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The discrete difference equations may then be solved iteratively to calculate a price for the option. [4] The approach arises since the evolution of the option value can be modelled via a partial differential equation (PDE), as a function of (at least) time and price of underlying; see for example the Black–Scholes PDE .
Stochastic differential equations originated in the theory of Brownian motion, in the work of Albert Einstein and Marian Smoluchowski in 1905, although Louis Bachelier was the first person credited with modeling Brownian motion in 1900, giving a very early example of a stochastic differential equation now known as Bachelier model.
In Itô calculus, the Euler–Maruyama method (also simply called the Euler method) is a method for the approximate numerical solution of a stochastic differential equation (SDE). It is an extension of the Euler method for ordinary differential equations to stochastic differential equations named after Leonhard Euler and Gisiro Maruyama. The ...
In mathematical finance, the Black–Derman–Toy model (BDT) is a popular short-rate model used in the pricing of bond options, swaptions and other interest rate derivatives; see Lattice model (finance) § Interest rate derivatives.
In mathematical finance, the asset S t that underlies a financial derivative is typically assumed to follow a stochastic differential equation of the form = +, under the risk neutral measure, where is the instantaneous risk free rate, giving an average local direction to the dynamics, and is a Wiener process, representing the inflow of randomness into the dynamics.
The finite difference method, on the other hand, experiences exponential growth in the number of computation grids with increasing dimensions, leading to significant computational and storage demands. This method is generally suitable for simple boundary conditions and low-dimensional BSDEs, but it is less effective in complex situations. [9]
The Black model (sometimes known as the Black-76 model) is a variant of the Black–Scholes option pricing model. Its primary applications are for pricing options on future contracts, bond options, interest rate cap and floors, and swaptions.
In finance, the Heston model, named after Steven L. Heston, is a mathematical model that describes the evolution of the volatility of an underlying asset. [1] It is a stochastic volatility model: such a model assumes that the volatility of the asset is not constant, nor even deterministic, but follows a random process.