116.83 116.83 116.83

B

100.00 100.00 100.00 100.00

A 85.59 85.59 85.59

73.26 73.26

62.71

0 1 2 3

Figure 3.7: State Space of a Trinomial Tree with Constant Volatility of 11%.

the value of sA = 100 “ today™s spot price. The next three nodes, at time t1 ,

are computed from equation (3.14): S1 = 116.83, S2 = 100.00 and S3 = 85.59,

respectively. In order to determine the transition probabilities, we need to know

the put price struck at S2 = 100 and expiring one year from now P (S2 , t1 ).

From the smile, the implied volatility of this option is 11%. We calculate

its price using a constant-volatility trinomial tree with the same state space,

and ¬nd it to be 0.987 EUR. The forward price corresponding to node A is

— —

FA = Se(r ’δ )∆t = 107.69. Here r— and δ — denote the continuous interest

70 3 Implied Trinomial Trees

rate r— = log(1 + r) and δ — = log(1 + δ), respectively. The down transition

probability is then computed from equation (3.23)

elog(1+0.12)—1 0.987 ’ Σ

qA = = 0.077.

1 — (100.00 ’ 85.59)

The summation term Σ in the numerator is zero in this case, because there are

no nodes with price lower than S3 at time t1 . The up transition probability pA

is computed from equation (3.24)

107.69 + 0.077 — (100.00 ’ 85.59) ’ 100

pA = = 0.523.

116.83 ’ 100.00

The middle transition probability is then equal to 1 ’ pA ’ qA = 0.6. As

one can see from equations (3.6) “ (3.11) the Arrow-Debreu prices turn out

to be just discounted transition probabilities: »1,1 = 0.467, »1,2 = 0.358 and

»1,3 = 0.069. Equation (3.25) then gives the value of the implied local volatility

at node A: σA = 9.5%. Let us show the computation of one node more. Have

a look at the node B in year 2 of Figure 3.7. The index level at this node is

sB = 116.83 and its forward price one year later is FB = 125.82. ¿From this

node, the index can move to one of three future nodes at time t3 = 3, with

prices s3,2 = 136.50, s3,3 = 116.83 and s3,4 = 100.00, respectively. The value

of a call option struck at 116.83 and expiring at year 3 is C(s3,3 , t3 ) = 8.87,

corresponding to the implied volatility of 10.83% interpolated from the smile.

The Arrow-Debreu price »2,2 is computed from equation (3.8):

»2,2 = e’ log(1+0.12)—1 (0.467 — (1 ’ 0.517 ’ 0.070) + 0.358 — 0.523) = 0.339.

The numerical values here are the already known values from the last level

t1 . The complete trees of Arrow-Debreu prices, transition probabilities and

local volatilities for our example are shown in Figures 3.8 “ 3.10. Now, using

equations (3.21) and (3.22) we can ¬nd the transition probabilities:

elog(1+0.12)—1 — 8.87 ’ Σ

p2,2 = = 0.515

0.339 — (136.50 ’ 116.83)

125.82 ’ 0.515 — (136.50 ’ 116.83) ’ 116.83

q2,2 = = 0.068,

100 ’ 116.83

where Σ contributes with the only term 0.215 — (147 ’ 116.83) (there is one

single node above SB whose forward price is equal to 147). Last from equation

(3.25), we ¬nd the implied local volatility at this node σB = 9.3%.

3.4 ITT™s and Their Construction 71

Upper Probabilities Middle Probabilities Lower Probabilities

0.431 0.060

0.508

0.413 0.417 0.070 0.068

0.517 0.515

0.523 0.523 0.521 0.401 0.401 0.404 0.077 0.077 0.075

0.538 0.534 0.368 0.375 0.094 0.090

0.571 0.296 0.133

Figure 3.8: Transition Probabilities when σimpl = 0.11 ’ ∆Strike — 0.001.

0.098

0.215 0.239

0.467 0.339 0.226

1.000 0.358 0.190 0.111

0.069 0.047 0.031

0.006 0.005

0.001

Figure 3.9: Tree of Arrow-Debreu prices when σimpl = 0.11 ’ ∆Strike — 0.001.

72 3 Implied Trinomial Trees

0.092

0.094 0.093

0.095 0.095 0.095

0.099 0.098

0.106

Figure 3.10: Implied Local Volatilities when σimpl = 0.11 ’ ∆Strike — 0.001.

3.4 ITT™s and Their Construction 73

As already mentioned in Section 3.4.4, the transition probabilities may fall out

of the interval (0, 1). If we slightly modi¬cate our previous example, we can

show such a case. Let us now assume that the Black-Scholes volatility from

our example increases (decreases) linearly 0.5 percentage points with every 10