The method of Implied Trinomial Trees represents an option pricing technique

that tries to ¬t the market volatility smile. It uses an inductive algorithm

constructing a possible evolution process of underlying prices from the current

market option data. This chapter recalls the procedure as described in Derman,

Kani and Chriss (1996) and shows its general use in XploRe.

3.1 Introduction

In recent decades option-like contracts have been through a giant evolution.

Options are ¬nancial derivatives speculating on the value of an underlying

asset. The boom in research on options and the use of them started after Black

and Scholes published the formula for options prices. Options are maybe more

common than one could think “ many issues of corporate securities (e.g. bonds

or stocks) also have option features. New ¬nancing techniques, for instance,

contingent value rights (CVR™s), are straightforward applications of options.

Liquidly traded options are important ¬nancial instruments used for hedging

“ they can be included into the portfolio to reduce risk. So option pricing has

become one of the basic techniques in ¬nance.

Unfortunately, option prices computed by the Black-Scholes formula and the

market options prices show a discrepancy. This deviation may be due to the

assumption of constant volatility in the Black-Scholes model which is hardly

ever ful¬lled in the market. Therefore, new approaches were examined to ¬nd

options prices consistent with the market. Binomial trees, discrete versions

of the Black-Scholes model, are intuitive and probably the most commonly

used basic procedure for option pricing. Derman and Kani (1994) presented

an extension of binomial trees called implied binomial trees that ¬t the market

data. Implied trinomial trees (ITT) are analogical extension of trinomial trees.

56 3 Implied Trinomial Trees

Trinomial trees converge in the continuous limit to the same result as binomial

trees, but the use of ITT™s give us a freedom in the choice of the possible path

of the underlying price, the so called state space. This freedom could be in

some circumstances desirable. Figure ??(a) shows a possible state space with

three periods over six years. An ITT with one-year time steps over ten years

is depicted in Figure ??(b).

3.2 Basic Option Pricing Overview 57

3.2 Basic Option Pricing Overview

Options are contingent claims on the value of an underlying asset, usually a

stock or a traded market index. The simplest type of options are European

options. Their importance is mainly theoretical. An European call (or put)

gives the owner the right to buy (or sell) an underlying asset at a ¬xed price

(the so called strike price) at a given date.

The famous Black-Scholes model of option pricing can be viewed as a continu-

ous generalization of duplication model. For an explanation of the duplication

model see Ross, Wester¬eld and Ja¬e (2002). Black and Scholes assume that

the underlying asset follows a geometric Brownian motion with a constant

volatility and is described by the stochastic di¬erential equation

dSt

= µdt + σdZt , (3.1)

St

where S denotes the underlying price, µ is the expected return and Z stands

for the standard Wiener process. As a consequence, the underlying distribu-

tion is lognormal. This is the crucial point “ volatility is the only parameter of

the Black-Scholes formula which is not explicitly observable from the market

data. When plugging the market option prices into the formula and solving for

volatility (the so called implied volatility), empirical data shows a variation

with both exercise (the skew structure) and expiration time (the term struc-

ture), both together are commonly called the volatility smile. Figure 3.1

illustrates this behavior. Panel 3.1(a) shows the decrease of implied volatility

σimp with the strike level of the put options on the DAX index with a ¬xed

expiration of 52 days, as observed on January 29, 1999. Panel 3.1(b) shows the

increase of σimp with the time to expiration of put options with a ¬xed strike

level 6000 DM.

The Skew Structure The Term Structure

55

36

50

35

Implied Volatility [%]

Implied Volatility [%]

45

34

40

33

32

35

31

3000 4000 5000 6000 7000 100 200 300 400 500

Strike Price [DM] Time to maturity [days]

Figure 3.1: Implied Volatilities of DAX Options on Jan 29, 1999.

58 3 Implied Trinomial Trees

Implied volatility of an option is the market™s estimate of the average future

underlying volatility during the life of that option, local volatility is the

market™s estimate of underlying volatility at a particular future time and market

level. Let us recall that as a consequence of complete markets, the Black-Scholes

model is risk-neutral.

The most natural simpli¬cation of the Black-Scholes model are binomial trees.

The binomial tree corresponding to the risk neutral underlying evaluation pro-

cess is the same for all options on this asset, no matter what the strike price

or time to expiration is. The tree cannot know which option we are valuating

on it. There are several alternative ways to construct a binomial tree. Here

we will recall the familiar Cox-Ross-Rubinstein (CRR) binomial tree. It has a

constant logarithmic spacing between nodes on the same level. This spacing

shows the future price volatility. Figure 3.2 shows a standard CRR tree. Start-

ing at a general node S, the price of the underlying asset can either increase

to the higher price Su with the probability p or decrease to the lower price Sd

with the probability 1 ’ p:

√

= Seσ ∆t

Su (3.2)

√

= Se’σ ∆t

Sd (3.3)

F ’ Sd

p = (3.4)

Su ’ Sd

where F denotes the forward price in the node S and σ is the constant volatility.

For other types of binomial trees see Derman, Kani and Chriss (1996) .

New models try to overcome the problem of varying volatility and to price

options consistent with the market prices “ to account for the volatility smile.