A frictionless market is an idealized concept in economics and finance where there are no transaction costs, taxes, barriers, or other impediments to trading. In such a market, buyers and sellers can exchange goods and services freely and efficiently. Here are some key features of a frictionless market: 1. **No Transaction Costs**: There are no fees associated with buying or selling assets, such as brokerage fees or commissions.

Stochastic calculus is a branch of mathematics that deals with processes that involve randomness or uncertainty. It extends classical calculus to include stochastic processes, which are mathematical objects that evolve over time in a probabilistic manner. Stochastic calculus is particularly useful in fields such as finance, economics, physics, and engineering, where systems are influenced by random factors. Key concepts and components of stochastic calculus include: 1. **Stochastic Processes**: These are mathematical objects that describe a collection of random variables indexed by time.

The Bochner–Kodaira–Nakano identity is a fundamental result in the study of the geometry of complex manifolds, particularly in the context of the study of Hermitian and Kähler metrics. This identity relates the curvature of a Hermitian manifold to the properties of sections of vector bundles over the manifold, and it plays a crucial role in several areas of differential geometry and mathematical physics.

Capelli's identity is a result in the field of algebra, specifically relating to determinants and matrices. It provides a way to express certain determinants, particularly those involving matrices formed by polynomial expressions. In its simplest form, Capelli's identity can be stated in terms of a square matrix whose entries are polynomials in variables. More formally, it relates the determinant of a matrix formed from the derivatives of polynomials to the determinant of a matrix derived from the polynomials themselves.

A Hawkes process is a type of point process that is used to model events that occur over time, where the occurrence of one event can increase the likelihood of subsequent events happening. It is particularly useful in fields like finance, seismology, neuroscience, and social sciences for modeling phenomena where events cluster in time.

The Heath–Jarrow–Morton (HJM) framework is a mathematical model used in finance to describe the evolution of interest rates over time. It is particularly useful for modeling the entire term structure of interest rates, which refers to the relationship between interest rates of different maturities. The HJM framework was developed by David Heath, Robert Jarrow, and Andrew Morton in the early 1990s.

Holding Period Return (HPR) is a measure of the total return on an investment over the period it is held. It considers both the income generated by the investment (such as dividends or interest) and any capital gains or losses realized during the holding period. HPR can be expressed as a percentage and is useful for investors to evaluate the performance of their investments over a specific timeframe.

Vladimir Zakharov is a prominent Russian mathematician known for his work in the fields of applied mathematics, particularly in the area of nonlinear wave processes, fluid dynamics, and mathematical physics. He is best known for contributing to the development of theories related to solitons and integrable systems, which are important in the study of wave propagation in various physical contexts.

The implied repo rate is a financial metric used to indicate the cost of financing a position with a security, typically in the context of futures contracts or options. It is derived from the difference between the spot price of the underlying asset and its futures price, taking into account the time until the contract's expiration.

Incomplete markets refer to a situation in an economy where not all risks can be completely insured or traded. In an incomplete market, individuals or entities do not have the opportunity to make transactions for every possible future state of the world, meaning that certain risks remain unhedged. This can lead to suboptimal consumption and investment decisions, as agents may not be able to fully insure against potential adverse outcomes.

Indifference price refers to the price at which an individual or an entity is indifferent between holding an asset and not holding it, meaning that the individual derives the same level of utility or satisfaction from both options. In a financial context, this concept is often applied to situations involving risky assets. For example, an investor might determine an indifference price for a stock based on their risk preferences, expected returns, and overall portfolio construction.

An interest rate is the percentage at which interest is charged or paid on the principal amount of a loan, investment, or deposit, typically expressed on an annual basis. It represents the cost of borrowing money or the return on investment for saving or lending funds. Interest rates can vary depending on several factors, including the type of financial product, the borrower's creditworthiness, inflation expectations, and the overall economic environment.

Itô calculus is a branch of mathematics that deals with the integration and differentiation of stochastic processes, particularly those that describe systems influenced by random forces. It is named after the Japanese mathematician Kiyoshi Itô, who developed these concepts in the context of stochastic analysis. At its core, Itô calculus provides tools for analyzing and solving stochastic differential equations (SDEs), which are differential equations in which one or more of the terms are stochastic processes.

Jamshidian's trick is a mathematical technique used primarily in the field of finance, particularly in the area of option pricing and the valuation of derivative securities. The trick simplifies the process of pricing certain types of options by transforming the problem into one that can be solved using standard tools like the risk-neutral pricing framework. The main idea behind Jamshidian's trick involves decomposing the pricing of a particular derivative into a series of simpler components that can be analyzed separately.

The Johansen test is a statistical method used to test for the presence of cointegration among a set of non-stationary time series variables. Cointegration refers to a relationship among two or more time series variables that move together over the long run, despite being individually non-stationary. The test helps to identify whether a linear combination of the non-stationary time series is stationary, indicating that the series are cointegrated.

Katashiro refers to a traditional practice in Japan where a straw figure or doll is made and used in Shinto rituals. The creation of these figures is often associated with the belief that they can absorb bad fortune or illness, acting as a surrogate for a person during ceremonies. Typically, katashiro are created at certain festivals or during specific times of the year, such as New Year's or during harvest festivals.

Kurtosis risk refers to the risk associated with extreme movements in the tails of a distribution, as indicated by the measure of kurtosis. In finance and investment, kurtosis is used to describe the shape of the probability distribution of asset returns, with a focus on the propensity for extreme events, or "fat tails.

A late fee is a charge incurred when a payment is not made by its due date. Late fees can apply to various types of payments, including bills, loans, rent, and credit card payments. Here are a few key points regarding late fees: 1. **Purpose**: Late fees are intended to encourage timely payments and compensate the creditor for the inconvenience and potential financial impact of delayed payments.

The Lattice model in finance refers to a method of pricing options and other derivatives using a discrete-time framework that represents the underlying asset's price dynamics as a lattice or tree. The most commonly known form of this model is the Binomial Lattice Model. ### Key Features of a Lattice Model: 1. **Discrete Time**: The model works over discrete time intervals, where asset prices can change at each time step.

Malliavin calculus is a branch of mathematics that extends calculus to the setting of stochastic processes, particularly in the study of stochastic differential equations (SDEs). It was developed by the French mathematician Paul Malliavin in the 1970s. The primary aim of Malliavin calculus is to provide tools for differentiating random variables that depend on stochastic processes and to study the smoothness properties of solutions to SDEs.