Quantitative trading - page 13
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Wall Street: The speed traders
Wall Street: The speed traders
Many people are unaware that the majority of stock trades in the United States are no longer executed by human beings but rather by robotic computers. These supercomputers are capable of buying and selling thousands of different securities in the blink of an eye. High-frequency trading, as it is known, has become prevalent on Wall Street in recent years and played a role in the mini market crash last spring when the Dow Jones Industrial Average plummeted 600 points in just 15 minutes.
The Securities and Exchange Commission and members of Congress have started raising tough questions about the usefulness, potential dangers, and suspicions of market manipulation through computer trading. The shift from human traders to machines has transformed the landscape of the New York Stock Exchange, which was once the center of the financial world. Now, less than 30% of trading occurs on the exchange floor, with the rest being conducted through electronic platforms and alternative trading systems.
Two electronic stock exchanges, BATS and Direct Edge, owned by big banks and high-frequency trading firms, have emerged and trade over a billion shares per day at astonishing speeds. High-frequency trading firms like Tradeworks, run by Manoj Narang and a team of mathematicians and scientists called quants (quantitative analysts), engage in this practice. They execute trades for fractions of a second, aiming to make a profit of a penny or less per trade. These firms rely on complex mathematical algorithms programmed into their computers to analyze real-time data and make split-second decisions.
One key aspect of high-frequency trading is that the computers have no understanding of the companies being traded. They do not know the value of the companies, their management, or any other qualitative factors. The trading decisions are purely based on quantitative factors, probability, and statistical analysis. This approach allows for capturing fleeting opportunities in the market but disregards fundamental factors.
High-frequency traders invest heavily in supercomputers and infrastructure to gain a speed advantage. The closer their computers are located to the stock exchange's servers, the quicker they receive critical market information. Even a few milliseconds of advantage can result in significant profits. Critics argue that high-frequency traders exploit this advantage to front-run orders, manipulate stocks, and extract money from the market without adding any real value.
While proponents claim that high-frequency trading increases market liquidity, reduces transaction costs, and tightens stock spreads, critics believe it undermines fairness and transparency. The high-speed nature of trading and the complexity of algorithms make it difficult for regulators to monitor and ensure a level playing field. The "flash crash" of 2010, when the Dow Jones plunged 600 points in a matter of minutes, exposed the potential risks associated with high-frequency trading and the lack of control.
Regulators and lawmakers have begun proposing reforms to address concerns related to high-frequency trading. The Securities and Exchange Commission is considering measures to track and identify high-frequency trades, and circuit breakers have been implemented to halt trading in cases of extreme price volatility. However, further changes are needed to restore confidence in the integrity of the market and provide transparency to average investors who feel that the system is rigged against them.
In recent years, high-frequency traders have expanded their activities into currency and commodity markets, further raising concerns about their impact on financial markets. The evolution of technology has outpaced the ability of regulators to keep up, and there is a growing call for reforms that strike a balance between innovation and market integrity.
"Mathematical Modeling and Computation in Finance: With Exercises and Python and MATLAB Computer Codes", by C.W. Oosterlee and L.A. Grzelak, World Scientific Publishing, 2019.
"Mathematical Modeling and Computation in Finance: With Exercises and Python and MATLAB Computer Codes" is an invaluable book that explores the intersection of mathematics, finance, and computer science. Written by experts in the field, it provides a comprehensive guide to understanding and implementing mathematical models in finance using popular programming languages like Python and MATLAB.
The book begins by introducing readers to the fundamental concepts of mathematical modeling in finance, including probability theory, stochastic calculus, and optimization techniques. It emphasizes the practical aspects of modeling and computation, highlighting the importance of numerical methods and simulation in solving real-world financial problems.
One of the standout features of this book is its inclusion of numerous exercises and computer codes in Python and MATLAB. These exercises allow readers to actively engage with the material, reinforce their understanding of the concepts, and develop their programming skills. By working through the exercises and implementing the provided codes, readers can gain hands-on experience in applying mathematical models to finance and enhance their proficiency in using these programming languages for financial analysis.
The book covers a wide range of topics relevant to finance, such as option pricing, portfolio optimization, risk management, and asset allocation. It delves into advanced topics like volatility modeling, interest rate modeling, and credit risk modeling, providing readers with a comprehensive understanding of the mathematical techniques used in financial modeling.
The authors strike a balance between theoretical rigor and practical application throughout the book. They provide clear explanations of the underlying mathematical concepts and algorithms, accompanied by real-world examples and case studies. This approach enables readers to grasp the theoretical foundations while also gaining insights into how these models can be applied to solve practical financial problems.
Furthermore, the book highlights the advantages and limitations of different modeling approaches, equipping readers with the critical thinking skills necessary to make informed decisions when choosing and implementing models in real-world scenarios.
"Mathematical Modeling and Computation in Finance: With Exercises and Python and MATLAB Computer Codes" is an excellent resource for students, researchers, and practitioners in the field of finance who are looking to deepen their understanding of mathematical modeling and computational methods. Its combination of theoretical explanations, practical exercises, and ready-to-use computer codes makes it an essential companion for anyone interested in applying mathematical techniques to solve financial problems.
https://github.com/LechGrzelak/Computational-Finance-Course
This course Computational Finance is based on the book: "Mathematical Modeling and Computation in Finance: With Exercises and Python and MATLAB Computer Codes"
Computational Finance: Lecture 1/14 (Introduction and Overview of Asset Classes)
This comprehensive lecture serves as an introduction to the fascinating fields of computational finance and financial engineering, covering a wide range of topics essential for understanding modern finance. The lecturer emphasizes the importance of theoretical models from mathematical and computational finance, which are utilized to create practical models for pricing derivatives under various scenarios.
In the course on computational finance, the students will delve into various topics that are crucial to understanding and applying practical financial methods. Led by the instructor, Leth Lag, the course will emphasize the implementation of efficient programming techniques using Python for simulation and option pricing. This comprehensive program is designed for individuals interested in finance, quantitative finance, and financial engineering. It will cover essential concepts such as implied volatilities, hedging strategies, and the fascinating realm of exotic derivatives.Computational finance is an interdisciplinary field situated between mathematical finance and numerical methods. Its primary objective is to develop techniques that can be directly applied to economic analysis, combining programming skills with theoretical models. Financial engineering, on the other hand, encompasses a multidisciplinary approach that employs financial theory, engineering methods, mathematical tools, and programming practices. Financial engineers play a critical role in creating practical models based on mathematical and computational finance, which can be utilized to price derivatives and handle complex financial contracts efficiently. These models must be theoretically sound and adaptable to diverse scenarios.
The course will shed light on different asset classes traded in computational finance, including stocks, options, interest rates, foreign exchange, credit markets, commodities, energy, and cryptocurrencies. Cryptocurrencies, in particular, offer exposure to various asset classes and can be employed for hedging purposes. Each asset class has its unique contracts used for risk control and hedging strategies. The Over-the-Counter (OTC) market, with its multiple counterparties, presents additional complexities that need to be understood.
The lecturer will explore the role of cryptocurrencies in finance, emphasizing their diverse features and the need for specific methodologies, models, and assumptions for pricing. Additionally, the market shares of different asset classes, such as interest rates, forex, equities, commodities, and credit default swaps (CDS), will be examined. While options represent a relatively small portion of the financial world, they offer a distinct perspective on financial and computational analysis.
The topic of options and speculation will be thoroughly discussed, highlighting how options provide an alternative to purchasing stocks by allowing individuals to speculate on the future direction of a stock with a relatively small capital investment. However, options have a maturity date and can lose value if the stock price remains unchanged, making timing a crucial factor in speculation. The course will provide an introduction to financial markets, asset classes, and the role of financial engineers in navigating these complex landscapes. Stocks, as the most popular asset class, will be explored in detail, emphasizing the concept of ownership and how stock value is influenced by company performance and future expectations.
The lecture will shed light on the stochastic nature of stock behavior in the market, influenced by factors such as supply and demand, competitors, and company performance. The expected value of a stock may differ from its actual value, leading to volatility. Volatility is a crucial element in modeling and pricing options as it determines the future fluctuations in stock prices. Additionally, the lecture will distinguish between two types of investors: those interested in dividend returns and those seeking growth opportunities.
The concept of dividends and dividend investing will be introduced, emphasizing how dividends provide a steady and certain investment as companies distribute payments to shareholders regularly. However, dividend payments can vary, and high dividend yields may indicate increased risk in a company's investments. The lecture will touch briefly on interest rates and money markets, acknowledging that these topics will be covered more extensively in a follow-up course.
Inflation and its impact on interest rates will be discussed, elucidating how central banks control inflation by adjusting interest rates. The lecture will explore the short-term benefits and long-term implications of lowering interest rates, as well as alternative strategies such as modern monetary theory or asset purchases by central banks. Moreover, the role of uncertainty among market participants in determining interest rates and the hidden tax effect of inflation on citizens will be explained. The lecture will conclude by delving into the topic of risk management in lending. The lecturer will highlight the potential risks faced by lenders, such as borrowers going bankrupt or defaulting on loans. To mitigate these risks, lenders often charge a risk premium to ensure they are adequately compensated for any potential losses.
Moving forward, the speaker will shift the focus to interest rates and their significance in finance. They will explain how interest rates affect various financial instruments, including savings accounts, mortgages, and loans. The concept of compounding interest will be introduced, emphasizing the notion that one unit of currency today is worth more than the same unit in the future due to factors like inflation. The two main methods of calculating interest rates, simple and compounded, will be discussed, with a detailed explanation of their differences and practical examples.
The speaker will then delve deeper into compounded interest rates, particularly for investments with a one-year maturity. They will explain the mathematical modeling of compounded rates using the exponential function, where one unit of currency is multiplied by e raised to the power of the interest rate. Furthermore, the speaker will describe how this mathematical representation aligns with the differential equations that govern savings accounts, leading to the determination of the multiplication factor used to discount future cash flows. However, the speaker will note that in reality, interest rates are not constant but vary over time, as evidenced by different instruments such as tenors and prices for currencies like the Euro and the USD.
The graphs representing interest rates and market liquidity for the Eurozone and the dollar will be discussed. Notably, the current state of the Eurozone reveals negative yields across all maturities up to 30 years, implying that investing in government bonds within the Eurozone could result in a loss of money. The speaker will suggest that individuals may prefer to exchange Euros for dollars and invest in US bonds, as they offer higher yields. Nevertheless, this approach carries risks, including potential losses due to foreign exchange rate fluctuations. The speaker will emphasize that interest rates are time-dependent and subject to market dynamics.
The lecturer will shed light on the concept of buying bonds, highlighting that bond buyers often pay more than the actual worth of the bond. Consequently, the value of money invested in bonds may depreciate over time, and inflation can erode the investment's value. Major buyers of bonds, such as pension funds and central banks, will be mentioned, underscoring their significant role in the bond market. Furthermore, the lecturer will touch upon the concept of volatility, which measures the variation in financial prices over time. Volatility is calculated using statistical measures like variance and provides insights into the tendency of a market or security to fluctuate, introducing uncertainty and risk.
The course will then shift its attention to asset returns and volatility, two crucial concepts in computational finance. Asset returns refer to the gains or losses of a security within a specific time period, while volatility measures the variance of these returns. A highly volatile market indicates significant price swings in a short span, resulting in heightened uncertainty and risk. The VIX index, an instrument that gauges market uncertainty, will be introduced. It utilizes out-of-the-money or put options and is commonly employed by investors to protect their capital in the event of market value decline. The importance of timing and predicting exposure times will be emphasized, as they can be challenging in practice.
The instructor will discuss the intricacies of analyzing the volatility of various indices, including the VIX index. They will acknowledge the difficulties in mathematically modeling volatility due to market circumstances and fluctuations. Additionally, European options, which serve as fundamental building blocks for derivative pricing based on volatility, will be introduced. The lecturer will provide a clear distinction between call options and put options, explaining that call options grant the holder the right to buy an asset at a predetermined price and date, while put options give the holder the right to sell an asset at a predetermined price and date, essentially acting as insurance.
With the foundation of options established, the lecturer will present an overview of options within different asset classes. They will emphasize the two key types of options: call options and put options. In the case of a call option, the buyer has the right to sell the underlying asset to the writer at a specified maturity date and strike price. This means that at maturity, the writer is obliged to buy the stock at the strike price if the buyer chooses to exercise the option. On the other hand, a put option grants the buyer the right to sell the underlying asset to the writer at a specified maturity date and strike price. At maturity, the writer must purchase the stock at the specified strike price if the buyer exercises the option.
To illustrate the potential profitability of options, the lecturer presents two graphical representations—one for call options and another for put options. These graphs depict the potential profit or loss based on the value of the underlying stock. By examining the graphs, viewers can gain insights into how changes in the stock's value can affect the profitability of options.
Throughout the course, the instructor will explore additional advanced topics related to computational finance, including modeling of derivatives, efficient programming implementation, and the use of Python for simulation and option pricing. They will program live during the sessions and analyze results collaboratively with the viewers, providing hands-on experience and practical insights.
The course is specifically designed for individuals interested in finance, quantitative finance, and financial engineering. It aims to bridge the gap between mathematical finance and numerical methods, offering interdisciplinary knowledge and skills required to tackle real-world financial problems. The concepts of implied volatilities, hedging strategies, and exotic derivatives will also be covered, providing a comprehensive understanding of computational finance and its applications in the financial industry.
By the end of the course, participants will have gained a solid foundation in computational finance, financial engineering, and the practical application of numerical methods. They will be equipped with the tools and knowledge to develop and implement models for pricing derivatives, managing risks, and analyzing financial data. This course serves as a stepping stone for those seeking to pursue careers in finance, quantitative analysis, or financial engineering, empowering them to make informed decisions and contribute to the ever-evolving field of computational finance.
Computational Finance: Lecture 2/14 (Stock, Options and Stochastics)
Computational Finance: Lecture 2/14 (Stock, Options and Stochastics)
The instructor begins by providing an overview of the course, emphasizing the importance of understanding trading confidence, hedging, and the necessity of mathematical models in finance. They delve into the topic of pricing put options and explain the concept of hedging. Stochastic processes and asset price modeling are also covered, with the introduction of Ito's lemma as a tool for solving stochastic differential equations.
To illustrate the practical application of these concepts, the instructor presents an example of a training strategy where an investor seeks to protect their investment from potential stock value decrease. They suggest buying insurance in the form of put options to ensure a minimum amount of money in a worst-case scenario.
Moving on to options trading, the lecturer focuses on the use of put options to protect against downward movements in stock prices. However, they note that buying put options can be expensive, particularly when the stock's volatility is high, as exemplified by Tesla. To reduce option costs, one can decrease the strike price, but this means accepting a lower price for the stock. The lecturer provides a screenshot from Reuters showcasing different types of options available in the market, categorized by maturity and strike price. They also explain the relationship between strike price and option prices for call and put options.
Implied volatility is introduced as a measure of market uncertainty. The lecturer explains that lower strike prices are associated with higher implied volatility. Delta, which measures an option's value dependence on the underlying asset, is also introduced. The video then delves into the concept of hedging and how a ratio can be established to achieve a risk-free portfolio, albeit potentially limiting gains if the stock does not increase in value. Hedging with options is discussed, highlighting its suitability for short-term investments, but noting its potential costliness during periods of high volatility.
Options trading is further explored as a means of hedging and risk reduction. The lecturer suggests that options are typically more desirable for short-term investments with a definite maturity, as they can be costly for long-term investments. The concept of hedging with calls is introduced, emphasizing how selling options can help reduce risk for investors holding a large portfolio of stocks. However, caution is advised against selling too many calls, as it can restrict potential upside and always carries a certain degree of risk.
The video then delves into commodities, explaining that they are raw materials used as hedges against inflation due to their unpredictable but often seasonal price patterns. Commodity trading is primarily conducted in the futures market, where deals are made to buy or sell commodities at a future date. The distinction between electricity markets and other commodities is highlighted, with electricity posing unique challenges due to its inability to be fully stored and its impact on derivative predictability and value.
The lecturer proceeds to discuss currency trading as an asset class, commonly referred to as the foreign exchange market. Unlike traditional buying or selling of a particular exchange rate, individuals exchange amounts of money between currencies. The lecturer emphasizes the role of the US dollar as the base currency and a reserve currency. They also touch upon the manipulation of exchange rates by Central Banks to strengthen or weaken currencies. Additionally, a small application of foreign exchange derivatives for hedging currency risks in international business is mentioned.
The speaker explains how banks and financial institutions can purchase or sell insurance against fluctuating exchange rates to manage investment uncertainties. Investing in different countries can introduce uncertainties due to varying currency strengths and monetary policies, leading to uncertain returns. Computational finance plays a crucial role in managing and calculating risks associated with such investments by modeling uncertainties and considering various factors. The speaker further notes that bitcoins can be considered foreign exchange rates and discusses their hybrid nature as a regulated commodity with value determined through exchange against the US dollar. The volatility of bitcoins makes their future value challenging to predict.
Furthermore, the speaker explores the concept of risk-neutral pricing, which is a fundamental principle in options pricing. Risk-neutral pricing assumes that in a perfectly efficient market, the expected return on an option should be equal to the risk-free rate. This approach simplifies the pricing process by considering the probabilities of different outcomes based on a risk-neutral measure, where the expected return on the option is discounted at the risk-free rate.
The speaker then introduces the Black-Scholes-Merton (BSM) model, which is a widely used mathematical model for pricing options. The BSM model incorporates various factors such as the current stock price, strike price, time to expiration, risk-free interest rate, and volatility of the underlying asset. It assumes that the underlying asset follows geometric Brownian motion and that the market is efficient.
The speaker explains the key components of the BSM model, including the formula for calculating the value of a European call or put option. They emphasize the importance of volatility in option pricing, as higher volatility increases the value of an option due to the potential for larger price fluctuations. The speaker also mentions the role of implied volatility, which is the market's expectation of future volatility implied by the option prices.
Next, the lecture delves into the concept of delta hedging, which is a strategy used to minimize risk by maintaining a neutral position in the underlying asset. Delta measures the sensitivity of an option's price to changes in the price of the underlying asset. By adjusting the number of shares held in the underlying asset, an investor can create a delta-neutral portfolio that is less affected by price movements.
The speaker explains the process of delta hedging using the BSM model and demonstrates how it can effectively reduce risk. They discuss the concept of dynamic hedging, where the hedge is continuously adjusted as the price of the underlying asset changes. This ensures that the portfolio remains delta-neutral and minimizes the exposure to market fluctuations.
In addition to delta hedging, the lecture covers other risk management techniques such as gamma hedging and vega hedging. Gamma measures the rate of change of delta, while vega measures the sensitivity of an option's price to changes in implied volatility. These techniques allow investors to manage and adjust their positions based on changing market conditions and risks.
Towards the end of the lecture, the speaker highlights the limitations and assumptions of the BSM model. They acknowledge that real-world markets may deviate from the model's assumptions, such as the presence of transaction costs, liquidity constraints, and the impact of market frictions. The speaker encourages a cautious approach and emphasizes the importance of understanding the limitations and uncertainties associated with option pricing models.
Overall, the lecture provides a comprehensive overview of trading confidence, hedging strategies, option pricing models, and risk management techniques. It equips learners with essential knowledge and tools to navigate the complex world of financial markets and make informed decisions in trading and investment activities.
Computational Finance: Lecture 3/14 (Option Pricing and Simulation in Python)
Computational Finance: Lecture 3/14 (Option Pricing and Simulation in Python)
In the lecture, the instructor delves into stock path simulation in Python and explores the Black-Scholes model for pricing options. They discuss two approaches to deriving the arbitrage-free price for options, namely hedging and martingales. The speaker demonstrates how to program martingales and simulate them, highlighting the connection between partial differential equations (PDEs) and Monte Carlo simulation in the pricing framework.
Using the Euler discretization method, the speaker explains how to simulate and generate graphs of stochastic processes. They start with a simple process and employ Ito's lemma to switch from S to X, the logarithm of S. The lecturer then introduces the Euler discretization method and demonstrates its implementation in Python. This method involves discretizing the continuous function and simulating the increments for both drift and Brownian motion, resulting in graphs of simulated paths.
From a computational perspective, the speaker discusses the simulation of paths for option pricing models. Instead of simulating each path individually, they explain the efficiency of performing time slicing and constructing a matrix where each row represents a specific path. The number of rows corresponds to the number of paths, while the number of columns corresponds to the number of time steps. The speaker explains the implementation of the discretization process using the standard normal random variable and emphasizes the importance of standardization for better convergence.
The lecture also covers the simulation of paths for geometric Brownian motion using Python. The speaker illustrates how to fix a random seed for stable simulations and introduces the Black-Scholes model, which involves a stochastic differential equation with drift and parameters such as mu and sigma for modeling asset prices. The speaker emphasizes that the Black-Scholes model is still widely used in the finance industry, particularly for pricing options on stocks. They discuss the concepts of real-world measure and risk-neutral measure, which aid in pricing options based on different outcome probabilities.
Furthermore, the lecture explores option pricing and simulation in Python. The speaker distinguishes between the real-world measure, estimated based on historical data without assuming arbitrage or risk-free conditions, and the risk-neutral measure, which requires certain conditions to hold. They present a trading strategy involving continuous trading in a stock and adjusting the option position to capture the underlying stock's movement. The speaker explains the dynamics of the portfolio using Ito's lemma and derives the stochastic nature of option values through this method.
The speaker also delves into techniques for constructing a hedging portfolio that is independent of Brownian motion. They discuss choosing a delta that nullifies the terms involving Brownian motion, ensuring a delta-neutral portfolio. The speaker highlights the importance of the portfolio yielding the same return as a savings account and introduces the concept of money settings accounts.
Additionally, the lecture addresses the derivation of partial differential equations (PDEs) for option valuation using the Black-Scholes model. The resulting PDE is a second-order derivative with boundary conditions that determine the fair value of an option. The speaker emphasizes that the Black-Scholes model's option pricing does not depend significantly on the drift parameter mu, which can be obtained from calibration or historical data. However, transaction costs for hedging are not considered in this model.
The lecture covers various important concepts within the Black-Scholes model and option pricing. It discusses the assumption of no arbitrage opportunities, leading to a risk-free scenario for the model's application. The speaker explains the concept of delta hedging and how it eliminates the largest random component of a portfolio. Additionally, the speaker introduces gamma as a measure of delta's behavior and emphasizes that every parameter in the model can be hedged. Finally, the lecture explores the determining factors of an option's value, such as time, strike, volatility, and market-related parameters.
In the lecture, the speaker further explores the Black-Scholes model and its application in option pricing. They discuss the assumptions and limitations of the model, including the assumption of constant volatility and the absence of transaction costs. Despite these limitations, the Black-Scholes model remains widely used in the financial industry due to its simplicity and effectiveness in pricing European call and put options.
The speaker introduces the concept of implied volatility, which is the market's expectation of future volatility derived from the current option prices. Implied volatility is a crucial parameter in the Black-Scholes model as it affects the pricing of options. The speaker explains how implied volatility can be obtained from market data using the model and discusses its significance in option trading strategies.
The lecture delves into various option trading strategies, such as delta hedging and gamma trading. Delta hedging involves continuously adjusting the portfolio's composition to maintain a neutral position in relation to changes in the underlying asset's price. Gamma trading focuses on exploiting changes in gamma, which measures how delta changes with respect to the underlying asset's price. These strategies aim to manage risk and maximize profitability in option trading.
The speaker also touches upon other important factors influencing option prices, including time decay (theta), interest rates (rho), and dividend yield. They explain how these factors impact option pricing and how traders can use them to make informed decisions.
Throughout the lecture, Python programming is utilized to demonstrate the implementation of various option pricing models and trading strategies. The speaker provides code examples and explains how to utilize libraries and functions to perform calculations and simulations.
In summary, the lecture provides a comprehensive overview of option pricing and simulation using the Black-Scholes model and related concepts. It emphasizes the practical application of these concepts in Python programming, making it a valuable resource for individuals interested in quantitative finance and options trading.
Computational Finance: Lecture 4/14 (Implied Volatility)
Computational Finance: Lecture 4/14 (Implied Volatility)
In this comprehensive lecture on computational finance, the concept of implied volatility takes center stage, shedding light on its significance in option pricing computations. While the Black-Scholes model serves as a foundation for calculating implied volatility, its limitations and inefficiencies are duly emphasized. The lecture delves into various methodologies for computing implied volatility, notably iterative processes such as the Newton-Raphson method. Additionally, the lecturer explores the challenges associated with modeling option prices and underscores the role of implied volatilities in reflecting market expectations. Throughout the lecture, the crucial importance of comprehending volatility in option pricing and constructing effective hedging portfolios remains a central theme.
The lecture extends its exploration by focusing on the relationship between option prices and implied volatility, with a specific emphasis on liquid out-of-the-money puts and calls. It examines different types of implied volatility skew, encompassing time-dependent volatility parameters and the influence of time dependency on the implied volatility smile. Furthermore, the lecture delves into the limitations of the Black-Scholes model and alternative approaches to handling volatility models, including local volatility models, jump models, and stochastic volatility models. The impact of option maturity on volatility is also elucidated, with shorter maturity options exhibiting a more concentrated distribution around the money level compared to longer maturities, where the smile effect becomes less pronounced.
The professor commences by summarizing the key concepts covered in previous sections, specifically relating to option pricing and volatility modeling. Implied volatility is introduced, highlighting its computation from market data and its role in measuring uncertainty. The algorithm for computing implied volatility is discussed in detail. Furthermore, the limitations and efficiencies of the Black-Scholes model are addressed, along with extensions such as incorporating time-dependent volatility parameters and generating implied volatility surfaces. The lecture also touches upon the downsides of relying solely on the Black-Scholes model and introduces alternative models like local volatility and stochastic volatility. Emphasis is placed on the need to specify an appropriate model for pricing contingent claims and the significance of constructing a hedging portfolio consisting of options and stocks to arrive at a pricing partial differential equation (PDE).
The speaker proceeds to explore the utilization of expectations in solving partial differential equations, specifically when dealing with a deterministic interest rate and the necessity of taking expectations under the risk-neutral measure. The pricing equation for European call and put options is presented, relying on an initial stock normal cumulative distribution function (CDF) evaluated at points d1, which depends on model parameters, along with an exponent involving the interest rate over the time to maturity. The lecture explains that this formula can be easily implemented in Excel.
Next, the lecturer elaborates on the parameters required for the Black-Scholes model, which serves as a tool for estimating option prices. These parameters encompass time to maturity, strike, interest rate, current stock value, and the volatility parameter, sigma, which needs to be estimated using market prices. The lecturer emphasizes the one-to-one correspondence between option price and volatility, highlighting that an increase in volatility implies a corresponding increase in option price, and vice versa. The concept of implied volatility is then discussed, emphasizing its calculation based on mid-price and its significance within the Black-Scholes model.
The lecture further delves into obtaining implied volatility from models with multiple parameters. It is noted that regardless of the chosen model, it must pass the Black-Scholes model's test. However, using the Black-Scholes model to price all options simultaneously becomes impractical due to differing implied volatilities for each strike. The lecture also points out that implied volatilities tend to increase with longer option maturities, signifying greater uncertainty. An example is provided to demonstrate the computation of implied volatility using market data and a standard call option on 100 shares.
The concept of implied volatility is further expounded upon by the lecturer. Historical data on an option is used to estimate its volatility using the Black-Scholes equation. However, the lecturer highlights that while this estimation provides a certain price for the option, the market may have priced it differently due to its forward-looking nature, contrasting with the backward-looking historical estimation. Despite this discrepancy, the relationship between the two volatilities is still utilized for investment purposes, although the lecturer advises caution against purely speculative reliance on this relationship. The lecture then proceeds to explain how to calculate implied volatility using the Black-Scholes equation given the market price and other specifications of an option. However, the lecturer acknowledges that the concept of implied volatility is inherently flawed as there is no definitive correct value, and the model used is an approximation rather than a true representation of option pricing.
The lecturer proceeds to explain the process of finding implied volatility by employing the Newton-Raphson method, an iterative approach. This method involves setting up a function based on the Black-Scholes equation and the market price to solve for sigma, the implied volatility. The lecturer highlights the use of a Taylor series expansion to calculate the difference between the exact solution and the iteration, with the objective of finding a function where the Black-Scholes implied volatility matches the market implied volatility. The ability to compute implied volatility rapidly in milliseconds is crucial for market makers to identify arbitrage opportunities and generate profits.
The concept of the iterative process for computing implied volatility using the Newton-Raphson method is introduced. The process entails multiple iterations until the function g approaches zero, with each new step estimated based on the previous one. The lecturer emphasizes the significance of the initial guess for the convergence of the Newton-Raphson method. Extreme out-of-the-money options or options close to zero can present challenges as the function becomes flat, resulting in a small gradient that hinders convergence. To overcome this issue, practitioners typically define a grid of initial guesses. The algorithm approximates the function using its tangent line and calculates the x-intercept, with steeper gradients leading to faster convergence.
Furthermore, the lecturer explains the implementation of the Newton-Raphson algorithm for calculating the implied volatility of an option. The algorithm relies on the Black-Scholes model, with input parameters including the market price, strike, time to maturity, interest rate, initial stock volume, and initial volatility parameter. The convergence of the algorithm is analyzed, and an error threshold is determined. The code is demonstrated using Python, with necessary methods and definitions prepared in advance, leveraging the NumPy and SciPy libraries.
The lecture elaborates on the computation of implied volatility, emphasizing the inputs required for this calculation, such as the option value and the derivative of the call price with respect to the volatility parameter, known as Vega. The core of the code involves the step-by-step process of computing implied volatility, with the lecturer providing explanations on the various parameters involved and their significance. The lecture concludes with a brief demonstration of the iterative process employed to compute implied volatility.
The speaker also addresses the topic of error in calculating implied volatility and how it is determined by the differences between iterations. The output chart showcases the implied volatility obtained for a call price, strike, maturity, and other parameters. The speaker illustrates how convergence varies with different initial guesses for volatility, underscoring the importance of this process in industry calibration. The initial guess must be close to the actual implied volatility for the model to converge successfully. Industry practitioners typically attempt different initial volatilities until a suitable convergence is achieved, and that particular volatility value is chosen.
the lecture dives deeper into the interpretation of implied volatilities. Implied volatilities can provide insights into market expectations and sentiment. When the implied volatility is high, it suggests that market participants anticipate significant price fluctuations, which may indicate uncertainty or perceived risk in the underlying asset. Conversely, low implied volatilities indicate expectations of relatively stable prices.
The lecture emphasizes that implied volatilities are not a measure of future volatility but rather a reflection of market pricing. Implied volatilities are influenced by various factors such as supply and demand dynamics, market sentiment, and market participants' risk appetite. Therefore, it is crucial to interpret implied volatilities in the context of other market indicators and fundamental analysis.
The lecturer also highlights the concept of implied volatility surfaces or volatility smiles. Implied volatility surfaces represent the relationship between implied volatilities and different strike prices and maturities. In certain market conditions, the implied volatilities of out-of-the-money options may be higher or lower than those of at-the-money options. This curvature in the implied volatility surface is known as the volatility smile or smirk. The lecture explains that the volatility smile indicates market participants' perception of the probability of extreme price movements, such as large downside risks or unexpected positive events.
Moreover, the lecture covers the concept of implied volatility term structures. Implied volatility term structures depict the relationship between implied volatilities and different maturities for a specific option. The lecturer explains that implied volatility term structures can exhibit different shapes, such as upward sloping (contango), downward sloping (backwardation), or flat curves. These term structures can provide insights into market expectations regarding future volatility over different time horizons.
Additionally, the lecture delves into the limitations and challenges associated with implied volatilities. It emphasizes that implied volatilities are derived from option prices, which are influenced by various factors and assumptions, including interest rates, dividend yields, and the efficient market hypothesis. Therefore, implied volatilities may not always accurately reflect the true underlying volatility.
Furthermore, the lecture discusses the concept of historical volatility and its comparison to implied volatility. Historical volatility is calculated based on past price movements of the underlying asset, while implied volatility is derived from option prices. The lecturer notes that historical volatility is backward-looking and may not fully capture future market expectations, while implied volatility incorporates forward-looking information embedded in option prices.
Lastly, the lecture concludes with a summary of the key points covered. It emphasizes the importance of understanding implied volatility, its calculation methods, and its interpretation in the context of option pricing and market expectations. The lecturer encourages further exploration and research in this area, given its significance in financial markets and investment decision-making.
where the volatility impact varies for different lengths of options. The video also shows how to compute implied volatility and generate paths with time-dependent volatility and how it affects the Black-Scholes implied volatility equation. The video also shows an example of fitting different volatility levels for two options with different maturities.
Computational Finance: Lecture 5/14 (Jump Processes)
Computational Finance: Lecture 5/14 (Jump Processes)
The lecture progresses to explore ways to enhance the Black-Scholes model by incorporating jumps in the stock process, transitioning from a diffusive model to a jump-diffusion model. The instructor begins by explaining the inclusion of jumps in the stock process and providing a definition of jumps. They then demonstrate a simple implementation of a jump process in Python, emphasizing the need to handle jumps in a stochastic process for stocks while ensuring the model remains under the q measure.
Furthermore, the lecture delves into the implications of introducing jumps in pricing and how it affects the pricing PDE (Partial Differential Equation), introducing additional integral terms. The discussion extends to the impact of different jump distributions on implied volatility shapes and the utilization of concepts such as expectation iterated expectations, the tower property of expectation, and characteristic functions for jump processes when dealing with complex expectations.
The lecturer emphasizes the practicality of jump processes in pricing options and calibrating models, highlighting their realism and ability to accommodate heavy tails, as well as control the kurtosis and asymmetry of lock and turn density. By incorporating a jump process, a better fit to the implied volatility smile or implied volatility skew can be achieved, making jump processes a more favorable alternative to the Black-Scholes model.
Shifting focus, the lecture introduces the concept of jump processes represented by a counting process, which are uncorrelated to Brownian motion. These processes are modeled using a random Poisson process, characterized by initial zero value and independent increments following a Poisson distribution. The rate of the Poisson process determines the average number of jumps in a specified time period. The lecture explains how to calculate the average number of jumps within a given interval for jump processes using notation and expectations.
In computational finance, the lecturer discusses the simulation of jump processes, noting that the jump magnitude cannot explode and outlining the technical assumptions associated with it. The process involves defining matrices and parameters for simulating independent increments using a Poisson distribution for each increment of the jump process. The lecture also covers the utilization of the Poisson process in the Ethos lemma to extend the dynamics of jump processes for stock pricing. Within the context of computational finance, the lecture introduces and explains the concept of jump processes. It defines the term "t-minus" as the time just before a jump occurs in a process and explores the dynamics of the process through the Ethos lemma and the calculation of derivatives with respect to time. The relationship between the jump size and the resulting adjustment in the function "g" is discussed, emphasizing the practical relevance of these concepts in modeling stochastic processes. The lecture also highlights the importance of considering the independence of jump processes and diffusive processes when modeling stock market behavior.
To derive the dynamics of a function "g" in a model incorporating both jump and diffusion processes, the lecture focuses on the behavior of high diffusion complexity and the application of Ito's lemma. Ito's lemma is used to handle cross terms, such as dxpt squared, in the context of increased model complexity. Once all the elements, including drift, diffusion, and jumps, are combined, the dynamics of "g" can be derived using Ito's lemma. The extension of the Ito table is also touched upon, emphasizing the differences between a Poisson process and Brownian motion. The lecture concludes by outlining the process of deriving the dynamics for a function "g" that incorporates both jump and diffusion processes.
Moving forward, the lecture describes the process of obtaining the dynamics of a stock with jump and Brownian motion under the Q measure. This process involves defining a new variable and determining its dynamics, ensuring that the expectation of the dynamics is zero. The jump component is assumed to be independent of all other processes, resulting in an expression that includes terms for drift, volatility, and expectation of J minus one. This expression is then substituted into the equation for the Q measure, ensuring that the dynamics of ST over the money savings account is a martingale.
The instructor proceeds to discuss how to derive a model with both diffusion and jumps, providing an example to illustrate the paths of a model with two components: diffusive and jump. The diffusive part represents continuous behavior, while the jump element introduces discontinuity, allowing for the representation of jump patterns observed in certain stocks. The instructor also covers the parameters for the jump and the volatility parameter for Brownian motion, along with the initial values for the stock and interest rates. To further enhance understanding, the instructor demonstrates how to program the simulation and plot the resulting paths.
The lecture then moves on to explain the expectation of e to the power of j, which is analytically calculated as the expectation of a log-normal distribution. The simulation of Poisson increments driven by c times pi times dt is performed, with z representing increments for a normal distribution and j representing the jump magnitude. The dynamics of the jump diffusion process involve both partial differential equations and integral differential equations, where the integral part represents the expectation of jump sizes. The pricing equation can be derived through portfolio construction or through the characteristic function approach, and the parameters need to be calibrated using option prices in the market.
In the context of portfolio construction, the lecture describes the process of constructing a portfolio comprising a sold option and a hedge with an underlying stock. By ensuring that the portfolio's dynamics increase at the same rate as the money savings account, a pricing differential equation can be derived. To achieve the desired dynamics, the stock divided by the money savings account must be a martingale. The lecture then derives the condition for mu, demonstrating that once the dynamics are established, the dynamics of v can be derived. This information is then used to compute expectations and derive the dynamics of v.
The lecturer further explores the equation for a first-order derivative with respect to time, which is also first-order with respect to x and includes an expectation for a value of a contract at time t with a jump. This leads to an integral term due to the presence of an expectation, resulting in a partial integral differential equation (PID) that is more challenging to solve than pure PDEs. The solution involves finding the analytical expression for the expected value, which may sometimes be expressed in terms of infinite series. The importance of boundary conditions and the transformation of PIDs into log transformations for improved convergence are also discussed.
Continuing the discussion on jump processes, the lecture focuses on the transformation of jump processes in the case of PID and PID under the deluxe option. The lecture presents two common approaches for specifying the jump magnitude, namely the classical merchants model and the non-symmetric double exponential. While the calibration of the model becomes more complicated with the addition of sigma j and mu j, practicality and industry acceptance often favor models with fewer parameters. The lecture also acknowledges that as the dynamics of jump processes become more complex, achieving convergence becomes challenging, necessitating advanced techniques such as Fourier space or analytical solutions for parameter calibration.
The lecture then proceeds to explain the process of pricing using Monte Carlo simulation for jump diffusion processes. Pricing involves computing the expectation of the future payoff by discounting its present value. While methods like PIDs and Monte Carlo simulation perform well in terms of computational complexity for simulations, they may not be ideal for pricing and model calibration due to the significant increase in the number of parameters when jumps are introduced. The lecture also delves into interpreting the distribution of jumps and intensity parameters and their impact on implied volatility smile and skew. A simulation experiment is conducted, varying parameters while keeping others fixed to observe the resulting effects on jumps and skew.
To analyze the effects of volatility and intensity of jumps on the shape of the implied volatility smile and level, the lecturer discusses their relationships. Increasing the volatility of a jump leads to a higher level of volatility, while the intensity of jumps also affects the level and shape of the implied volatility smile. This information is crucial for understanding the behavior of option prices and calibrating models to real-market data.
The lecture then introduces the concept of the Tower Property and its application in simplifying problems in finance. By conditioning on a path from one process to compute the expectation or price of another process, problems with multiple dimensions in stochastic differential equations can be simplified. The Tower Property can also be applied to problems in Black-Scholes equations with volatility parameters and accounting processes, which often become summations when dealing with jump integrals. The lecturer emphasizes the need for making assumptions regarding parameters in these applications.
Next, the lecturer discusses the use of Fourier techniques for solving pricing equations in computational finance. Fourier techniques rely on the characteristic function, which can be found in analytical form for some special cases. The lecturer walks through an example using Merton's model and explains how to find the characteristic function for this equation. By separating expectation terms involving independent parts, the lecturer demonstrates how to express the summation in terms of expectations, allowing for the determination of the characteristic function. The advantage of using Fourier techniques is their ability to enable fast pricing computations, which are crucial for model calibration and real-time evaluation.
Next, the lecturer discusses the use of Fourier techniques for solving pricing equations in computational finance. Fourier techniques rely on the characteristic function, which can be found in analytical form for some special cases. The lecturer walks through an example using Merton's model and explains how to find the characteristic function for this equation. By separating expectation terms involving independent parts, the lecturer demonstrates how to express the summation in terms of expectations, allowing for the determination of the characteristic function. The advantage of using Fourier techniques is their ability to enable fast pricing computations, which are crucial for model calibration and real-time evaluation.
Throughout the lecture, the instructor emphasizes the importance of understanding and incorporating jump processes in computational finance models. By including jumps, models can better capture the behavior of real-world stock prices and provide more accurate pricing and calibration results. The lecture also highlights the challenges associated with jump processes, such as the complexity of solving integral differential equations and the need for careful parameter calibration. However, with the appropriate techniques and methodologies, jump processes can significantly enhance the accuracy and realism of computational finance models.
Computational Finance: Lecture 6/14 (Affine Jump Diffusion Processes)
Computational Finance: Lecture 6/14 (Affine Jump Diffusion Processes)
The lecturer provides insights into the selection of pricing models within financial institutions, focusing on the distinction between the front office and the back office. The front office handles trading activities and initiates trades, which are then transferred to the back office for trade maintenance and bookkeeping. The lecturer emphasizes the need to consider various factors, including calibration, risk assessment, pricing accuracy, and computational efficiency when choosing a pricing model. Additionally, the concept of characteristic functions and affine jump diffusion processes is introduced as model classes that allow for efficient pricing evaluation. These models are capable of fast pricing calculations, making them suitable for real-time trading. The lecture also delves into topics such as currency function derivation, framework extension through jump incorporation, and the workflow of pricing and modeling in financial institutions.
The importance of understanding jump processes and their impact on pricing accuracy is highlighted throughout the lecture, along with the challenges involved in solving integral differential equations and calibrating model parameters. By leveraging appropriate techniques and methodologies, computational finance models can be enhanced to better reflect real-world stock price behavior and improve pricing and calibration results.
Furthermore, the speaker emphasizes the role of the front office in financial institutions, particularly in designing and pricing financial products for clients. The front office is responsible for selecting the appropriate pricing models for these products and ensuring that the trades are booked correctly. Collaboration with the back office is crucial to validate and implement the chosen models, ensuring their suitability for the institution's risks and trades. The primary objective of the front office is to strike a balance between providing competitive prices to clients and managing risks within acceptable limits while ensuring a steady flow of profits.
The speaker outlines the essential steps involved in successful pricing, starting with the specification of the financial product and the formulation of stochastic differential equations to capture the underlying risk factors. These risk factors play a critical role in determining the pricing model and the subsequent calculation of prices. Proper specification and modeling of these risk factors are crucial for accurate pricing and risk management.
During the lecture, different methods of pricing are discussed, including exact and semi-exact solutions, as well as numerical techniques such as Monte Carlo simulation. The speaker highlights the importance of model calibration, where the pricing model's parameters are adjusted to match market observations. Fourier techniques are introduced as a faster alternative for model calibration, allowing for efficient computation of model parameters.
The lecture also compares two popular approaches for pricing in computational finance: Monte Carlo simulation and partial differential equations (PDEs). Monte Carlo simulation is widely used for high-dimensional pricing problems, but it can be limited in accuracy and prone to sampling errors. PDEs, on the other hand, offer advantages such as the ability to calculate sensitivities like delta, gamma, and vega at a low cost and smoothness in the solutions. The speaker mentions that Fourier-based methods will be covered in future lectures as they offer faster and more suitable pricing approaches for simple financial products.
The concept of characteristic functions is introduced as a key tool for bridging the gap between models with known analytical probability density functions and those without. By using characteristic functions, it becomes possible to derive the probability density function of a stock, which is essential for pricing and risk assessment.
Throughout the lecture, the importance of calibration is emphasized. Liquid instruments are used as references for calibration, and their parameters are then applied to price more complex derivative products accurately. The lecturer highlights the need to continuously improve and refine pricing models and techniques to adapt to evolving market conditions and achieve reliable pricing results.
In summary, the lecture provides insights into the process of choosing pricing models in financial institutions, focusing on the front office's role, model calibration, and considerations of risk, efficiency, and accuracy. It also introduces various techniques such as Monte Carlo simulation, PDEs, and Fourier-based methods for pricing and model calibration. The concept of characteristic functions and their significance in deriving probability density functions is discussed, along with the challenges and importance of model refinement and adaptation to real-world conditions.
Computational Finance: Lecture 7/14 (Stochastic Volatility Models)
Computational Finance: Lecture 7/14 (Stochastic Volatility Models)
In the lecture, we delve into the concept of stochastic volatility models as an alternative to Black-Scholes models, which may have their limitations. The speaker emphasizes that stochastic volatility models belong to the class of affine diffusion models, which require advanced techniques to efficiently obtain prices and implied volatilities. The motivation behind incorporating stochastic volatility is explained, and the two-dimensional stochastic volatility model of Heston is introduced.
One important aspect covered is the calibration of models to the entire implied volatility surface rather than just a single point. This is particularly crucial when dealing with path-dependent payoffs and strike direction dependency. Practitioners typically calibrate models to liquid instruments such as calls and puts and then extrapolate to the prices of exotic derivatives. Stochastic volatility models are popular in the market as they allow calibration to the entire volatility surface, despite their inherent limitations.
The lecture also highlights the significance of volatility surfaces in the stock market and the need for appropriate models. If the volatility surface exhibits a steep smile, models incorporating jumps or stochastic volatility are often preferred. Different measures used for pricing options, including the P measure and risk-neutral measure, are discussed. It is noted that while making interest rates time-dependent does not improve smiles or skew, introducing stochastic or local volatility can aid in calibration. The Hassel model, which utilizes mean-reverting square root processes to model volatility, is introduced as well.
The lecture explores the concept of stochastic volatility models in detail. Initially, a normal process and Brownian motion are used to define a stochastic differential equation, but it is acknowledged that this approach fails to accurately capture volatility, especially as it can become negative. The benefits of the Box Inverse Process, also known as the CIR process, are explained as it exhibits fat tails and remains non-negative, making it a suitable model for volatility. The Heston model, with its stochastic volatility structure, is introduced, and the variance (VT) is shown to follow a non-central chi-square distribution. It is clarified that this distribution is a transition distribution, and the Feller's condition is mentioned as a critical technical condition to be checked during model calibration.
The conditions for stochastic volatility models to avoid paths hitting zero, referred to as the Feller's condition, are discussed. The condition is satisfied when two times the product of the kappa parameter and the long-term mean is greater than or equal to gamma squared, the volatility squared. When the condition is not met, paths can hit zero and bounce back, leading to an attainable boundary condition. The properties of non-central chi-squared distributions and their relation to CIR processes are explained. Variance paths and density graphs are provided to illustrate the effects of satisfying or not satisfying the Feller's condition.
The significance of fat-tailed distributions in stochastic volatility models is emphasized, as they are often observed after calibrating models to market data. It is noted that if a model's Feller's condition is not satisfied, Monte Carlo paths may hit zero and remain at zero. The inclusion of correlation in models via Brownian motion is explained, and it is mentioned that jumps are typically considered to be independent. The lecture concludes with a graph depicting the impact of the Feller's condition on density.
The lecture focuses on correlation and variance in Brownian motion. The speaker explains that when dealing with correlated Brownian motions, a certain relation must hold true, and the same applies to increments. The technique of Cholesky decomposition is introduced as a means to correlate two Brownian motions using a positive definite matrix and the multiplication of two lower triangular matrices. This method is helpful in formulating the two processes discussed later in the lecture.
The construction of lower triangular matrix multiplication with independent Brownian motions is discussed, resulting in a vector containing a combination of independent and correlated processes.
Furthermore, the lecturer explains that the characteristic function of the Heston model provides valuable insights into efficient and fast pricing. By deriving the characteristic function, it becomes apparent that all the terms involved are explicit, eliminating the need for complex analytical or numerical computations to solve the ordinary differential equations. This simplicity is considered one of the significant advantages of the Heston model, making it a practical and powerful tool for pricing derivatives.
The speaker emphasizes that understanding the characteristics and implications of each parameter in the Heston model is crucial for effectively managing risks associated with volatility. Parameters such as kappa, the long-term mean, volatility, correlation, and the initial value of the variance process all have distinct impacts on volatility dynamics and the implied volatility surface. By calibrating these parameters to the market and analyzing their effects, practitioners can gain valuable insights into implied volatility smiles and skews, enabling more accurate pricing and risk management.
The lecture highlights the importance of calibrating stochastic volatility models to the entire implied volatility surface rather than just a single point. Path-dependent payoffs and strike direction dependencies necessitate a comprehensive calibration approach to capture the full complexity of market data. Typically, practitioners calibrate the models to liquid instruments such as calls and puts and then extrapolate to exotic derivatives' prices. While stochastic volatility models allow for calibration to the entire volatility surface, it is acknowledged that the calibration process is not perfect and has its limitations.
To further enhance the understanding of stochastic volatility models, the lecturer delves into the concept of fat-tailed distributions, which are often observed when calibrating models to market data. The speaker explains that if a model's feller condition is not satisfied, the Monte Carlo paths may hit zero and remain at zero, affecting the model's accuracy. Additionally, the inclusion of jumps and the independent consideration of correlations in stochastic volatility models are discussed. The lecture provides insights into how these elements influence volatility dynamics and pricing.
The lecture concludes by comparing the Heston model to the Black-Scholes model. While the Heston model offers greater flexibility and stochasticity in modeling volatility, the Black-Scholes model remains a benchmark for pricing derivatives. Understanding the implications of different parameter changes on implied volatility smiles and skews is essential for practitioners to choose the appropriate model for their specific needs. Through comprehensive calibration and analysis, stochastic volatility models such as Heston's can provide valuable insights into pricing and risk management in financial markets.
In addition to discussing the Heston model, the lecture addresses the importance of correlation and variance in Brownian motion. The speaker explains that when dealing with correlated Brownian motions, certain relationships and conditions must hold true, including the use of Cholesky decomposition. This technique allows for the correlation of two Brownian motions using a positive definite matrix and the multiplication of two lower triangular matrices. The lecture emphasizes that this method is essential for formulating processes in multi-dimensional cases and achieving the desired correlation structure.
Furthermore, the lecturer focuses on the construction and representation of independent and correlated Brownian motions in stochastic volatility models. While Cholesky decomposition is a useful tool for correlating Brownian motions, the lecture points out that for practical purposes, it is not always necessary. Instead, Ito's lemma can be applied to incorporate correlated Brownian motions effectively. The lecture provides examples of constructing portfolios of stocks with correlated Brownian motions and demonstrates how to apply Ito's lemma to determine the dynamics of multi-dimensional functions involving multiple variables.
The lecture also covers the pricing partial differential equation (PDE) for the Heston model using a martingale approach. This approach involves ensuring that a specific quantity, called pi, which represents the ratio of volatility over the long-term mean, is a martingale. By applying Ethos Lemma, the lecture derives the equation for the martingale, which involves derivatives and the variance process. The pricing PDE allows for the determination of fair prices for derivative contracts and the use of the risk-neutral measure in pricing.
Moreover, the speaker discusses the impact of different parameters on the implied volatility shape in stochastic volatility models. Parameters such as gamma, correlation, and the speed of mean reversion (kappa) are shown to influence the curvature, skewness, and term structure of implied volatilities. Understanding the effects of these parameters helps in accurately calibrating the models and capturing the desired volatility dynamics.
Throughout the lecture, the speaker emphasizes the importance of model calibration, particularly to the entire implied volatility surface. Calibrating to liquid instruments and extrapolating to exotic derivatives is a common practice among practitioners. Stochastic volatility models, including the Heston model, provide the flexibility to calibrate to the entire volatility surface, enabling better accuracy in pricing and risk management. However, it is acknowledged that model calibration is not without limitations and that subtle differences between models, such as the Heston and Black-Scholes models, should be carefully examined to ensure appropriate pricing and risk assessment.
The lecture provides a comprehensive overview of stochastic volatility models, focusing on the Heston model, its parameter implications, calibration techniques, and the role of correlation and variance in Brownian motion. By understanding and effectively applying these concepts, practitioners can enhance their ability to price derivatives, manage risks, and navigate the complexities of financial markets.
Computational Finance: Lecture 8/14 (Fourier Transformation for Option Pricing)
Computational Finance: Lecture 8/14 (Fourier Transformation for Option Pricing)
During the lecture on Fourier Transformation for option pricing, the instructor delves into the technique's application and various aspects. They begin by explaining that Fourier Transformation is utilized to compute the density and efficiently price options for models falling under the class of fine diffusion models. The technique involves computing an integral over the real axis, which can be computationally expensive. However, by employing the inversion lemma, the instructor elucidates how the domain for "u" can be reduced, enabling the computation of the real part of the integral. This approach helps minimize the computational burden associated with expensive computations.
The lecturer further discusses the improvement of this representation using fast Fourier transformation (FFT), which significantly enhances implementation efficiency. By leveraging the properties of FFT, the computational workload is reduced, making option pricing more efficient and faster. The session concludes with a comparison between the Fourier transformation method and the cost method, providing insights into their respective implementation details.
Moving forward, the lecturer delves into the first step in deriving a fast way to calculate density using the Fourier transformation. This step involves dividing the domain into two and extracting the real part, which is a computationally inexpensive operation. Additionally, the lecturer explores the division of complex numbers and the importance of taking the conjugate, as it facilitates more efficient calculations of the characteristic function. The construction of a grid to obtain the density for each "x" value is also discussed, highlighting the significance of selecting appropriate domains and defining boundaries.
The lecture proceeds with an explanation of the calculation of the density of "x" using a Fourier transformation integral and a grid comprising "n" grid points. The instructor emphasizes the need to perform density calculations for multiple "x" values simultaneously. Once the grids are defined, a new integral involving a function named "gamma" is introduced, and trapezoidal integration is employed to approximate the discrete integral. To illustrate this process, the lecturer provides an example of performing trapezoidal integration for a function with an equally spaced grid.
The speaker then delves into the process of configuring parameters to define the grid for Fourier transformation. These parameters encompass the number of grid points, the maximum value of "u," and the relationship between delta "x" and delta "u." Once these parameters are established, integrals and summations can be substituted, enabling the derivation of a function for each "x" value. The lecture includes an equation incorporating trapezoidal integration and characteristic functions evaluated at the boundary nodes of the trapezoid.
The representation of the integral and the importance of employing fast Fourier transformation (FFT) in option pricing are discussed in detail. The speaker explains that by defining a function suitable for input into FFT, practitioners can take advantage of the fast evaluation and implementation capabilities already present in most libraries. The lecturer proceeds to explain the steps involved in computing this transformation and how it can be utilized to calculate integrals. Overall, the lecture underscores the significance of FFT in computational finance and its usefulness in option pricing.
In addition to the aforementioned topics, the lecture explores various aspects related to Fourier transformation for option pricing. These include the use of interpolation techniques to ensure accurate calculations for a discrete number of points, the relationship between the Taylor series and the characteristic function, the application of the cosine expansion method for even functions, and the use of truncated domains to approximate density. The lecture also covers the recovery of density, the numerical results obtained using Fourier expansion, and the pricing representation in the form of matrices and vectors.
Throughout the lecture, the instructor emphasizes the practical implementation of the Fourier transformation method, discusses the impact of different parameters, and highlights the advantages and limitations of the approach. By providing comprehensive explanations and numerical experiments, the lecture equips learners with the knowledge and tools necessary to apply Fourier transformation for option pricing in real-world scenarios.
The lecturer proceeds to discuss the recovery of density function in Fourier Transformation for option pricing. They emphasize the importance of selecting a sufficiently large number of points (denoted as "n") in the transformation to achieve high accuracy density calculations. The lecturer introduces the complex number "i" to define the domain and maximum, with "u_max" determined by the distribution. Furthermore, the lecturer explains the need for interpolation, particularly using cubic interpolation at the grid points "x_i" to ensure accurate calculation of the output density function, even for inputs that do not lie on the grid.
The speaker further explores the benefits of interpolation and its relevance to option pricing using Fourier transformation. While Fourier transformation is advantageous for larger grids, interpolation may be preferred when dealing with larger numbers, as it is comparatively less computationally expensive than FFT. The speaker demonstrates how interpolation works through code examples, highlighting that by adjusting parameters, it becomes possible to calculate sensitivities and obtain Greeks at no additional cost. This feature makes the cosine expansion technique ideal for pricing more exotic derivatives such as barrier and Bermuda options.
Additionally, the lecturer discusses the relationship between the Taylor series and the characteristic function in computational finance. The lecture showcases the one-to-one correspondence between the series and the characteristic function, allowing for direct relations without requiring additional integrals. The lecturer then describes the "cos method" for option pricing, which employs a Fourier cosine expansion to represent even functions around zero. This method involves calculating integrals and coefficients, with the crucial note that the first term of the expansion should always be multiplied by half.
The lecture takes a closer look at the process of changing the domain of integration for function "g" to achieve a finite support range from "a" to "b". The speaker explains the importance of the Euler formula in simplifying the expression and shows how substituting "u" with "k pi divided by b-a" leads to a simpler expression involving the density. The truncated domain is denoted by a hat symbol, and specific values for parameters "a" and "b" are chosen based on the problem being solved. The speaker emphasizes that this is an approximation technique and that heuristic choices are involved in selecting the values of "a" and "b".
Furthermore, the lecture explores the relationship between Fourier expansion and the recovery of density. By taking the real parts of both sides of the equation, the lecture demonstrates the Euler formula that allows expressing the integral of the density as a real part of the characteristic function. This elegant and fast method facilitates finding the relations between integrals of the target function and the characteristic function by using the definition of the characteristic function. The cost method aims to discover these relations to calculate expansion coefficients and recover the density. Although the method introduces errors from infinite summation and the truncation domain, these errors are easy to control.
The lecture then focuses on summarizing the Fourier cosine expansion, which can achieve high accuracy even with a small number of terms. A numerical experiment involving a normal probability density function (PDF) is conducted to examine error generation based on the number of terms, with time measurement included. The code experiment is structured to generate density using the cosine method, defining error as the maximum absolute difference between the density recovered using the cosine method and the exact normal PDF. The cosine method requires only a few lines of code to recover density using the characteristic function, which lies at the heart of the method.
Additionally, the speaker discusses the numerical results of the Fourier expansion, which can be efficiently performed using matrix notation. The error decreases as the number of expansion terms increases, with an error as low as 10^-17 achieved with 64 terms. Using a smaller number of terms can result in oscillations or a poorer fit. The speaker notes that parameters such as the domain and number of expansion terms should be carefully tuned, especially for heavily tailed distributions. Furthermore, the lecture highlights that the log-normal density can also be modeled using the normal characteristic function.
Moving forward, the lecturer delves into the log-normal case and explains how its density differs from the normal distribution. Due to the log-normal distribution, a higher number of expansion terms is typically required. The lecturer emphasizes the importance of choosing an appropriate number of terms for a specific type of distribution and domain.
The lecture emphasizes that the cost method is particularly useful for recovering density and is commonly employed for derivative pricing, such as European-type options that only have a payment at maturity. The lecturer proceeds to explain how pricing works, involving the integration of the product of a density and payoff function under the risk-neutral measure.
As the lecture progresses, the speaker discusses more exotic options, where a connectivity function can be derived and cosines can be used. The term "transition densities" is introduced, referring to the distributions that describe the transition from one point on the time axis to another. The initial value is given in terms of the distribution of a random variable. The presentation further explores truncation of density, where the density is limited to a specified interval. The Gaussian quadrature method is explained, which involves integrating a summation of the real parts of a characteristic function multiplied by some exponent.
The lecture introduces the concept of the adjusted log asset price, which is defined as the logarithm of the stock at maturity divided by a scaling coefficient. An alternative representation of the payoff is presented, and the speaker notes that the choice of "v" directly impacts the coefficient "h_n." This approach can be used for evaluating payoffs for multiple strikes, providing a convenient method for pricing options at various strike prices simultaneously.
Next, the speaker delves into the process of computing the integral of a payoff function multiplied by the density using exponential and cosine functions in Fourier transformation for option pricing. A generic form for the two integrals involved is provided, and different coefficients are selected to calculate various payoffs. The speaker emphasizes the importance of being able to implement this technique for multiple strikes, allowing for the pricing of all strikes at once, which saves time and reduces computational expenses. Finally, the pricing representation is presented in the form of a matrix multiplied by a vector.
The implementation formula for Fourier transformation in option pricing is discussed, involving the vectorization of elements and matrix manipulations. The lecture explains the process of taking "k" as a vector and creating a matrix with "n_k" strikes. Real parts are calculated to handle complex numbers. The characteristic function is of high importance as it does not depend on "x" and plays a key role in achieving efficient implementations for multiple strikes. The accuracy and convergence of the implementation depend on the number of terms, and a sample comparison is shown.
Additionally, the speaker delves into the code used for the Fourier transformation method in option pricing and explains the different variables involved. They introduce the concept of a range for coefficients "a" and "b," typically kept at 10 or 8 for jump diffusion models. The code includes a lambda expression for the characteristic function, which is a generic function adaptable to different models. The speaker emphasizes the significance of measuring time by conducting multiple iterations of the same experiment and calculating the average time. Finally, they illustrate the cost method and how it utilizes the integration range to assume a large volatility.
The lecture continues with an explanation of the process of defining strikes and calculating coefficients for the Fourier transform method of option pricing. The lecturer emphasizes that while tuning the model parameters can lead to better convergence and require fewer terms for evaluation, it is generally safe to stick with standard model parameters. They detail the steps of defining a matrix and performing matrix multiplication to obtain the discounted strike price, comparing the resulting error against that of the exact solution. The lecture highlights that the error depends on the number of terms and the chosen strike range.
The speaker then presents a comparison of different methods for option pricing, including the Fast Fourier Transform (FFT) method and the Cosine method. They explain that the FFT method is more suitable for a large number of grid points, while the Cosine method is more efficient for a smaller number of grid points. The lecturer demonstrates the calculation of option prices using both methods and compares the results.
Moreover, the lecture covers the application of Fourier-based methods in other areas of finance, such as risk management and portfolio optimization. The lecturer explains that Fourier-based methods can be used to estimate risk measures such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). By combining Fourier methods with optimization techniques, it is possible to find optimal portfolio allocations that minimize risk or maximize returns.
The lecture concludes by summarizing the main points discussed throughout the presentation. Fourier transformation techniques provide a powerful tool for option pricing and other financial applications. The Cosine method allows for efficient and accurate pricing of options by leveraging the characteristic function and Fourier expansion. The choice of parameters, such as the number of terms and the domain, impacts the accuracy and convergence of the method. Additionally, Fourier-based methods can be extended to various financial problems beyond option pricing.
Overall, the lecture provides a comprehensive overview of Fourier transformation techniques in option pricing, covering topics such as the recovery of density, interpolation, the cos method, log-normal distributions, multiple strikes, implementation considerations, and comparisons with other pricing methods. The lecturer's explanations and code examples help illustrate the practical application of these techniques in finance and highlight their benefits in terms of accuracy and efficiency.