Search results
Results from the WOW.Com Content Network
These are the same solutions (up to time translation) that were obtained by Fischer Black in 1976. [6] Reverting ,, to the original set of variables yields the above stated solution to the Black–Scholes equation. The asymptotic condition can now be realized.
From the parabolic partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price given the risk of the security and its expected return (instead replacing the ...
When deriving further properties of GBM, use can be made of the SDE of which GBM is the solution, or the explicit solution given above can be used. For example, consider the stochastic process log(S t). This is an interesting process, because in the Black–Scholes model it is related to the log return of the stock price.
If we know that (,) satisfies an equation (like the Black–Scholes equation) we are guaranteed that we can make good use of the equation in the derivation of the equation for a new function (,) defined in terms of the old if we write the old V as a function of the new v and write the new and x as functions of the old t and S.
Itô's lemma can be used to derive the Black–Scholes equation for an option. [2] Suppose a stock price follows a geometric Brownian motion given by the stochastic differential equation dS = S(σdB + μ dt). Then, if the value of an option at time t is f(t, S t), Itô's lemma gives
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. Once in this form, a ...
Nikolay Borisovich Sukhomlin (Russian: Николай Борисович Сухомлин; April 1945, in Leningrad – 12 January 2010, in Haiti) was a Russian scientist who discovered new solutions and symmetry for the Black-Scholes equation.
A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena in, i.a., engineering science, quantum mechanics and financial mathematics. Examples include the heat equation, time-dependent Schrödinger equation and the Black–Scholes ...