abilities for all tree nodes (n, i) at each tree level n. Let C(K, tn+1 ) and

P (K, tn+1 ) denote today™s price for a standard European call and put option,

respectively, struck at K and expiring at tn+1 . These values can be obtained

by interpolating the smile surface at various strike and time points. Then the

trinomial tree value of an option is the sum over all nodes (n + 1, j) of the

discounted probability of reaching that node multiplied by the pay-o¬ func-

tion, i.e. max(Sj ’ K, 0) for the call and max(K ’ Sj , 0) for the put option

respectively, where Sj denotes the spot price at the node (n + 1, j):

C (K, tn+1 ) = e’r∆t {pj »j + (1 ’ pj’1 ’ qj’1 )»j’1 + qj’2 »j’2 } (Sj ’ K)+

j

(3.19)

3.4 ITT™s and Their Construction 67

P (K, tn+1 ) = e’r∆t {pj »j + (1 ’ pj’1 ’ qj’1 )»j’1 + qj’2 »j’2 } (K ’ Sj )+

j

(3.20)

If we set the strike price K to the value Si+1 , the stock price at the node

(n + 1, i + 1), rearrange the terms in the sum and use equation ( 3.12) we

can compute from equation ( 3.19) the transition probabilities pi , qi for all the

nodes above the central node:

i’1

er∆t C(Si+1 , tn+1 ) ’ »j (Fj ’ Si+1 )

j=1

pi = (3.21)

»i (Si ’ Si+1 )

Fi ’ pi (Si ’ Si+1 ) ’ Si+1

qi = (3.22)

Si+2 ’ Si+1

Similarly, we compute the transition probabilities from equation ( 3.20) for all

the nodes below (and including) the center node (n + 1, n) at time tn :

2n’1

er∆t P (Si+1 , tn+1 ) ’ »j (Si+1 ’ Fj )

j=i+1

qi = (3.23)

»i (Si+1 ’ Si+2 )

Fi ’ qi (Si+2 ’ Si+1 ) ’ Si+1

pi = (3.24)

Si ’ Si+1

Derivation of these equations is described in detail in Komor´d (2002). Finally

a

the implied local volatilities are approximated from equation (3.13):

. pi (Si ’ Fi )2 + (1 ’ pi ’ qi )(Si+1 ’ Fi )2 + qi (Si+2 ’ Fi )2

2

σi = . (3.25)

Fi2 ∆t

3.4.4 Possible pitfalls

Equations (3.21) - (3.24) can unfortunately result in transition probabilities

which are negative or greater than 1. This is inconsistent with rational option

prices and allows arbitrage. We must face two forms of this problem. First, we

must watch that no forward price Fn,i of a node (n, i) falls outside the range of

daughter nodes at the level n + 1: Fn,i ∈ (sn+1,i+2 , sn+1,i ). Figure 3.6 shows

this type of problem. This is not very di¬cult to overcome, since the state

68 3 Implied Trinomial Trees

(a) (b)

123.50

114.61

113.60

113.08

106.82

106.34

105.22

100.46

100.00 100.00 100.01

95.04

94.04

88.43

87.27

80.97

0 1 3 6 0 1 2 3 4 5 6 7 8 9 10

Figure 3.6: Two Kinds of the Forward Price Violation.

space is our free choice “ we can overwrite the nodes causing this problem. The

second problem refers to extreme values of options prices, which would imply

an extreme value of local volatility. This may not be possible to obtain with

probabilities between (0, 1). In such case we have to overwrite the option price

which produces the unacceptable probabilities. In practice it means that we

overwrite even these probabilities with numbers from the interval (0, 1) This

is always possible if our state space does not violate the forward condition.

From the great number of possible ways to overwrite wrong probabilities with

numbers between 0 and 1, we chose to set:

Fi ’ Si+1 Fi ’ Si+2 Si ’ Fi

1 1

pi = + , qi = (3.26)

Si ’ Si+1 Si ’ Si+2 Si ’ Si+2

2 2

if Fi ∈ (Si+1 , Si ) and

Fi ’ Si+2 Si+1 ’ Fi S i ’ Fi

1 1

pi = , qi = + (3.27)

Si ’ Si+2 Si+1 ’ Si+2 Si ’ Si+2

2 2

if Fi ∈ (Si+2 , Si+1 ). In either case the middle transition probability is equal to

1 ’ pi ’ q i .

3.4.5 Illustrative examples

To illustrate the construction of an implied trinomial tree, let us now consider

that the volatility varies only slowly with the strike price and time to expiration.

3.4 ITT™s and Their Construction 69

Assume that the current index level is 100 EUR, the annual riskless interest rate

r = 12%, dividend yield is δ = 4%. The annual Black-Scholes implied volatility

is assumed to be σ = 11%, moreover let us assume that the Black-Scholes

implied volatility increases (decreases) linearly by 10 basis points (=0.1%) with

every 10 unit drop (rise) in the strike price, i.e. σimpl = 0.11 ’ ∆Strike — 0.001.

To keep the example easy, we consider one-year steps over three years. First,

we construct the state space “ a constant-volatility trinomial tree as described

in Section 3.4.2. The ¬rst node at time t0 = 0, labeled A in Figure 3.7, has

159.47

136.50 136.50