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Two Sigma Presents Deep Learning for Sequences in Quantitative Finance David Kriegman
Two Sigma Presents Deep Learning for Sequences in Quantitative Finance David Kriegman
During the presentation, the speaker introduces the event and provides background information on Two Sigma, a renowned financial sciences company that applies scientific methods to the field of finance. They highlight that Two Sigma operates across multiple businesses within the financial sector, including quantitative hedge funds, broker-dealer services, private investments, insurance, and investor solutions. The speaker emphasizes the diversity of backgrounds among the audience, indicating that the lecture will cater to individuals at all levels of expertise, showcasing how deep learning can be applied effectively in quantitative finance. Notably, they mention that Two Sigma employs approximately 1600 professionals worldwide, with 600 of them holding advanced degrees and over 200 possessing Ph.Ds.
Moving on, the speaker introduces the concept of deep learning for sequences and illustrates its impact on various applications over the past decade. They provide examples such as sentiment classification, video activity recognition, and machine translation. The speaker explains that sequence processing tasks involve taking sequences as input and generating sequences as output, which may vary in length. Specifically, they discuss the application of deep learning in predicting stock market values using historical sequences. The speaker underscores the significance of predicting both the high and low points in order to maximize profitability.
Next, the speaker delves into the typical quantitative investment pipeline in finance, which encompasses a sequence of processes involved in making investment decisions. They outline the two key stages of the pipeline: alpha modeling and feature extraction. Alpha modeling involves predicting the direction of stock prices using mean reversion models or momentum models. Feature extraction focuses on extracting technical features from the market, such as price, volume, and bid-ask spread. The speaker emphasizes that these processes eventually lead to decisions regarding buying or selling in the markets, with the ultimate goal of generating profits and minimizing losses. They stress the importance of avoiding emotional decision-making and highlight the significance of diversifying portfolios in the realm of finance.
Subsequently, David Kriegman from Two Sigma takes the stage to discuss the factors that play a crucial role in making informed decisions in stock trading. The first factor highlighted is gathering fundamental data, which can be obtained through direct reports from companies or inferred from publicly available information. Furthermore, sentiment analysis can be conducted by interpreting unstructured data derived from sources like news, social media, and analyst comments. The speaker introduces the idea of utilizing non-traditional sources, such as the number of cars in a parking lot or the congestion of container ships in a harbor, to gather information that may indicate the performance of a specific stock. After employing the alpha model to make predictions about stock performance, the next step in the pipeline involves portfolio optimization. This step often entails solving large-scale optimization problems and considering factors such as the current stock holdings, confidence in the forecasts, diversification requirements, and associated trading costs. Finally, the execution phase involves making decisions about order size, placement, and type, with the aid of a model to understand the potential impact of these actions.
Returning to the topic of deep learning, the speaker highlights the sequential nature of the quantitative finance decision-making pipeline. They then shift focus to deep learning, describing it as a type of neural network characterized by multiple layers. The speaker discusses significant developments in neural networks since their initial introduction in the 1950s, including the emergence of new network architectures, the availability of massive training datasets, and advancements in parallel computation. To illustrate the basic idea behind a single perceptron, the speaker explains how it takes inputs, computes a weighted sum, and passes the result through a non-linear function. They mention that the traditional activation function, a threshold, has been replaced by an alternative called rectified linear unit (ReLU), which outputs zero for values below a threshold and the actual value for higher values.
Continuing with the topic of neural networks, the speaker introduces the concept of a multi-layer perceptron. In this architecture, each circle represents a perceptron with its own activation function and set of weights. This can be represented by a pair of weight matrices, allowing for the creation of larger networks. The speaker goes on to discuss the application of neural networks for alpha modeling, specifically in predicting stock prices based on historical performance. The network is trained using a set of training data that includes both features and price data, with the optimization goal of minimizing the total loss. This training process involves various techniques such as backpropagation and stochastic gradient descent.
To enhance the alpha model further, the speaker explains the importance of incorporating multiple features rather than relying on a single signal, such as price or past history. By combining all relevant features, a more powerful and accurate model can be created. However, utilizing a fully connected network with this approach can lead to a problem known as the curse of dimensionality, as the number of weights becomes extremely large and not all of them can be effectively trained. To overcome this challenge, the speaker introduces another class of sequence processing networks called recurrent neural networks (RNNs). These networks introduce a memory aspect and feed information backward, building a state with each time instant. As a result, the issue of having an excessive number of weights is mitigated. In RNNs, the weights are shared between each element, making the network deep and providing a tractable solution.
The speaker highlights the difficulties of training deep networks and how gated networks, such as Gated Recurrent Units (GRUs) and Long Short-Term Memory (LSTM) networks, address these challenges. Gated networks incorporate analog gates that control the flow of information and enable the updating of previous states with potential new states. The components of these networks are differentiable, allowing them to be trained using backpropagation. Compared to LSTMs, GRUs have longer memory capabilities.
Discussing various architectures used in deep learning for sequences, the speaker introduces LSTM and GRU networks, as well as more recent developments such as convolutional neural networks (CNNs), attention mechanisms, and transformers. They also touch upon reinforcement learning, which optimizes sequential decision-making processes like those involved in trading and market interactions. While reinforcement learning has shown success in games, applying it to finance requires suitable simulators, robust software infrastructure, and significant computational resources. Overall, the speaker emphasizes that the different architectures and models discussed represent powerful tools for quantitative finance, each with its own advantages and challenges.
Returning to David Kriegman's contribution, he sheds light on the pipeline employed in quantitative finance and how deep neural networks can be trained to implement different parts of it. He highlights Two Sigma's extensive operations, which involve trading in thousands of equities and making hundreds of millions of decisions every day. Handling such vast amounts of data necessitates substantial computational power, robust software infrastructure, and a team of creative individuals. Addressing concerns about the lack of explainability and interpretability associated with deep neural networks and their impact on strategy development, Kriegman explains that certain architectures can introduce interpretable representations. He also emphasizes that in rapidly changing trading scenarios, different distributions are required. Additionally, Two Sigma incorporates human traders who monitor and implement systems during extreme market events.
The speaker discusses how deep learning approaches can interact with the hypothesis of an efficient market in quantitative finance. While the market is generally considered efficient, deep learning can facilitate quicker response to information and offer alternative methods of assimilating data, potentially identifying inefficiencies and investment opportunities. They also highlight the relevance of computer vision techniques in sequential modeling within finance, particularly during the initial stages of extracting features from unstructured data. Two Sigma actively seeks individuals for engineering and modeling roles, and while different roles align with different teams, the application of deep learning pervades the entire organization. Recent college graduates and MSc-level applicants are encouraged to apply through the Two Sigma website.
During the Q&A session, the speaker addresses several challenges associated with applying deep learning to quantitative finance. One major challenge is the lack of stationarity in financial time series, as deep learning models perform best when the future resembles the past. To tackle this issue, the speaker emphasizes the importance of simulating and predicting to introduce domain transfer methods, enabling the models to adapt to changing market conditions. Additionally, the speaker notes that the error rate in quantitative finance is generally higher compared to other fields, and even being slightly better than 50% can provide a significant edge in trading.
When asked about promising implications for quantitative finance, the speaker mentions that nearly every research area in deep learning and neural networks holds promising implications. They specifically highlight reinforcement learning and domain transfer as areas of interest. Furthermore, they acknowledge the data storage challenges in finance and suggest that data compression techniques can be helpful in addressing these issues.
Expanding on the topic of the engineering team responsible for implementing deep learning models in quantitative finance, the speaker explains that the team works on various tasks, including storage management, physical systems, and the layers built on top of those systems. They emphasize that both deep learning models and statistical modeling have their roles depending on the specific use case. However, they note that if a deep model is reduced to a degenerate form of linear regression, it loses its intrinsic interest and power.
The presentation emphasizes the application of deep learning in quantitative finance, particularly in the context of sequence processing and decision-making pipelines. It highlights the challenges and opportunities that arise when utilizing deep neural networks in this domain, including the need for interpretability, addressing non-stationarity, and leveraging diverse architectures. Throughout the presentation, Two Sigma is presented as a prominent company actively incorporating deep learning techniques in its operations and actively seeking talented individuals to join their team.
Two Sigma Presents: Machine Learning Models of Financial Data
Two Sigma Presents: Machine Learning Models of Financial Data
Justin Ceriano from Two Sigma Securities delivers a comprehensive presentation on the integration of machine learning models in the field of finance. He begins by highlighting the increasing interest of financial companies in leveraging machine learning to enhance their predictive capabilities and decision-making processes. Specifically, machine learning algorithms can be utilized to predict future prices of financial instruments and determine optimal trading strategies.
Ceriano introduces the concept of reinforcement learning, which falls under a class of methods capable of learning decision policies directly from available data in order to maximize an appropriate objective function. Reinforcement learning proves particularly valuable in finance, where the goal is to optimize outcomes based on historical data.
One of the fundamental aspects discussed is the application of machine learning models to analyze limit order books in electronic markets. In this system, buyers and sellers submit orders specifying the prices at which they are willing to buy or sell a particular asset. These orders are then matched based on the best available ask or bid price. Ceriano emphasizes that the order book data, which represents the visible supply and demand for a stock, forms a high-dimensional sequence that can be effectively utilized to predict future price changes using machine learning models.
Moreover, Ceriano emphasizes the significance of considering non-zero spreads in trading strategies. These spreads can impact the profitability of price predictions, thereby necessitating careful evaluation and adjustment.
To demonstrate the practical implementation of machine learning models, Ceriano explains the construction of a recurrent neural network designed to predict price changes using high-frequency financial data. The model is trained to forecast whether the next price change will be positive or negative, and its performance is compared to a linear recurrent model. The dataset employed consists of three years of event-by-event high-frequency data for approximately 1,000 stocks. The objective is to assess whether non-linear machine learning models, such as recurrent networks, outperform linear statistical models in capturing non-linear relationships within the data. The optimization of the models' predictions is achieved through the backpropagation algorithm, minimizing the prediction error. To reduce computational costs, the truncated backpropagation through time algorithm is utilized.
Challenges related to optimizing recurrent networks, particularly the well-known vanishing gradient problem, are addressed in the presentation. The vanishing gradient problem refers to the issue of gradients becoming extremely small as they propagate through the lower layers of the network. Consequently, this can hinder training speed and make it difficult for the network to retain information from distant parts of the sequence. Ceriano introduces the Long Short-Term Memory (LSTM) network, one of the most popular types of recurrent networks, which has been specifically designed to address this problem by efficiently updating the memory state, thus enabling the model to retain relevant information from far in the past.
The presentation proceeds to discuss the training and evaluation of machine learning models using high-frequency order book data. The authors compare the accuracy of a linear model with that of an LSTM recurrent network, and the results clearly indicate the superior performance of the deep learning model when tested on roughly 500 stocks over a three-month out-of-sample period. The discussion also delves into the universal nature of the relationship between order book data and price movements, suggesting the existence of a universal price formation model applicable across multiple stocks. This finding holds significant practical implications, such as reducing computational costs and the ability to improve a model for one stock using data from another.
The experiment aims to train a universal model by pooling data from numerous stocks and evaluating its accuracy in comparison to stock-specific models. The results consistently demonstrate the superiority of the universal model, indicating shared universality in the order book dynamics across different stocks. This not only reduces overfitting but also enhances the accuracy of the model. Furthermore, the universal model exhibits stability for over a year and scalability with the aid of high-performance computing, utilizing 25 GPUs with asynchronous stochastic gradient descent.
The presentation also explores the application of reinforcement learning to optimize order submission strategies for optimal execution. The focus is on developing policies for market orders or limited orders of one share, aiming to maximize expected rewards and cost savings within discrete time intervals. By utilizing historical order book data, the reinforcement learning model is trained to simulate executed prices for small orders. The model determines whether to submit a market order immediately or wait for the best ask price to decrease, using the limit order book data as input. The performance of the model is evaluated using one year of data and then tested on a separate six-month dataset.
Simulation results across a universe of 100 stocks are presented, considering time horizons of 10 and 60 seconds for both a market order-only reinforcement learning strategy and a simple limit order strategy. The results consistently indicate positive cost savings achieved by the reinforcement learning model across the 50 stocks, although with some variability. Moreover, the cost savings tend to increase with longer time horizons. The presentation introduces the concept of using historical order book data to simulate whether a submitted limit order will be executed within a specific time interval. The reinforcement learning model is trained to dynamically select the optimal time to maximize expected cost savings. While the cost savings vary across different stocks, the reinforcement learning strategy consistently yields positive results, with some stocks exhibiting significantly higher cost savings than others.
The presentation concludes by addressing the need for developing advanced optimization methods and deep learning architectures specifically tailored for financial data. It emphasizes the ongoing challenges in merging reinforcement learning with accurate simulations for larger order sizes to further enhance the application of machine learning in finance. To effectively grasp the concepts discussed, Ceriano recommends gaining hands-on experience by implementing machine learning techniques on large-scale datasets. He highlights the importance of understanding the underlying mathematical theory and having proficiency in deep learning libraries such as TensorFlow and PyTorch. Additionally, high-performance computing skills for parallelizing model training are emphasized.
Furthermore, the presenters discuss Two Sigma's hiring policies and remote work opportunities. While a full-time remote work policy is not in place, Two Sigma hires individuals from various countries worldwide and operates an online team called Alpha Studio for remote work. They emphasize the importance of acquiring knowledge in quantitative finance, probability, and statistics through multiple courses for those interested in pursuing machine learning in finance. The presentation also mentions the utilization of deep learning libraries such as TensorFlow and PyTorch in Two Sigma's codebase.
The hiring process at Two Sigma is discussed, with emphasis on recruitment taking place throughout the year, particularly during the summer. Exceptions are made for fall and spring hires, and the company encourages interested individuals to start as early as possible, even if it means beginning in December. The presenters suggest that impressive projects involve identifying patterns and trends in real data and applying machine learning approaches to solve real-world problems. Ownership of the project and highlighting one's contributions within the project are emphasized as valuable qualities sought by recruiters. The fundamental equity research team at Two Sigma, which collaborates closely with engineers and data scientists, is also briefly mentioned.
The distinction between a data scientist and a quant researcher at Two Sigma is elucidated. While both positions involve modeling and trading, data science focuses primarily on the data science aspect and feature engineering, whereas quant researchers consider the complete trading process from start to finish. The presenters touch upon office culture and meetings at Two Sigma, describing meetings as primarily informal and offering whiteboards for collaborative discussions. Prepared presentations are occasionally required for specific meetings.
Finally, the benefits of employing a universal model versus stock-specific models are highlighted. The universal model's ability to leverage transfer learning and mitigate overfitting issues is emphasized as a key advantage. The presentation concludes by mentioning that the recorded session will be made available on Two Sigma's YouTube channel and highlighting the company's global hiring practices, with the majority of hires being based in the United States.
Keys to Success in Algorithmic Trading | Podcast | Dr. E.P. Chan
Keys to Success in Algorithmic Trading | Podcast | Dr. E.P. Chan
Quantitative trading, or trading in general, is considered one of the most challenging professions to break into and succeed in. Dr. D.E. Shaw, a pioneer in quantitative trading and founder of a multi-billion dollar hedge fund in New York, has acknowledged that the field has become increasingly challenging with each passing year. This sentiment is echoed by many experienced traders in the industry.
Despite its difficulty, quantitative trading is still worth pursuing for those who are passionate about it. Just like becoming a successful actor, singer, model, or fiction writer, achieving success in algorithmic trading requires dedication and perseverance. While not everyone can reach the level of renowned traders like D.E. Shaw or Renaissance Technologies, aspiring traders should not be discouraged. It is important to be prepared for failure as success in this field is an outlier.
For individuals who are not already in the financial industry, it is advisable to not quit their day job immediately after graduating and starting their first trading strategy. It is recommended to have at least two profitable trading strategies running live for a period of two years before considering full-time trading. This advice is based on personal experience and the experiences of other successful traders.
Traders often make the mistake of being overly optimistic about the past performance of a strategy, leading them to leverage too high. It is crucial to avoid excessive leverage, as it can quickly wipe out an account's equity. Additionally, strategy performance does not usually continue trending in the same manner. Allocating capital based solely on past performance is a common mistake. Instead, a risk parity allocation, where capital is allocated inversely proportional to a strategy's volatility, is generally a better approach.
Another common mistake is failing to invest profits into data equipment and personnel during good times. It is essential to reinvest a portion of profits to improve data infrastructure and hire skilled personnel, as this can help prevent future drawdowns.
On a positive note, starting with simple strategies that have intuitive justification is recommended. It is wise to understand and improve upon existing strategies before delving into more complex approaches like recurrent neural networks or deep learning. By starting with simple strategies, traders can better understand the reasons behind successes or failures, attributing them to specific factors.
In conclusion, quantitative trading is a challenging yet potentially rewarding profession. It requires perseverance, continuous learning, and careful decision-making. While there are pitfalls to avoid, there are also valuable lessons to be learned from experienced traders. By starting with simple strategies, managing risk, and investing in infrastructure and personnel, aspiring traders can increase their chances of success in the field of quantitative trading.
"Basic Statistical Arbitrage: Understanding the Math Behind Pairs Trading" by Max Margenot
"Basic Statistical Arbitrage: Understanding the Math Behind Pairs Trading" by Max Margenot
Welcome to the Quanto Peon Algorithmic Trading Meetup, an event dedicated to exploring the world of quantitative finance. I'm Max Margit, a data scientist at Quanto Peon, and today I'll be delving into the fascinating topic of statistical arbitrage and the fundamental statistical concepts associated with it.
Before we dive into the theoretical aspects, let me provide you with a brief introduction to Quanto Peon. Our main objective is to make quantitative finance accessible to everyone by offering free open-source tools that empower individuals to research and develop their own algorithmic trading strategies. Algorithmic trading involves using instructions to execute trades automatically in financial markets, ranging from simple rules like buying Apple stocks every day at 10:00 AM to more sophisticated quantitative analysis using statistical models.
Statistical arbitrage, the focus of today's discussion, revolves around exploiting market inefficiencies using statistical analysis instead of relying on physical imbalances. This approach aims to identify and capitalize on statistical imbalances in asset prices. To better comprehend this concept, it's crucial to understand some fundamental statistical concepts.
One of the key concepts we'll explore is stationarity, particularly in the context of time series data. Stationarity refers to a series of data points where each sample is drawn from the same probability distribution with consistent parameters over time. In simpler terms, it means that the mean and standard deviation of the data remain constant over time. This is important because many statistical models used in finance assume stationarity. By ensuring stationarity, we can trust the results obtained from these models.
To illustrate the concept of stationarity, let's generate some data points. I'll use a basic function called "generate_data_point" to create a set of samples from a standard normal distribution. These samples represent a stationary time series often referred to as white noise. In this case, the mean is zero, and the standard deviation is one. When we plot this data, we observe a random pattern resembling white noise.
However, not all time series data exhibit stationarity. If we introduce a trend in the mean, the time series becomes non-stationary. In finance, non-stationarity can be much more complex than this simple example. Descriptive statistics, such as the mean, become meaningless for non-stationary data as they do not accurately represent the entire time series.
Now, how do we determine if a time series is stationary or not? This is where hypothesis tests come into play, such as the augmented Dickey-Fuller test commonly used in stationarity analysis. This test helps us assess the probability of a given time series being non-stationary.
Let's apply the augmented Dickey-Fuller test to our generated time series data. The test provides a p-value, which indicates the likelihood of rejecting the null hypothesis that the time series is non-stationary. In our first example, where the data was deliberately generated as stationary, the p-value is close to zero. This allows us to reject the null hypothesis and conclude that the time series is likely stationary. On the other hand, in the second example with the introduced trend, the p-value exceeds the threshold (0.01), and we fail to reject the null hypothesis, indicating that the time series is likely non-stationary.
However, it's important to note that hypothesis tests have limitations. False positives can occur, especially when dealing with subtle or complex relationships within financial data. Therefore, it is essential to exercise caution and not solely rely on hypothesis tests for determining stationarity.
Now, let's shift our focus to pairs trading. If I want to engage in pairs trading, I need to consider multiple pairs and place independent bets on each of them. Instead of relying on a single pair, diversifying my portfolio by trading 100, 200, or even 300 pairs allows me to leverage any edge I may have in each pair, thereby increasing my overall chances of success.
Trading pairs requires a robust framework to manage and monitor the trades effectively. This involves continuously updating the relationship between the pairs and adjusting positions accordingly. Since the beta values, which represent the relationship between the pairs, can change over time, I need a system that dynamically adapts to these changes.
Additionally, having a clear exit strategy for each trade is crucial. I must determine when to close a position if the pair is no longer exhibiting the expected behavior or if the relationship between the pairs breaks down. This requires constant monitoring of the spread and having predefined criteria for exiting a trade.
Moreover, risk management plays a significant role in pairs trading. It's essential to carefully calculate the position sizes for each pair based on factors such as volatility, correlation, and overall portfolio exposure. By diversifying my trades and managing risk effectively, I can minimize the impact of adverse market conditions and maximize potential profits.
To implement pairs trading strategies effectively, traders often rely on advanced quantitative techniques and develop sophisticated algorithms. These algorithms automatically scan the market for potential pairs, evaluate their cointegration and statistical properties, and generate trading signals based on predefined criteria.
In conclusion, understanding stationarity and conducting appropriate tests are crucial when building statistical models for algorithmic trading. By grasping the concept of stationarity and using tests like the augmented Dickey-Fuller test, traders can assess the likelihood of non-stationarity in time series data. Pairs trading, as a statistical arbitrage strategy, allows traders to exploit temporary deviations from the historical relationship between two correlated securities. However, successful implementation requires robust frameworks, continuous monitoring, risk management, and the use of advanced quantitative techniques.
At Quanto Peon, we strive to bridge the gap between finance and technology by offering free lectures on statistics and finance through our Quanto Peon Lecture Series. Our mission is to democratize quantitative finance and provide individuals with the tools and knowledge to develop their algorithmic trading strategies.
Brownian Motion for Financial Mathematics | Brownian Motion for Quants | Stochastic Calculus
Brownian Motion for Financial Mathematics | Brownian Motion for Quants | Stochastic Calculus
Hello, YouTube, and welcome back to the ASX Portfolio channel. My name is Jonathan, and today we're going to delve into the fascinating world of Brownian motion, specifically in the context of financial mathematics. This is a crucial topic as it forms the foundation of stochastic processes and stochastic calculus, which are essential in the field of financial mathematics. Brownian motion is the basis of Ito integrals and holds great significance, so understanding it is of utmost importance. In future videos, we will explore the mathematics further, covering topics such as geometric Brownian motion, its applications, and Ito integrals. Make sure to hit the subscribe button if you want to stay tuned for those upcoming videos.
In this video, we will run through a Jupyter notebook that I have prepared to explain what Brownian motion is and how it arises. So, let's jump right in. We will begin by considering a symmetric random walk and then move on to a scaled random walk, demonstrating how they converge to Brownian motion. Throughout this explanation, we will be using notation and examples from Steven Shreve's book, "Stochastic Calculus for Finance II."
First and foremost, it's crucial to understand that the main properties of Brownian motion are as follows: it is a martingale, meaning the expectation is solely based on the current position of the particle or stock price. Additionally, it is a Markov process, and it accumulates quadratic variation. Quadratic variation is a unique concept in stochastic calculus, setting it apart from ordinary calculus. In this episode, we will delve into what quadratic variation entails.
If you want to follow along with the code, it is available on my website. I have imported the necessary dependencies that we will need for this demonstration. It's important to note that Brownian motion is a stochastic process, and for our purposes, we will consider a filtered probability space with outcomes and a filtration F, along with a probability space P. Here, we have a set of real outcomes within the interval from 0 to time T.
Brownian motion always has an initial value of zero. It has independent increments, follows a Gaussian distribution, and exhibits continuous sample paths almost surely. We will explain all these properties in detail.
Let's start with the simplest example: a symmetric random walk. If you are unfamiliar with the concept of a random walk, think of it as a sequence of successive coin tosses. Each outcome, represented by the variable omega, can be either a head or a tail. We will use the variable X_j to represent each outcome, taking on a value of 1 for heads and -1 for tails.
If we define a process with m_0 equal to zero, then m_k will be the summation along all possible coin toss paths for k tosses. In this case, we have a random walk where the process can move up by 1 or down by 1, and we sum these increments over the paths. I have written a script to generate 10 sample paths over a time horizon of 10 years. The plot demonstrates how the random walk moves up or down by 1 at each time step along the paths.
This example reveals some interesting properties. First, the increments between time periods, such as m_k+1 - m_k, are independent. Furthermore, the expectation of these independent increments is zero, and the variance is equal to the difference in time or the distance between the time steps (k_i+1 - k_i). Variance accumulates at a rate of one per unit time.
Additionally, the symmetric random walk is a martingale. This means that the conditional expectation of the next value, given the current position, is equal to the current position. In the context of a symmetric random walk, the expectation of the
Continuing from where we left off, in the next video, we will explore how to create samples of geometric Brownian motion using Python. Geometric Brownian motion is a stochastic process commonly used in financial mathematics to model stock prices. It is an essential concept to understand in the field.
But before we dive into that, let's recap some of the key properties of Brownian motion. Brownian motion is a stochastic process characterized by several properties:
Independent Increments: The increments of Brownian motion are independent, meaning that the change between any two points in time is unrelated to the change between any other two points.
Gaussian Distribution: The increments of Brownian motion follow a Gaussian or normal distribution. This distribution describes the probability of various outcomes and is a fundamental concept in probability theory.
Continuous Sample Paths: Brownian motion has continuous sample paths, which means that it is non-differentiable at each time period. This property makes it suitable for modeling various phenomena with random fluctuations.
Quadratic Variation: Quadratic variation is a unique property of Brownian motion in stochastic calculus. It measures the accumulated fluctuations over time and is crucial for understanding the behavior of stochastic processes.
Now, let's discuss geometric Brownian motion. Geometric Brownian motion is an extension of Brownian motion that incorporates exponential growth. It is commonly used to model the behavior of financial assets such as stock prices. Geometric Brownian motion has the following form:
dS(t) = μS(t)dt + σS(t)dW(t)
Here, S(t) represents the asset price at time t, μ is the expected return or drift rate, σ is the volatility or standard deviation of returns, dt is a small time interval, and dW(t) is a standard Brownian motion increment.
To simulate geometric Brownian motion, we can discretize the process using numerical methods like Euler's method or the Itô integral. These methods allow us to approximate the continuous process using a sequence of discrete steps.
In the upcoming video, we will explore the mathematical details of geometric Brownian motion and its applications in financial mathematics. We will also provide practical examples and code snippets in Python to simulate and visualize geometric Brownian motion.
If you're interested in learning more about this topic, make sure to subscribe to our channel and stay tuned for the next video. We look forward to sharing more insights with you. Thank you for your attention, and see you in the next video!
Simulating Geometric Brownian Motion in Python | Stochastic Calculus for Quants
Simulating Geometric Brownian Motion in Python | Stochastic Calculus for Quants
Good day, YouTube, and welcome back to the ASX Portfolio Channel. My name is Jonathan, and today we're going to simulate geometric Brownian motion in Python. In this tutorial, we won't be going through the derivation of the dynamics of geometric Brownian motion or covering Ito calculus, Ito integrals, and stochastic processes. However, we will explore those topics in detail in the following tutorial. If you're interested in learning more about them, please subscribe to our channel and hit the notification bell so you can be notified when that video is released.
Let's jump into the simulation. I'll be using this Jupyter notebook for demonstration purposes. First, we'll define the parameters for our simulation. The drift coefficient, mu, is set to 0.1 or 10% over a year. We'll define the number of time steps as "n" and set it to 100 for a granular simulation. Time will be measured in years, denoted as "T". The number of simulations will be denoted as "m" and set to 100. The initial stock price, S0, is set to 100, and the volatility, sigma, is set to 30. Let's import the necessary dependencies: numpy as np and matplotlib.pyplot as plt.
Now let's simulate the geometric Brownian motion paths. To calculate the time step, we divide T by n. Next, we'll use numpy arrays to perform the simulation in one step instead of iterating over paths. We'll define an array called "st" and use numpy's exponential function. Inside the function, we'll define the components: mu minus sigma squared divided by 2, multiplied by dt. Then, we'll multiply sigma by numpy's random.normal function, which samples from the normal distribution, and multiply it by the square root of dt. The size of this array will be m by n, representing the number of simulations and time steps, respectively. Since we want the simulation for each time step, we'll take the transpose of this array.
To include the initial point for each simulation, we'll use numpy's vstack function to stack a numpy array of ones with the st simulation array. This will ensure that each simulation starts with the initial value. Finally, we'll multiply the stacked array by the initial value to account for the daily changes in terms of the drift, variance, and stochastic component. This will give us the time step implementations. To accumulate these values over time, we'll use numpy's cumulative product function along each simulation path, specifying axis 1. This will calculate the cumulative product for each path.
Now that we have the simulated paths, let's consider the time intervals in years. We'll use numpy's linspace function to generate evenly spaced time steps from 0 to T, with n+1 spaces. This will give us an array called "time". Next, we'll create a numpy array called "fill" with the same shape as st so that we can plot the function. We'll use numpy's full function and set the fill_value to time. Taking the transpose of this vector, we can plot the graph with years along the x-axis and the stock price along the y-axis, taking into account the dispersion resulting from the 30% volatility and 10% increase in the mean or drift over this geometric Brownian motion.
Geometric Brownian motion is a useful model for option pricing theory and various financial mathematics applications. I hope you found value in this tutorial. In the next video, we'll dive deeper into financial mathematics, Ito calculus, Ito integrals, and explore how to increase the complexity of stochastic differential equations by adding different parameters. If you want to learn more, be sure to subscribe to our channel and hit the notification bell so you can be notified when that video is released next week. Until then, stay tuned for more valuable content. Thank you for watching, and see you in the next video.
Stochastic Calculus for Quants | Understanding Geometric Brownian Motion using Itô Calculus
Stochastic Calculus for Quants | Understanding Geometric Brownian Motion using Itô Calculus
Good day, YouTube, and welcome back to ASX Portfolio. Today, we're going to discuss why Brownian motion is an inappropriate choice for modeling financial markets. It's quite obvious that Brownian motion would result in negative stock prices, which is not realistic. Instead, we need a way to preserve some of the stochastic properties of Brownian motion and incorporate them into our models. This can be achieved by using Ito processes, which allow us to add the source of risk from Brownian motion.
One well-known Ito process is Geometric Brownian Motion (GBM), which many of you might be familiar with. We can leverage the properties of Brownian motion to develop new models that better align with real-life examples. To accomplish this, we employ a special type of calculus known as Ito calculus, which is commonly used in financial stochastic mathematics.
Today, we'll focus on understanding the Ito integral and how it can help us solve complex problems. We'll discuss Ito's lemma, which serves as the identity in Ito calculus and aids in rule derivation. Additionally, we'll explore the Ito-Dobelin formula and the derivation of the dynamics of Geometric Brownian Motion.
To dive deeper into these concepts, I highly recommend Stephen Shreve's second book, "Continuous-Time Models for Stochastic Calculus." Chapter 4 covers the material we'll be discussing today.
Now, let's begin by understanding what an Ito integral is. It's essential to remember that all the mathematics we'll be discussing are based on a filtered probability space. This space encompasses the outcomes, filtrations, and probability measures. Filtration refers to a sigma-algebra that contains all the information up to time t. Although probability theory is complex, we'll only briefly touch upon it today. For a more in-depth understanding, I recommend referring to the first three chapters of Shreve's book.
The Ito integral is represented by the symbol ∫δdW, where δ is a stochastic process and dW is the Wiener process. To grasp its meaning, let's imagine partitioning the time period from 0 to T into small intervals. We can denote the stochastic process δ to the power of n, where n represents the number of time intervals. This process is adapted, meaning its values are determined by the outcomes of coin flips at each time interval.
Now, consider the integral as the limit of a sum as the number of intervals approaches infinity. Each summand consists of the stochastic process δ multiplied by the change in the Wiener process between intervals. As the intervals become smaller, we converge on the Ito integral. However, for this limit to exist, two conditions must be satisfied: the process δ must be adapted to the filtration, and it must be square integrable.
Now that we understand the notation, let's move on to general Ito processes. These processes occur in the same time domain with the same outcome space. They involve time-based integrals and Ito integrals with respect to the Wiener process. The time-based integral is similar to a regular Riemann integral, while the Ito integral captures the stochastic nature of the process. These processes can be divided into drift and diffusion terms.
An example of an Ito process is Geometric Brownian Motion (GBM). It comprises a drift term and a diffusion term. The drift is determined by a constant μ, while the diffusion is controlled by a volatility parameter σ. The dynamics of GBM can be expressed using integrals, as shown in the equation.
Expanding on this, we can also consider the integral of an Ito process. For example, the integral of the Ito process may represent the trading profit and loss (P&L).
In the Itô-Doob decomposition, we have this generic process represented by the integral of the drift term, the integral of the diffusion term, and the Itô integral term. Now, the Itô-Doob formula provides a way to calculate the differential of a function of the process. It states that the differential of the function is equal to the partial derivative of the function with respect to time, plus the partial derivatives of the function with respect to the state variables multiplied by the drift terms, plus the partial derivatives of the function with respect to the state variables multiplied by the diffusion terms, plus the integral of the partial derivatives of the function with respect to the state variables multiplied by the Itô integral term.
This formula allows us to calculate the change in the value of a function as the process evolves over time. It is a fundamental tool in Itô calculus and is used extensively in stochastic analysis and mathematical finance.
Moving on to geometric Brownian motion (GBM), it is a specific type of Itô process commonly used to model the dynamics of stock prices and other financial assets. GBM incorporates both drift and diffusion components. The drift term represents the expected rate of return on the asset, while the diffusion term captures the volatility or randomness in the asset's price movements.
The dynamics of GBM can be derived using the Itô calculus. By applying the Itô formula to the logarithm of the asset price, we obtain an expression that describes the change in the logarithm of the price over time. This change is equal to the drift term multiplied by the time increment, plus the diffusion term multiplied by the Itô integral. By exponentiating both sides of the equation, we recover the dynamics of the asset price itself.
Understanding the dynamics of GBM is crucial in options pricing and risk management. It allows us to model the stochastic behavior of asset prices and estimate the probabilities of various outcomes. GBM has been widely used in financial mathematics and has served as a foundation for many pricing models, such as the Black-Scholes model for option pricing.
In summary, Itô calculus provides a powerful framework for modeling and analyzing stochastic processes in finance. By incorporating Itô integrals and applying Itô's lemma and the Itô-Doob formula, we can derive the dynamics of various financial variables and develop models that capture the stochastic properties of real-world markets. Itô calculus has revolutionized the field of mathematical finance and continues to be an essential tool for understanding and managing financial risk.
Stochastic Calculus for Quants | Risk-Neutral Pricing for Derivatives | Option Pricing Explained
Stochastic Calculus for Quants | Risk-Neutral Pricing for Derivatives | Option Pricing Explained
In this video, we will delve into the financial mathematics behind valuing a financial derivative using Monte Carlo simulation and risk-neutral pricing. We will answer questions such as why Monte Carlo simulation is used, what risk-neutral pricing is, and why the stock growth rate doesn't enter into the derivative model.
Risk-neutral pricing is a methodology where the value of an option is the discounted expectation of its future payoffs. In other words, it is the expected value of all the possible payoffs of a derivative, discounted back to the present time. The underlying stock growth rate does not impact the option price in the risk-neutral pricing framework. This is because the derivative and the underlying stock have a perfect correlation, allowing for replication and the creation of a risk-free portfolio.
There are several benefits to using the risk-neutral pricing approach over other valuation methods. Firstly, with complex derivative formulations, closed-form solutions may not be feasible. In such cases, using replication methods and solving partial differential equations (PDEs) can be computationally expensive. Risk-neutral pricing, on the other hand, allows for easy approximation of the option value using Monte Carlo simulation, which is less computationally expensive.
To explain risk-neutral pricing, we start by considering the one-period binomial model. In this model, the stock can either go up or down, and the option value depends on these two possible outcomes. By constructing a portfolio of the underlying stock and a risk-free asset, we can replicate the option's payoff. Using the principle of no arbitrage, the option value at time zero must be equal to the portfolio's value at time zero. By solving the linear equations, we can obtain a formula that represents the discounted expectation in the binomial model.
We introduce the concept of a risk-neutral probability measure, denoted as q, which allows us to shift from the physical probabilities of the stock price to the risk-neutral probabilities. This shift is accomplished by reweighting the physical probabilities by a random variable called the random-nickdem derivative. This derivative enables us to translate the option value from the risk-neutral pricing world to the physical probability world.
The objective of risk-neutral pricing is to identify the random-nickdem derivative process, denoted as Zt, that ensures all the discounted stock prices are martingales under the risk-neutral probability measure q. By performing a change of measure, we can convert the original Brownian motion under the physical probability measure to a new Brownian motion under the risk-neutral probability measure. This new Brownian motion is a martingale process, indicating that its expectation remains constant over time.
To apply these concepts, we consider the geometric Brownian motion model, which represents the dynamics of a non-dividend-paying stock. The model consists of a deterministic component and a stochastic component, representing the volatility. However, the original stock dynamics are not a martingale under the physical probabilities due to the deterministic component. To make the dynamics a martingale, we introduce the Radon-Nikodym derivative, which removes the drift term and transforms the stock dynamics into a martingale process under the risk-neutral probability measure.
In summary, risk-neutral pricing and Monte Carlo simulation provide a valuable framework for valuing financial derivatives. The risk-neutral pricing approach offers benefits such as simplicity, computationally efficiency, and the ability to handle complex derivative structures. By using the random-nickdem derivative and changing the measure from physical probabilities to risk-neutral probabilities, we can accurately value derivatives and replicate their payoffs in a risk-free manner.
Trading stock volatility with the Ornstein-Uhlenbeck process
Trading stock volatility with the Ornstein-Uhlenbeck process
At the beginning of 2020, the S&P 500 experienced a significant increase in volatility as prices sharply declined. Within a span of one month, the index plummeted by nearly a thousand points. Concurrently, the expectation of future volatility, based on traded index options, also surged during this period, reaching a peak of 66. It became apparent that during periods of market volatility when the index value decreased, the VIX (Volatility Index) rose. The VIX serves as a future estimate of volatility. This phenomenon led market makers and trading professionals to anticipate that the realized volatility would persist.
In this video, we aim to explain the market characteristics of volatility and discuss a methodology to model volatility by fitting the Ornstein-Uhlenbeck formula to a specific volatility index. We will use the maximum likelihood estimation method to calibrate the three parameters of the model to market data. Subsequently, we will simulate this process in Python, allowing us to comprehend and analyze the dynamics of volatility over time.
To accomplish this, we will import various dependencies such as time, math, numpy, pandas, datetime, scipy, matplotlib, pandas_datareader, and the plot_acf function from the stats module. The data we will utilize is the S&P 500 data from 2003 onwards. To study volatility clustering and its properties in financial time series, we will refer to the research paper "Volatility Clustering in Financial Markets" by Ramacant (2005), which explores the statistical properties of financial time series. The three significant properties we will focus on are excess volatility, heavy tails, and volatility clustering.
Volatility clustering refers to the observation that large changes in prices tend to be followed by other large changes, regardless of their direction, while small changes are often followed by small changes. This quantitative manifestation suggests that although returns may be uncorrelated, the absolute returns or their squares display a small positive correlation that gradually decreases over time. To analyze this, we examine log returns, which represent the logarithm of price changes over time. By visually examining the log returns of the S&P 500, we can observe clusters of high magnitude during specific periods, such as the significant clusters in 2008-2009 and 2020.
Next, we evaluate the correlation between lagged log returns. Notably, we find no statistically significant autocorrelation in log returns over the specified data range. However, when we square the log returns to focus on the absolute magnitude, we observe a strong positive correlation that extends even to lagged days and weeks. This implies that during periods of high volatility, it is likely to persist, and during low volatility periods, the trend is also likely to continue. This phenomenon is known as volatility clustering.
To visualize the rolling volatility over a specific number of days, we select a trading window and compute the standard deviation over that window. To annualize the volatility, we take the square root of the number of trading days in a year, which is typically 252. This approach allows us to observe significant upsurges in realized volatility during certain periods.
To model this realized volatility process, we turn to the Ornstein-Uhlenbeck formula. This formula, also known as the Vasicek model in financial mathematics, considers three parameters: kappa, which represents the mean reversion rate; theta, the average volatility around which prices fluctuate; and sigma, the volatility itself. We aim to find parameter values that maximize the likelihood of the observed data adhering to this distribution.
To achieve this, we employ the maximum likelihood estimation (MLE) method, which applies to random samples and probability density functions. In the case of the normal distribution, the likelihood function is the product of the individual sample probabilities given the parameters. By taking the log of the likelihood function, we can convert
Now that we have derived the expectation and variance of the Ornstein-Uhlenbeck process, we can proceed to model the volatility using this framework. To do so, we will calibrate the model parameters to market data using the maximum likelihood estimation (MLE) method.
First, we import the necessary dependencies, including libraries such as time, math, numpy, pandas, datetime, scipy, matplotlib, pandas_datareader, and the plot_acf function from the stats module. We also import the S&P 500 data since 2003, which will serve as our market data.
Next, we explore the concept of volatility clustering in financial time series. Volatility clustering refers to the phenomenon where large changes in prices tend to be followed by other large changes, and small changes tend to be followed by small changes. We observe this clustering effect visually when plotting the log returns of the S&P 500. We can see that during periods of market volatility, the magnitude of log returns clusters together, indicating a correlation between large price movements. For example, we can see clusters during the financial crisis in 2008-2009 and the volatility spike in 2020.
To quantify the correlation between log returns, we calculate the autocorrelation function (ACF). While the log returns themselves show no significant autocorrelation, the squared log returns (representing the absolute magnitude) display a small positive correlation that slowly decays over time. This autocorrelation of the absolute magnitude confirms the presence of volatility clustering, where periods of high volatility tend to persist, while periods of low volatility also tend to persist.
To further analyze volatility, we compute rolling volatility over a specified number of days by computing the standard deviation and annualizing it using the square root of the number of trading days in a year. By plotting the rolling volatility, we can observe periods of increased volatility, indicated by significant upswings in realized volatility.
Now, we introduce the Ornstein-Uhlenbeck (OU) formula, which is used to model volatility. The OU model incorporates mean reversion, average level, and volatility around the average price. The parameters of the model include kappa (rate of mean reversion), theta (average level), and sigma (volatility). To estimate these parameters, we apply the maximum likelihood estimation (MLE) method, which involves finding the parameter values that maximize the likelihood of the observed data coming from the OU distribution.
We begin by discussing the likelihood function, which is the joint probability density function (pdf) of the observed data given the parameters. In the case of the normal distribution, the likelihood function is the product of the individual pdf values. Taking the logarithm of the likelihood function simplifies the calculations, as it transforms the product of probabilities into the sum of logarithms. By finding the maximum likelihood estimator (MLE) of the parameters, we can determine the values that maximize the likelihood of the observed data.
In the case of the OU process, we need to use numerical methods to find the maximum likelihood estimates due to the non-differentiability of the log-likelihood function. We utilize the scipy.optimize.minimize function to minimize the negative log-likelihood, as it provides a numerical solution to the maximization problem. By defining the log-likelihood function, initial parameters, and constraints, we can estimate the parameters that maximize the likelihood of the observed data.
Once we have estimated the parameters of the OU process, we can simulate the process using Python. We can simulate the process either by discretizing the time steps and obtaining a path over time or by simulating it as a continuous-time Itô process. The latter method provides a more accurate representation of volatility dynamics at specific times.
In conclusion, the text discusses the volatility characteristics observed in the S&P 500 during periods of market volatility. It introduces the concept of volatility clustering and demonstrates its presence using log returns and squared log returns. The Ornstein-Uhlenbeck (OU) model is then introduced as a framework to model volatility, and the maximum likelihood estimation (MLE) method is used to estimate the model parameters. Finally, the simulation of the OU process is explained, allowing for the analysis and understanding of volatility dynamics over time.
The Magic Formula for Trading Options Risk Free
The Magic Formula for Trading Options Risk Free
In this video, you will learn how to use the Breeden-Litzenberger formula to derive risk-neutral probability density functions from option prices. This technique is extremely useful when computing option prices becomes time-consuming and computationally intensive, especially for complex dynamics and high-dimensional scenarios. The Breeden-Litzenberger formula allows us to compute complex derivatives once for different strikes and time-to-maturity values, resulting in a risk-neutral probability distribution function that simplifies the calculation of various complex derivatives.
To begin, let's understand the concept of risk-neutral probability. The Feynman-Kac analysis enables us to define risk-neutral probability as a measure (Q) of the terminal risk-neutral probability at time (t). The cumulative probability distribution function (F) represents the risk-neutral probability distribution. Pricing a European call option at time (t) with a strike (k) and time-to-maturity (tau) can be done by taking the risk-neutral discounted expectation of the payoff. This can be expressed as the integral of (S_t - k) multiplied by the risk-neutral density function (pdf) between the strike (k) and infinity, discounted by the risk-free rate.
To calculate the risk-neutral probability directly from this formula, we can use the Breeden-Litzenberger formula from 1978. It states that the first derivative of the integral with respect to the strike (k) is equal to minus the exponential discount factor multiplied by (1 - F), where F is the cumulative density function. The second derivative of the integral centered around the strike (k) extracts the pdf, which is the discount factor multiplied by the risk-neutral pdf.
Now, let's discuss how to apply this formula in Python. We need to import libraries such as NumPy, SciPy, Pandas, and Matplotlib. For the example, we will consider a European call option with stochastic volatility under the Heston model. The Heston model provides the dynamics of the underlying asset and its volatility. We initialize the necessary parameters, such as the stock price, strike, time-to-maturity, risk-free rate, and Heston model parameters like mean reversion rate, long-term variance, initial volatility, correlation, and volatility of volatility.
Using the Breeden-Litzenberger formula, we can determine the risk-neutral probability distribution function. By approximating the second derivative using finite difference approximation, we calculate the risk-neutral distribution for different strikes and time-to-maturity values. We construct a 2D pdf for a particular time-to-maturity.
To calculate option prices under the Heston model, we use the characteristic function and perform numerical integration using rectangular integration. We define the characteristic function and compute the complex integral over a specified domain using rectangular integration. The step size chosen for the integration affects the precision, especially for out-of-the-money options.
We compare the results obtained using rectangular integration with the QuantLib library, which is implemented in C and provides more accurate numerical integration. Although there are some differences between the two approaches, the mean squared error (MSE) is small. The discrepancies are mainly due to rounding errors caused by the binary representation of decimal values in Python.
After obtaining the discrete approximate pdf, we multiply it by the forward factor. We use interpolation to smooth out the curve and create a continuous risk-neutral distribution function. Finally, we can use this risk-neutral distribution to price various complex derivatives easily.
In conclusion, the Breeden-Litzenberger formula allows us to derive risk-neutral probability density functions from option prices. By approximating the second derivative using finite difference approximation and performing numerical integration, we can calculate the risk-neutral distribution for different strikes and time-to-maturity values. This enables us to price complex derivatives efficiently.