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Bibliography

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±t-Sahalia, Y. and Lo, A. W. (2000). Nonparametric Risk management and
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Empirical Tests, Journal of Finance Vol. LIII, 6, pp. 2059“2106.
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a
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Hafner, R. and Wallmeier, M. (2001). The Dynamics of DAX Implied Volatil-
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Muirhead, R. J. (1982). Aspects of Multivariate Statistics, Wiley Series in
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o o
Roll, R. (1977). A Critique of the Asset Pricing Theory™s Tests: Part I, Journal
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Stochastic Systems, LNS 147, Springer Verlag, New York, pp. 131“147.
7 How Precise Are Price
Distributions Predicted by
Implied Binomial Trees?
Wolfgang H¨rdle and Jun Zheng
a


In recent years, especially after the 1987 market crash, it became clear that
the prices of the underlying asset do not exactly follow the Geometric Brow-
nian Motion (GBM) model of Black and Scholes. The GBM model with con-
stant volatility leads to a log-normal price distribution at any expiration date:
All options on the underlying must have the same Black-Scholes (BS) implied
volatility, and the Cox-Ross-Rubinstein (CRR) binomial tree makes use of this
fact via the construction of constant transition probability from one node to
the corresponding node at the next level in the tree. In contrast, the implied bi-
nomial tree (IBT) method simply constructs a numerical procedure consistent
with the volatility smile. The empirical fact that the market implied volatil-
ities decrease with the strike level, and increase with the time to maturity of
options is better re¬‚ected by this construction. The algorithm of the IBT is a
data adaptive modi¬cation of the CRR method.
An implied tree should satisfy the following principles:

• It must correctly reproduce the volatility smile.
• negative node transition probabilities are not allowed.
• The branching process must be risk neutral (forward price of the asset
equals to the conditional expected value of it) at each step .

The last two conditions also eliminate arbitrage opportunities.
The basic purpose of the IBT is its use in hedging and calculations of implied
probability distributions (or state price density (SPD)) and volatility surfaces.
146 7 How Precise Are Price Distributions Predicted by IBT?


Besides these practical issues, the IBT may evaluate the future stock price dis-
tributions according to the BS implied volatility surfaces which are calculated
from currently observed daily market option prices.
We describe the construction of the IBT and analyze the precision of the pre-
dicted implied price distributions. In Section 7.1, a detailed outline of the IBT
algorithm for a liquid European-style option is given. We follow ¬rst the Der-
man and Kani (1994) algorithm, discuss its possible shortcomings, and then
present the Barle and Cakici (1998) construction. This method is character-
ized by a normalization of the central nodes according to the forward price.
Next, we study the properties of the IBT via Monte-Carlo simulations and
comparison with simulated conditional density from a di¬usion process with a
non-constant volatility. In Section 7.3, we apply the IBT to a DAX index data
set containing the underlying asset price, strike price, interest rate, time to
maturity, and call or put option price from the MD*BASE database (included
in XploRe), and compare SPD estimated by historical index price data with
those predicted by the IBT. Conclusions and discussions on practical issues are
presented in the last section.


7.1 Implied Binomial Trees
A well known model for ¬nancial option pricing is a GBM with constant volatil-
ity, it has a log-normal price distribution with density,
2
® 
σ2
ST
ln St ’ (r ’ 2 )„ 
1
√ exp °’
p(St , ST , r, „, σ) = », (7.1)

2σ 2 „
ST 2πσ 2 „

at any option expiration T , where St is the stock price at time t, r is the riskless
interest rate, „ = T ’t is time to maturity, and σ the volatility. The model also
has the characteristic that all options on the underlying must have the same
BS implied volatility.
However, the market implied volatilities of stock index options often show ”the
volatility smile”, which decreases with the strike level, and increases with the
time to maturity „ . There are various proposed extensions of this GBM model
to account for ”the volatility smile”. One approach is to incorporate a stochas-
tic volatility factor, Hull and White (1987); another allows for discontinuous
jumps in the stock price, Merton (1976). However, these extensions cause sev-
eral practical di¬culties. For example, they violate the risk-neutral condition.
7.1 Implied Binomial Trees 147


The IBT technique proposed by Rubinstein (1994), Derman and Kani (1994),
Dupire (1994), and Barle and Cakici (1998) account for this phenomenon.
These papers assume the stock prices in the future are generated by a modi¬ed
random walk where the underlying asset has a variable volatility that depends
on both stock price and time. Since the implied binomial trees allow for non-
constant volatility σ = σ(St , t), they are in fact modi¬cations of the original
Cox, Ross and Rubinstein (1979) binomial trees. The IBT construction uses
the observable market option prices in order to estimate the implied distribu-
tion. It is therefore nonparametric in nature. Alternative approaches may be
based on the kernel method, A¨ ±t-Sahalia, and Lo (1998), nonparametric con-
strained least squares, H¨rdle and Yatchew (2001), and curve-¬tting methods,
a
Jackwerth and Rubinstein (1996).
The CRR binomial tree is the discrete implementation of the GBM process
dSt
= µdt + σdZt , (7.2)
St
where Zt is a standard Wiener process, and µ and σ are constants. Similarly,
the IBT can be viewed as a discretization of the following model in which the
generalized volatility parameter is allowed to be a function of time and the
underlying price,
dSt
= µt dt + σ(St , t)dZt , (7.3)
St
where σ(St , t) is the instantaneous local volatility function. The aim of the
IBT is to construct a discrete approximation of the model on the basis of the
observed option prices yielding the variable volatility σ(St , t). In addition, the
IBT may re¬‚ect a non-constant drift µt .


7.1.1 The Derman and Kani (D & K) algorithm

In the implied binomial tree framework, stock prices, transition probabilities,
and Arrow-Debreu prices (discounted risk-neutral probabilities, see Chapter 8)
at each node are calculated iteratively level by level.
Suppose we want to build an IBT on the time interval [0, T ] with equally spaced
levels, t apart. At t = 0, S0 = S, is the current price of the underlying, and
there are n nodes at the nth level of the tree. Let sn,i be the stock price of
the ith node at the nth level, s1,1 = S and Fn,i = er t sn,i the forward price
at level n + 1 of sn,i at level n, and pn,i the transition probability of making
148 7 How Precise Are Price Distributions Predicted by IBT?

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Figure 7.1. Construction of an implied binomial tree



a transition from node (n, i) to node (n + 1, i + 1). Figure 7.1 illustrates the
construction of an IBT.
We assume the forward price Fn,i satis¬es the risk-neutral condition:

Fn,i = pn,i sn+1,i+1 + (1 ’ pn,i )sn+1,i . (7.4)

Thus the transition probability can be obtained from the following equation:
Fn,i ’ sn+1,i
pn,i = . (7.5)
sn+1,i+1 ’ sn+1,i

The Arrow-Debreu price »n,i , is the price of an option that pays 1 unit payo¬
in one and only one state i at nth level, and otherwise pays 0. In general,
7.1 Implied Binomial Trees 149


Arrow-Debreu prices can be obtained by the iterative formula, where »1,1 = 1
as a de¬nition.
±
e’r t {(1 ’ pn,1 )»n,1 } ,
»n+1,1 =

= e’r t {»n,i pn,i + »n,i+1 (1 ’ pn,i+1 )} , 2 ¤ i ¤ n, (7.6)
»
 n+1,i+1
»n+1,n+1 = e’r t {»n,n pn,n } .



We give an example to illustrate the calculation of Arrow-Debreu prices in a
CRR Binomial tree. Suppose that the current value of the underlying S = 100,
time to maturity T = 2 years, t = 1 year, constant volatility σ = 10%, and
riskless interest rate r = 0.03, and „ = T . The Arrow-Debreu price tree can be
calculated from the stock price tree:

stock price
122.15
110.52
100.00 100.00
90.48
81.88


Arrow-Debreu price

0.37
0.61
1.00 0.44
0.36
0.13

For example, using the CRR method, s2,1 = s1,1 e’σ t = 100 — e’0.1 = 90.48,
and s2,2 = s1,1 eσ t = 110.52, the transition probability p1,1 = 0.61 is obtained
by the formula (7.5), then according to the formula (7.6), »2,1 = e’r t (1 ’
p1,1 ) = 0.36. At the third level, calculate the stock prices according to the
corresponding nodes at the second level, For example, s3,1 = s2,1 · e’σ t =
122.15, s3,2 = s1,1 = 100.
150 7 How Precise Are Price Distributions Predicted by IBT?


Option prices in the Black-Scholes framework are given by:
+∞
’r„
max(ST ’ K, 0) p(St , ST , r, „ )dST ,
C(K, „ ) =e (7.7)
0
+∞
= e’r„ max(K ’ ST , 0) p(St , ST , r, „ )dST ,
P (K, „ ) (7.8)
0

where C(K, „ ) and P (K, „ ) are call option price and put option price respec-
tively, and K is the strike price. In the IBT, option prices are calculated
analogously for „ = n t,
n+1
»n+1,i max(sn+1,i ’ K, 0),
C(K, n t) = (7.9)
i=1
n+1
»n+1,i max(K ’ sn+1,i , 0).
P (K, n t) = (7.10)
i=1

Using the risk-neutral condition (7.4) and the discrete option price calculation
from (7.9) or (7.10), one obtains the iteration formulae for constructing the
IBT.
There are (2n + 1) parameters which de¬ne the transition from the nth to
the (n + 1)th level of the tree, i.e., (n + 1) stock prices of the nodes at the
(n + 1)th level, and n transition probabilities. Suppose (2n ’ 1) parameters
corresponding to the nth level are known, the sn+1,i and pn,i corresponding to
the (n + 1)th level can be calculated depending on the following principles:
We always start from the center nodes in one level, if n is even, de¬ne sn+1,i =
s1,1 = S, for i = n/2 + 1, and if n is odd, start from the two central nodes
sn+1,i and sn+1,i+1 for i = (n + 1)/2, and suppose sn+1,i = s2 /sn+1,i+1 =
n,i
2
S /sn+1,i+1 , which adjusts the logarithmic spacing between sn,i and sn+1,i+1
to be the same as that between sn,i and sn+1,i . This principle yields the
calculation formula of sn+1,i+1 , see Derman and Kani (1994),

S{er t C(S, n t) + »n,i S ’ ρu }
sn+1,i+1 = for i = (n + 1)/2. (7.11)
»n,i Fn,i ’ er t C(S, n t) + ρu

Here ρu denotes the following summation term
n
»n,j (Fn,j ’ sn,i ),
ρu = (7.12)
j=i+1
7.1 Implied Binomial Trees 151


C(K, „ ) is the interpolated value for a call struck today at strike price K and
time to maturity „ . In the D & K construction, the interpolated option price
entering (7.11) is based on a CRR binomial tree with constant parameters
σ = σimp (K, „ ), where the BS implied volatility σimp (K, „ ) can be calculated
from the known market option prices. Calculating interpolated option prices
by the CRR method has a drawback, it is computational intensive.
Once we have the initial nodes™ stock prices, according to the relationships
among the di¬erent parameters, we can continue to calculate those at higher
nodes (n + 1, j), j = i + 2, . . . n + 1 and transition probabilities one by one
using the formula:

sn,i {er t
t) ’ ρu } ’ »n,i sn,i (Fn,i ’ sn+1,i )
C(sn,i , n
sn+1,i+1 = , (7.13)
{er t C(s t) ’ ρu } ’ »n,i (Fn,i ’ sn+1,i )
n,i , n

where the de¬nition of ρu is the same as (7.12).

Similarly, we are able to continue to calculate the parameters at lower nodes
(n + 1, j), j = i ’ 1, . . . , 1 according to the following recursion:

sn,i+1 {er t P (sn,i , n t) ’ ρl } ’ »n,i sn,i (Fn,i ’ sn+1,i+1 )
sn+1,i = , (7.14)
{er t P (sn,i , n t) ’ ρl } + »n,i (Fn,i ’ sn+1,i+1 )
i’1
j=1 »n,j (sn,i ’ Fn,j ), and P (K, „ ) is similar to
where ρl denotes the sum
C(K, „ ), again these option prices are obtained by the CRR binomial tree
generated from market options prices.


7.1.2 Compensation

In order to avoid arbitrage, the transition probability pn,i at any node should
lie between 0 and 1, it makes therefore sense to limit the estimated stock prices

Fn,i < sn+1,i+1 < Fn,i+1 . (7.15)
If the stock price at any node does not satisfy the above inequality, we rede¬ne
it by assuming that the di¬erence of the logarithm of the stock prices between
this node and its adjacent node is equal to the corresponding two nodes at the
previous level, i.e.,

log(sn+1,i+1 /sn+1,i ) = log(sn,i /sn,i’1 ).
152 7 How Precise Are Price Distributions Predicted by IBT?


Sometimes, the obtained price still does not satisfy inequality (7.15), then we
choose the average of Fn,i and Fn,i+1 as a proxy for sn+1,i+1 .
In fact, the product of the Arrow-Debreu prices »n,i at the nth level with the
in¬‚uence of interest rate er(n’1) t can be considered as a discrete estimation
of the implied distribution, the SPD, p(ST , St , r, „ ) at „ = (n ’ 1) t. In the
case of the GBM model with constant volatility, this density is corresponding
to (7.1).
After the construction of an IBT, we know all stock prices, transition proba-
bilities, and Arrow-Debreu prices at any node in the tree. We are thus able
to calculate the implied local volatility σloc (sn,i , m t) (which describes the
structure of the second moment of the underlying process) at any level m as a
discrete estimation of the following conditional variance at s = sn,i , „ = m t.
Under the risk-neutral assumption
2
Var(log St+„ |St = s)
σloc (s, „ ) =

(log St+„ ’ E log St+„ )2 p(St+„ |St = s) dSt+„
=

(log St+„ ’ E log St+„ )2 p(St , St+„ r, „ ) dSt+„ . (7.16)
=


In the IBT construction, the discrete estimation can be calculated as:
sn+1,i+1
pn,i (1 ’ pn,i ) log
σloc (sn,i , t) = .
sn+1,i

Analogously, we can calculate the implied local volatility at di¬erent times. In
general, if we have calculated the transition probabilities pj , j = 1, . . . , m from
the node (n, i) to the nodes (n + m, i + j), j = i, . . . , m, then with
m
mean = E(log(S(n+m’1) t )|S(n’1) = sn,i ) = pj log(sn+m,i+j ),
t
j=1

m
2
pj (log(sn+m,i+j ) ’ mean)) . (7.17)
σloc (sn,i , m t) =
j=1


Notice that the instantaneous volatility function used in (7.3) is di¬erent from
the BS implied volatility function de¬ned in (7.16), but in the GBM they are
identical.
7.1 Implied Binomial Trees 153


If we choose t small enough, we obtain the estimated SPD at ¬xed time to
maturity, and the distribution of implied local volatility σloc (s, „ ). Notice that
the BS implied volatility σimp (K, „ ) (which assumes Black-Scholes model is
established (at least locally)) and implied local volatility σloc (s, „ ) is di¬erent,
they have di¬erent parameters, and describe di¬erent characteristics of the
second moment.


7.1.3 Barle and Cakici (B & C) algorithm

Barle and Cakici (1998) proposed an improvement of the Derman and Kani
construction. The major modi¬cation is the choice of the stock price of the
central nodes in the tree: their algorithm takes the riskless interest rate into
account. If (n + 1) is odd, then sn+1,i = s1,1 er n t = Ser n t for i = n/2 + 1,
if (n + 1) is even, then start from the two central nodes sn+1,i and sn+1,i+1
2
for i = (n + 1)/2, and suppose sn+1,i = Fn,i /sn+1,i+1 . Thus sn+1,i can be
calculated as:


»n,i Fn,i ’ {er t
C(Fn,i , n t) ’ ρu }
sn+1,i = Fn,i for i = (n + 1)/2, (7.18)
»n,i Fn,i + {er t C(F
n,i , n t) ’ ρu }


where C(K, „ ) is de¬ned as in the Derman and Kani algorithm, and the ρu is
n
»n,j (Fn,j ’ Fn,i ).
ρu = (7.19)
j=i+1


After stock prices of the initial nodes are obtained, then continue to calculate
those at higher nodes (n + 1, j), j = i + 2, . . . n + 1 and transition probabilities
one by one using the following recursion:


sn+1,i {er t C(Fn,i , n t) ’ ρu } ’ »n,i Fn,i (Fn,i ’ sn+1,i )
sn+1,i+1 = , (7.20)
{er t C(Fn,i , n t) ’ ρu } ’ »n,i (Fn,i ’ sn+1,i )

where ρu is as in (7.19), pn,i is de¬ned as in (7.5).

Similarly, continue to calculate the parameters iteratively at lower nodes (n +
154 7 How Precise Are Price Distributions Predicted by IBT?


1, j), j = i ’ 1, . . . 1.

»n,i Fn,i (sn+1,i+1 ’ Fn,i ) ’ sn+1,i+1 {er t P (Fn,i , n t) ’ ρl }
sn+1,i = , (7.21)
»n,i (sn+1,i+1 ’ Fn,i ) ’ {er t P (Fn,i , n t) ’ ρl }
i’1
where ρl denotes the sum j=1 »n,j (Fn,i ’ Fn,j ). Notice that in (7.13) and
(7.14), C(K, „ ) and P (K, „ ) denote the Black-Scholes call and put option prices,
this construction makes the calculation faster than the interpolation technique
based on the CRR method.
The balancing inequality (7.15) and a rede¬nition are still used in the Barle
and Cakici algorithm for avoiding arbitrage: the algorithm uses the average of
Fn,i and Fn,i+1 as the re-estimation of sn+1,i+1 .


7.2 A Simulation and a Comparison of the SPDs
The example used here to show the procedure of generating the IBT, is taken
from Derman and Kani (1994). Assume that the current value of the stock is
S = 100, the annually compounded riskless interest rate is r = 3% per year
for all time expirations, the stock has zero dividend. The annual BS implied
volatility of an at-the-money call is assumed to be σ = 10%, and the BS
implied volatility increases (decreases) linearly by 0.5 percentage points with
every 10 point drop (rise) in the strike. From the assumptions, we see that
σimp (K, „ ) = 0.15 ’ 0.0005 K.
In order to investigate the precision of the SPD estimation obtained from the
IBT, we give a simulation example assuming that the stock price process is
generated by the stochastic di¬erential equation model (7.3), with an instan-
taneous local volatility function σ(St , t) = 0.15 ’ 0.0005 St , µt = r = 0.03. We
may then easily compare the SPD estimations obtained from the two di¬erent
methods.


7.2.1 Simulation using Derman and Kani algorithm

With the XploRe quantlet XFGIBT01.xpl, using the assumption on the BS
implied volatility surface, we obtain the following one year stock price implied
binomial tree, transition probability tree, and Arrow-Debreu price tree.
7.2 A Simulation and a Comparison of the SPDs 155



XFGIBT01.xpl



Derman and Kani one year (four step) implied binomial tree

stock price
119.91
115.06
110.04 110.06
105.13 105.13
100.00 100.00 100.00
95.12 95.12
89.93 89.92
85.22
80.01

transition probability
0.60
0.58
0.59 0.59
0.56 0.56
0.59 0.59
0.54
0.59

Arrow-Debreu price
0.111
0.187
0.327 0.312
0.559 0.405
1.000 0.480 0.343
0.434 0.305
0.178 0.172
0.080
0.033


This IBT is corresponding to „ = 1 year, and t = 0.25 year, which shows the
stock prices, and the elements at the jth column are corresponding to the stock
156 7 How Precise Are Price Distributions Predicted by IBT?


prices of the nodes at the (j ’ 1)th level in the tree. The second one, its (n, j)
element is corresponding to the transition probability from the node (n, j) to
the nodes (n + 1, j + 1). The third tree contains the Arrow-Debreu prices of
the nodes. Using the stock prices together with Arrow-Debreu prices of the
nodes at the ¬nal level, a discrete approximation of the implied distribution
can be obtained. Notice that by the de¬nition of the Arrow-Debreu price, the
risk neutral probability corresponding to each node should be calculated as the
product of the Arrow-Debreu price and the factor er„ .
If we choose small enough time steps, we obtain the estimation of the implied
price distribution and the implied local volatility surface σloc (s, „ ). We still use
the same assumption on the BS implied volatility surface as above here, which
means σimp (K, „ ) = 0.15 ’ 0.0005 K, and assume S0 = 100, r = 0.03, T = 5
year.

XFGIBT02.xpl



Two ¬gures are generated by running the quantlet XFGIBT02.xpl, Figure 7.2
shows the plot of the SPD estimation resulting from ¬tting an implied ¬ve-year
tree with 20 levels. The implied local volatilities σloc (s, „ ) in the implied tree at
di¬erent time to maturity and stock price levels is shown in Figure 7.3, which
obviously decreases with the stock price and increases with time to maturity
as expected.


7.2.2 Simulation using Barle and Cakici algorithm

The Barle and Cakici algorithm can be applied in analogy to Derman and
Kani™s. The XploRe quantlets used here are similar to those presented in
Section 7.2.1, one has to replace the quantlet IBTdk by IBTdc. The following
¬gure displays the one-year (four step) stock price tree, transition probability
tree, and Arrow-Debreu tree. Figure 7.4 presents the plot of the estimated SPD
by ¬tting a ¬ve year implied binomial tree with 20 levels to the volatility smile
using Barle and Cakici algorithm, and Figure 7.5, shows the characteristics of
the implied local volatility surface of the generated IBT, decreases with the
stock price, and increases with time.
7.2 A Simulation and a Comparison of the SPDs 157


Estimated Implied Distribution
0.15
probability*0.1
0.1
0.05
0




50 100 150
stock price

Figure 7.2. SPD estimation by the Derman and Kani IBT.


Implied Local Volatility Surface
 




5.50
4.25
¡




0.23
3.00 0.16
1.75 0.08


50.00 75.00 100.00 125.00



Figure 7.3. Implied local volatility surface estimation by the Derman
and Kani IBT.


Barle and Cakici one year implied binomial tree

stock price
158 7 How Precise Are Price Distributions Predicted by IBT?


123.85
117.02
112.23 112.93
104.84 107.03
100.00 101.51 103.05
96.83 97.73
90.53 93.08
87.60
82.00


transition probability
0.46
0.61
0.38 0.48
0.49 0.49
0.64 0.54
0.36
0.57


Arrow-Debreu price
0.050
0.111
0.185 0.240
0.486 0.373
1.000 0.619 0.394
0.506 0.378
0.181 0.237
0.116
0.050




7.2.3 Comparison with Monte-Carlo Simulation

We now compare the SPD estimation at the ¬fth year obtained by the two IBT
methods with the estimated density function of the Monte-Carlo simulation
of St , t = 5 generated from the model (7.3), where σ(St , t) = 0.15 ’ 0.0005 St ,
7.2 A Simulation and a Comparison of the SPDs 159


Estimated Implied Distribution
0.15
probability*0.1
0.1
0.05
0




50 100 150
stock price

Figure 7.4. SPD estimation by the Barle and Cakici IBT.




Implied Local Volatility Surface


5.50
4.25 0.24
3.00 0.16
1.75 0.08

50.00 75.00 100.00 125.00




Figure 7.5. Implied local volatility surface by the Barle and Cakici
IBT.


µt = r = 0.03. We use the Milstein scheme, Kloeden, Platen and Schurz (1994)
to perform the discrete time approximation in (7.3). It has strong convergence
rate δ 1 . We have set the time step with δ = 1/1000 here.
160 7 How Precise Are Price Distributions Predicted by IBT?


In order to construct the IBT, we calculate the option prices corresponding
to each node at the implied tree according to their de¬nition by Monte-Carlo
simulation.

XFGIBT03.xpl XFGIBTcdk.xpl XFGIBTcbc.xpl


Estimated State Price Density
0.15
probability*0.1
0.1
0.05
0




50 100 150 200
stock price
Figure 7.6. SPD estimation by Monte-Carlo simulation, and its 95%
con¬dence band, the B & C IBT, from the D & K IBT (thin), level
=20, T = 5 year, t = 0.25 year.


Here we use the quantlets XFGIBTcdk.xpl and XFGIBTcbc.xpl. These
two are used to construct the IBT directly from the option price function,
not starting from the BS implied volatility surface as in quantlets IBTdk and
IBTbc. In the data ¬le ”IBTmcsimulation20.dat”, there are 1000 Monte-Carlo
simulation samples for each St in the di¬usion model (7.3), for t = i/4 year,
i = 1, ...20, from which we calculate the simulated values of the option prices
according to its theoretical de¬nition and estimate the density of St , T = 5
year as the SPD estimation at the ¬fth year.
From the estimated distribution shown in the Figures 7.2.3, we observe their
deviation from the log-normal characteristics according to their skewness and
kurtosis. The SPD estimation obtained from the two IBT methods coincides
with the estimation obtained from the Monte-Carlo simulation well, the di¬er-
ence between the estimations obtained from the two IBTs is not very large.
On the other hand, we can also estimate the implied local volatility surface
from the implied binomial tree, and compare it with the one obtained by the
7.2 A Simulation and a Comparison of the SPDs 161


simulation. Compare Figure 7.7 and Figure 7.8 with Figure 7.9, and notice that
in the ¬rst two ¬gures, some edge values cannot be obtained directly from the
¬ve-year IBT. However, the three implied local volatility surface plots all actu-
ally coincide with the volatility smile characteristic, the implied local volatility
of the out-the-money options decreases with the increasing stock price, and
increase with time.

Implied Local Volatility Surface




5.50
4.25 0.22
3.00 0.14
1.75 0.07


50.00 75.00 100.00 125.00



Figure 7.7. Implied local volatility surface of the simulated model,
calculated from D& K IBT.


We use the data ¬le ”IBTmcsimulation50.dat” to obtain an estimated BS
implied volatility surface. There are 1000 Monte-Carlo simulation samples for
each St in the di¬usion model (7.3), for t = i/10 year in it, i = 1, ...50, because
we can calculate the BS implied volatility corresponding to di¬erent strike
prices and time to maturities after we have the estimated option prices corre-
sponding to these strike price and time to maturity levels. Figure 7.10 shows
that the BS implied volatility surface of our example re¬‚ects the characteris-
tics that the BS implied volatility decrease with the strike price. But this BS
implied volatility surface does not change with time a lot, which is probably
due to our assumption about the local instantaneous volatility function, which
only changes with the stock price.
XFGIBT04.xpl
162 7 How Precise Are Price Distributions Predicted by IBT?


Implied Local Volatility Surface




5.50
4.25 0.20
3.00 0.13
1.75 0.07


50.00 75.00 100.00 125.00



Figure 7.8. Implied local volatility surface of the simulated model,
calculated from B& C IBT.




7.3 Example “ Analysis of DAX data
We now use the IBT to forecast the future price distribution of the real stock
market data. We use DAX index option prices data at January 4, 1999, which
are included in MD*BASE, a database located at CASE (Center for Applied
Statistics and Economics) at Humboldt-Universit¨t zu Berlin, and provide some
a
dataset for demonstration purposes. In the following program, we estimate the
BS implied volatility surface ¬rst, while the quantlet volsurf, Fengler, H¨rdle
a
and Villa (2001), is used to obtain this estimation from the market option
prices, then construct the IBT using Derman and Kani method and calculate
the interpolated option prices using CRR binomial tree method. Fitting the
function of option prices directly from the market option prices is hardly ever
attempted since the function approaches a value of zero for very high strike
prices and option prices are bounded by non-arbitrage conditions.
7.3 Example “ Analysis of DAX data 163


Implied Local Volatility Surface
 




5.00
3.88 0.28
2.75 0.20
1.62 0.13

50.00 75.00 100.00 125.00



Figure 7.9. Implied local volatility surface of the simulated model,
calculated from Monte-Carlo simulation.



XFGIBT05.xpl



Figure 7.11 shows the price distribution estimation obtained by the Barle and
Cakici IBT, for „ = 0.5 year. Obviously, the estimated SPD by the Derman
and Kani IBT can be obtained similarly. In order to check the precision of
the estimated price distribution obtained by the IBT method, we compare it
to use DAX daily prices between January 1, 1997, and January 4. 1999. The
historical time series density estimation method described in A¨
±t-Sahalia, Wang
and Yared (2000) is used here. Notice that Risk-neutrality implies two kinds
of SPD should be equal, historical time series SPD is in fact the conditional
density function of the di¬usion process. We obtain the historical time series
SPD estimation by the following procedure:

1. Collect stock prices time series
164 7 How Precise Are Price Distributions Predicted by IBT?


Implied Volatility Surface




strike
76.00
92.72
109.44
126.16
142.88
159.60
0.13

0.11

0.08
vola
0.05

0.03
5.12
4.14
3.15
2.16
0.00
1.18
0.19 maturity




Figure 7.10. BS implied volatility surface estimation by Monte-Carlo
simulation.


2. Assume this time series is a sample path of the di¬usion process
dSt
= µt dt + σ(St , t)dZt ,
St
where dZt is a Wiener process with mean zero and variance equal to dt.
3. Estimate di¬usion function σ(·, ·) in the di¬usion process model using
nonparametric method from stock prices time series
4. Make Monte-Carlo simulation for the di¬usion process with drift function
is interest rate and estimated di¬usion function
7.3 Example “ Analysis of DAX data 165


Estimated Implied Distribution
0.03
0.03
probability*E-2
0.02
0.01
0.01
0
0




5000 10000
stock price

Figure 7.11. The estimated stock price distribution, „ = 0.5 year.



5. Estimate conditional density function g = p(ST |St , µ, σ ) from Monte-
ˆˆ
Carlo simulated process

From Figure 7.12 we conclude that the SPD estimated by the Derman and Kani
IBT and the one obtained by Barle and Cakici IBT can be used to forecast fu-
ture SPD. The SPD estimated by di¬erent methods sometimes have deviations
on skewness and kurtosis. In fact the detection of the di¬erence between the
historical time series SPD estimation and the SPD recovered from daily option
prices may be used as trading rules, see Table 7.1 and Chapter 9. In Table 7.1,
SPD estimated from daily option prices data set is expressed by f and the time
series SPD is g. A far out of the money (OTM) call/put is de¬ned as one whose
exercise price is 10% higher (lower) than the future price. While a near OTM
call/put is de¬ned as one whose exercise price is 5% higher (lower) but 10%
lower(higher)than the future price. When skew(f ) < skew(g), agents appar-
ently assign a lower probability to high outcomes of the underlying than would
be justi¬ed by the time series SPD (see Figure 7.13). Since for call options
only the right ˜tail™ of the support determines the theoretical price the latter is
smaller than the price implied by di¬usion process using the time series SPD.
That is we buy calls. The same reason applies to put options.
166 7 How Precise Are Price Distributions Predicted by IBT?


State Price Density Estimation




0.05 0.1 0.15 0.2 0.25 0.3
probability*E-3
0




5000 10000
stock price

Figure 7.12. SPD estimation by three methods, by historical estima-
tion, and its 95% con¬dence band (dashed), by B & C IBT, and by D
& K IBT (thin), „ = 0.5 year.

Trading Rules to exploit SPD di¬erences
Skewness (S1) skew(f )< skew(g) sell OTM put,
buy OTM call
Trade (S2) skew(f ) > skew(g) buy OTM put
sell OTM call
Kurtosis (K1) kurt(f )> kurt(g) sell far OTM and ATM
buy near OTM options
Trade (K2) kurt(f ) < kurt(g) buy far OTM and ATM,
sell near OTM options

Table 7.1. Trading Rules to exploit SPD di¬erences.


From the simulations and real data example, we ¬nd that the implied binomial
tree is an easy way to assess the future stock prices, capture the term structure
of the underlying asset, and replicate the volatility smile. But the algorithms
still have some de¬ciencies. When the time step is chosen too small, negative
transition probabilities are encountered more and more often. The modi¬cation
of these values loses the information about the smile at the corresponding
nodes. The Barle and Cakici algorithm is a better choice when the interest
rate is high.Figure 7.15 shows the deviation of the two methods under the
7.3 Example “ Analysis of DAX data 167


Skewness Trade
g
0.4




f=SPD
probability*0.1
0.3
0.2




sell put
0.1




¡




buy call
0




1 2 3 4 5 6 7
 




stock price
Figure 7.13. Skewness Trade, skew(f )< skew(g).


Kurtosis Trade
0.5




f
0.4




g
probability*0.1
0.3
0.2
0.1




sell sell

buy sell buy
0




-1 0 1 2 3 4 5
 




stock price
Figure 7.14. Kurtosis Trade, kurt(f )> kurt(g).



situation that r = 0.2. When the interest rate is a little higher, Barle and
Cakici algorithm still can be used to construct the IBT while Derman and
Kani™s cannot work any more. The times of the negative probabilities appear
are fewer than Derman and Kani construction (see Jackwerth (1999)).
168 7 How Precise Are Price Distributions Predicted by IBT?


Estimated State Price Density




0.3
probability*0.1
0.2
0.1
0




100 120 140 160
stock price
Figure 7.15. SPD estimation by Monte-Carlo simulation, and its 95%
con¬dence band (dashed), the B & C IBT, from the D & K IBT (thin),
level =20, „ = 1 year, r = 0.20.



Besides its basic purpose of pricing derivatives in consistency with the market
prices, IBT is useful for other kinds of analysis, such as hedging and calculating
of implied probability distributions and volatility surfaces. It estimate the
future price distribution according to the historical data. On the practical
application aspect, the reliability of the approach depends critically on the
quality of the estimation of the dynamics of the underlying price process, such
as BS implied volatility surface obtained from the market option prices.
The IBT can be used to produce recombining and arbitrage-free binomial trees
to describe stochastic processes with variable volatility. However, some serious
limitations such as negative probabilities, even though most of them appeared
at the edge of the trees. Overriding them causes loss of the information about
the smile at the corresponding nodes. These defects are a consequence of the
requirement that a continuous di¬usion is approximated by a binomial process.
Relaxation of this requirement, using multinomial trees or varinomial trees is
possible.
7.3 Example “ Analysis of DAX data 169


Bibliography

±t-Sahalia, Y. and Lo, A. (1998). Nonparametric Estimation of State-Price
Densities Implicit in Financial Asset Prices, Journal of Finance, 53: 499“
547.


±t-Sahalia, Y. , Wang, Y. and Yared, F.(2001). Do Option Markets Correctly
Price the Probabilities of Movement of the Underlying Asset? Journal of
Econometrics, 102: 67“110.
Barle, S. and Cakici, N. (1998). How to Grow a Smiling Tree The Journal of
Financial Engineering, 7: 127“146.
Bingham, N.H. and Kiesel, R. (1998). Risk-neutral Valuation: Pricing and
Hedging of Financial Derivatives, Springer Verlag, London.
Cox, J., Ross, S. and Rubinstein, M. (1979). Option Pricing: A simpli¬ed
Approach, Jouranl of Financial Economics 7: 229“263.
Derman, E. and Kani, I. (1994). The Volatility Smile and Its Implied Tree
http://www.gs.com/qs/
Derman, E. and Kani, I. (1998). Stochastic Implied Trees: Arbitrage Pric-
ing with Stochastic Term and Strike Structure of Volatility, International
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Dupire, B. (1994). Pricing with a Smile, Risk 7: 18“20.
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a
Volatilities: A Common Principal Components Approach, Discussion pa-
per 38, Sonderforschungsbereich 373, Humboldt-Universit¨t zu Berlin.
a
H¨rdle, W., Hl´vka, Z. and Klinke, S. (2000). XploRe Application Guide,
a a
Springer Verlag, Heidelberg.
H¨rdle,W. and Yatchew, A. (2001). Dynamic Nonparametric State price Den-
a
sity Estimation using Constrained least Squares and the Bootstrap, Dis-
cussion paper 1, Sonderforschungsbereich 373, Humboldt-Universit¨t zu
a
Berlin.
Hull, J. and White, A. (1987). The Pricing of Options on Assets with Stochastic
Volatility, Journal of Finance 42: 281“300.
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Jackwerth, J. (1999). Optional-Implied Risk-Neutral Distributions and Implied
Binomial Trees: A Literature Review, Journal of Finance 51: 1611“1631.

Jackwerth, J. and Rubinstein, M. (1996). Recovering Probability Distributions
from Option Prices, Journal of Finance 51: 1611“1631.
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Through Computer Experiments, Springer Verlag, Heidelberg.
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continuous, Journal of Financial Economics January-March: 125“144.

Rubinstein, M. (1994). Implied Binomial Trees. Journal of Finance 49: 771“
818.
8 Estimating State-Price Densities
with Nonparametric Regression
Kim Huynh, Pierre Kervella and Jun Zheng




8.1 Introduction
Derivative markets o¬er a rich source of information to extract the market™s
expectations of the future price of an asset. Using option prices, one may derive
the whole risk-neutral probability distribution of the underlying asset price at
the maturity date of the options. Once this distribution also called State-Price
Density (SPD) is estimated, it may serve for pricing new, complex or illiquid
derivative securities.
There exist numerous methods to recover the SPD empirically. They can be
separated in two classes:

• methods using option prices as identifying conditions
• methods using the second derivative of the call pricing function with
respect to K

The ¬rst class includes methods which consist in estimating the parameters of a
mixture of log-normal densities to match the observed option prices, Melick and
Thomas (1997). Another popular approach in this class is the implied binomial
trees method, see Rubinstein (1994), Derman and Kani (1994) and Chapter 7.
Another technique is based on learning networks suggested by Hutchinson, Lo
and Poggio (1994), a nonparametric approach using arti¬cial neural networks,
radial basis functions, and projection pursuits.
The second class of methods is based on the result of Breeden and Litzen-
berger (1978). This methodology is based on European options with identical
172 8 Estimating State-Price Densities with Nonparametric Regression


time to maturity, it may therefore be applied to fewer cases than some of the
techniques in the ¬rst class. Moreover, it also assumes a continuum of strike
prices on R+ which can not be found on any stock exchange. Indeed, the
strike prices are always discretely spaced on a ¬nite range around the actual
underlying price. Hence, to handle this problem an interpolation of the call
pricing function inside the range and extrapolation outside may be performed.
In the following, a semiparametric technique using nonparametric regression of
the implied volatility surface will be introduced to provide this interpolation
task. A new approach using constrained least squares has been suggested by
Yatchew and H¨rdle (2002) but will not be explored here.
a
The concept of Arrow-Debreu securities is the building block for the analysis of
economic equilibrium under uncertainty. Rubinstein (1976) and Lucas (1978)
used this concept as a basis to construct dynamic general equilibrium models
in order to determine the price of assets in an economy. The central idea of this
methodology is that the price of a ¬nancial security is equal to the expected
net present value of its future payo¬s under the risk-neutral probability density
function (PDF). The net present value is calculated using the risk-free interest
rate, while the expectation is taken with respect to the weighted-marginal-rate-
of-substitution PDF of the payo¬s. The latter term is known as the state-price
density (SPD), risk-neutral PDF, or equivalent martingale measure. The price
of a security at time t (Pt ) with a single liquidation date T and payo¬ Z(ST )
is then:

E— [Z(ST )]
’rt,„ „ ’rt,„ „
Z(ST )ft— (ST )dST
Pt = e =e (8.1)
t
’∞

where E— is the conditional expectation given the information set in t under the
t
equivalent martingale probability, ST is the state variable, rt,„ is the risk-free
rate at time t with time to maturity „ , and ft— (ST ) is the SPD at time t for
date T payo¬s.
Rubinstein (1985) shows that if one has two of the three following pieces of
information:

• representative agent™s preferences
• asset price dynamics or its data-generating process
• SPD

then one can recover the third. Since the agent™s preferences and the true data-
8.2 Extracting the SPD using Call-Options 173


generating process are unknown, a no-arbitrage approach is used to recover the
SPD.


8.2 Extracting the SPD using Call-Options
Breeden and Litzenberger (1978) show that one can replicate Arrow-Debreu
prices using the concept of butter¬‚y spread on European call options. This
spread entails selling two call options at exercise price K, buying one call option
at K ’ = K ’ ∆K and another at K + = K + ∆K, where ∆K is the stepsize
between the adjacent call strikes. These four options constitute a butter¬‚y
spread centered on K. If the terminal underlying asset value ST is equal to K
1
then the payo¬ Z(·) of ∆K of such butter¬‚y spreads is de¬ned as:


u1 ’ u2
Z(ST , K; ∆K) = P (ST ’„ , „, K; ∆K)|„ =0 = =1 (8.2)
∆K ST =K,„ =0


where

u1 = C(ST ’„ , „, K + ∆K) ’ C(ST ’„ , „, K),
u2 = C(ST ’„ , „, K) ’ C(ST ’„ , „, K ’ ∆K).

C(S, „, K) denotes the price of a European call with an actual underlying price
S, a time to maturity „ and a strike price K. Here, P (ST ’„ , „, K; ∆K) is the
1
corresponding price of this security ( ∆K — butterf ly spread(K; ∆K)) at time
T ’ „.
As ∆K tends to zero, this security becomes an Arrow-Debreu security paying 1
if ST = K and zero in other states. As it is assumed that ST has a continuous
distribution function on R+ , the probability of any given level of ST is zero
and thus, in this case, the price of an Arrow-Debreu security is zero. However,
1
dividing one more time by ∆K, one obtains the price of ( (∆K)2 — butterf ly
spread(K; ∆K)) and as ∆K tends to 0 this price tends to f — (ST )e’rt,„ for
ST = K. Indeed,

P (St , „, K; ∆K)
= f — (ST )e’rt,„ .
lim (8.3)
∆K
∆K’0
K=ST
174 8 Estimating State-Price Densities with Nonparametric Regression


This can be proved by setting the payo¬ Z1 of this new security
1
(∆K ’ |ST ’ K|)1(ST ∈ [K ’ ∆K, K + ∆K])
Z1 (ST ) =
(∆K)2

in (8.1) and letting ∆K tend to 0. Indeed, one should remark that:
K+∆K
(∆K ’ |ST ’ K|)dST = (∆K)2 .
∀(∆K) :
K’∆K


If one can construct these ¬nancial instruments on a continuum of states (strike
prices) then at in¬nitely small ∆K a complete state pricing function can be
de¬ned.
Moreover, as ∆K tends to zero, this price will tend to the second derivative of
the call pricing function with respect to the strike price evaluated at K:
u1 ’ u2
P (St , „, K; ∆K)
lim = lim
∆K’0 (∆K)2
∆K
∆K’0

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