Use

The binomial Black-Scholes-Richardson model is used to price standard American options.

Integration and calculation bases

The following parameters are used to call the price calculator for standard American options:

Scope of functions and valuation

If the maturity date of the option has been reached, the intrinsic value of the option (difference between the spot and strike) is shown.

If the option term has not expired, the price is calculated as follows:

Starting from the end nodes, the exercise price and the holding value of the option are calculated backwards, for each node. The larger of the two is the option price at each node. A node value results from the total of the previous node values, weighted according to the likelihood of its occurrence.

lograte = log( 1 + domestic_rate / 100 ).

logforeign = log( 1 + foreign_rate / 100 ).

d_time = days / 365 / steps."time in years per step"

disc = 1 / exp( lograte * d_time ).

up = exp( vola * sqrt( d_time ) / 100 ).

uplog = log( up ).

down = 1 / up

ha = exp( ( lograte - logforeign ) * d_time ).

prob = ( ha - down ) / ( up - down ).

Holding value _ij =

(Price of the larger predecessor * Probability of upward movement+

Price of the smaller predecessor * Probability of a downward movement ) * Discount factor for one time step

= (P _i+1,j * prob + P _i+1,j+1 * ( 1 - prob ) ) * disc

Exercise price _ij =

Underlying price at time point i - Strike

= spot * exp( uplog * ( 2 * i - j ) ) - strike

P _ij = max(exercise price _ij | holding value _ij )

P ₀₁ = Price of the option

To improve the convergence, in the last time step before the exercise date the system calculates as the holding value the Black-Scholes price of the option on each calculation node in terms of the residual validity period.It also applies Richardson extrapolation. This adjusts not only the oscillation of the option price based on the number of steps in the binomial tree, but also corrects the systematic deviation of the mean of the oscillation [see also Broadie, M. and Detemple, J.: "American Option Valuation: New Bounds, Approximations, and a Comparison of Existing Methods, Review of Financial Studies 9.4 (1996), S.1211-1250].