Bibliography

A¨

±t-Sahalia, Y. and Lo, A. W. (1998). Nonparametric Estimation of State-Price

Densities Implicit in Financial Assets, Journal of Finance Vol. LIII, 2,

pp. 499“547.

A¨

±t-Sahalia, Y. and Lo, A. W. (2000). Nonparametric Risk management and

implied risk aversion, Journal of Econometrics 94, pp. 9“51.

Dumas, B., Fleming, J. and Whaley, R. E. (1998). Implied Volatility Functions:

Empirical Tests, Journal of Finance Vol. LIII, 6, pp. 2059“2106.

Fengler, M. R., H¨rdle, W. and Villa, Chr. (2001). The Dynamics of Implied

a

Volatilities: A Common Principal Components Approach, SfB 373 Discus-

sion Paper No. 2001/38, HU Berlin.

Flury, B. (1988). Common Principle Components Analysis and Related Multi-

variate Models, Wiley Series in Probability and Mathematical Statistics,

John Wiley & Sons, New York.

Flury, B. and Gautschi, W. (1986). An Algorithm for simultaneous orthogonal

transformation of several positive de¬nite symmetric matrices to nearly

diagonal form SIAM Journal on Scienti¬c and Statistical Computing,7,

pp. 169“184.

H¨rdle, W.(1990). Applied Nonparametric Regression, Econometric Society

a

Monographs 19, Cambridge University Press.

H¨rdle, W., M¨ller, M., Sperlich, S. and Werwartz, A. (2002). Non- and

a u

Semiparametric Modelling, Springer, e-book http://www.xplore-stat.de

H¨rdle, W. and Schmidt, P. (2002). Common Factors Governing VDAX Move-

a

ments and the Maximum Loss, Financial Markets and Portfolio Manage-

ment, forthcoming.

Hafner, R. and Wallmeier, M. (2001). The Dynamics of DAX Implied Volatil-

ities, International Quarterly Journal of Finance,1, 1, pp. 1“27.

144 6 The Analysis of Implied Volatilities

Muirhead, R. J. (1982). Aspects of Multivariate Statistics, Wiley Series in

Probability and Mathematical Statistics, John Wiley & Sons, New York.

Redelberger, T. (1994). Grundlagen und Konstruktion des VDAX-Volatilit¨ts-

a

index der Deutsche B¨rse AG, Deutsche B¨rse AG, Frankfurt am Main.

o o

Roll, R. (1977). A Critique of the Asset Pricing Theory™s Tests: Part I, Journal

of Financial Economics,4, pp. 129“176.

Taleb, N. (1997). Dynamic Hedging: Managing Vanilla and Exotic Options,

John Wiley & Sons, New York.

Villa, C. and Sylla, A. (2000). Measuring implied surface risk using PCA

in Franke, J., H¨rdle, W. and Stahl, G.: Measuring Risk in Complex

a

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?

¤¢¢¢

£¡

¥ ¥ © !" " ¥ ¥

¦ # ¦"

"

" ¥ §¥ ©

¥¨§¥

¦ ¦

¥ §¥

¦ !"

¥ ¥ © ! (¥ §¥

¦ # " % ¦

""

" % ¦

(§¥ ¥ ©

'&$§¥

% ¥¦

(¥ ¥

%¦

(¥ ¢()¥ © !

% ¦

(40 ¦¢¥ © !" " 0 §¥

# ¦"

"

" 0 ¥ ©

01¥

¦ ¦

0 ¥

¦ ! 0 §¥ 32

¦ %

0 ¢()¥ ©

¦

6 ¥ © !" " 5 ¥

¦ # ¦"

"

" 5 §¥ ©

5¨§¥

¦ ¦

5 ¥

¦ !" 5 ¥ ©

¦

! " ¥

# ¦"

"

" §¥ ©

1§¥

¦ ¦

¥

¦ ! ¥ ©

¦

7 ¤9$¤ 7 ¥ A@¥

8

G

¤ E DB

C ¥ F

¥F

FH

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

A¨

±t-Sahalia, Y. and Lo, A. (1998). Nonparametric Estimation of State-Price

Densities Implicit in Financial Asset Prices, Journal of Finance, 53: 499“

547.

A¨

±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

Journal of Theroetical and Applied Finance 1: 7“22.

Dupire, B. (1994). Pricing with a Smile, Risk 7: 18“20.

Fengler, M. R., H¨rdle, W. and Villa, Chr. (2001). The Dynamics of Implied

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.

170 7 How Precise Are Price Distributions Predicted by IBT?

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.

Kloeden, P., Platen, E. and Schurz, H. (1994). Numerical Solution of SDE

Through Computer Experiments, Springer Verlag, Heidelberg.

Merton, R. (1976). Option Pricing When Underlying Stock Returns are Dis-

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