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Feynman–Kac Formula
The Feynman–Kac formula, named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations and stochastic processes. In 1947, when Kac and Feynman were both faculty members at Cornell University, Kac attended a presentation of Feynman's and remarked that the two of them were working on the same thing from different directions. The Feynman–Kac formula resulted, which proves rigorously the real-valued case of Feynman's path integrals. The complex case, which occurs when a particle's spin is included, is still an open question. It offers a method of solving certain partial differential equations by simulating random paths of a stochastic process. Conversely, an important class of expectations of random processes can be computed by deterministic methods. Theorem Consider the partial differential equation \fracu(x,t) + \mu(x,t) \fracu(x,t) + \tfrac \sigma^2(x,t) \fracu(x,t) -V(x,t) u(x,t) + f(x,t) = 0, defined for all x \in \mathbb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Feynman
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist. He is best known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfluidity of supercooled liquid helium, and in particle physics, for which he proposed the parton model. For his contributions to the development of quantum electrodynamics, Feynman received the Nobel Prize in Physics in 1965 jointly with Julian Schwinger and Shin'ichirō Tomonaga. Feynman developed a pictorial representation scheme for the mathematical expressions describing the behavior of subatomic particles, which later became known as Feynman diagrams and is widely used. During his lifetime, Feynman became one of the best-known scientists in the world. In a 1999 poll of 130 leading physicists worldwide by the British journal ''Physics World'', he was ranked the seventh-greatest physicist of all time. He assisted in the Manhatt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Integral
Functional integration is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space, but a space of functions. Functional integrals arise in probability, in the study of partial differential equations, and in the path integral approach to the quantum mechanics of particles and fields. In an ordinary integral (in the sense of Lebesgue integration) there is a function to be integrated (the integrand) and a region of space over which to integrate the function (the domain of integration). The process of integration consists of adding up the values of the integrand for each point of the domain of integration. Making this procedure rigorous requires a limiting procedure, where the domain of integration is divided into smaller and smaller regions. For each small region, the value of the integrand cannot vary much, so it may be replaced by a single value. In a functional integral the domain of integration is a space of functions. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kunita–Watanabe Inequality
In stochastic calculus, the Kunita–Watanabe inequality is a generalization of the Cauchy–Schwarz inequality to integrals of stochastic processes. It was first obtained by Hiroshi Kunita and Shinzo Watanabe and plays a fundamental role in their extension of Ito's stochastic integral to square-integrable martingales. Statement of the theorem Let ''M'', ''N'' be continuous local martingales and ''H'', ''K'' measurable In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts hav ... processes. Then : \int_0^t \left, H_s \ \left, K_s \ \left, \mathrm \langle M,N \rangle_s \ \leq \sqrt \sqrt where the angled brackets indicates the quadratic variation and quadratic covariation operators. The integrals are understood in the Lebesgue–Stieltjes sense. References * {{DEFAULTSORT:Kunita- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffusion Monte Carlo
Diffusion Monte Carlo (DMC) or diffusion quantum Monte Carlo is a quantum Monte Carlo method that uses a Green's function to calculate low-lying energies of a quantum many-body Hamiltonian. Introduction and motivation of the algorithm Diffusion Monte Carlo has the potential to be numerically exact, meaning that it can find the exact ground state energy for any quantum system within a given error, but approximations must often be made and their impact must be assessed in particular cases. When actually attempting the calculation, one finds that for bosons, the algorithm scales as a polynomial with the system size, but for fermions, DMC scales exponentially with the system size. This makes exact large-scale DMC simulations for fermions impossible; however, DMC employing a clever approximation known as the fixed-node approximation can still yield very accurate results. To motivate the algorithm, let's look at the Schrödinger equation for a particle in some potential in one dimensi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schrödinger Equation
The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after Erwin Schrödinger, an Austrian physicist, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics. Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of the wave function, the quantum-mechanical characterization of an isolated physical system. The equation was postulated by Schrödinger based on a postulate of Louis de Broglie that all matter has an associated matter wave. The equati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Chemistry
Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions to physical and chemical properties of Molecule, molecules, Material, materials, and solutions at the atomic level. These calculations include systematically applied approximations intended to make calculations computationally feasible while still capturing as much information about important contributions to the computed Wave function, wave functions as well as to observable properties such as structures, spectra, and Thermodynamics, thermodynamic properties. Quantum chemistry is also concerned with the computation of quantum effects on molecular dynamics and chemical kinetics. Chemists rely heavily on spectroscopy through which information regarding the Quantization (physics), quantization of energy on a molecular scale can be obtained ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Option Style
In finance, the style or family of an option is the class into which the option falls, usually defined by the dates on which the option may be exercised. The vast majority of options are either European or American (style) options. These options—as well as others where the payoff is calculated similarly—are referred to as " vanilla options". Options where the payoff is calculated differently are categorized as "exotic options". Exotic options can pose challenging problems in valuation and hedging. American and European options The key difference between American and European options relates to when the options can be exercised: * A European option may be exercised only at the expiration date of the option, i.e. at a single pre-defined point in time. * An American option on the other hand may be exercised at any time before the expiration date. For both, the payoff—when it occurs—is given by * \max\, for a call option * \max\, for a put option where K is the strike ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Term Structure Model
An affine term structure model is a financial model that relates zero-coupon bond prices (i.e. the discount curve) to a spot rate model. It is particularly useful for deriving the yield curve – the process of determining spot rate model inputs from observable bond market data. The affine class of term structure models implies the convenient form that log bond prices are linear functions of the spot rate (and potentially additional state variables). Background Start with a stochastic short rate model r(t) with dynamics: : dr(t)=\mu(t,r(t)) \, dt + \sigma(t,r(t)) \, dW(t) and a risk-free zero-coupon bond maturing at time T with price P(t,T) at time t. The price of a zero-coupon bond is given by:P(t,T) = \mathbb^\left\where T=t+\tau, with \tau being is the bond's maturity. The expectation is taken with respect to the risk-neutral probability measure \mathbb. If the bond's price has the form: :P(t,T)=e^ where A and B are deterministic functions, then the short rate model is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero-coupon Bond
A zero-coupon bond (also discount bond or deep discount bond) is a bond in which the face value is repaid at the time of maturity. Unlike regular bonds, it does not make periodic interest payments or have so-called coupons, hence the term zero-coupon bond. When the bond reaches maturity, its investor receives its par (or face) value. Examples of zero-coupon bonds include US Treasury bills, US savings bonds, long-term zero-coupon bonds, and any type of coupon bond that has been stripped of its coupons. Zero coupon and deep discount bonds are terms that are used interchangeably. In contrast, an investor who has a regular bond receives income from coupon payments, which are made semi-annually or annually. The investor also receives the principal or face value of the investment when the bond matures. Some zero coupon bonds are inflation indexed, and the amount of money that will be paid to the bond holder is calculated to have a set amount of purchasing power, rather than a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Valuation Of Options
In finance, a price (premium) is paid or received for purchasing or selling options. The calculation of this premium will require sophisticated mathematics. Premium components This price can be split into two components: intrinsic value, and time value (also called "extrinsic value"). Intrinsic value The ''intrinsic value'' is the difference between the underlying spot price and the strike price, to the extent that this is in favor of the option holder. For a call option, the option is in-the-money if the underlying spot price is higher than the strike price; then the intrinsic value is the underlying price minus the strike price. For a put option, the option is in-the-money if the ''strike'' price is higher than the underlying spot price; then the intrinsic value is the strike price minus the underlying spot price. Otherwise the intrinsic value is zero. For example, when a DJI call (bullish/long) option is 18,000 and the underlying DJI Index is priced at $18,050 then the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Black–Scholes Equation
In mathematical finance, the Black–Scholes equation, also called the Black–Scholes–Merton equation, is a partial differential equation (PDE) governing the price evolution of derivatives under the Black–Scholes model. Broadly speaking, the term may refer to a similar PDE that can be derived for a variety of options, or more generally, derivatives. Consider a stock paying no dividends. Now construct any derivative that has a fixed maturation time T in the future, and at maturation, it has payoff K(S_T) that depends on the values taken by the stock at that moment (such as European call or put options). Then the price of the derivative satisfies :\begin \frac + \frac\sigma^2 S^2 \frac + rS\frac - rV = 0 \\ V(T, s) = K(s) \quad \forall s \end where V(t, S) is the price of the option as a function of stock price ''S'' and time ''t'', ''r'' is the risk-free interest rate, and \sigma is the volatility of the stock. The key financial insight behind the equation is that, under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
