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In mathematical finance, the Black–Scholes equation, also called the Black–Scholes–Merton equation, ... where H(x) is the Heaviside step function.
In finance, the binomial options pricing model (BOPM) provides a generalizable numerical method for the valuation of options.Essentially, the model uses a "discrete-time" (lattice based) model of the varying price over time of the underlying financial instrument, addressing cases where the closed-form Black–Scholes formula is wanting.
In fact, the Black–Scholes formula for the price of a vanilla call option (or put option) can be interpreted by decomposing a call option into an asset-or-nothing call option minus a cash-or-nothing call option, and similarly for a put—the binary options are easier to analyze, and correspond to the two terms in the Black–Scholes formula.
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 finite difference model can be derived, and the valuation obtained. [2]
The Black model extends Black-Scholes from equity to options on futures, bond options, swaptions, (i.e. options on swaps), and interest rate cap and floors (effectively options on the interest rate). The final four are numerical methods, usually requiring sophisticated derivatives-software, or a numeric package such as MATLAB.
In finance, Black's approximation is an approximate method for computing the value of an American call option on a stock paying a single dividend. It was described by Fischer Black in 1975. [1] The Black–Scholes formula (hereinafter, "BS Formula") provides an explicit equation for the value of a call option on a non-dividend paying stock. In ...
It is based on the iteration of a two step procedure: First, a backward induction process is performed in which a value is recursively assigned to every state at every timestep. The value is defined as the least squares regression against market price of the option value at that state and time (-step). Option value for this regression is ...
Further enhancements are designed to achieve stability relative to Black-Scholes as the number of time-steps changes. More recent models, in fact, are designed around direct convergence to Black-Scholes. [9] A variant on the Binomial, is the Trinomial tree, [10] [11] developed by Phelim Boyle in 1986.