# THE LAYMAN’S QUANT LEXICON

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A

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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:

dXt = μdt + σdWt

Here,

- dXt denotes the change in the variable at time t,

- μ represents the drift coefficient,

- σ is the volatility, and

- dWt is the increment of a Wiener process or Brownian motion at time t.

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Actuarial Analysis

Actuarial analysis is the discipline that 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.

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Affine Differential Equations

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)

- dx(t) is the change in the process x at time t,

- a(x(t)) is a function representing the drift term, which is linear in x(t),

- b(x(t)) is the function associated with the diffusion or volatility term, linear in x(t), determining how the stochastic process x(t) reacts to random shocks or noise,

- dW(t) is the increment of the Wiener process or Brownian motion at time t.

Affine models often allow for analytical expressions for the price of derivatives, simplifying calculations and analyses.

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This refers to a refined method of executing financial trades with advanced mathematical models and algorithms steering the decision-making process, enabling a pace and precision unattainable by human traders. It’s the quintessence of technology meeting finance, where millions of transactions can be executed in fractions of a second, often aiming to capitalize on small price discrepancies across different trading platforms or moments in time.

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« Almost surely »

Almost surely is a term in probability theory, often applied in quantitative finance, to indicate that an event will occur with probability one. However, it doesn't guarantee the event's occurrence. In the context of stochastic processes or random variables, stating that an event happens almost surely means that the set of sample points for which the event does not occur has probability zero. In practical terms, it’s used to describe outcomes that are virtually certain in probabilistic models of financial markets and risk.

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Alternative Hypothesis

This hypothesis is contrary to the null hypothesis. It suggests there is a significant effect or difference present. The objective of hypothesis testing is to determine which hypothesis is supported by the sample data.

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American Option

An American option is a type of options contract that can be exercised at any time up to its expiration date. This is in contrast to a European option, which can only be exercised at expiration. The flexibility of exercise time can result in American options having a higher premium than their European counterparts.

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Analytical Method

A technique that provides exact solutions to mathematical problems using algebraic, calculus, and other mathematical theories. In finance, analytical methods are often used to derive closed-form solutions for option pricing, investment valuation, and other financial modeling where explicit mathematical expressions can be obtained.

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ANOVA Table

A table used in analysis of variance (ANOVA) that displays the sources of variation in a statistical model, their degrees of freedom, sum of squares, mean squares, and the F-statistic. It is employed to assess the significance of different factors or variables in explaining the variation observed in a dataset.

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Arbitrage

Arbitrage involves the simultaneous purchase and sale of an asset to profit from an imbalance in price. It is a trade that profits by exploiting the price differences of identical or similar financial instruments on different markets or in different forms.

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Argmin function

In quantitative finance, as well as in mathematics and computer science, "argmin" refers to the argument of the minimum. Specifically, it is used to denote the input, or argument, at which a given function attains its minimum value. The "argmin" function is expressed mathematically as follows:

argminx∈Sf(x)argmin  x∈S  f(x). This expression is read as "the argument X over the set SS that minimizes the function f(x) where  f(x) is a function of the variable X, and S is the set over which the minimization is performed. The output of the "argmin" function is the value of X for which f(x) is minimized.

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ARIMA (AutoRegressive Integrated Moving Average)

A popular statistical method for time series forecasting. ARIMA models capture different aspects of temporal structure in data, including trends, seasonality, and noise, to make future predictions.

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Asset Allocation

Asset allocation is the practice of spreading investments among various asset categories, such as bonds, stocks, and cash equivalents, to maximize return for a given level of risk, based on an individual's or institution’s financial goals and risk tolerance.

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In quantitative finance, an Asset Swap Spread is a measure used to evaluate the relative value of a fixed-income security, typically a corporate bond or asset-backed security, in comparison to a risk-free government bond. It represents the additional yield or spread that an investor can earn by exchanging the fixed-income security for a package of assets that includes the security itself and interest rate swaps.

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Asset-Backed Securities (ABS)

Asset-Backed Securities are financial securities whose income payments, and hence value, are derived from and collateralized (or "backed") by a specified pool of underlying assets which are often illiquid and can't be sold individually.

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Asian Options

A type of exotic option whose payoff is determined by the average price of the underlying asset over a specified time period, rather than the price at the option's expiration. Two common types are the Asian call option, where the payoff is based on the positive difference between the average asset price and the strike price, and the Asian put option, where the payoff is derived from the positive difference between the strike price and the average asset price. These options are known for reducing price volatility and typically cost less than their European or American counterparts.

Payoff (Asian Call) = max(0, (1/T * ∑St from t=1 to T) - K)

Payoff (Asian Put) = max(0, K - (1/T * ∑St from t=1 to T))

Here:

- T is the total number of observation points,

- St is the asset price at time t,

- K is the strike price.

These formulas give the average price of the underlying asset over a series of specified times, which is then compared to the strike price to determine the option’s payoff.

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Asymmetric Risk

Asymmetric risk refers to the scenario where the potential upside and downside of an investment opportunity are not equal. It quantifies a situation where the prospects of gains and losses are misaligned, typically resulting in a risk-reward profile that is skewed to either the upside or downside.

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Automated Valuation Model (AVM)

Automated Valuation Model is a technology-based system that utilizes mathematical modeling to value properties, drawing on a database of recent property sales, property characteristics, and price trends.

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Autoregressive Models

Time series models where the value of a variable at a certain time is a linear function of its previous values.

B

Basis Point

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.

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Basis Risk

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.

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Barrier Option

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.

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Bayesian inference

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. Bayesian inference is an important technique in statistics, and especially in mathematical statistics.

P(A|B) = (P(B|A) * P(A)) / P(B) Where: - P(A|B) is the probability of A occurring, given that B has occurred (posterior probability). - P(B|A) is the probability of B occurring, given that A has occurred (likelihood). - P(A) is the initial probability of A occurring (prior probability). - P(B) is the total probability of B occurring (evidence).

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Bermudan Option

An option that can be exercised at several predetermined times, offering flexibility in execution.

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Beta

Beta is a measure of a security's or portfolio's systematic risk or volatility in comparison to the overall market or a specific benchmark index. A beta greater than 1 indicates higher volatility than the market, while a beta less than 1 signifies lower volatility. It is a critical component in the Capital Asset Pricing Model (CAPM) to estimate the expected return of an asset based on its beta and expected market returns.

Beta = Covariance(Return of Asset, Return of Market) / Variance(Return of Market)

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Binomial Distribution

The Binomial Distribution describes the probability of a fixed number of successful outcomes in a fixed number of independent Bernoulli trials. It's often used in options pricing and risk management.

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Binomial Option Pricing Model

This model offers a computational approach to valuing options by dividing the option’s life into a finite number of time intervals and computing possible prices at each stage. It is versatile and accommodates various types of options, including American options, and considers the probability of price movements at each interval to derive the option's overall value.

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Binomial Tree

A graphical representation used in finance to model the possible evolution of an asset's price over time, dividing time into discrete intervals. Each node represents a possible price of the asset at a given point in time, allowing for the valuation of financial derivatives, including options.

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Black-Litterman Model

Combines subjective views with market returns to form an updated return distribution for portfolio optimization.

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Black-Scholes Model

The Black-Scholes Model is a mathematical framework for pricing European-style options and calculating the theoretical value of a financial derivative based on the asset’s current price, expected volatility, time until expiration, and risk-free interest rate. It serves as a foundational element in the field of financial engineering for option pricing and risk management.

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Boltzmann equation

The classical Boltzmann equation is represented as:

∂f/∂t + v • ∇f + F • ∇vf = Q(f,f)

Here:

- f(x, v, t) is the distribution function, depending on position x, velocity v, and time t.

- v is the velocity of the particle.

- F is the force acting on the particle.

- Q(f,f) is the collision term representing the rate of change of f due to particle collisions.

In a financial context, while the Boltzmann equation isn't directly applied, a similar approach could hypothetically be used to model the evolution of asset prices influenced by various market factors. The equation would need to be adapted to reflect financial variables and forces, aiming to describe how the distribution of asset prices changes over time.

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Bond Yield

Bond yield refers to the return an investor realizes on a bond. It is often articulated as an annual percentage, reflecting the interest or dividends collected by the investor. There are various measures of bond yield, including current yield, yield to maturity, and yield to call, each offering insights into the bond's anticipated performance under different scenarios and assumptions.

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Borel Measurable Function

A function associated with the real numbers that is measurable with respect to the Borel sigma-algebra. In simpler terms, it is a function for which you can measure the probability of all its possible outcomes, making it particularly crucial in probability theory and mathematical statistics. These functions ensure the viability of integration and measure, underpinning advanced concepts like Lebesgue integration.

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Bootstrapping

Refer to building a yield curve by sequentially estimating zero-coupon yields, ensuring each bond's price matches its theoretical price. It’s also used to assess the accuracy of sample estimates by generating multiple simulated samples.

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Butterfly

In finance, a butterfly refers to a neutral option trading strategy that involves buying and selling options of the same type (calls or puts) with the same expiration date but different strike prices. The strategy aims to earn a profit from minimal price movement in the underlying asset. It consists of three legs: purchasing one lower strike option (in-the-money), selling two middle strike options (at-the-money), and buying one higher strike option (out-of-the-money).

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Butterfly Volatility

Butterfly volatility refers to a measure that illustrates the convexity or "smile" effect of the implied volatility skew for options at different strike prices. It's often calculated as the sum of the implied volatilities of out-of-the-money and in-the-money options, minus twice the implied volatility of at-the-money options for a particular expiration. Butterfly volatility helps traders and analysts gauge the market's expectation of future price uncertainty and the potential for extreme price movements.

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Buy side refers to firms and institutions that purchase investment services, including mutual funds, pension funds, and insurance companies. These entities are involved in the acquisition of investment assets and securities. The buy side's primary focus is on research and analysis to make informed investment decisions, aiming to maximize returns for their clients or their own portfolios. They often work with sell side firms that offer advisory, underwriting, and brokerage services, but buy side entities make the final investment decisions.

C

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Call Option

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.

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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.

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Chain Rule of Differentiation

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.

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Cholesky decomposition

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 in the following format:

A = L * L^T

Where:

- A is the original positive definite matrix.

- L is the lower triangular matrix.

- L^T is the transpose of L.

The Cholesky decomposition is valuable for various applications, including solving systems of linear equations, generating correlated random variables, and performing numerical optimizations efficiently. It is particularly useful when working with symmetric and positive definite matrices.

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CIR Model

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. The model ensures that interest rates remain non-negative, addressing a limitation in earlier models like Vasicek’s. The CIR model is often applied in the fields of finance and economics, especially in the risk management and derivative pricing contexts.

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Closed-Form Solution

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.

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Conditional Value at Risk (CVaR)

A risk assessment measure that quantifies the expected losses occurring beyond the Value at Risk (VaR) threshold level.

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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.

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Concatenation

In finance, "concatenation" typically refers to the process of combining or linking financial data, such as strings of numbers or text, together. It can be used to create complex formulas or data structures. For example, you might concatenate the values of several cells in a spreadsheet to create a single string of data for analysis or reporting purposes. Concatenation is often used in financial modeling and data analysis to manipulate and organize financial information.

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Correlation Coefficient

The correlation coefficient is a statistical measure that quantifies the degree and direction of the linear relationship between two quantitative variables. In finance, it’s often used to measure the correlation between asset returns, with values ranging from -1 (perfect negative correlation) to +1 (perfect positive correlation).

r = [Σ(xi - X̄)(yi - Ȳ)] / sqrt([Σ(xi - X̄)^2][Σ(yi - Ȳ)^2])

Here:

r is the correlation coefficient

xi, yi are individual data points

X̄ is the mean of all X data points

Ȳ is the mean of all Y data points

Σ denotes the sum

The numerator represents the covariance of the variables, and the denominator is the product of their standard deviations. The correlation coefficient ranges from -1 (perfect negative correlation) to 1 (perfect positive correlation). A correlation of 0 indicates no linear relationship between the variables.

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Covered Call

A covered call is an options strategy where an investor holds a long position in an asset and sells or "writes" call options on that same asset to generate additional income. This strategy is employed to enhance returns on the underlying asset, though it caps the upside potential to the strike price of the written option.

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Commodity Futures

Commodity futures are standardized contracts that obligate the buyer to purchase, and the seller to sell, a specific quantity of a commodity at a predetermined price on a specified date. They are employed for speculation, hedging, and price discovery purposes on commodities like metals, energy, and agricultural products.

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Counterparty Risk

Counterparty risk, or default risk, pertains to the likelihood that one of the parties involved in a financial transaction may fail to fulfill their contractual obligations. It is a pivotal consideration in over-the-counter (OTC) transactions, including derivatives trading, where no centralized exchange exists to guarantee trade execution.

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Conditional Expectation

It refers to the expected value of a random variable given the value of another variable. In finance, it is often used in the context of stochastic processes and time series analysis to predict future values based on known or observed values.

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Continuous Compounding

The process where interest earned on an asset is calculated and reinvested continuously, leading to exponential growth. It is expressed mathematically as A = P e^(rt), where (A) is the amount of money accumulated after (n) years, including interest.

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Copula

It's a function that joins or "couples" multiple univariate distributions to form a multivariate distribution. In finance, copulas are used to analyze and model the dependency structure among various assets or factors, playing a crucial role in areas like portfolio optimization, risk management, and derivative pricing.

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Corridor Variance Swap

A type of variance swap, a volatility hedge instrument, where the realized variance is measured only when the underlying asset's price is within a certain range (corridor).

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Cointegration

A statistical property of time series variables that share a long-term equilibrium relationship despite being non-stationary individually. In finance, cointegration is essential for pairs trading and other arbitrage strategies, as it helps identify assets that move together, enabling profit from temporary price dislocations.

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Covariance

Covariance measures the joint variability of two random variables. It indicates the direction of the linear relationship between the variables.

The formula for calculating the covariance between two variables X and Y is:

cov(X, Y) = Σ (xi - x̄) * (yi - ȳ) / (n - 1)

Here:

xi and yi are individual data points of variables X and Y, respectively.

x̄ is the mean of X and ȳ is the mean of Y.

n is the total number of data points.

Σ denotes the sum of the products of the differences between each data point and their respective means.

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CPD (Cumulative Probability Distribution)

The CPD gives the probability that a random variable is less than or equal to a particular value. It's used to describe the distribution of random variables in statistical modeling and is integral for risk management, option pricing, and various other applications in finance.

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Credit Risk

Credit risk refers to the risk of loss resulting from the failure of a borrower or counterparty to honor their financial obligations, primarily in lending, investment, and hedging contexts. Managing credit risk involves assessing the creditworthiness of borrowers and counterparties and mitigating exposure through diversification, collateral requirements, and other mechanisms.

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Cross-Sectional Analysis

Analyzing and comparing different assets at a single point in time, often used to identify the factors affecting asset prices.

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Cubic Spline Interpolation

A type of interpolation where a series of piecewise third-order polynomials is used to interpolate a curve through a set of points. In finance, cubic spline interpolation can be used for yield curve smoothing, option pricing, and other applications where an accurate estimation between data points is needed.

D

Degree of Freedom

In statistics, the term "degree of freedom" refers to the number of independent values or quantities which can vary in the calculation of a statistic or a statistical test. In finance, it's essential in various contexts like hypothesis testing, confidence intervals, and regression analysis. Degrees of freedom are used to estimate population parameters, and they influence the shape of distributions like the chi-square and t-distribution, which are often used in hypothesis testing to account for sample variability.

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Delta

Delta is a key concept in the derivatives and options trading space, representing the rate of change in the price of an option relative to a one-unit change in the price of the underlying asset. It quantifies the sensitivity and is used in risk management, valuation models, and trading strategies.

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Delta Hedging

A risk management strategy used in options trading to minimize or eliminate the directional risk (price movement risk) of a portfolio of options or other derivatives. It involves adjusting the position in the underlying asset (usually stocks or futures) to offset the price changes in the options or derivatives. The goal of delta hedging is to maintain a neutral or zero delta, which means that changes in the underlying asset's price will have minimal impact on the overall portfolio value. Delta represents the sensitivity of an option's price to changes in the underlying asset's price.

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25-Delta

25-delta refers to an options term that describes an option that has a 25% probability of expiring in-the-money. In the context of foreign exchange (forex), a 25-delta call option implies a 25% probability that the option will finish in-the-money, meaning it will have intrinsic value at expiration. Similarly, a 25-delta put option has a 25% chance of finishing in-the-money. This term is often used in the context of risk reversals and butterfly spreads to gauge market sentiment and volatility.

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Default Probability

In credit risk modeling, default probability is a critical metric. It quantifies the likelihood of a borrower defaulting on their debt obligations within a given time frame. It is often modeled using statistical and mathematical tools and is integral to the pricing of credit derivatives like credit default swaps.

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Derman-Kani Model

This is a quantitative tool used in financial engineering for option pricing. It extends the Black-Scholes Model to incorporate implied volatility surfaces, enabling the more accurate pricing of derivative products.

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Dirac measure

A Dirac measure is a type of measure that is uniquely determined by a specific point in a given space, assigning all "mass" or measure to that particular point. In mathematical terms, if you have a set A, the Dirac measure δx(A) equals 1 if the specified point x is in A and 0 if x is not in A. This concept is commonly used in probability theory to represent a distribution where a random variable is certain to take one specific value.

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Discrete-Time Model

In quantitative finance, a discrete-time model is used for financial modeling where the system transitions from one state to another in distinct, non-overlapping intervals. This is in contrast to continuous-time models where state transitions are continuous and instantaneous.

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Down-and-Out Option

A type of barrier option that becomes null and void when the price of the underlying asset falls below a certain level. This structure is often used in quantitative finance to create option contracts that are tailored to specific risk-return requirements.

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Dual-Beta Model

The Dual-Beta Model accounts for both the market and downside risk by estimating two separate beta coefficients. It differentiates between up and down market conditions, offering a more nuanced understanding of an asset's or portfolio's risk and performance characteristics.

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Dynamic Replication

Dynamic replication refers to a strategy used in financial engineering to hedge the risk of a derivative or option by dynamically adjusting a mixed portfolio of the underlying asset and risk-free investments to replicate the derivative’s payoffs.

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Derivatives

Financial securities whose value is derived from underlying assets like stocks, bonds, or commodities.

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Derivative Pricing

This refers to the complex process of determining the fair market price of a derivative instrument. It involves mathematical models and computational techniques, taking into account various factors like the price of the underlying asset, volatility, time to expiration, and interest rates.

These terms are entrenched in the specialized field of quantitative finance, reflecting a range of complex concepts and strategies used by professionals in the industry.

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Dependent Variable

A variable often denoted as y, its value depends on the independent variable(s). In statistical modeling, it's the outcome or response variable being studied.

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Deterministic Term

In stochastic differential equations and financial modeling, the deterministic term is the predictable, non-random part of the equation. It can represent expected trends or drifts in financial data, unaffected by random fluctuations.

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Differentiation

The process of determining the rate at which a quantity changes. In stochastic calculus, differentiation becomes more nuanced to accommodate the inherent randomness and variability in financial markets.

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Diffusion Term

This term is associated with the stochastic or random component in a differential equation used for modeling financial or physical systems. In the financial context, it typically accounts for random asset price fluctuations and is often linked to Brownian motion or similar stochastic processes.

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Digital Option

A type of option where the payoff is fixed after the underlying stock exceeds the predetermined threshold or strike price. It is valued using complex quantitative methods and is often part of structured financial products.

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Discretization

In quantitative finance, discretization refers to the process of converting continuous models or variables into a discrete form. This involves breaking down continuous data, functions, or equations into discrete, finite sets that can be analyzed or computed numerically. For example, a continuous-time stochastic process, like Geometric Brownian Motion (GBM) used to model stock prices, is often discretized for numerical simulations.

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Drift Term

In stochastic calculus and finance, the drift term refers to the deterministic trend that affects the expected value of a stochastic process or financial model over time. It represents the consistent and predictable change, contrasting with the random fluctuations introduced by the diffusion term.

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Duration Gap

A measure used in interest rate risk management that quantifies the difference in duration between assets and liabilities in a bank’s or financial institution’s balance sheet. It helps in assessing the sensitivity of the value of a portfolio to changes in interest rates.

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Dynamic Hedging

Dynamic hedging is a risk management strategy where the hedging activities are continuously adjusted, often through the use of derivatives, to counteract changes in the value of an underlying asset. It’s particularly common in the trading of options and other complex financial instruments.

E

Econometrics

Econometrics combines statistical methods with economic theory to estimate and test economic relationships. It involves the application of statistical techniques to analyze financial and economic data, assisting in decision-making and forecasting future trends and patterns.

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Efficient Market Hypothesis (EMH)

The theory that asset prices fully reflect all available information.

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Eigenvalue

In linear algebra, an eigenvalue is a scalar associated with a linear system of equations, specifically square matrices. It indicates the factor by which an eigenvector is scaled during the transformation described by the matrix. In finance, they are crucial in portfolio optimization, stress testing, and risk modeling, offering insights into the volatility and stability of financial systems.

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Eigenvector

An eigenvector is a non-zero vector that only changes by a scalar factor when a linear transformation is applied to it, meaning it retains its direction. It's associated with a specific eigenvalue which indicates the magnitude of this scaling. In finance, eigenvectors are used in various applications including risk analysis, portfolio optimization, and factor analysis to identify principal components or key drivers of asset returns and volatilities. They are essential in understanding the underlying structures and correlations in financial data.

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Elliptical distributions

Elliptical distributions are a family of probability distributions extending the multivariate normal distribution, offering flexibility in modeling various shapes of distribution, including "fat tails" and skewness. They are pivotal in finance, especially in portfolio optimization, risk management, and derivative pricing, due to their ability to model dependencies between multiple asset returns while maintaining mathematical tractability. Examples include the multivariate normal and Student’s t-distributions. These distributions provide insights into tail risks, facilitating nuanced risk assessments and efficient computations in the complex financial landscape.

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Entropy

In the context of information theory, entropy measures the uncertainty associated with random variables. In finance, it can be applied to quantify the unpredictability or the information content associated with price movements or the returns on an asset.

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Entropy Pooling

The process employs the principles of information theory, particularly the concept of cross-entropy, to measure the dissimilarity between probability distributions. The goal is to find a consensus distribution that is as close as possible to the original distributions, subject to the constraints that represent the information or beliefs to be pooled.

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Euler Method

The Euler method is a numerical technique for solving ordinary differential equations (ODEs) with a given initial value. It is named after Leonhard Euler who introduced the method.

In this method, the continuous differential equation is approximated by a discrete difference equation.

Consider a general ODE:

dy/dt = f(t, y),  y(t0) = y0

The Euler method updates the solution of the ODE in discrete steps as follows:

y(n+1) = y(n) + f(tn, y(n)) Δt

Here, Δt is the step size, y(n) is the approximation of the solution at time tn, and f(tn, y(n)) is the derivative of y at tn, providing the slope of the solution curve at tn.

In the stochastic context, for instance, the Geometric Brownian Motion (GBM), described by the SDE:

dSt = μSt dt + σSt dWt can be discretized using the Euler method as:

S(t+1) = St + μSt Δt + σSt ΔWt

Here, St is the asset price, μ is the expected return, σ is the volatility, and ΔWt is the increment of the Wiener process over the interval Δt.

This method offers an easy-to-implement and computationally efficient way to approximate solutions to differential equations, making it a popular choice in various fields, including quantitative finance.

However, it might not be the best choice for highly volatile or path-dependent options due to potential inaccuracies stemming from the linear approximation and fixed time step.

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Exotic Option

A type of option that is more complex than commonly traded options, having features making it more complex to value. These features can include choices where payoffs depend on multiple underlying assets, for example.

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Exotic Option Hedging

Exotic option hedging involves the implementation of strategies to mitigate risks associated with trading complex financial derivatives known as exotic options. These strategies can include the use of standard options, dynamic trading, and other financial derivatives to counter the unique risks presented by exotic options. Advanced mathematical models and computer algorithms are often employed for effective risk management. Path Dependency in Exotic Option Hedging

Because of the path dependency features inherent in many exotic options, hedging strategies must consider not only the current price of the underlying asset but also its historical prices. Path-dependent options, like Asian or Barrier options, have payoffs that depend on the entire price path of the underlying asset. Hedging these requires continuous monitoring and adjustments to the hedge positions as the price of the underlying asset evolves over time. This often involves complex mathematical modeling and risk management techniques to effectively mitigate the risk of adverse price movements and ensure profitability.

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Exponential Distribution

The Exponential Distribution models the time between events in a Poisson process, making it suitable for modeling waiting times or durations in finance, like the time between trades.

F

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Fama-MacBeth Regression

The Fama-MacBeth Regression is a two-step method used in finance to analyze the relationship between asset returns and predictive variables. Initially, individual assets are regressed on predictive variables through a time-series regression. Then, the obtained coefficients are analyzed across all assets using a cross-sectional regression to derive average risk premiums for each variable. This approach is crucial for testing asset pricing models, informing portfolio construction, and developing investment strategies. It's valued for its efficiency with large datasets and ability to mitigate multi-collinearity, though it assumes linear relationships and requires extensive data preparation.

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Fat Tails

In statistics and finance, "fat tails" refer to the presence of extreme and infrequent events or outliers in a probability distribution or dataset. These tails represent the likelihood of rare and significant deviations from the mean, which can have a disproportionate impact on risk assessments and financial markets. Understanding fat tails is crucial in risk management because traditional models often assume normal (Gaussian) distributions, which underestimate the potential for extreme events. Fat tails are associated with higher volatility and the potential for unexpected market crashes or large losses.

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Feynman-Kac Formula

A representation that connects partial differential equations (PDEs) with stochastic processes. In finance, it's instrumental in connecting the Black-Scholes PDE with the stochastic differential equation of geometric Brownian motion, facilitating option pricing.

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Filtration (in Probability Theory)

In probability theory and stochastic processes, a Filtration is a sequence of increasing σ-algebras {𝓕_t} over a set of events Ω, typically associated with a random process or stochastic system indexed by time t. Mathematically, a Filtration is defined as:

- {𝓕_t, t ≥ 0}

Where:

- 𝓕_t is a σ-algebra representing the set of events observed or known up to time t.

- t is a non-negative real number or a discrete time index.

A Filtration is used to formalize the notion of "information available up to a certain time" in the context of stochastic processes. It describes the evolution of information as time progresses and plays a fundamental role in understanding conditional expectations, martingales, and other stochastic concepts in probability and mathematical finance.

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Financial Engineering

Application of mathematical methods to solve problems in finance.

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First Difference

A transformation of a time series dataset where the difference between consecutive data points is computed. It is commonly used to stabilize the mean of a time series by removing seasonality and trends, making the series stationary.

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Finite Difference Method

The Finite Difference Method (FDM) is a popular technique used to approximate solutions to differential equations, particularly useful in solving partial differential equations in various fields including quantitative finance, like the Black-Scholes equation for option pricing.

In FDM, derivatives are approximated using finite differences. For instance, the first derivative approximation using a forward difference is calculated as:

f'(x) ≈ (f(x + Δx) - f(x)) / Δx

Here, Δx is a small increment in x. Backward and central differences are other schemes used to estimate derivatives.

The second derivative can be approximated using a central difference as:

f''(x) ≈ (f(x + Δx) - 2f(x) + f(x - Δx)) / (Δx)^2

In the context of the Black-Scholes equation, the FDM transforms the continuous problem into a discrete problem by discretizing the time and asset price movements. Finite differences are then used to approximate the derivatives, transforming the partial differential equation into a system of algebraic equations that can be solved iteratively.

It's essential to handle challenges related to stability and convergence to ensure that the solutions obtained are accurate and reliable.

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Fleming–Viot process

In the context of finance, the Fleming–Viot process isn't a standard or commonly referenced model. However, if we are to consider its potential applications, it could theoretically be used to model the random fluctuations of financial instruments or assets within a particular market.

This stochastic process could help to understand how the composition of assets in a market changes over time, similar to how it models the changes in allele frequencies in a population in biology. It could account for various random factors affecting the assets, including their "birth" and "death" analogous to buying and selling, mutations equivalent to sudden changes in value, and selection similar to preference trends among investors.

It would be an advanced and complex method to study financial markets, likely involving a significant adaptation of the original process to make it relevant and applicable to financial data and scenarios.

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Future

A standardized financial contract obligating the buyer to purchase, and the seller to sell, a specific quantity of an underlying asset at a predetermined price on a specified future date. Futures are traded on organized exchanges and are used for hedging, speculation, and arbitrage.

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Fokker-Planck Equation (or Kolmogorov Forward Equation)

The Fokker-Planck equation describes the evolution of the probability density function of a stochastic process. It's particularly useful in physics and mathematics to predict how systems influenced by random forces change over time.

In simpler terms, the equation helps to understand how the distribution of a particle's position changes over time due to both systematic forces (drift) and random forces (diffusion).

The general form of the Fokker-Planck equation can be written as:

∂P/∂t = -∂(AP)/∂x + ∂²(BP)/∂x²

The equation consists of two main parts:

1. The drift term: -∂(AP)/∂x represents the deterministic movement of particles due to systematic forces.

2. The diffusion term: ∂²(BP)/∂x² represents the spread of particles due to random forces.

Here:

- P is the probability density function of the particle’s position.

- A(x, t) represents the systematic, deterministic forces causing the particle to drift in a particular direction.

-The diffusion term B(x,t) represents random motion and is often given by the formula B(x,t) = kT/γ. Here, k is the Boltzmann constant, T is the temperature, and γ is the friction coefficient. This term is integral in the Fokker-Planck equation to model the influence of random forces on the system being studied.

- t is time, and x is the position of the particle.

The Fokker-Planck equation is widely used in many areas of physics, including statistical mechanics and quantum mechanics, as well as in financial mathematics for modeling the time evolution of complex systems influenced by both deterministic and stochastic forces.

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Fourier transform

The Fourier transform is a mathematical operation that converts a signal or function of time (or space) into a function of frequency. It helps in analyzing the frequency content of the signal. The Fourier transform of a function f(x) is given by the integral:

F(ω) = ∫[from -∞ to ∞] f(x) * exp(-iωx) dx

Here:

- F(ω) is the Fourier transform of f(x),

- f(x) is the original signal or function,

- exp(-iωx) is the complex exponential function, which represents the frequencies,

- i is the imaginary unit,

- ω is the angular frequency,

- dx indicates integration with respect to x.

The Fourier transform is used in finance for options pricing, time series analysis, and risk management. It helps in analyzing complex financial data by converting it into the frequency domain, making it easier to identify patterns, analyze risk, and price complex options. The Fast Fourier Transform (FFT) is particularly useful for efficiently handling large datasets.

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Forward

A customized, over-the-counter financial contract where two parties agree to buy or sell an asset at a specified future date for a price that is determined today. Unlike futures, forward contracts are not traded on exchanges and can be tailored to meet the specific requirements of the parties involved.

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FRA (Forward Rate Agreement)

An FRA is a financial derivative instrument in which two parties agree on an interest rate for a specified period starting on a future date. It serves as a hedge against interest rate changes and is used by corporations and banks to manage interest rate risk.

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Fractal

A fractal is a complex structure showing self-similarity, meaning its small details, visible at any scale, mirror the overall shape. In finance, fractals indicate recurring patterns in asset price movements and are used in technical analysis to anticipate future price directions.

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Fractional Differentiation

A generalization of ordinary differentiation where the order of differentiation can be a non-integer, allowing for more complex and nuanced behaviors. In finance, fractional differentiation can be used to transform non-stationary time series data into stationary while preserving memory, which is crucial for modeling and forecasting financial time series that exhibit long-term dependencies.

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F-Test

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.

G

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Gamma

The rate of change in an option's delta for a one-unit change in the price of the underlying asset.

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Gamma Function

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 it's defined by an integral expression with properties and behaviors similar to those of the factorial operation. In quantitative finance, it might be used in analytical expressions and calculations related to option pricing and risk management.

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GARCH Model (Generalized Autoregressive Conditional Heteroskedasticity)

Used in quantitative finance to model financial time series data's changing volatility over time. It is particularly useful for forecasting future volatility and is used extensively in financial market risk management.

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Gaussian Process

A type of stochastic process where every finite collection of those random variables has a multivariate normal distribution. It’s characterized by its mean function and covariance function. In finance, Gaussian processes are used for tasks like optimization, regression, and machine learning, offering a flexible and efficient method to model unknown functions, capture uncertainty, and make predictions.

https://www.finance-tutoring.fr/the-gaussian-process-regression-in-layman’s-terms/

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Geometric Brownian Motion (GBM)

A continuous-time stochastic process used to model variables that follow a lognormal distribution, such as stock prices. It ensures that prices remain positive and is integral in option pricing models like Black-Scholes.

The formula for GBM is:

dSt = μSt dt + σSt dWt

Here,

- dSt represents the change in stock price at time t,

- μ is the expected return,

- σ signifies the volatility, and

- dWt is the increment in the Wiener process or Brownian motion at time t.

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Greeks

Measures of the sensitivity of an option’s price to various factors including changes in the price of the underlying asset, interest rates, volatility, and time decay.

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Girsanov Theorem

A principle in the theory of stochastic processes that describes how the probability measure changes when a new drift term is added to a stochastic differential equation. It's crucial in finance for changing from the physical measure to the risk-neutral measure, aiding in the pricing of derivatives.

H

Heston Model

A mathematical model describing the evolution of volatility over time. The Heston model is characterized by two stochastic differential equations:

1. dS = µS dt + √v S dW1

2. dv = κ(θ - v) dt + σ √v dW2

Here:

- S is the asset price

- v is the instantaneous variance

- µ is the expected asset return

- κ is the speed of reversion to the mean variance

- θ is the long-term mean variance

- σ is the volatility of volatility

- dW1 and dW2 are standard Wiener processes that are correlated with a coefficient ρ

In this model, the volatility is a variable, allowing it to change over time and making the Heston model a popular choice for pricing options due to its ability to capture the volatility smile.

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Heat Equation (in Quantitative Finance)

The Heat Equation in quantitative finance is a partial differential equation (PDE) used to describe the gradual diffusion or spreading of financial quantities over time. It is adapted from the heat diffusion equation in physics and is applied to represent how financial instrument prices or values evolve and spread.

In one dimension, the Heat Equation is expressed as:

∂u/∂t = α * ∂²u/∂x²

Where:

- ∂u/∂t represents the rate of change of the financial quantity with respect to time (e.g., the change in the price of a financial asset).

- ∂²u/∂x² denotes the second spatial derivative, modeling the diffusion or dispersion effect.

- α is the diffusion coefficient, which quantifies the level of randomness or volatility in the financial process.

In quantitative finance, the Heat Equation is a fundamental mathematical tool for understanding how financial information and pricing changes diffuse through markets. It provides insights into how securities' prices or values spread and converge, making it valuable for option pricing, risk management, and the study of market dynamics.

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Hedging Portfolio (in the Black-Scholes SDE)

A Hedging Portfolio, within the context of the Black-Scholes stochastic differential equation (SDE), is a strategically constructed combination of the underlying asset (S), cash (C), and an option position (V) designed to replicate the behavior of the option. This mathematical construct helps maintain a risk-neutral position by ensuring that changes in the option's value are offset by corresponding adjustments in the value of the portfolio. The portfolio's delta Delta_t  is pivotal, representing the sensitivity of the option's value to changes in the underlying asset price. Hedging portfolios play a fundamental role in option pricing and risk management, enabling market participants to efficiently manage risk associated with options positions.

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Heath-Jarrow-Morton (HJM) Model

Models the evolution of interest rate curves, particularly forward rate curves, essential in pricing interest rate derivatives.

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Heteroskedasticity

It's the opposite of homoskedasticity, indicating a scenario where the variance of errors differs across various levels of the explanatory variables. In such cases, the spread or dispersion of the data varies, becoming either narrower or wider at different data levels.

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Homoskedasticity

This refers to the assumption that the variance of the errors is constant across all levels of the explanatory variables. In regression analysis, it implies that the data's spread remains consistent across the entire range of data.

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Hurst Exponent

Used to quantify the long-term memory of time series, helpful in diagnosing whether a financial time series is trending, mean-reverting, or a random walk.

I

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Implied Correlation

A measure inferred from the prices of options on multiple assets, reflecting the market’s expectation of the correlation between those assets.

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Implicit Volatility Surface

A three-dimensional plot featuring strike price, time-to-maturity, and implied volatility, often used in the derivatives market.

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Implied Volatility

A measure of how much the markets expect the price of an asset to move, implied from option prices.

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Independent Variable

A variable often denoted as x, used in statistical modeling to predict or explain variations in the dependent variable.

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Integrand

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 a fundamental concept in calculus and is heavily used in various areas of quantitative finance like derivative pricing, risk management, and investment strategy optimization. The integrand, together with the limits of integration, defines the integral, which represents the accumulated total or area under the curve of the integrand function over a specified interval.

Integration

Integration in quantitative finance refers to a mathematical process used to calculate the accumulated value of a financial quantity, such as price or rate of return, over a specified interval or time period. It's a fundamental tool in calculus, employed extensively in various areas of finance including the pricing of derivative securities, risk management, and investment analysis. Integration helps in understanding how financial quantities change and accumulate over time, aiding in informed decision-making.

∫_a^b f(x) dx

This represents the total accumulated value of the function f(x) from x=a to x=b.

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Interest Rate Swap

A contract to exchange fixed or floating rate interest payments.

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Interpolation

A mathematical and statistical technique used to estimate unknown values between two known values in a data set. In finance, interpolation is often used to estimate a curve that fits into a set of data points, which can be useful for predicting and understanding trends, valuations, and pricing financial instruments.

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Ito's Lemma

A fundamental concept in stochastic calculus that provides a formula to determine the differential of a function of a stochastic process, particularly Brownian motion. It’s essential for option pricing and understanding complex financial derivatives.

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Ito Calculus Multiplication Rules

- dW_t^2 = dt : The square of the increment of a standard Brownian motion over a small time interval equals the time increment. This is a key result that distinguishes Brownian motion from regular deterministic processes.

- dt^2 = 0: The square of an infinitesimal time increment is negligible.

- dW_t dt = 0: The product of the increment of Brownian motion and the time increment is negligible.

These rules are essential for manipulating stochastic differential equations and deriving results like Ito's lemma. They highlight the non-intuitive nature of stochastic processes and the care that must be taken when dealing with such equations in quantitative finance and related fields.

J

Jensen's Inequality

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 φ, Jensen's Inequality is expressed as:

φ(E[X]) ≤ E[φ(X)]

and for a concave function, it becomes

φ(E[X]) ≥ E[φ(X)]

Here, E[X] is the expected value of the random variable X. This inequality is pivotal in areas of finance like portfolio theory and risk management. In finance, this inequality is used in portfolio theory, option pricing, and risk management for deriving inequalities or approximations and establishing theoretical results involving expectations and variances.

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Jump Diffusion Model

Incorporates sudden and significant changes in asset prices, "jumping" from one level to another.

Jensen's Alpha: A risk-adjusted performance measure that represents the average return on a portfolio over and above that predicted by CAPM.

K

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Kalman Filter

A mathematical method to estimate the state of a linear dynamic system from a series of incomplete and noisy measurements.

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Kurtosis

A statistical measure used to describe the distribution of returns.

L

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Latent Variable Models

Statistical models that involve variables that are not directly observed but are rather inferred from other observed variables.

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Lebesgue Integral

This integral, introduced by Henri Lebesgue, surpasses the Riemann integral's limitations by focusing on "how much" a function takes on specific values rather than "where" those values are taken. It measures using horizontal slices instead of Riemann's vertical slices, allowing the integration of a broader class of functions.

The integral is integral to advanced topics like Ito’s calculus and the valuation of derivative securities under stochastic volatility models or jump-diffusion models, where payoffs or paths of underlying assets can be irregular or complex.

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Leverage

The use of various financial instruments or borrowed capital to increase the potential return of an investment.

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Lévy Flight

A stochastic process used to model random movement or displacement, particularly in the context of asset price movements in finance. In a Lévy flight, the steps taken are characterized by power-law-distributed step sizes, meaning there is a possibility of rare, large jumps in the process. This concept is named after French mathematician Paul Lévy. Lévy flights are used to capture the occasional extreme price movements or jumps observed in financial markets, providing a more accurate representation of asset price dynamics than traditional random walks or Brownian motion models.

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Lévy Process

A type of stochastic process that starts at zero, has stationary and independent increments, and is continuous in probability. It’s a generalization of the Poisson process and Brownian motion, often used to model the behavior of financial data.

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The LIBOR Market Model (LMM)

The LIBOR Market Model (LMM) is a financial model used for interest rate derivatives, in which each forward LIBOR rate is modeled as a log-normal process driven by a number of Brownian motions under the risk-neutral measure.

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Linear Algebra

A branch of mathematics concerning linear equations, linear functions, and their representations with matrices and vector spaces. In finance, linear algebra is essential for portfolio construction, risk assessment, and various other computations and optimizations involving large datasets.

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Liquidity

The ease at which an asset can be quickly bought or sold without affecting its price.

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Local Volatility Model

The local volatility model is essential in the field of quantitative finance for calculating the implied volatility of financial derivatives, especially options. Developed by Bruno Dupire, it addresses the limitations of the Black-Scholes model by considering that volatility changes over time and varies with the asset price.

In this model, the option's price depends on both time and the asset’s spot price. It is represented by the partial differential equation:

dF / dt + 0.5 * (localVol * S)^2 * d^2F / dS^2 + r * S * dF / dS - r * F = 0

Here, F is the option price, t is time, S is the spot price of the underlying asset, localVol is the local volatility (a function of both time and the asset price), and r is the risk-free interest rate.

The localVol term can be calculated from the market prices of vanilla options, making the model particularly adept at more accurately pricing complex, exotic, and path-dependent options.

While providing more precise option pricing and a detailed understanding of volatility, the model is computationally intensive but remains a crucial tool for pricing and risk management in modern finance.

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Logistic Regression

Used for binary classification problems; it estimates the probability that a given input point belongs to a certain class.

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Lognormal Distribution

The Lognormal Distribution is commonly used to model asset prices in quantitative finance. It's characterized by a positively skewed shape and is suitable for modeling phenomena where prices can only increase.

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Lookback Option

An exotic option that allows holders to "look back" over time to determine the option’s payoff, typically allowing the holder to choose the most favorable price at which to exercise the option.

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Log Return

The natural logarithm of the ratio of an asset’s ending price to its beginning price. Used in finance for its ease of mathematical manipulation and it allows returns to be additive.

M

Martingale

A type of stochastic process characterized by the property that the expected future value, given all past values, is equal to the current value. It suggests no advantage in predicting future movements based on past behavior, akin to a "fair game."

https://www.finance-tutoring.fr/bridging-martingale-markov-and-brownian-motion-in-layman’s-terms…/

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Markov Chain

A statistical model that predicts future states of a system based on the current state, without considering the events that occurred before the present state. Every state depends only on the previous state, embodying the Markov property. In finance, Markov chains are used in credit scoring, stock price modeling, and other applications where future states are probabilistically determined but dependent only on the current state. The model's transitions between states are represented by a matrix of probabilities, providing a stochastic yet structured approach to modeling dynamic systems.

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Market Microstructure

The detailed processes and mechanisms governing trading and price formation within financial markets, including rules, infrastructure, and the behavior of market participants.

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Markov Processes

A type of stochastic process that has the Markov property, meaning the probability distribution of future states depends only on the current state, not on the sequence of preceding states (melorylessness).

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Mean Reversion

A theory suggesting asset prices and returns eventually move back towards their mean or average levels.

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Moment Generating Function

This is a statistical function used to derive the moments (such as mean, variance, skewness, and kurtosis) of a probability distribution. It helps in characterizing the distribution and is defined as the expected value of the exponential of the random variable multiplied by a parameter.

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Monte Carlo Simulation

A problem-solving technique used to approximate the probability of certain outcomes by running multiple trial runs, called simulations, using random variables.

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Moran interaction

In finance, Moran's interaction is a tool used to analyze spatial patterns and correlations within financial data sets that have a geographical component. It helps in identifying clusters or dispersions in data, such as investment values, market performance, or asset prices across different locations, aiding in more informed decision-making.

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Multicollinearity

In statistics and econometrics, multicollinearity occurs when two or more independent variables in a multiple regression model are highly correlated, meaning one can be linearly predicted from the others with significant accuracy. In the context of finance, multicollinearity can make it difficult to isolate the effect of each individual variable, leading to unreliable and unstable estimates of regression coefficients. This condition complicates the interpretation and validation of statistical models used in financial analysis, forecasting, and decision-making. Addressing multicollinearity often involves techniques like variable selection, dimensionality reduction, or regularization methods.

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Multivariate

In statistics and econometrics, multivariate refers to techniques and analyses involving multiple variables or datasets simultaneously, often used to understand complex relationships among variables.

N

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Neural Network Architectures

Models like LSTM (Long Short-Term Memory) and GRU (Gated Recurrent Units) used in deep learning for tasks like time series forecasting

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Newton-Raphson Method

It's an iterative numerical method used for finding approximations to the roots (or zeros) of a real-valued function. In the context of finance, it's particularly useful for options pricing and other applications where closed-form solutions are unattainable or complicated. The formula for the method is expressed as x(n+1) = x(n) - f(x(n))/f'(x(n)), where x(n+1) is the next approximation, x(n) is the current approximation, f(x(n)) is the value of the function at x(n), and f'(x(n)) is the derivative at x(n). The iterations continue until the estimates reach the desired level of accuracy.

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Risk that arises when traders who are not fully rational influence prices and trade in the market.

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Null Hypothesis

The null hypothesis is a statement in statistical inference that there is no significant effect or difference, serving as the default or initial assumption. It is tested against the alternative hypothesis.

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Numeraire

In the context of finance and mathematical economics, the numeraire is a unit of account used for relative valuation of different assets or securities.

For instance, in the Black-Scholes-Merton model, the numeraire is often the risk-free asset, and all other assets’ values are expressed relative to this. In interest rate modeling, especially in LIBOR Market Model (LMM) and certain FX models, different numeraires can be selected to change the measure under which the model is evaluated, aiding in pricing derivatives by essentially making the derivative being priced a martingale under the chosen numeraire, thereby simplifying the pricing expression.

Moreover, in risk-neutral valuation, the concept of a numeraire is vital as it's closely related to the change of measure, which is a fundamental technique used to price derivatives by transferring the valuation to a risk-neutral world where all assets are expected to grow at the risk-free rate when valued against an appropriate numeraire.

https://www.finance-tutoring.fr/from-replicating-portfolios-to-numeraires-bridging-black-scholes-and-interest-rate-options-in-layman’s-terms…/

https://www.finance-tutoring.fr/2023/08/26/numéraire-and-change-of-numéraire-in-layman-s-terms/

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Numerical Method

A technique used to solve mathematical problems through numerical approximation, often employed when analytical solutions are impractical or impossible. In finance, numerical methods like finite difference, Monte Carlo simulation, and binomial trees are crucial for valuing financial derivatives, risk management, and other quantitative analyses.

https://www.finance-tutoring.fr/the-difference-between-numerical-and-analytical-methods-in-layman’s-terms…/

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No Arbitrage principle

A principle stating that it is impossible to have a risk-free profit in excess of the risk-free interest rate by exploiting price differences of similar or identical financial instruments on different markets or in different forms. No-arbitrage is a fundamental concept underpinning the pricing of financial derivatives and building financial models.

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No-Arbitrage Term Structure Models

A subset of term structure models that ensure the absence of arbitrage opportunities when pricing interest rate derivatives and fixed-income securities. By preventing risk-free profits from discrepancies in the model and market prices, these models ensure consistency with observed market prices and behaviors. The Heath-Jarrow-Morton (HJM) and the Libor Market Model (LMM) are prominent examples that operate under the no-arbitrage condition, ensuring the model-generated forward rate curves are in alignment with the current market term structure.

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Non-Differentiability

A mathematical property describing a function or curve that is not differentiable at one or more points within its domain. In calculus, a differentiable function is one for which a derivative (rate of change) exists at every point. When a function is non-differentiable, it means that the derivative is undefined or discontinuous at certain points, often due to sharp corners, cusps, or singularities in the function.

In finance and economics, non-differentiability may be relevant when dealing with discontinuous payoffs or situations where traditional mathematical models break down. Incorporating non-differentiability into the nature of stochastic processes can be done by considering specific types of stochastic processes that exhibit such properties.

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Nonlinear Time Series Analysis

A specialized area of time series analysis that deals with modeling and analyzing time-varying data where the relationships between variables are not linear. It employs nonlinear mathematical models to capture complex and often non-smooth patterns and behaviors in time series data, offering a more accurate representation of real-world phenomena. This field is crucial for understanding systems with intricate dynamics, including those found in finance, environmental science, and engineering.

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Normal Distribution (Gaussian Distribution)

The Normal Distribution is a symmetric probability distribution characterized by a bell-shaped curve. It's used in quantitative finance to model a wide range of naturally occurring phenomena and is central to concepts like the Central Limit Theorem.

O

Ocone Martingale

It refers to a martingale component constructed from a given semimartingale and its natural filtration, often employed in advanced financial mathematics for option pricing and hedging in incomplete markets. The Ocone Martingale is associated with the Clark-Ocone theorem, providing an explicit formula for the martingale representation of functionals of Brownian motion and other semimartingales.

https://www.finance-tutoring.fr/the-ocone-martingale-in-layman’s-terms…/

https://www.finance-tutoring.fr/the-origin-of-martingale-in-layman’s-terms…/

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Option

Derivative contract that give the holder the right, but not the obligation, to buy or sell an asset at a predetermined price before or at the expiration of the contract.

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Ornstein-Uhlenbeck Process

This process is known for modeling mean-reverting behavior. The mathematical representation is:

dX(t) = θ(μ - X(t))dt + σdW(t)

Here:

- dX(t) is the change in the process X at time t,

- θ (theta) is the rate of mean reversion,

- μ (mu) is the long-term mean that the process reverts to,

- X(t) is the value of the process at time t,

- σ (sigma) is the volatility, and

- dW(t) is the increment of a Wiener process or Brownian motion at time t.

It's widely used in financial modeling, especially for interest rates and volatility that have a tendency to revert to a mean value over time.

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Overfitting

Overfitting occurs when a statistical model is too complex, capturing noise in the training data instead of the underlying pattern. It performs well on the training data but poorly on new, unseen data because it's too tailored to the specific dataset and fails to generalize. Overfit models have excessive parameters relative to the amount of data available, leading to an overly flexible model. Avoiding overfitting involves techniques like cross-validation, regularization, and pruning.

P

Pair trading is a market-neutral trading strategy that matches a long position with a short position in two highly correlated stocks or other financial instruments. The idea is that when the two securities deviate in their relative pricing, a trading opportunity is created. When one security is relatively cheaper than the other, you buy the cheaper one (go long) and sell the expensive one (go short), expecting the prices to revert back to their historical or predicted relative norms, earning a profit from the convergence. This strategy is particularly popular in equity markets and is often employed by hedge funds and other institutional investors.

Path-Dependent Options

Financial derivatives whose payoff depends not only on the final price of the underlying asset at expiration but also on the specific path the asset's price takes over a defined time period. Unlike standard or "European-style" options, whose payoffs are determined solely by the asset's price at a fixed maturity date, path-dependent options consider the entire price trajectory. Common examples include Asian options, barrier options, and lookback options.

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Physical Measure

Refers to the actual or real-world probabilities of different outcomes occurring. In finance, it contrasts with the risk-neutral measure and is essential for evaluating real-world scenarios and risk assessments.

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Poisson Distribution

The Poisson Distribution models the number of events occurring in a fixed interval of time or space. It's applied in finance to analyze rare events such as defaults or rare market movements.

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Poisson Process

Understood. Here is a simplified version with plain text math.

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Poisson Process

A Poisson process is a model used in finance to represent random events happening over time, like the arrival of orders in a trading system or jumps in asset prices. It is characterized by a rate parameter, lambda (λ), which indicates the average number of events occurring in a unit time interval.

Key Properties

1. Events in different time intervals are independent.

2. Only one event can happen at a given instant.

3. The average rate of events is constant over time.

The probability of n events in time t is calculated as

P(N(t) = n) = (e^(-λt) * (λt)^n) / n!

where N(t) is the total number of events by time t.

The time between events is exponentially distributed with a mean of 1/λ. Its probability density is expressed as:

f_T(t) = λ * e^(-λt) for t >= 0.

The Poisson process is often used to model jumps in asset prices, combining continuous price movements with sudden jumps. The asset price S(t) at time t can be modeled with a jump-diffusion equation:

dS(t) = μ * S(t) * dt + σ * S(t) * dW(t) + J(t) * dq(t)

Here,

- μ is the drift, or average return.

- σ is the volatility.

- dW(t) is the Wiener process, representing continuous price changes.

- J(t) is the price jump at time t.

- dq(t) is the Poisson process, capturing the random arrival of jumps.

Each jump J(t) is typically assumed to follow a certain distribution like log-normal, depending on the specific financial model used.

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Portfolio Optimization

The process of selecting the best portfolio from the set of all portfolios being considered according to some criteria.

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Principal Component Analysis (PCA)

A dimensionality reduction technique used in data analysis and statistics to transform a dataset into a new coordinate system where the variables (or features) are uncorrelated and arranged in order of decreasing importance, called principal components. PCA helps identify underlying patterns and reduce the complexity of data while retaining the most significant information. It is commonly used in quantitative finance for risk modeling, portfolio optimization, and factor analysis.

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Probability Distribution

A function that describes the likelihood of obtaining the possible values of a random variable. In finance, different probability distributions like normal, binomial, and Poisson are used to model various types of financial data and scenarios.

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Probability Measure

A fundamental concept in quantitative finance that assigns a real number, typically between 0 and 1, to each possible outcome of a random experiment, indicating the likelihood of that outcome. All possible outcomes have their probabilities sum up to one. In the context of finance, probability measures are crucial for tasks like pricing derivatives, managing risk, and making informed investment decisions. Specific types of probability measures, such as the Risk-Neutral Measure and Physical Measure, are used for different applications. The Risk-Neutral Measure assumes that all assets grow at the risk-free rate, simplifying the pricing of derivatives. In contrast, the Physical Measure is used to evaluate the real-world probabilities of different outcomes, playing a crucial role in risk assessment and management.

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Probability Space

This term refers to a mathematical model for random experiments, defined by a sample space, a σ-algebra, and a probability measure. It provides a formal framework for calculating probabilities and is the foundation of probability theory, underpinning many concepts in quantitative finance.

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Put-Call Parity

This principle is a fundamental concept in options pricing that establishes a relationship between the prices of European call options and put options of the same class with the same strike prices and expiration dates. It is expressed as:

C - P = S - X * (1 / (1 + r)^t)

- C is the price of the European call option,

- P is the price of the European put option,

- S is the current stock price,

- X is the strike price of the options,

- r is the risk-free interest rate,

- t is the time until the options' expiration, expressed in years.

Put-call parity helps in pricing options and understanding the relationships between various types of options. It also creates arbitrage opportunities if the principle doesn't hold.

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P-Value

A p-value is a measure that helps in hypothesis testing to determine the strength of the evidence against the null hypothesis. A lower p-value indicates that the observed data would be highly unlikely under the null hypothesis, leading to its rejection in favor of the alternative hypothesis.

Q

Quantitative Analysis

The use of mathematical and statistical techniques to study behavior and predict outcomes.

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Quadratic variation for a Brownian motion process is given by the formula:

QV(T) = lim (n→∞) Σ [B(ti) - B(ti-1)]^2 for i=1 to n

Here, T is the time period, n is the number of partitions as they approach infinity, and B(t) is the value of the Brownian motion at time t. Each term in the sum is the square of the difference in the Brownian motion's value at two subsequent times, summed over all partitions. For standard Brownian motion, the quadratic variation over a finite interval equals the length of the time interval, meaning QV(T) = T.

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The Quadratic Rough Heston Volatility Model

The Quadratic Heston Volatility Model is a sophisticated mathematical framework used in quantitative finance to describe and predict the behavior of financial market volatility, particularly in the context of options pricing and risk management. Here's a simplified explanation:

Imagine you want to understand how the volatility, or the degree of price fluctuations, of financial assets like stocks or currencies evolves over time. The Quadratic Heston Model helps you achieve this by incorporating complex dynamics into the modeling of volatility.

Key concepts in the Quadratic Heston Volatility Model:

1. Volatility Process: The model assumes that volatility follows a stochastic process. In other words, it recognizes that volatility is not a fixed value but varies over time in a probabilistic manner.

2. Quadratic Variation: Quadratic variation is a mathematical concept that quantifies the cumulative impact of irregularities or roughness in volatility. It provides a way to measure how volatility evolves over time and how it affects option prices.

3.  Heston Parameters: The model involves several parameters, including the speed of mean reversion (how quickly volatility returns to its long-term average), the long-term average volatility, and the volatility of volatility. These parameters are calibrated to historical data to match observed market behavior.

4. Complex Dynamics: Unlike simpler models, the Quadratic Heston Model captures the complex interplay between asset prices and volatility. It considers how changes in asset prices influence volatility and vice versa.

5. Option Pricing: One of the primary applications of this model is in pricing options more accurately. By modeling volatility realistically, it can provide more precise pricing estimates for various option contracts.

In essence, the Quadratic Heston Volatility Model is a powerful tool for capturing the intricate relationship between asset prices and volatility in financial markets. It enables quants and financial analysts to better understand and predict market behavior, especially in the context of derivatives pricing and risk management.

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Quant

A colloquial abbreviation for a "quantitative analyst," a professional in the field of quantitative finance. Quants use mathematical and statistical models, as well as computer programming, to analyze financial data, develop trading strategies, and manage risk. They are skilled in applying quantitative methods to gain insights into financial markets and make data-driven decisions for investment and trading purposes. Quantitative analysts play a crucial role in various aspects of finance, including asset management, risk assessment, and the development of complex financial instruments.

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Quantitative Analyst

Professionals specialized in quantitative finance who employ mathematical, statistical, and computational methods to analyze financial data, design trading strategies, manage risk, optimize investment portfolios, price complex derivatives, conduct research to enhance trading strategies, and assess strategy performance through backtesting. Quants are pivotal figures in various financial domains, including hedge funds, investment banks, and asset management firms, leveraging their quantitative expertise to inform decision-making and gain competitive advantages in financial markets.

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Quant Developer

A professional who combines expertise in quantitative finance with software development skills to design, implement, and maintain the computational infrastructure and algorithms used by quantitative analysts (quants) and researchers. Quant developers play a crucial role in the financial industry by creating and optimizing software tools and systems for data analysis, risk management, algorithmic trading, and modeling. Their responsibilities include coding, debugging, and ensuring the reliability and efficiency of quantitative models and trading strategies. They bridge the gap between finance and technology, contributing to the success of quantitative trading firms, hedge funds, and investment banks.

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Quantitative Finance (Quant Finance):

A specialized field within finance that applies mathematical, statistical, and computational techniques to analyze financial data, model financial markets, and develop strategies for trading and risk management. Professionals in quantitative finance, known as quants, use mathematical models to price financial instruments, optimize portfolios, and design trading algorithms. This discipline plays a crucial role in modern finance, encompassing derivative pricing, risk assessment, algorithmic trading, and financial engineering.

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Quantitative Researcher

A specialized role within the field of quantitative finance, a quantitative researcher is responsible for conducting in-depth research and analysis using mathematical and statistical techniques to develop trading strategies, optimize portfolios, and gain insights into financial markets. Quantitative researchers often work closely with quantitative analysts (quants) and use their expertise to generate data-driven insights, construct financial models, and improve trading algorithms. Their work involves designing experiments, backtesting strategies, and assessing the viability of quantitative approaches to financial decision-making. Quantitative researchers are essential for the development and success of quantitative trading firms and asset management companies.

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Quasi-Monte Carlo Methods

A method for numerical integration and solving some PDEs, similar to the Monte Carlo method but employing quasi-random sequences.

R

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A tool in measure theory for defining the derivative of one measure with respect to another, crucial in finance for measure changes.

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Rainbow Option

A rainbow option is a type of exotic option that is linked to two or more underlying assets. The option's payoff depends on the performance comparison of these assets. They are called "rainbow options" because they involve multiple assets, like the multiple colors of a rainbow. These options are often used to speculate on the relative performance of assets or to hedge against the risk associated with price discrepancies between correlated assets. They can be complex to value given the multiple underlying assets and factors affecting their prices.

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Random Walk

A stochastic process where a variable's value changes in discrete steps, with each step being random. In finance, it's often used to model stock prices and exchange rates, underpinning the theory that asset prices move randomly and cannot be precisely predicted.

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Reflection Principle

In the context of stochastic processes, it implies that if a process, such as Brownian motion, reaches a certain level, the future path of the process mirrors itself around that level. In simpler terms, if B(t) represents Brownian motion and T is the time when it first hits a level a, then for every t >= T, B(t) - 2a is also a Brownian motion and is not dependent on the process before time T.

https://www.finance-tutoring.fr/the-reflection-principle-in-layman’s-terms…/

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Regression

A statistical method to model the relationship between a dependent (target) and one or more independent (predictor) variables. In finance, regression is widely used for prediction and forecasting.

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Riemann Integral

A type of integral introduced by Bernhard Riemann, used for measuring the area under a curve. It sums up values of a function at infinite points over an interval to calculate the total area. In finance, it's utilized for continuous compounding, options pricing, and other applications requiring integration over a continuous interval. The formula generally involves partitioning the integration interval, approximating the area using rectangles, and taking the limit as the partition becomes infinitely fine.

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Risk-Neutral Probability

A probability measure used in mathematical finance in which each share or investment earns at the risk-free rate. Under this measure, the expected value of an investment equals the present value of its cash flows discounted at the risk-free rate.

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Risk Reversal

Risk reversal refers to a strategy in options trading where a trader simultaneously buys a call option and sells a put option, or vice versa, with the same expiry date. It is used to hedge or speculate on directional movements in the underlying asset's price. In the context of forex markets, it can also indicate the implied volatility spread between similar out-of-the-money call and put options, serving as a gauge for the market’s sentiment on a currency’s future direction.

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Risk Reversal Volatility

It refers to the volatility skew between similar out-of-the-money (OTM) call and put options. In the context of foreign exchange (FX) and options trading, it's used as a measure of the market's expectation of a future move in the price of an underlying asset, reflecting the balance of sentiment among options traders. Mathematically, it's often calculated as the implied volatility of the call option minus the implied volatility of a comparable put option. The sign of the risk reversal volatility can indicate whether the market sentiment is bullish or bearish on the underlying asset.

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Rho

The sensitivity of an option's or options portfolio's value to a change in interest rates.

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Rough Volatility Model

The Rough Volatility Model is a mathematical framework used in quantitative finance to describe and predict the behavior of financial market volatility, especially in the context of options pricing and risk management. Here's a simplified explanation:

Imagine you want to understand how volatile or unpredictable the prices of financial assets, such as stocks or currencies, will be in the future. The Rough Volatility Model helps you do this by considering the "roughness" or irregularity in the way prices change over time.

Key concepts in the Rough Volatility Model:

1. Roughness: The model acknowledges that financial market prices often exhibit rough and irregular patterns. Instead of assuming smooth price movements, it captures the inherent choppiness or roughness in asset price paths.

2. Stochastic Variance: It incorporates a stochastic (random) component for volatility. This means it recognizes that volatility itself is not constant but changes over time due to market dynamics and uncertainty.

3. Hurst Exponent: The model uses a parameter called the Hurst exponent (often denoted as "H") to quantify the roughness of price paths. A higher Hurst exponent indicates more irregular and volatile price movements.

4. Predictive Power: By considering roughness and stochastic volatility, the Rough Volatility Model aims to provide more accurate predictions of future asset price movements, which is crucial for pricing options and managing risk.

In essence, the Rough Volatility Model takes into account the real-world complexity and irregularity of financial markets to improve the modeling of volatility. It's particularly relevant in situations where traditional models, like the Black-Scholes model, may not accurately capture the intricacies of price dynamics.

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The SABR model

The SABR quant model is a mathematical tool used in quantitative finance to estimate the volatility and pricing of financial derivatives, such as options. It's named after the initials of its creators: Stochastic Alpha, Beta, Rho.

Imagine you're trying to predict how the price of an option will change as time passes and market conditions fluctuate. The SABR model helps you do this by considering three key factors:

1. Alpha (α): This represents the sensitivity of the option's volatility to changes in the underlying asset's price. In simpler terms, it tells you how fast the option's volatility can change in response to market movements.

2. Beta (β): Beta is related to the correlation between the asset's price and its volatility. It helps in understanding whether the option's volatility moves in sync with the underlying asset or not.

3. Rho (ρ): Rho measures the sensitivity of the option's value to changes in interest rates. It tells you how interest rate changes can impact the option's price.

By using these three parameters, the SABR model allows quantitative analysts (quants) to make more accurate predictions about option prices and how they might change in different market conditions. It's a valuable tool for managing risk and making informed investment decisions in the world of finance.

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Scalar

A scalar is a single numerical value, often real or complex, used to scale vectors and matrices. In mathematical operations, it adjusts the magnitude without altering the direction of a vector. In finance, scalars can represent various elements, including the size of an investment, a rate of return, or a factor by which values are adjusted for calculation simplicity or model calibration. Scalars are fundamental in quantitative finance, particularly in linear algebra applications.

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Schrödinger Equation

A mathematical formula in quantum mechanics that describes the changes over time of a physical system, represented by its wave function. Named after Erwin Schrödinger, who introduced it in 1926, the equation governs the behavior of particles at the atomic and subatomic levels, providing insights into their probability distributions and energy states.

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SDE (Stochastic Differential Equation)

An equation that describes how a stochastic process evolves over time, combining deterministic and stochastic elements. SDEs are foundational in quantitative finance, particularly for option pricing and other financial derivatives.

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Semi-Martingale

A semi-martingale is a mathematical concept in stochastic calculus that generalizes martingales and encompasses a broader class of stochastic processes. In finance, it’s a useful model for asset prices and rates where the process has both a deterministic trend and a stochastic element, capturing more complex price dynamics than a traditional martingale.

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Sell Side

The sell side consists of investment banks, brokers, and dealers who create, promote, and sell financial instruments like stocks, bonds, and other securities. They act as intermediaries between the issuers of securities and the investing public, often providing research, recommendations, and order execution services to buy side entities. Sell side firms focus on underwriting new securities issues, promoting and selling them to investors, and market making to facilitate trading and liquidity.

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Sharpe Ratio

This is a measure for calculating risk-adjusted return. It's expressed as:

Sharpe Ratio = (Rp - Rf) / σp

Here:

- Rp is the expected return of the portfolio,

- Rf is the risk-free rate of return,

- σp (sigma p) is the standard deviation of the portfolio's return, a measure of the portfolio's risk.

The Sharpe ratio helps in understanding the return of an investment compared to its risk. A higher value indicates better risk-adjusted returns.

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Short Rate Models

A class of financial models that describe the stochastic evolution of the short-term interest rate over time. Essential for pricing interest rate derivatives and understanding interest rate dynamics. Notable models include the Vasicek, Cox-Ingersoll-Ross (CIR), and Hull-White. Central to these models are principles of mean reversion and assumptions about volatility and behavior of interest rates.

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Sigma Algebra (σ-algebra)

In the context of Quantitative Finance, a sigma-algebra (or σ-algebra) is a mathematical concept used in probability theory to define a collection of sets that is closed under countable unions, countable intersections, and complements. It forms the foundation for defining a measurable space, which is essential for constructing probability measures and integrating functions. This concept is crucial for the rigorous development of probability theory and its applications in finance, such as the modeling of random processes and the valuation of complex financial derivatives.

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Sigma Square

Sigma square (σ^2) typically denotes the variance in statistics and finance. It is a measure of the dispersion or spread of a set of values, offering insights into the volatility or risk associated with a particular asset or investment portfolio.

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Skewness

A statistical measure that describes the asymmetry of a distribution around its mean. Positive skewness indicates a distribution with an asymmetric tail extending towards more positive values, while negative skewness indicates a tail extending towards more negative values.

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Stationarity

A characteristic of a process where statistical properties like mean and variance are constant over time, crucial in time series analysis.

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Statistical arbitrage

Statistical arbitrage involves the use of quantitative models to identify and exploit statistical mispricings between related financial instruments. Traders use complex algorithms and high-speed computers to identify assets that are priced inefficiently. The strategy typically involves taking a long position in an undervalued asset and a short position in an overvalued asset with the expectation that the prices will converge to a mean over time. It’s often considered a high-frequency trading strategy, requiring rapid execution of trades to capitalize on small price discrepancies.

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Stochastic Calculus

A specialized branch of mathematics focusing on the analysis and application of random variables and stochastic processes. It extends the concepts of classical calculus to understand and model systems affected by randomness and uncertainty.

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Stopping Time

A concept in stochastic processes and probability, representing a specific type of random time when a predetermined condition is fulfilled. In finance, it can model the time to sell an asset based on various stochastic factors. It is defined with respect to a filtration (F_t), satisfying the condition {τ ≤ t} ∈ F_t for all t, indicating the event of the stopping time being less than or equal to any given t is known at that time.

https://www.finance-tutoring.fr/the-stopping-time-in-layman’s-terms…/

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Student's t-Distribution

Student's t-Distribution is used in hypothesis testing and risk assessment. It accounts for the uncertainty of sample estimates when population parameters are unknown, making it valuable in finance for situations with limited data.

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A straddle involves buying or selling both a call option and a put option with the same strike price and expiration date. This strategy is used when an investor expects a significant price movement but is unsure of the direction. Buying a straddle benefits from high volatility, resulting in potential profit if the asset's price moves dramatically in either direction. Selling a straddle, on the other hand, is profitable in a stable market with low volatility, where the asset’s price remains relatively unchanged.

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Swap Pricing

Swap pricing refers to the process of determining the value or price of a swap contract. Swaps are derivative contracts in which two parties agree to exchange cash flows or other financial instruments over a specified time period. The pricing of swaps depends on multiple factors including the time value of money, the credit risk of the parties involved, and the expected future cash flows. In the context of interest rate swaps, for example, the swap rate is often calculated to make the present value of future cash flows from the fixed leg equal to the present value of expected future floating cash flows, ensuring that the swap has an initial net value of zero.

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Swaption

A financial derivative contract that gives the holder the option but not the obligation to enter into an interest rate swap at a specified future date and at predetermined terms. Swaptions provide flexibility to hedge against or speculate on changes in interest rates. They come in two primary forms: payer swaptions, where the holder has the option to pay fixed and receive floating rates, and receiver swaptions, where the holder has the option to receive fixed and pay floating rates. Swaptions are widely used in interest rate risk management and hedging strategies.

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Symmetric Random Walk

A type of stochastic process where an entity has an equal probability of moving in either of two directions in each time step. In a symmetric random walk, the steps are often referred to as being +1 or -1, occurring with a 50/50 chance, making the process unbiased. In finance, it’s frequently used to model stock prices or other financial variables, providing insights into the random behavior and fluctuations in financial markets. This model is a special case of the broader category of random walks, characterized by its specific step probabilities and step sizes.

T

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Taylor Expansion

It is a method used to approximate a function as an infinite sum of terms, calculated from the function's derivatives at a single point. In finance, it's frequently employed for the valuation of financial instruments and risk management. The general formula is expressed as f(x) ≈ f(a) + f'(a)(x - a) + f''(a)(x - a)^2 / 2! + f'''(a)(x - a)^3 / 3! + ..., where f'(a), f''(a), etc., are the first, second, and higher order derivatives of the function f(x) at the point x = a.

https://www.finance-tutoring.fr/the-taylor-expansion-in-layman’s-terms…/

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Term Structure Models

Financial models that depict the evolution of the entire yield curve over time, rather than focusing solely on the short-term interest rate. These models account for the relationship between yields and varying maturities, providing insights into the future dynamics of interest rates. Renowned models encompass the Nelson-Siegel, Svensson, and Heath-Jarrow-Morton (HJM) frameworks. Essential for analyzing and pricing a wide range of fixed-income securities and interest rate derivatives, they often incorporate assumptions about market factors, risk premia, and the nature of rate movements.

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Time Series

A sequence of numerical data points in successive order, typically occurring at uniformly spaced time intervals. In finance, time series analysis is crucial for forecasting, risk management, and investment decisions, helping professionals analyze and predict future prices, returns, and trends based on historical data.

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Time-Series Analysis

Methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data.

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Theta

The rate at which the price of a derivative loses its value as the expiration date approaches.

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T-Test

A statistical test used to compare the means of two groups and determine if they are significantly different from each other. The t-test is based on the t-statistic, and its interpretation depends on a comparison of the calculated t-value to a critical value from the t-distribution.

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Transition Probability Function

In the context of stochastic processes and Markov chains, the transition probability function quantifies the probability of transitioning from one state to another within a given time frame. It’s a crucial element in various fields including finance, for pricing derivatives and understanding credit risk, and in statistics and mathematics for studying complex systems and networks.

Mathematically, it can be represented as Pij(t), where Pij(t) is the probability of transitioning from state i to state j at time t. In a continuous-time Markov chain, this function is often described by a differential equation or a matrix of transition intensities.

In finance, particularly, understanding the transition probabilities can be crucial for assessing the likely future states of financial markets and the risks associated with various investments.

U

Uniform Distribution

The Uniform Distribution assigns equal probability to all values within a specified range. It's used in finance for scenarios where all outcomes within a given range are equally likely, such as random market fluctuations.

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Unit Root

A characteristic of a time series where trends can be random and non-stationary. In the context of econometrics, having a unit root implies the time series is non-stationary, with statistical properties that vary over time.

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Utility Theory

A mathematical representation in economics to model preferences, satisfaction, or usefulness experienced by a consumer from consuming goods or services.

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Value at Risk (VaR)

A measure of the risk of loss for investments.

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Vanna

Vanna measures the rate of change in delta with respect to changes in the volatility of the underlying asset. It's essential for understanding how an option’s price sensitivity to the underlying asset changes as market volatility shifts.

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Variance

A statistical measurement of the spread between numbers in a data set, indicating how far the numbers lie from the mean. In finance, it helps quantify the dispersion of returns, indicating the riskiness of an asset or portfolio.

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Variance Swap

A financial derivative contract that allows investors to speculate on or hedge against future volatility in an underlying asset's price. In a variance swap, one party agrees to pay the other party an amount equal to the difference between the realized and pre-specified future variance (volatility) of the asset's returns. It is commonly used by traders and investors to gain exposure to volatility without buying or selling the underlying asset.

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Vasicek Model

A mean-reverting model describing the change in interest rates, used in the valuation of interest rate derivatives.

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Vega

The rate of change in an option’s price relative to a one percentage point increase in implied volatility.

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Volatility

A statistical measure of the dispersion of returns around the mean for a given security or market index.

It helps investors gauge the degree of fluctuation in asset prices.

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It's a correction factor applied to the volatility measure of a financial instrument to better align with empirical data and market realities. It helps in refining pricing models, making them more reflective of actual market behaviors and ensuring that they are responsive to changes in market conditions.

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Volatility Skew

A term used in options trading and finance to describe the unevenness or slope in the implied volatility levels for options with the same expiration date but different strike prices. It typically manifests as higher implied volatility for either out-of-the-money (OTM) call options or out-of-the-money put options. A positive skew implies greater concern about downside risks, while a negative skew suggests a focus on potential upside gains in the market. Traders and investors analyze the volatility skew to make informed decisions regarding options pricing and trading strategies.

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Volatility Smile

A graphical representation of implied volatilities for options with the same expiration date but different strike prices. The term "smile" refers to the U-shaped curve that typically forms on the graph. In a volatility smile, implied volatilities tend to be higher for out-of-the-money (OTM) options (both call and put) compared to at-the-money (ATM) options. The smile indicates that market participants expect a higher likelihood of extreme price movements (higher volatility) for the underlying asset. It is often used as an indicator of market sentiment and can influence option pricing and trading strategies.

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Volga

Volga refers to the rate of change in vega with respect to the volatility of the underlying asset. It measures the sensitivity of the option's vega to changes in volatility and is used to manage the gamma risk of options portfolios.

W

Weibull Distribution

The Weibull Distribution is employed in finance to model extreme events, particularly in risk management and insurance. Its shape parameter allows for flexibility in modeling different types of risks.

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Wiener Process

Also known as Brownian motion, it is a real-valued continuous-time stochastic process.

Y

Yield Curve

A line that plots the interest rates, at a set point in time, of bonds having equal credit quality but differing maturity dates.

Z

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Zero-Coupon Bond

A debt security that doesn't pay interest but is traded at a deep discount, rendering profit at maturity when the bond is redeemed for its full face value.

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In quantitative finance, the Z-Spread, short for Zero-Volatility Spread, represents the additional yield or spread over the risk-free yield curve that an investor can earn from a fixed-income security, such as a bond, taking into account all sources of risk, including credit risk and interest rate risk. It is a measure used to assess the relative value of a bond, considering its unique risk profile. The Z-Spread is an essential metric for bond pricing and portfolio management, providing insights into how a bond compensates investors for the added risks compared to risk-free assets.

FINANCE TUTORING

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FINANCE TUTORING

Registered Training Organization No. 24280185328

Contact: Florian CAMPUZAN Phone: 0680319332 Email:fcampuzan@finance-tutoring.fr