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ABM (Arithmetic Brownian Motion)
It is a linear stochastic process where the value of a financial variable changes linearly with time, influenced by random movements. It is characterized by a constant mean and variance rate of change.
The formula for ABM is:
$$ dX_t = \mu dt + \sigma dW_t $$
Here, - \( dX_t \): Change in the variable at time t. - \( \mu \): Drift coefficient. - \( \sigma \): Volatility. - \( dW_t \): Increment of a Wiener process or Brownian motion at time t.
Actuarial analysis applies mathematical and statistical methods to assess risk in the insurance and finance industries. Actuaries use this analysis to develop models that calculate the financial impact of uncertain future events.
These are a type of stochastic differential equations that are linear and are commonly used in quantitative finance to model the dynamics of financial derivatives and other assets.
They can be expressed as:
$$ dx(t) = a(x(t))dt + b(x(t))dW(t) $$
Where: - \( dx(t) \): Change in process x at time t. - \( a(x(t)) \): Drift term, linear in x(t). - \( b(x(t)) \): Diffusion term, linear in x(t). - \( dW(t) \): Increment of a Wiener process.
This refers to executing trades with advanced mathematical models and algorithms, enabling precision and speed unattainable by humans. It capitalizes on small price discrepancies across platforms.
Almost surely is a term in probability theory to indicate that an event occurs with probability one, although it doesn't guarantee the event's occurrence.
This hypothesis opposes the null hypothesis and suggests a significant effect or difference is present. Hypothesis testing determines which hypothesis is supported by data.
An American option can be exercised at any time up to its expiration date, providing flexibility and often resulting in higher premiums than European options.
A technique providing exact solutions to mathematical problems using algebra, calculus, and mathematical theories, often for pricing and valuation models in finance.
Used in analysis of variance (ANOVA), it displays sources of variation, degrees of freedom, sum of squares, and F-statistic to evaluate variables' significance.
Arbitrage exploits price differences in identical or similar assets across markets to earn profits, often simultaneously buying and selling.
"Argmin" finds the input where a function attains its minimum value. Expressed mathematically as:
$$ \text{argmin}_{x \in S} f(x) $$
- \( f(x) \): Function of variable X. - \( S \): Set over which minimization is performed.
ARIMA (AutoRegressive Integrated Moving Average)
A statistical method for time series forecasting that captures trends, seasonality, and noise for predictions.
The practice of diversifying investments among asset classes like bonds, stocks, and cash to maximize returns relative to risk tolerance.
Measures the relative value of fixed-income securities compared to risk-free bonds, considering yield differences and swaps.
Securities backed by pools of assets, offering returns based on cash flows from collateralized assets.
Exotic options with payoffs based on the average price of the underlying asset over time, reducing volatility risk.
$$ \text{Payoff (Call)} = \max(0, \frac{1}{T} \sum_{t=1}^T S_t - K) $$
- \( T \): Observation points. - \( S_t \): Asset price at t. - \( K \): Strike price.
Refers to unequal potential for gains and losses, resulting in skewed risk-reward profiles.
A basis point is a unit of measure for interest rates and financial percentages, equivalent to one-hundredth of a percent (0.01%). It is commonly used to convey changes in interest rates and financial ratios, providing a nuanced view of variations that are less perceptible when expressed in percentage terms.
Basis risk refers to the risk associated with the unexpected variance in the spread between the spot price of an asset and the futures price of a contract written on the same asset. It represents the mismatch between the performance or behavior of a hedged position and the corresponding hedge, leading to incomplete risk mitigation.
A type of option whose existence or features depend upon the underlying asset's price reaching a certain level. It becomes active or inactive when crossing specific thresholds, known as barriers.
Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available.
$$ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} $$
Where: - \( P(A|B) \): Posterior probability of A given B. - \( P(B|A) \): Likelihood of B given A. - \( P(A) \): Prior probability of A. - \( P(B) \): Total probability of B.
An option that can be exercised at several predetermined times, offering flexibility in execution.
Beta measures a security's or portfolio's systematic risk or volatility relative to the market or a benchmark index. A beta greater than 1 indicates higher volatility than the market, while a beta less than 1 signifies lower volatility.
$$ \beta = \frac{\text{Cov}(R_a, R_m)}{\text{Var}(R_m)} $$
Where: - \( R_a \): Return of the asset. - \( R_m \): Return of the market.
The binomial distribution describes the probability of achieving a fixed number of successes in a fixed number of independent Bernoulli trials. It is widely used in options pricing and risk management.
This model offers a computational approach to valuing options by dividing the option’s life into discrete intervals and calculating potential prices at each stage. It accommodates various types of options, including American options, and derives values based on probabilities of price movements.
The Black-Scholes model is a mathematical framework for pricing European options and calculating the theoretical value of derivatives based on asset price, volatility, time to expiration, and interest rates. It is foundational in financial engineering for option pricing and risk management.
The classical Boltzmann equation models particle distribution based on time, position, and velocity:
$$ \frac{\partial f}{\partial t} + v \cdot \nabla f + F \cdot \nabla_v f = Q(f, f) $$
Where: - \( f(x, v, t) \): Distribution function. - \( v \): Particle velocity. - \( F \): Force acting on particles. - \( Q(f, f) \): Collision term.
In finance, a similar approach can hypothetically model asset price evolution influenced by market factors.
Bond yield represents the return an investor earns on a bond, expressed as an annual percentage. Different measures include current yield, yield to maturity, and yield to call, providing insights into performance under various scenarios.
Bootstrapping refers to constructing a yield curve by sequentially estimating zero-coupon yields to match bond prices. It is also used for statistical resampling techniques to validate estimates through simulated samples.
A neutral options trading strategy involving buying and selling options with different strike prices. It profits from minimal price movement and combines in-the-money, at-the-money, and out-of-the-money options.
Buy-side firms purchase investment assets and services, including mutual funds and pension funds. Their focus is research and analysis to maximize returns, often working with sell-side firms for advisory and brokerage services.
A call option is a financial derivative contract granting the option holder the right, but not the obligation, to purchase a specified amount of an underlying asset at a predetermined price within a specified time frame. It’s often employed for speculative activities or hedging against potential price increases of underlying assets.
Capital Asset Pricing Model (CAPM)
The Capital Asset Pricing Model is a theoretical framework employed in finance to delineate the relationship between the expected return of an asset and its systematic risk, quantified by beta. It posits that the expected return of an asset or portfolio equals the risk-free rate plus the asset's beta multiplied by the expected market risk premium.
In calculus, this rule helps find the derivative of a composite function. It expresses the derivative of the composite function in terms of the derivatives of its constituent functions.
Cholesky decomposition is a numerical method used in linear algebra to decompose a positive definite matrix into the product of a lower triangular matrix and its transpose. It is often used in numerical simulations, optimization, and solving linear systems of equations. The result of Cholesky decomposition is expressed as:
\[ A = L \cdot L^T \]
Where:
- \(A\) is the original positive definite matrix.
- \(L\) is the lower triangular matrix.
- \(L^T\) is the transpose of \(L\).
Named after Cox, Ingersoll, and Ross, the CIR model is a mathematical description of interest rate movements. It is a type of one-factor model, stochastic and mean-reverting, used to predict interest rate changes and the pricing of bonds.
A solution expressed as an explicit function or formula, derived analytically from mathematical equations. In finance, closed-form solutions are highly valued for their accuracy and efficiency, often used in options pricing models and other financial calculations to obtain precise results directly.
Conditional Value at Risk (CVaR)
A risk assessment measure that quantifies the expected losses occurring beyond the Value at Risk (VaR) threshold level.
Collateralized Debt Obligation (CDO)
A CDO is a structured financial product that pools together cash flow-generating assets, such as mortgages and bonds, and repackages them into discrete tranches that can be sold to investors. Each tranche has a distinct risk profile and rating, influencing its yield and payment priority.
The correlation coefficient measures the linear relationship between two variables. The formula is:
\[ r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \cdot \sum (y_i - \bar{y})^2}} \]
Where: - \( r \) is the correlation coefficient.
- \( x_i, y_i \) are data points.
- \( \bar{x}, \bar{y} \) are means.
Covariance measures the joint variability of two random variables. The formula is:
\[ \text{Cov}(X, Y) = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{n - 1} \]
Where:
- \(x_i, y_i\) are data points of variables X and Y.
- \(\bar{x}, \bar{y}\) are the means of X and Y.
- \(n\) is the total number of data points.
The degree of freedom refers to the number of independent values or quantities that can vary in the calculation of a statistic or test. It's essential in hypothesis testing, confidence intervals, and regression analysis, influencing the shape of distributions like chi-square and t-distribution.
Delta measures the rate of change in the price of an option relative to a one-unit change in the underlying asset. It quantifies sensitivity and is widely used in valuation models, risk management, and trading strategies.
Delta hedging is a risk management strategy used in options trading to minimize directional risk. It involves adjusting positions in the underlying asset to offset changes in options prices, aiming to maintain a neutral or zero delta.
The term 25-delta describes options with a 25% probability of expiring in-the-money. It is often used in forex for risk reversals and butterfly spreads to assess market sentiment and volatility.
Default probability quantifies the likelihood of a borrower defaulting on debt obligations within a specified time frame, critical for pricing credit derivatives like credit default swaps.
The Derman-Kani Model extends the Black-Scholes model by incorporating implied volatility surfaces, enabling more accurate derivative pricing.
A Dirac measure assigns all measure to a single point. Mathematically, δx(A) = 1 if x is in A, otherwise 0. It's used in probability theory to represent distributions focused on a specific value.
A discrete-time model represents financial systems transitioning between states in distinct intervals, contrasting with continuous-time models.
A down-and-out option is a barrier option that expires worthless if the underlying asset's price falls below a set level, often used for tailored risk-return structures.
The dual-beta model estimates separate beta coefficients for up and down markets, offering insights into risk and performance.
Dynamic replication involves continuously adjusting a portfolio of assets and risk-free investments to replicate a derivative’s payoffs.
Derivative pricing calculates fair values using models that consider asset prices, volatility, expiration time, and interest rates.
The degree of freedom refers to the number of independent values or quantities that can vary in the calculation of a statistic or test. It's essential in hypothesis testing, confidence intervals, and regression analysis, influencing the shape of distributions like chi-square and t-distribution.
Delta measures the rate of change in the price of an option relative to a one-unit change in the underlying asset. It quantifies sensitivity and is widely used in valuation models, risk management, and trading strategies.
Delta hedging is a risk management strategy used in options trading to minimize directional risk. It involves adjusting positions in the underlying asset to offset changes in options prices, aiming to maintain a neutral or zero delta.
The term 25-delta describes options with a 25% probability of expiring in-the-money. It is often used in forex for risk reversals and butterfly spreads to assess market sentiment and volatility.
Default probability quantifies the likelihood of a borrower defaulting on debt obligations within a specified time frame, critical for pricing credit derivatives like credit default swaps.
The Derman-Kani Model extends the Black-Scholes model by incorporating implied volatility surfaces, enabling more accurate derivative pricing.
A Dirac measure assigns all measure to a single point. Mathematically, δx(A) = 1 if x is in A, otherwise 0. It's used in probability theory to represent distributions focused on a specific value.
A discrete-time model represents financial systems transitioning between states in distinct intervals, contrasting with continuous-time models.
A down-and-out option is a barrier option that expires worthless if the underlying asset's price falls below a set level, often used for tailored risk-return structures.
The dual-beta model estimates separate beta coefficients for up and down markets, offering insights into risk and performance.
Dynamic replication involves continuously adjusting a portfolio of assets and risk-free investments to replicate a derivative’s payoffs.
Derivative pricing calculates fair values using models that consider asset prices, volatility, expiration time, and interest rates.
The F-Test is a statistical test used to compare the variances of two or more groups to determine if they are significantly different. It is often employed in analysis of variance (ANOVA) and regression analysis to assess the significance of model parameters or group differences.
The Fama-MacBeth Regression is a two-step method used in finance to analyze the relationship between asset returns and predictive variables.
In statistics and finance, "fat tails" refer to the presence of extreme and infrequent events or outliers in a probability distribution or dataset.
The Feynman-Kac Formula connects partial differential equations (PDEs) with stochastic processes.
Filtration (in Probability Theory)
In probability theory, Filtration is a sequence of increasing σ-algebras {𝓕_t} associated with a stochastic process. It formalizes the notion of "information available up to a certain time."
Financial Engineering involves the application of mathematical methods to solve problems in finance.
The Finite Difference Method (FDM) approximates solutions to differential equations, particularly for option pricing in quantitative finance.
The first derivative is approximated as: \[ f'(x) \approx \frac{f(x + \Delta x) - f(x)}{\Delta x} \] The second derivative is approximated as: \[ f''(x) \approx \frac{f(x + \Delta x) - 2f(x) + f(x - \Delta x)}{(\Delta x)^2} \]
First Difference transforms a time series by calculating the difference between consecutive data points, making the series stationary.
The Fleming–Viot Process could theoretically model fluctuations in financial markets, analogous to population genetics processes.
The Fokker-Planck Equation describes the evolution of a probability density function in stochastic systems.
It is expressed as: \[ \frac{\partial P}{\partial t} = -\frac{\partial (AP)}{\partial x} + \frac{\partial^2 (BP)}{\partial x^2} \]
Forward contracts are over-the-counter agreements to buy or sell an asset at a future date for a predetermined price.
An FRA is a financial derivative where two parties agree on an interest rate for a future period, hedging against interest rate changes.
A Fractal is a self-similar pattern observed at different scales. In finance, fractals are used to detect recurring patterns in asset price movements.
Fractional Differentiation generalizes ordinary differentiation to non-integer orders, preserving memory and improving financial time series modeling.
A Future is a standardized financial contract to buy or sell an asset at a set price on a future date, traded on exchanges for hedging and speculation.
Gamma measures the rate of change in an option's delta for a one-unit change in the price of the underlying asset.
The Gamma function is a mathematical concept often used in calculus, probability theory, and complex number theory. It extends the factorial function to complex and real number inputs and is defined by an integral expression.
Applications in finance include:
GARCH Model (Generalized Autoregressive Conditional Heteroskedasticity)
The GARCH model is used in quantitative finance to model changing volatility in financial time series data. It is widely employed for:
A Gaussian process is a stochastic process where every finite collection of variables has a multivariate normal distribution. It’s characterized by:
In finance, Gaussian processes are used for:
Geometric Brownian Motion (GBM)
Geometric Brownian Motion (GBM) is a continuous-time stochastic process used to model stock prices that follow a lognormal distribution. It ensures prices remain positive and supports option pricing models like Black-Scholes.
The formula for GBM is: \[ dS_t = \mu S_t dt + \sigma S_t dW_t \]
Where:
Greeks measure the sensitivity of an option’s price to various factors, including:
Girsanov Theorem describes how the probability measure changes when a new drift term is added to a stochastic differential equation.
Applications in finance include:
Heat Equation (in Quantitative Finance)
The Heat Equation models the gradual diffusion or spreading of financial quantities over time. Adapted from physics, it’s often used in quantitative finance for option pricing and risk management.
In one dimension, it is expressed as:
\[ \frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2} \]
Where:
Hedging Portfolio (in the Black-Scholes SDE)
A Hedging Portfolio combines the underlying asset (\(S\)), cash (\(C\)), and an option position (\(V\)) to replicate the behavior of the option and maintain a risk-neutral position.
Key components:
Hedging portfolios are essential for managing risks in option pricing and financial markets.
Heath-Jarrow-Morton (HJM) Model
The HJM model describes the evolution of forward rate curves, used in pricing interest rate derivatives and managing interest rate risks.
The Heston Model describes the evolution of volatility over time. It is defined by two stochastic differential equations:
Where:
The Heston model allows volatility to vary over time, making it a popular tool for pricing options as it captures the volatility smile effectively.
Heteroskedasticity occurs when the variance of errors in a regression model changes across levels of explanatory variables, leading to inconsistencies such as:
It contrasts with homoskedasticity, where variances remain constant.
High-Frequency Trading involves executing large volumes of orders at extremely high speeds using algorithms, often exploiting small price movements.
Key features:
Homoskedasticity refers to a condition where the variance of errors is constant across levels of explanatory variables.
Assumptions:
Deviations indicate heteroskedasticity, requiring adjustments for unbiased estimations.
The Hurst Exponent quantifies the long-term memory of time series, helping determine whether a financial series exhibits:
It is widely used in fractal analysis, market efficiency studies, and risk modeling.
Implied correlation is a measure inferred from the prices of options on multiple assets, reflecting the market’s expectation of the correlation between those assets.
Implied volatility is a measure of how much the markets expect the price of an asset to move, inferred from option prices.
An implicit volatility surface is a three-dimensional plot featuring strike price, time-to-maturity, and implied volatility, often used in the derivatives market.
An independent variable, often denoted as \(x\), is used in statistical modeling to predict or explain variations in the dependent variable.
In quantitative finance, an integrand refers to the function that is to be integrated in the context of a mathematical or computational model. It is fundamental in:
Together with limits of integration, it defines the integral, representing the accumulated value under the curve.
Integration is a mathematical process used in quantitative finance to calculate accumulated values, such as rates of return, over a specified interval or time period.
The formula is:
\[ \int_a^b f(x) \, dx \]
Where:
Applications include:
An interest rate swap is a contract where two parties exchange fixed or floating rate interest payments, often used for hedging interest rate risks.
Interpolation is a mathematical and statistical technique used to estimate unknown values between two known values in a data set.
Applications in finance include:
Ito's Lemma is a fundamental concept in stochastic calculus that determines the differential of a function of a stochastic process, particularly Brownian motion.
Applications include:
Ito Calculus Multiplication Rules
Rules in stochastic calculus for handling products of increments:
These rules are vital for stochastic differential equations, distinguishing stochastic processes from deterministic ones.
Jensen's Inequality states that for a convex function, the function of an expectation is always less than or equal to the expectation of the function.
For a convex function \( \phi \), it is expressed as:
\[ \phi(E[X]) \leq E[\phi(X)] \]
For a concave function:
\[ \phi(E[X]) \geq E[\phi(X)] \]
Where:
Applications in finance:
Jensen's Alpha is a risk-adjusted performance measure representing the average return of a portfolio over and above that predicted by the Capital Asset Pricing Model (CAPM).
It evaluates whether the portfolio has outperformed or underperformed its expected return given its risk level, measured by beta (\( \beta \)).
The formula for Jensen's Alpha:
\[ \alpha = R_p - [R_f + \beta(R_m - R_f)] \]
Where:
A positive alpha indicates outperformance, while a negative alpha signals underperformance relative to the expected returns.
The Jump Diffusion Model incorporates sudden and significant changes, or "jumps," in asset prices, combining both continuous and discontinuous movements.
The model modifies standard Brownian motion with a jump component to better capture observed market behaviors like crashes or rallies.
The equation is:
\[ dS_t = \mu S_t dt + \sigma S_t dW_t + J_t dN_t \]
Where:
Applications include:
A mathematical method to estimate the state of a linear dynamic system from a series of incomplete and noisy measurements.
A statistical measure describing the tailedness of a probability distribution, based on its fourth central moment. It evaluates the likelihood of extreme values in a dataset.
Formula:
\[ K = \frac{n \sum (x_i - \bar{x})^4}{\left( \sum (x_i - \bar{x})^2 \right)^2} \]
A normal distribution has a kurtosis of 3. Values greater than 3 indicate fat tails (leptokurtic). Values less than 3 indicate thin tails (platykurtic).
Statistical models that involve variables that are not directly observed but are rather inferred from other observed variables.
Introduced by Henri Lebesgue, this integral generalizes the Riemann integral by measuring "how much" a function takes specific values rather than "where" those values occur.
Applications include:
The use of financial instruments or borrowed capital to increase the potential return of an investment.
A stochastic process modeling random movements with occasional large jumps, useful for describing extreme price movements in financial markets.
A type of stochastic process that starts at zero, has stationary and independent increments, and is continuous in probability. It generalizes Brownian motion and Poisson processes to model financial data.
A model for pricing interest rate derivatives where forward LIBOR rates follow a log-normal process driven by Brownian motions under the risk-neutral measure.
A branch of mathematics focusing on linear equations, matrices, and vector spaces, widely used in finance for portfolio optimization, risk assessment, and managing large datasets.
The ease at which an asset can be quickly bought or sold without affecting its price.
Developed by Bruno Dupire, this model calculates implied volatility as a function of time and asset price, improving accuracy over the Black-Scholes model.
The equation is:
\[ \frac{\partial F}{\partial t} + 0.5 \cdot (localVol \cdot S)^2 \cdot \frac{\partial^2 F}{\partial S^2} + r \cdot S \cdot \frac{\partial F}{\partial S} - r \cdot F = 0 \]
Applications:
A statistical method used for binary classification problems, estimating the probability that an input belongs to a particular class.
A distribution used to model asset prices, characterized by its positive skewness and suitability for modeling non-negative values like stock prices.
An exotic option allowing holders to "look back" over time to select the most favorable price for exercising the option.
The natural logarithm of the ratio between an asset’s ending price and its starting price:
\[ R = \ln\left(\frac{P_t}{P_0}\right) \]
This approach simplifies mathematical manipulation and allows returns to be additive over time.
A statistical model that predicts future states of a system based on the current state, without considering past states.
Applications in finance:
A stochastic process where the expected future value, given all past values, equals the current value. It represents a "fair game" without predictive advantage.
A stochastic process with the Markov property, where the probability distribution of future states depends only on the current state, not on previous states.
The detailed processes and mechanisms governing trading and price formation within financial markets, including rules, infrastructure, and participant behavior.
A theory suggesting that asset prices and returns eventually move back towards their historical average or mean levels over time.
A statistical function used to derive the moments (mean, variance, skewness, kurtosis) of a probability distribution. It is defined as:
\[ M(t) = E\left[e^{tX}\right] \]
Applications:
A problem-solving technique that estimates probabilities by running multiple simulations using random variables.
Applications in finance:
Used in finance to analyze spatial patterns and correlations in data sets with geographical components. It identifies clusters or dispersions in investment values, aiding decision-making.
A condition in regression models where two or more independent variables are highly correlated, making it difficult to isolate their individual effects.
Solutions:
Techniques and analyses involving multiple variables or datasets simultaneously, often used to study relationships between variables in complex systems.
Models like LSTM (Long Short-Term Memory) and GRU (Gated Recurrent Units) used in deep learning for tasks like time series forecasting.
An iterative numerical method used to approximate the roots of a real-valued function.
Formula:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]
Applications in finance:
A principle stating that it is impossible to achieve risk-free profits above the risk-free interest rate by exploiting price discrepancies in financial instruments.
No-Arbitrage Term Structure Models
Term structure models that ensure the absence of arbitrage opportunities in pricing interest rate derivatives and fixed-income securities. Examples include:
Risk arising when irrational traders influence prices and trading patterns in financial markets.
A property where a function lacks a derivative at certain points, often due to sharp corners or discontinuities.
Applications in finance:
Nonlinear Time Series Analysis
Methods for modeling time series data with complex, non-linear relationships to capture patterns not explained by linear models.
Normal Distribution (Gaussian Distribution)
A symmetric probability distribution characterized by its bell-shaped curve, used extensively in finance.
Formula for probability density function:
\[ f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{(x - \mu)^2}{2 \sigma^2}} \]
Where:
Applications:
A unit of account used to value assets relatively, often set as the risk-free asset in pricing models.
Applications:
Techniques for solving mathematical problems via approximation, essential for derivative pricing, risk management, and quantitative analysis.
A default assumption in statistical tests asserting no significant effect or difference, tested against an alternative hypothesis.
A martingale component derived from a semimartingale and its natural filtration, often used in option pricing and hedging in incomplete markets.
Applications:
A derivative contract giving the holder the right, but not the obligation, to buy or sell an asset at a predetermined price before or at expiration.
Types:
A stochastic process used to model mean-reverting behavior in finance. It is represented by the equation:
\[ dX(t) = \theta(\mu - X(t))dt + \sigma dW(t) \]
Where:
Applications:
Overfitting occurs when a statistical model is too complex, fitting noise in training data instead of underlying patterns.
Characteristics:
Methods to prevent overfitting:
A market-neutral trading strategy that pairs a long position with a short position in two highly correlated securities, profiting from their price convergence.
Financial derivatives whose payoff depends on the path taken by the underlying asset's price over time, not just its final value at expiration. Examples include:
The actual probabilities of outcomes occurring in real-world scenarios, contrasting with the risk-neutral measure used in derivative pricing.
Models the number of events occurring in a fixed interval, commonly applied in finance for rare events like defaults or market jumps.
Formula:
\[ P(N(t) = n) = \frac{e^{-\lambda t} (\lambda t)^n}{n!} \]
Where:
The process of selecting the best portfolio according to predefined criteria, such as maximizing returns or minimizing risk.
Principal Component Analysis (PCA)
A dimensionality reduction technique used to transform data into uncorrelated variables (principal components) ranked by importance.
Applications:
A function describing the likelihood of different outcomes for a random variable, including distributions like:
Assigns a probability (between 0 and 1) to outcomes of a random process.
Types in finance:
A mathematical framework for modeling random experiments, consisting of:
A fundamental relationship between European call and put options, expressed as:
\[ C - P = S - \frac{X}{(1 + r)^t} \]
Where:
A statistical measure in hypothesis testing that indicates the likelihood of observing data as extreme as the sample, assuming the null hypothesis is true.
A small p-value suggests rejecting the null hypothesis in favor of the alternative hypothesis.
A colloquial abbreviation for a "quantitative analyst," a professional who uses mathematical and statistical models, programming, and computational techniques to analyze data, develop trading strategies, and manage risk.
A professional who designs and maintains the computational infrastructure and algorithms used in quantitative finance for data analysis, risk management, and algorithmic trading.
The application of mathematical and statistical techniques to study behavior, predict outcomes, and make data-driven decisions in finance.
Specialists in quantitative finance who analyze financial data, optimize portfolios, design trading strategies, and manage risk using mathematical models and statistical methods.
A field applying mathematical, statistical, and computational methods to price financial instruments, optimize trading strategies, and manage risk in modern financial markets.
Professionals conducting research to develop trading strategies, optimize portfolios, and improve financial models through data-driven insights, backtesting, and statistical techniques.
A numerical method similar to Monte Carlo simulations, but it uses quasi-random sequences to improve accuracy and convergence when solving partial differential equations and performing numerical integration.
A mathematical measure of variability for stochastic processes like Brownian motion.
Formula:
\[ QV(T) = \lim_{n \to \infty} \sum_{i=1}^{n} [B(t_i) - B(t_{i-1})]^2 \]
Where:
For standard Brownian motion, the quadratic variation over time \(T\) equals \(T\).
The Quadratic Rough Heston Volatility Model
A mathematical framework modeling volatility dynamics for options pricing and risk management.
Key Features:
Applications:
A tool in measure theory for defining the derivative of one measure with respect to another, crucial in finance for measure changes.
An exotic option linked to two or more underlying assets, with a payoff based on their relative performance.
Applications:
A stochastic process where values change in random steps, often used to model stock prices and exchange rates.
A concept in stochastic processes where the path of a Brownian motion mirrors itself after hitting a specific level.
A statistical method for modeling relationships between a dependent variable and one or more independent variables, widely used in forecasting and predictions.
A type of integral used to calculate areas under curves, applied in finance for continuous compounding and options pricing.
A probability measure in which all investments grow at the risk-free rate, simplifying derivative pricing and valuation.
An options strategy involving simultaneous buying of a call and selling of a put (or vice versa) to hedge or speculate on asset price movements.
Measures the volatility skew between out-of-the-money call and put options, reflecting market sentiment on price movements.
Sensitivity of an option's value to changes in interest rates.
A mathematical framework capturing irregular patterns in price volatility for improved options pricing and risk management.
Key Features:
A mathematical model used to estimate volatility and price financial derivatives, particularly options.
Key Parameters:
A single numerical value used to scale vectors and matrices, often representing rates, returns, or factors in financial calculations.
A fundamental equation in quantum mechanics describing particle behavior and probability distributions.
SDE (Stochastic Differential Equation)
An equation modeling stochastic processes over time, combining deterministic and random components.
Applications:
A generalization of martingales used in stochastic calculus, modeling asset prices with both deterministic trends and stochastic variations.
Investment banks, brokers, and dealers that create, promote, and sell financial instruments, facilitating trading and liquidity.
A measure of risk-adjusted returns:
\[ Sharpe\ Ratio = \frac{R_p - R_f}{\sigma_p} \]
Where:
Models describing the evolution of short-term interest rates, such as:
A mathematical framework in probability theory for defining measurable spaces, crucial for modeling stochastic processes in finance.
Represents variance, measuring the spread or volatility of asset returns.
Describes the asymmetry of a distribution around its mean:
A property of time series where mean and variance remain constant over time, essential for statistical modeling and forecasting.
A trading strategy using quantitative models to identify and exploit mispricings between related financial instruments.
Extends classical calculus to handle randomness and uncertainty, forming the basis for modeling asset prices and derivative pricing.
A concept in stochastic processes defining the time when a specific condition is fulfilled, used in trading strategies and risk management.
A probability distribution used in hypothesis testing, accounting for sample size uncertainty, particularly valuable for small datasets.
An options strategy involving buying or selling a call and put option with the same strike price and expiration, profiting from volatility.
The process of valuing swap contracts based on expected future cash flows and present value calculations.
An option to enter into an interest rate swap, providing flexibility for hedging or speculating on interest rate changes.
A stochastic process with equal probabilities of upward or downward movement, modeling random behavior in asset prices.
A method for approximating functions as infinite sums of terms derived from derivatives at a specific point.
Formula:
\[ f(x) \approx f(a) + f'(a)(x - a) + \frac{f''(a)(x - a)^2}{2!} + \frac{f'''(a)(x - a)^3}{3!} + \dots \]
Applications:
Models describing the evolution of the yield curve over time, essential for pricing fixed-income securities and derivatives.
Examples:
Measures the rate at which the price of a derivative decreases as it nears expiration, reflecting time decay.
A statistical test comparing the means of two groups to determine if they are significantly different.
Formula for t-statistic:
\[ t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \]
Where:
A sequence of data points recorded at successive time intervals, widely used in finance for forecasting and trend analysis.
Statistical techniques for analyzing time-series data to identify patterns, trends, and cycles, enabling forecasting and decision-making.
Transition Probability Function
Quantifies the probability of transitioning from one state to another over time in stochastic processes and Markov chains.
Formula:
\[ P_{ij}(t) = P(X(t) = j \mid X(0) = i) \]
Applications:
A probability distribution where all outcomes within a specified range have equal probability.
Formula (continuous case):
\[ f(x) = \begin{cases} \frac{1}{b - a} & \text{if } a \leq x \leq b, \\ 0 & \text{otherwise.} \end{cases} \]
Applications:
A property of a time series indicating non-stationarity, where statistical properties like mean and variance change over time.
Mathematical representation:
\[ X_t = \rho X_{t-1} + \epsilon_t \]
Where:
Implications:
A mathematical framework in economics and finance to model preferences, satisfaction, or usefulness derived from consuming goods or services.
Utility Function:
\[ U(x) = x^\alpha \]
Where:
Applications:
A measure of the risk of loss for investments, estimating the maximum potential loss over a specified time period at a given confidence level.
Applications:
Measures the sensitivity of an option’s delta to changes in implied volatility, helping to assess how price sensitivity changes with market volatility.
A measure of the spread of data points around the mean, indicating risk or volatility in financial returns.
Formula:
\[ \text{Variance} = \frac{\sum (X_i - \bar{X})^2}{N} \]
Where:
A financial derivative contract used to hedge or speculate on future volatility without owning the underlying asset.
Features:
A mean-reverting stochastic model used to describe interest rate dynamics, assuming rates revert to a long-term mean.
Formula:
\[ dr_t = a(b - r_t) dt + \sigma dW_t \]
The sensitivity of an option's price to a 1% change in implied volatility, used to evaluate volatility risk.
A measure of the degree of variation in an asset’s price, reflecting uncertainty and risk.
Formula:
\[ \sigma = \sqrt{\frac{\sum (X_i - \bar{X})^2}{N}} \]
A correction factor applied to volatility measures to improve accuracy in pricing models and reflect actual market behavior.
The uneven distribution of implied volatilities for options with the same expiration date but different strike prices.
Observations:
A U-shaped pattern of implied volatilities across different strike prices, suggesting higher expected volatility for extreme price movements.
Uses:
Measures the sensitivity of an option’s vega to changes in volatility, used to manage gamma risks in options portfolios.
The Weibull Distribution is commonly used in finance to model extreme events, particularly in risk management and insurance.
Formula (Probability Density Function):
\[ f(x; \lambda, k) = \frac{k}{\lambda} \left( \frac{x}{\lambda} \right)^{k - 1} e^{-(x/\lambda)^k} \]
Where:
Applications:
Also known as Brownian motion, the Wiener process is a real-valued, continuous-time stochastic process widely used in finance.
Properties:
Formula:
\[ W(t) = W(0) + \mu t + \sigma B(t) \]
Applications:
A graphical representation of interest rates on bonds with the same credit quality but different maturities, showing the relationship between yields and time.
Types of yield curves:
Applications:
A bond that doesn't pay periodic interest (coupons) but is issued at a discount and redeemed at face value upon maturity.
Pricing Formula:
\[ P = \frac{F}{(1 + r)^t} \]
Where:
Applications:
Z-Spread (Zero-Volatility Spread)
A measure of the yield spread over the risk-free yield curve, reflecting additional compensation for risk in fixed-income securities.
Formula:
\[ P = \sum_{t=1}^{T} \frac{C_t}{(1 + r_t + Z)^t} + \frac{F}{(1 + r_T + Z)^T} \]
Where:
Applications: