tion, for instance, incorporating a stochastic volatility factor or discontinuous

jumps in the underlying price. But these methods bring other problems. In

the next section we will pay attention to another approach of Black-Scholes

extension developed in Derman and Kani (1994) “ to the implied trees.

3.3 Trees and Implied Models 59

Su

p

S

1-p

Sd

Figure 3.2: CRR Binomial Tree.

3.3 Trees and Implied Models

While the Black-Scholes attitude assumes that the underlying asset follows

geometric Brownian motion from equation (??) with a constant volatility,

implied theories assume that the stock or index price follows a process, whose

volatility σ(S, t) varies with the spot price and time. This process can be

expressed by the following stochastic di¬erential equation:

dSt

= µdt + σ(S, t)dZt . (3.5)

St

This approach ensures that the valuation of the option remains preference-free

(all uncertainty is in the spot price and thus we can hedge options using the

underlying). Derman and Kani (1994) shows that it is possible to determine

σ(S, t) directly from the market prices of liquidly traded options.

The most natural discrete representation of a non-lognormal evolution process

of the underlying prices is an implied binomial tree (IBT). It is similarly like the

60 3 Implied Trinomial Trees

CRR binomial tree is a discrete version of the Black-Scholes constant-volatility

process. For an exhaustive explanation on IBT™s see Derman and Kani (1994).

Generally we can use any (higher) multinomial tree as a discrete development

from the Black-Scholes model. However, all of them converge, as the time

step tends towards zero, to the same continuous result “ the constant-volatility

process. IBT™s are, from the set of all implied multinomial trees, minimal, i.e.

they have only one degree of freedom “ the arbitrary choice of the central node

at each level of the tree. In this sense one may feel that they are su¬cient

and no more complicated trees are necessary. Despite the limiting similarity of

multinomial trees, one form of them could be more convenient. Sometimes one

may wish more ¬‚exibility in the discrete world “ to get transition probabilities

and probability distributions which vary as smoothly as possible across the tree.

This is important especially when the market option prices are not very precise

(e.g. because of ine¬ciency, market frictions, etc.). It is useful to recall the

concept of Arrow-Debreu prices before the following derivation in Section 3.4.

Arrow-Debreu price »n,i at node (n, i) is computed as the sum over all paths

starting in the root of the tree leading to node (n, i) of the product of the

risklessly discounted transition probabilities. Arrow-Debreu price of the root

is equal to one. Arrow-Debreu prices at the ¬nal level of a (multinomial) tree

are a discrete approximation of the implied distribution. Notice that they are

discounted and thus the risk-neutral probability corresponding to each node

(at the ¬nal level) should be calculated as the product of the Arrow-Debreu

price and the factor er .

3.3 Trees and Implied Models 61

Possible paths from the root to A

A

Figure 3.3: Computing Arrow-Debreu Price in a Binomial Tree.

62 3 Implied Trinomial Trees

3.4 ITT™s and Their Construction

3.4.1 Basic insight

A trinomial tree with N levels is a set of nodes sn,i , n = 1, . . . , N , i =

1, . . . , 2n’1. The underlying price can move from each node sn,i to one of three

nodes: with probability pi to the upper node, value sn+1,i , with probability qi

to the lower node, value sn+1,i+2 and with probability 1 ’ pi ’ qi to the middle

node, value sn+1,i+1 . Let us denote the nodes in the new level with capital

letters: Si (=sn+1,i ), Si+1 (=sn+1,i+1 ) and Si+2 (=sn+1,i+2 ) respectively.

(a) (b)

S1

s1 S2

s2 S3

Si

pi

S4

sn,i 1 - pi- q i

S i+1

S 2n-2

qi

S i+2 S 2n-1

s 2n-2

S2n

s 2n-1

S2n+1

Figure 3.4: General Nodes in a Trinomial Tree.

Figure 3.4(a) shows a single node of a trinomial tree. Starting in the node sn,i

at time tn there are ¬ve unknown parameters: the two transition probabilities

pn,i , qn,i and the three new node prices Si , Si+1 , and Si+2 . Let Fi denote the

known forward price of the spot sn,i and »i the known Arrow-Debreu price

at node (n, i). In particular, Arrow-Debreu prices for a trinomial tree can be

3.4 ITT™s and Their Construction 63

obtained by the following iterative formulas:

»1,1 = 1 (3.6)

e’r∆t »n,1 pn,1

»n+1,1 = (3.7)

’r∆t

(»n,1 (1 ’ pn,1 ’ qn,1 ) + »n,2 pn,2 )

»n+1,2 = e (3.8)

’r∆t

(»n,i’1 qi’1 + »n,i (1 ’ pn,i ’ qn,i ) + »n,i+1 pn,i+1 )

»n+1,i+1 = e (3.9)

e’r∆t (»n,2n’1 (1 ’ pn,2n’1 ’ qn,2n’1 ) + »n,2n’2 qn,2n’2 )

»n+1,2n = (3.10)

e’r∆t »n,2n’1 qn,2n’1

»n+1,2n+1 = (3.11)