Finite difference methods (FDM) are numerical techniques used to solve partial differential equations (PDEs) that arise in various fields, particularly in financial mathematics for option pricing. These methods are particularly useful for pricing options when the underlying asset follows a stochastic process governed by a PDE, such as the Black-Scholes equation. ### Overview of Finite Difference Methods Finite difference methods involve discretizing a continuous domain into a grid (or lattice), allowing the approximation of derivatives using finite differences.
The Fisher equation is an important concept in economics that describes the relationship between nominal interest rates, real interest rates, and inflation. It is named after the American economist Irving Fisher.
The Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of forces, such as random fluctuations or deterministic forces. It is commonly used in various fields, including statistical mechanics, diffusion processes, and financial mathematics, to model systems that exhibit stochastic behavior.
"Forward measure" is a concept used in financial mathematics and quantitative finance, particularly in the context of modeling and pricing derivatives. It generally refers to a particular probability measure under which certain processes, like asset prices or tradeable instruments, exhibit specific properties over time. In mathematical finance, different measures are used to analyze stochastic processes, especially when it comes to pricing options and other derivatives.
Forward volatility refers to the expected volatility of an asset's return over a future period, as implied by the pricing of options or other derivatives. It is an essential concept in finance, particularly in options pricing models. ### Key Points of Forward Volatility: 1. **Forward Contracts vs. Spot Contracts:** Forward volatility is related to the idea of forward contracts, which are agreements to buy or sell an asset at a future date at a price agreed upon today.
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.
Future value (FV) is a financial concept that represents the value of an investment or cash flow at a specific point in the future, taking into account a specified rate of return or interest rate. It helps individuals and businesses determine how much an investment made today will grow over time.
Girsanov's theorem is a fundamental result in the theory of stochastic processes, particularly in the field of stochastic calculus and quantitative finance. It provides a way to change the probability measure under which a stochastic process is defined, transforming it into another process that may have different characteristics. This is particularly useful in financial mathematics for pricing derivatives and in risk management. ### Key Concepts: 1. **Stochastic Processes**: A stochastic process is a collection of random variables indexed by time or space.
Good-deal bounds are a concept in financial economics, particularly in the context of pricing and arbitrage bounds for derivatives and financial instruments. The main idea behind good-deal bounds is to establish a range of prices for an asset that reflects a balance between two competing elements: the desire to avoid arbitrage opportunities and the willingness to accept potential mispricings due to risk preferences.
The Graham number is a specific large number named after mathematician Ronald Graham. It is an upper bound for a certain problem in Ramsey theory, which is a branch of combinatorial mathematics. The Graham number itself arises in connection with the properties of hypercubes and is famously known for being enormously large—much larger than numbers typically encountered in mathematics.
In finance, particularly in the context of options trading and derivatives, "Greeks" refer to a set of metrics used to measure the sensitivity of an option's price to changes in various underlying factors. Each Greek represents a different dimension of risk and can help traders understand how different variables can affect the value of options and other derivatives.
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.
The Heston model is a mathematical model used to describe the evolution of financial asset prices, particularly in the context of options pricing. Developed by Steven Heston in 1993, this model is notable for its incorporation of stochastic volatility, which allows for the volatility of the asset price to change over time in a random manner, as opposed to assuming it is constant, which is a limitation of the classic Black-Scholes model.
High frequency data refers to datasets that are collected and recorded at very short intervals, often in real time. This type of data is commonly used in various fields, including finance, economics, and environmental monitoring. Here are some key characteristics and applications of high frequency data: ### Characteristics: 1. **Time Interval**: High frequency data is typically collected at intervals of seconds, minutes, or even milliseconds, as opposed to traditional datasets that may be updated daily, weekly, or monthly.
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.
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.
Implied volatility (IV) is a measure used in the financial markets to indicate the market's expectation of the future volatility of an asset, usually associated with options pricing. Unlike historical volatility, which measures past price fluctuations, implied volatility reflects the market's forecast of how much an asset's price is likely to move in the future.
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.