the volatility is ¬ve times as steep as before “ illustrated in Figure 3.11). Using

the same state space, we ¬nd unadmissable transition probabilities at nodes C

and D of Figures 3.12 “ 3.14. To overwrite them with plausible values, we used

the set up described in equations (3.26) and (3.27).

0.13

0.12

Probability

0.11

0.1

0.09

50 100 150

Index level

Figure 3.11: Skew structure from the examples. Thick line for σimpl = 0.11 ’

∆Strike—0.001 and thin line when σimpl = 0.11’∆Strike—0.005.

74 3 Implied Trinomial Trees

Upper Probabilities Middle Probabilities Lower Probabilities

0.271 0.146

0.582

C C

C

0.467 0.482 0.041 0.033

0.492 0.485

0.523 0.523 0.514 0.401 0.401 0.420 0.077 0.077 0.066

0.605 0.605 0.222 0.222 0.173 0.173

0.582 0.271 0.146

D D D

Figure 3.12: Transition Probabilities when σimpl = 0.11 ’ ∆Strike — 0.005.

0.107

0.205 0.206

C

0.467 0.362 0.266

1.000 0.358 0.182 0.099

0.069 0.038 0.024

0.011 0.008

D

0.001

Figure 3.13: Tree of Arrow-Debreu prices when σimpl = 0.11’∆Strike—0.005.

3.5 Computing Implied Trinomial Trees

3.5.1 Basic skills

The simplest way to invoke implied trinomial trees in XploRe is to use the

interactive menus, which guide you through the computation procedure:

{Ttree, P, Q, AD, LocVol, Onodes, Oprobs, Time} = ITT (volafunc)

3.5 Computing Implied Trinomial Trees 75

0.108

C

0.087 0.086

0.095 0.095 0.093

0.113 0.113

0.108

D

Figure 3.14: Implied Local Volatilities when σimpl = 0.11 ’ ∆Strike — 0.005.

There is only one parameter that has to be inserted “ the name of the volatility

function(s) volafunc. The output parameters will be discussed later on. First,

we consider only slow volatility variation. Then volafunc is one string con-

taining the name of the volatility function de¬ned by the user. This function

has the following form:

sigma = volafunc (S, K, tau)

where S denotes the spot price, K the exercise price and tau is the time to

maturity on each level. For our ¬rst example we will assume that the implied

volatility is 15% for all expirations and that it increases (decreases) 0.5% with

every 10 points drop (rise) in the strike price:

library("finance") ; load the library

proc(sigma)=volafunc(S, K, tau) ; define the volatility function

sigma=0.015 + (S-K)/10 * 0.05

endp

ITT("volafunc") ; ITT™s with interactive menus

STFitt01.xpl

76 3 Implied Trinomial Trees

XploRe asks you to insert the strike price, the annual interest rate and the

divided yield. Subsequently, you must insert all the time points, when the

steps of the ITT occurred. At this stage let us recall that the tree is supposed

to start at time t0 = 0. To end the insertion of the time steps, type any non-

positive value. For instance, in our example from subsection 3.4.5, we typed

consecutively: 0, 1, 2, 3 and 0. Once the ITT is computed, you can display

the results “ a menu asks you to choose between the state space, the tree of

the transition probabilities, the tree of Arrow-Debreu prices, the tree of local

volatilities and the state price density. If you choose the state space of the tree

you can specify whether you want to describe the nodes with underlying prices

or the arrows with transition probabilities respectively. While in this plot the

scale of the y-axe corresponds to the underlying price, the tree in any other

graph is only auxiliary and the vertical spacing does not mean anything.

Next, let us consider signi¬cant term and skew structure. In such cases the

input parameter volafunc is a string vector with three rows containing names

of functions with the following syntax (the names are arbitrary):

sigma = term (t)

sigmader = termder (t)

sigma = skew (S)