process of underlying prices can be seen discrete through an implied binomial

tree. Implied trinomial tree (ITT) is another discrete representation of the

same process. What we desire from an ITT, and in general from any implied

tree, is:

1. the tree reproduces correctly the volatility smile,

2. the tree is risk-neutral,

3. all the transition probabilities are from the interval (0, 1).

To ful¬ll the risk-neutrality condition, the expected value of the underlying

price sn,i in the next period tn+1 must be its known forward price:

Esn,i = pi Si + (1 ’ pi ’ qi )Si+1 + qi Si+2 = Fi = e(r’δ)∆t sn,i , (3.12)

where r denotes the continuous interest rate, δ the dividend yield and ∆t is

the time step from tn to tn+1 . Derman, Kani and Chriss (1996) say that the

node prices and transition probabilities also satisfy:

pi (Si ’Fi )2 +(1’pi ’qi )(Si+1 ’Fi )2 +qi (Si+2 ’Fi )2 = Fi2 σi ∆t+o(∆t), (3.13)

2

where σi is the stock or index price volatility during this time period and o(∆t)

denotes a term of higher order than ∆t. Therefore we have two constraints in

equations (3.12) and (3.13) for ¬ve unknown parameters. As a consequence,

there is no unique implied trinomial tree, we can thus choose one of the large

set of equivalent ITT™s. Equivalent in the sense, that as the time spacing ∆t

tends towards zero, all of these trees tend to give the same results. A common

1 Here we slightly modi¬cated the notation of Derman, Kani and Chriss (1996) to be con-

sistent with the program code.

64 3 Implied Trinomial Trees

method is to choose freely the underlying prices and then to use these two

equations and solve them for the transition probabilities pi , qi . Afterward we

must make sure that these probabilities do not violate the condition 3 above.

Since in general all multinomial trees converge to the same theoretical result

(namely to the continuous process of evolution in the underlying asset), the

reason why we use ITT™s instead of IBT™s is that we gain these three degrees

of freedom. This freedom may allow us to ¬t the smile better, especially when

inconsistent or arbitrage-violating market options prices make a consistent tree

impossible. Even though the constructed tree is consistent, other di¬culties

can arise when its local volatility and probability distributions are jagged and

”implausible”.

3.4.2 State space

There are several methods we can use to construct an initial state space. As

already mentioned, all of them converge to the same theory (constant-volatility

Black-Scholes model), in the continuous limit. In other words all these models

are equivalent discretizations of the constant volatility di¬usion process. We

chose a constant-volatility Cox-Ross-Rubinstein (CRR) binomial tree. Then

we combined two steps of this tree into a single step of our new trinomial tree.

This is illustrated in Figure 3.5. Using equations (3.2) and (3.3) we can derive

the following formulas for the nodes in the new trinomial tree:

√

σ 2∆t

Si = sn+1,i = sn,i e (3.14)

Si + 1 = sn+1,i+1 = sn,i (3.15)

√

= sn,i e’σ 2∆t

Si + 2 = sn+1,i+2 (3.16)

Now, summing up the probabilities of reaching one of the three nodes, using for-

mula (3.4), we get the relationship for the up and down transition probabilities

in the trinomial tree (the middle transition probability is equal to 1 ’ pi ’ qi ):

√ 2

’σ ∆t/2

r∆t/2

’e

e

√ √

pi =

’σ ∆t/2

σ ∆t/2

’e

e

√ 2

σ ∆t/2 r∆t/2

’e

e

√ √

qi =

’σ ∆t/2

σ ∆t/2

’e

e

Derman, Kani and Chriss (1996) describes two other methods for building a

constant-volatility trinomial tree.

3.4 ITT™s and Their Construction 65

Figure 3.5: Building a Constant-Volatility Tree by Combining Two Steps of a

CRR Binomial Tree.

When the implied volatility varies only slowly with strike and expiration, the

regular state space with uniform mesh size is adequate for constructing ITT

models. But if the volatility varies signi¬cantly with strike or time to maturity,

we should choose a state space re¬‚ecting this properties. Construction of a

trinomial space with proper skew and term structure proceeds in four steps.

• First, we build a regular trinomial lattice with constant time spacing ∆t

and constant level spacing, assuming all interest rates and dividends equal

to zero. Such a lattice can be constructed in the way described above.

• Secondly, we modify ∆t at di¬erent times. Let us denote the known

equally spaced time points t0 = 0, t1 , . . . , tn = T , then we can ¬nd the

˜ ˜ ˜

unknown scaled times t0 = 0, t1 , . . . , tn = T by solving the following set

66 3 Implied Trinomial Trees

of non-linear equations:

n’1 k

1 ˜ 1 =T 1

˜ ; k = 1, . . . , n ’ 1.

tk + tk 2 (3.17)

˜ ˜

σ 2 (ti ) σ 2 (t)

σ (T )

i=1 i=1

• Subsequently, we change ∆S at di¬erent levels. Denoting S1 , . . . , S2n+1

˜ ˜

the original known underlying prices and S1 , . . . , S2n+1 the new scaled

unknown underlying prices respectively, we solve the following equation:

˜

Sk c Sk

= exp ln k = 2, . . . , 2n + 1 (3.18)

˜ σ(Sk ) Sk’1

Sk’1

for some constant c (recommended to be some typical value of the local

volatility). There are 2n equations for 2n + 1 unknown parameters. Here

we always suppose that the new central node is the same as the original

˜

central node: Sn+1 = Sn+1 . For a more elaborate explanation of the

theory behind equations (3.17) and (3.18), set in the continuous frame,

see Derman, Kani and Chriss (1996).

• Lastly, you can grow all the node prices by a su¬ciently large growth fac-

tor which removes forward prices violations (explained in Subsection 3.4.4).

Multiplying all zero-rate node prices at time tn by e(r’δ)(tn ’tn’1 ) should

be always su¬cient.

3.4.3 Transition probabilities