly match actual market fluctuations and conditions, and the model itself has some limitations and restrictions.
Algorithm used in the Black-Scholes model
In the Black-Scholes model, several mathematical methods and algorithms are used to calculate option prices. The main algorithms are described below.
- Analytical solution of the Black-Scholes equation: The Black-Scholes model uses a partial differential equation called the Black-Scholes equation to calculate option prices. This equation links the option price to relevant factors (stock price, strike price, time to maturity, volatility, interest rate, etc.), and the analytical solution of this equation may be given in closed form under certain conditions.
- Numerical Solution: Numerical solution methods are used when an analytical solution to the Black-Scholes equation is not available or when complex conditions are required. Numerical methods solve the equations numerically using discretization and approximation techniques. Some well-known numerical methods include the finite difference method, Monte Carlo simulation, and the Binomial Tree Method.
- Finite difference method: The finite difference method is a method for numerically solving differential equations and partial differential equations. The finite difference method begins by dividing the continuous domain of the differential equation into a discrete grid (grids), and at each point in the domain, a difference equation is constructed to compute an approximation of the derivative, thereby converting the differential equation into a difference equation.
- Monte Carlo Simulation: Monte Carlo simulation can be a method of estimating option prices using the generation of random numbers based on probability distributions. In this method, a huge number of random stock price scenarios are generated, option prices are calculated for each scenario, and the final option price is obtained from the mean, median, etc. of these scenarios.
- Binary tree method: The binary tree method is a type of data structure and algorithm used to efficiently perform operations such as data exploration and sorting. A binary tree is a tree structure in which each node has at most two child nodes, known in particular as a binary search tree.
Below we will explore the Black-Scholes equation in a little more depth.
Black-Scholes equation
The Black-Scholes equation is a partial differential equation used to calculate option prices and was published by Fisher Black and Merton Scholes in 1973.
The Black-Scholes equation is expressed as follows.
\[∂V/∂t + 1/2 σ^2 S^2 ∂^2V/∂S^2 + rS ∂V/∂S – rV = 0\]
where V is the option price function, t is time, S is the stock price, σ is volatility (a measure of price variation), and r is the interest rate.
This equation relates changes in option prices over time to changes in stock prices. The meaning of each term is as follows
- ∂V/∂t: the rate of change in the option price over time (the rate of decrease in the option’s time value)
- \(1/2 σ^2 S^2 ∂^2V/∂S^2\): Rate of change in option prices related to changes in stock prices (effect of volatility)
- rS ∂V/∂S: Rate of change in option price related to changes in stock price (growth rate of stock price)
- rV: Discounted present value of the option (effect of interest rate)
The Black-Scholes equation is used to obtain option prices analytically, and under certain conditions, an analytical solution may be obtained. For example, in the case of the European Call Option, the analytical solution of the Black-Scholes equation is given by.
\[C = S₀e^{(-qt)N(d₁)} – Xe^{(-rt)N(d₂)}\]
where C is the option price, S₀ is the current stock price, X is the strike price, t is the time to maturity, r is the risk-free interest rate, q is the dividend yield, N is the cumulative distribution function of the standard normal distribution, and d₁ and d₂ are the variables used in the calculation.
Although the Black-Scholes equation has become a fundamental tool widely applied in financial engineering for option pricing and risk management, there are certain restrictions on the scope of application of the equation and it may not perfectly match the real conditions of the market. Therefore, more sophisticated models and methods are being developed.
Next, specific implementations of the Black-Scholes model are presented.
Example python implementation of the Black-Scholes model
Below is an example implementation of the Black-Scholes model using Python. In this example, the price of a European call option is computed.
import math
def black_scholes_call(S, X, T, r, sigma):
d1 = (math.log(S / X) + (r + (sigma**2) / 2) * T) / (sigma * math.sqrt(T))
d2 = d1 - sigma * math.sqrt(T)
call_price = S * math.exp(-r * T) * norm_cdf(d1) - X * math.exp(-r * T) * norm_cdf(d2)
return call_price
def norm_cdf(x):
return (1.0 + math.erf(x / math.sqrt(2.0))) / 2.0
# Parameter Setting
S = 100 # stock prices
X = 100 # exercise price
T = 1.0 # Time to maturity (years)
r = 0.05 # risk-free interest rate
sigma = 0.2 # volatility
# European Call Option Price Calculation
call_price = black_scholes_call(S, X, T, r, sigma)
print("Call Option Price:", call_price)In this implementation, the black_scholes_call function calculates the European Call Option Price based on the Black-Scholes model. norm_cdf function is used to calculate the cumulative distribution function of the standard normal distribution. The result of the execution is the European call option price calculated as Call Option Price.
This code is a simple example and will require error handling and implementation of additional functionality when used for actual trading and risk management. Libraries and packages (e.g. NumPy, SciPy) can also be used for more efficient implementation and additional functionality.
Financial Engineering and Artificial Intelligence Technology
Financial engineering and artificial intelligence technologies play an important role in the modern financial industry, with the following specific applications
- Forecasting and Analysis: Artificial intelligence techniques are used in forecasting financial market volatility and analyzing risk. Machine learning and deep learning are used to analyze historical market data and economic indicators to build models that can predict future price trends and risks.
- High-Frequency Trading: Artificial intelligence technologies play an important role in high-speed, high-volume trading in financial markets. Algorithmic trading and high-frequency trading use automated trading systems that utilize artificial intelligence to detect subtle market patterns and trends and execute trades efficiently.
- Risk Management: Risk management is essential for financial institutions. Artificial intelligence technology can be used to predict customer credit risk and monitor market risk. In addition, artificial intelligence technology is used to detect fraud and money laundering.
- Natural Language Processing: In the financial industry, there are large amounts of textual data such as news articles, reports, and contracts. Natural language processing, a part of artificial intelligence technology, is used to analyze these data, extract information, analyze sentiment, and summarize.
The combination of financial engineering and artificial intelligence technology has had a significant impact on improving efficiency and decision-making in the financial industry by improving the ability to analyze data and the accuracy of predictions, enabling more rational and effective financial transactions and risk management.
The Black-Scholes Model and Artificial Intelligence Technology
One application of artificial intelligence techniques to the Black-Scholes model is to use artificial intelligence techniques (especially machine learning) to estimate parameters and forecast prices in the Black-Scholes model. Financial markets are very complex and accurate parameter estimation and price forecasting is difficult, and training machine learning algorithms using historical market data and relevant information can improve volatility estimation and price forecasting. Machine learning models have the ability to process large amounts of data and extract patterns and correlations, which can help complement the limiting assumptions and premises of the Black-Scholes model.
The other will use artificial intelligence technology for advanced risk management and portfolio optimization. Financial institutions and investors need to consider various factors and constraints in risk management and portfolio optimization, and artificial intelligence technology can analyze large amounts of data and market trends to build predictive and decision models to support decision making for risk management and portfolio optimization It is possible to do this. This includes, for example, portfolio management using reinforcement learning and risk prediction using deep learning.
In addition, artificial intelligence techniques have been applied to automation and acceleration in financial trading and investment. In areas such as algorithmic trading and high-frequency trading, artificial intelligence techniques are used to analyze market data, develop trading strategies, and execute trades automatically.
References and Bibliography
The application of artificial intelligence technology to economic fields such as financial engineering is discussed in “Economics, Business, and Artificial Intelligence Technology. See also “On Economics, Business, and Artificial Intelligence Technology. The application of probability and statistics is discussed in “Intuitive Methods in Economic Mathematics: A Note on Probability and Statistics.
Reference book is “Dark Pools: The Rise of the Machine Traders and the Rigging of the U.S. Stock Market”
“Flash Boys”
“The Physics of Wall Street: A Brief History of Predicting the Unpredictable”
“Clojure for Finance”
“Dark Pools and High Frequency Trading For Dummies”
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