compare Figure ??(a) with the result of nicemat: ITTnicemat

State Space Matrix STFitt05.xpl

Last in this section we will introduce a quantlet, which actually belongs into the

next section, but it is also possible to use it separately. ITTcrr is called by ITT

to compute the option price. It builds up a constant-volatility trinomial tree

(combining two steps of a standard CRR binomial tree), its formal de¬nition

is:

{ttree,optprice} = ITTcrr (S, K, r, sigma, time, opt, div)

where S is the the spot price of the underlying, K is the exercise price, r is

the continuous interest rate (∈ (0, 1)), sigma is the constant volatility, time

is the vector of time points, opt = 1 speci¬es a call option and opt = 0 a

put option. div is the dividend rate (∈ (0, 1)). The output parameters are

ttree and optprice, the constant-volatility trinomial tree and the price of

given option respectively.

84 3 Implied Trinomial Trees

3.5.3 What is hidden

This subsection comments on auxiliary quantlets which are not intended for a

direct usage. First we present the quantlet ITTnewnodes, which computes a

new level of an implied trinomial tree. It also checks that the forward condition

from Section 3.4.4 is ful¬lled.

{newnodes,overwritten} = ITTnewnodes (old, r, sigma, deltat, div)

old is a vector containing the underlying prices on last known level of the

tree. r and div are the continuous interest rate and dividend yield, sigma

is the volatility and deltat is the time step between the last and the new

level ∆t = tn+1 ’ tn The outputs are two vectors: the vector with the prices

in the new level newnodes and the vector overwritten. This corresponds to

newnodes, it is ¬rstly ¬lled with NaN™s and afterward, if a node is overwritten,

the corresponding node in overwritten contains its new value. It is worth

to mention how the implausible price will be overwritten: If the highest new

underlying price S1 is lower than the forward price F1 at the node sn,1 , it will

be set up one percent higher than the forward, and similarly, any lowest new

underlying price S2n+1 which is higher than the forward price F2n’1 of the

node sn,2n’1 will be set 1% lower than this forward price.

The Arrow-Debreu prices will be computed stepwise with the same philosophy,

like the state space, with the di¬erence that no overwrites are necessary.

newad = ITTad (up, down, ad, r, deltat)

ITTad computes the Arrow-Debreu prices of an Implied Trinomial Tree on one

particular level. up and down are vectors of up and down transition probabilities

going from the last level of known Arrow-Debreu prices ad. The new computed

prices are then returned in the vector newad.

fx = ITTterm (t)

3.5 Computing Implied Trinomial Trees 85

ITTterm represents the non-linear equation system (3.17) “ it computes its

value in time points given by the vector t. The equation system is solved

with the quantlet nmnewton. The Jacobian matrix necessary for the numerical

solution is returned by quantlet ITTtermder:

jac = ITTtermder (t)

86 3 Implied Trinomial Trees

Bibliography

Derman, E. and Kani, I. (1994). The Volatility Smile and Its Implied Tree.

Quantitative Strategies Research, http://gs.com/qs, Goldman Sachs.

Derman, E., Kani, I. and Chriss, N. (1996). Implied Trinomial Trees of

the Volatility Smile. Quantitative Strategies Research, http://gs.com/qs,

Goldman Sachs.

H¨rdle, W., Hl´vka, Z. and Klinke, S. (2000). XploRe Application Guide.

a a

Springer Verlag, Heidelberg.

H¨rdle, W., Kleinow, T. and Stahl, G. (2002). Applied Quantitative Finance.

a

Springer-Verlag, Berlin.

Hull, J (1989). Options, Futures and Other Derivatives. Prentice-Hall, Engle-

wood Cli¬s, New Jersey.

Komor´d, K (2002). Implied Trinomial Trees and Their Implementation with

a

XploRe. Bachelor Thesis, HU Berlin.

Ross, S., Wester¬eld, R. and Ja¬e, J. (2002). Corporate Finance. Mc Graw-Hill.

88 Bibliography

4 Functional data analysis

Michal Benko, Wolfgang H¨rdle

a

4.1 Introduction

90 4 Functional data analysis

5 Nonparametric Productivity

Analysis

Wolfgang H¨rdle, Seok-Oh Jeong

a

5.1 Introduction

Productivity analysis is an important part of classical microeconomic the-

ory. Farrel (1957) studied the problem of measuring e¬ciency in an empiri-

cal way. Shephard (1970) formed the foundation of the modern theory of cost

and frontier function. F¨re, Grosskopf, and Lovell (1985) and F¨re, Grosskopf,

a a

and Lovell (1994) provide a comprehensive survey of the problem of measuring

e¬ciency and the theoretical formulations of frontier function.

The activity of production units such as banks, universities, governments and

hospitals may be described and formalized by the production set of feasible

input-output points (x, y):

p+q

Ψ = {(x, y) ∈ R+ | x can produce y}.

In many cases y is one-dimensional, so Ψ can be characterized by a function g

called the frontier function or the production function:

Ψ = {(x, y) ∈ Rp+1 | y ¤ g(x)}.

+

The input (requirement) set X(y) is de¬ned by:

X(y) = {x ∈ Rp | (x, y) ∈ Ψ},

+

which is the set of all input vectors x ∈ Rp that yield at least the output vector

+

y. The output (correspondence) set Y (x) de¬ned by:

Y (x) = {y ∈ Rq | (x, y) ∈ Ψ},

+

92 5 Nonparametric Productivity Analysis

which is the set of all output vectors y ∈ Rq that is obtainable from the input

+

vector x.

In productivity analysis one is interested in the input and output isoquants or

e¬cient boundaries, denoted by ‚X(y) and ‚Y (x) respectively. They consist

of the minimal attainable boundary in a radial sense: for y = 0,