XploRe Help : ITT

Keywords - Function groups - @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Library:	finance
See also:	grITTcrr grITTstsp grITTspd ITTad ITTcrr ITTnewnodes ITTnicemat ITTterm ITTtermder plotITT IBTdk bitree nmnewton

Quantlet:		ITT
Description:		main function for the Derman/Kani/Chriss method of implied trinomial trees (ITT). It computes the nodes of the ITT, the probability matrices, the Arrow-Debreu prices and the local volatility matrix.

Reference(s):

Derman, E., Kani, I. and Chriss, N. (1996). Implied Trinomial Trees of the Volatility Smile. Journal of Derivatives, 4, pp. 7-22. Komorad, K. (2002). Implied Trinomial Trees and Their Implementation with XploRe.

Link:

Tutorial: ITT

Usage:	`{Ttree,P,Q,AD,LocVol,Onodes,Oprobs,Time}=ITT(S,r,div,time,volafunc{,skewconst}) or {Ttree,P,Q,AD,LocVol,Onodes,Oprobs,Time}=ITT(volafunc)`
Input:
	`S`	scalar; spot price of the underlying
	`r`	scalar; interest rate from interval (0,1)
	`div`	scalar; the dividend rate from interval (0,1)
	`time`	t x 1 vector; time points (in years) when the levels of the tree occurred (they are supposed to increase).
	`volafunc`	string; if there is only a slow variation in the implied volatilities, volafunc has only one row containing the name of the volatility function. This function has three input parameters: (S, K, tau), where S is the spot price of the underlying, K denotes the exercise price and tau the time to maturity on each level. If the volatility varies significantly with strike or time to expiration, volafunc can have up to 3 rows: the first row contains the name of the volatility function for the term structure: term(t). The second row represents the name of the first derivative of this volatility function: term'(t). Both of these two functions have only one parameter time and it is necessary that they can use vectors. The third row corresponds to the name of the function for the skew structure: skew(t). If there is no term structure, but significant skew structure, insert volafunc[1]="".
	`skewconst`	optional vector; the skew constant c (see the reference), default is 0.1
Output:
	`Ttree`	matrix; the Implied Trinomial Tree (the root is found in the upper left corner; elements which do not belong to the tree are NaN's)
	`P`	matrix; the up transition probabilities
	`Q`	matrix; the down transition probabilities
	`AD`	matrix; the Arrow-Debreu prices for the computed tree
	`LocVol`	matrix; the local volatilities
	`Onodes`	matrix; the overwritten nodes of the Implied Trinomial Tree, not overwritten elements represented as NaN's. If there are no overwritten nodes Onodes = NaN (only scalar).
	`Oprobs`	((1+4*k) x 2) matrix; coordinates of the nodes where overwritten probabilities occurred. The first row is only auxiliary, but after that each of the following four rows corresponds to one overwritten probability. The first element of this quartet represents the parental node, the three other elements are daughter nodes. If there are no overwritten probabilities Oprobs = NaN (only scalar).
	`Time`	vector; useful only when there is a significant term structure - only in this case it differs from the input parameter time.

Note:

(2) Interactive menus will be invoked if "volafunc" is the only parameter.

Example:

library("finance") proc(sigma)=volafunc(S,K,time) sigma=0.15 +(K-S)/10 * 0.005 endp S = 100 ; current index level r = 0.1 ; interest rate div = 0.05 ; dividend yield time = 0|0.5|1 ; 2 half-year periods t=ITT(S, r, div, time, "volafunc") t.Ttree

Result:

Contents of Ttree

[1,]      100   116.18   134.99
[2,]     +NAN      100   116.18
[3,]     +NAN   86.071      100
[4,]     +NAN     +NAN   86.071
[5,]     +NAN     +NAN   74.082

Example:

library("finance") proc(sigma)=volafunc(S,K,time) sigma=0.15 +(K-S)/10 * 0.005 endp t=ITT("volafunc")

Result:

several interactive menus are invoked to let the user
type the input values

Example:

library("finance") library("nummath") ; now define the term volatility function and its first derivative proc(fx)=t1(t) T=3 q=0.3 k=-q/(T+2) fx = sqrt(k*t+q) endp proc(fder)=t1der(t) T=3 q=0.3 k=-q/(T+2) fder = k / 2 / sqrt(k*t+q) endp ; define the skew volatility function proc(sigma)=skew(S) K=100 ; exercise price sigma =(S/(3*K+S))^2 endp ; and compute S = 100 ; current index level r = 0.1 ; compounded riskless interest rate div = 0.05 ; dividend yield time = 0|1|2|3 ; time vector volaf = "t1"|"t1der"|"skew" t=ITT(S, r, div, time, volaf) plotITT(t,1) t.LocVol

Result:

--------------------------------
             Warning!
================================
 probabilities of 6 node(s)
 had to be overwritten in order
 to avoid the arbitrage
--------------------------------

Contents of LocVol
[1,]  0.044084  0.043385  0.038446
[2,]     +NAN  0.031908  0.02952
[3,]     +NAN  0.032168  0.027637
[4,]     +NAN     +NAN  0.031259
[5,]     +NAN     +NAN  0.042086

Author: K. Komorad, W. Haerdle, 20020319 license MD*Tech