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в€‚ 2 Ct (В·)
= . (8.4)
в€‚K 2

Equating (8.3) and (8.4) across all states yields:
в€‚ 2 Ct (В·)
= eв€’rt,П„ П„ ftв€— (ST )
в€‚K 2 K=ST

where rt,П„ denotes the risk-free interest rate at time t with time to maturity П„
and ftв€— (В·) denotes the risk-neutral PDF or the SPD in t. Therefore, the SPD
is deп¬Ѓned as:
2
rt,П„ П„ в€‚ Ct (В·)
в€—
ft (ST ) = e . (8.5)
в€‚K 2 K=ST
This method constitutes a no-arbitrage approach to recover the SPD. No as-
sumption on the underlying asset dynamics are required. Preferences are not
restricted since the no-arbitrage method only assumes risk-neutrality with re-
spect to the underlying asset. The only requirements for this method are
that markets are perfect (i.e. no sales restrictions, transactions costs or taxes
and that agents are able to borrow at the risk-free interest rate) and that
C(В·) is twice diп¬Ђerentiable. The same result can be obtained by diп¬Ђerentiat-
ing (8.1) twice with respect to K after setting for Z the call payoп¬Ђ function
Z(ST ) = (ST в€’ K)+ .
8.2 Extracting the SPD using Call-Options 175

8.2.1 Black-Scholes SPD

The Black-Scholes call option pricing formula is due to Black and Scholes (1973)
and Merton (1973). In this model there are no assumptions regarding prefer-
ences, rather it relies on no-arbitrage conditions and assumes that the evolution
of the underlying asset price St follows a geometric Brownian motion deп¬Ѓned
through
dSt
= Вµdt + ПѓdWt . (8.6)
St

Here Вµ denotes the drift and Пѓ the volatility assumed to be constant.
The analytical formula for the price in t of a call option with a terminal date
T = t + П„ , a strike price K, an underlying price St , a risk-free rate rt,П„ , a
continuous dividend yield Оґt,П„ , and a volatility Пѓ, is:
в€ћ
= eв€’rt,П„ в€—
max(ST в€’ K, 0)fBS,t (ST )dST
CBS (St , K, П„, rt,П„ , Оґt,П„ ; Пѓ)
0
в€’Оґt,П„ П„
О¦(d1 ) в€’ Keв€’rt,П„ П„ О¦(d2 )
= St e
where О¦(В·) is the standard normal cumulative distribution function and
1
log(St /K) + (rt,П„ в€’ Оґt,П„ + 2 Пѓ 2 )П„
в€љ
d1 = ,
ПѓП„
в€љ
= d1 в€’ Пѓ П„ .
d2

As a consequence of the assumptions on the underlying asset price process the
Black-Scholes SPD is a log-normal density with mean (rt,П„ в€’ Оґt,П„ в€’ 1 Пѓ 2 )П„ and
2
variance Пѓ 2 П„ for log(ST /St ):
в€‚ 2 Ct
в€—
fBS,t (ST ) = ert,П„ П„
в€‚K 2 K=ST
[log(ST /St ) в€’ (rt,П„ в€’ Оґt,П„ в€’ 1 Пѓ 2 )П„ ]2
1 2
в€љ exp в€’
= .
2Пѓ 2 П„
2П„
ST 2ПЂПѓ
The risk measures Delta (в€†) and Gamma (О“) are deп¬Ѓned as:
в€‚CBS
def
в€†BS = = О¦(d1 )
в€‚St
2
def в€‚ CBS О¦(d1 )
в€љ
О“BS = =
2
в€‚St St Пѓ П„
176 8 Estimating State-Price Densities with Nonparametric Regression

The Black-Scholes SPD can be calculated in XploRe using the following quant-
let:

bsspd = spdbs(K,s,r,div,sigma,tau)
estimates the Black-Scholes SPD

The arguments are the strike prices (K), underlying price (s), risk-free interest
rate (r), dividend yields (div), implied volatility of the option (sigma), and
the time to maturity (tau). The output consist of the Black-Scholes SPD
(bsspd.fbs), в€† (bsspd.delta), and the О“ (bsspd.gamma) of the call options.
Please note that spdbs can be applied to put options by using the Put-Call
parity.
However, it is widely known that the Black-Scholes call option formula is not
valid empirically. For more details, please refer to Chapter 6. Since the Black-
Scholes model contains empirical irregularities, its SPD will not be consistent
with the data. Consequently, some other techniques for estimating the SPD
without any assumptions on the underlying diп¬Ђusion process have been devel-
oped in the last years.

8.3 Semiparametric estimation of the SPD

8.3.1 Estimating the call pricing function

The use of nonparametric regression to recover the SPD was п¬Ѓrst investigated
by AВЁД±t-Sahalia and Lo (1998). They propose to use the Nadaraya-Watson esti-
mator to estimate the historical call prices Ct (В·) as a function of the following
state variables (St , K, П„, rt,П„ , Оґt,П„ ) . Kernel regressions are advocated because
there is no need to specify a functional form and the only required assumption
is that the function is smooth and diп¬Ђerentiable, HВЁrdle (1990). When the re-
a
gressor dimension is 5, the estimator is inaccurate in practice. Hence, there is
a need to reduce the dimension or equivalently the number of regressors. One
method is to appeal to no-arbitrage arguments and collapse St , rt,П„ and Оґt,П„
into the forward price Ft = St e(rt,П„ в€’Оґt,П„ )П„ in order to express the call pricing
function as:

C(St , K, П„, rt,П„ , Оґt,П„ ) = C(Ft,П„ , K, П„, rt,П„ ). (8.7)
8.3 Semiparametric estimation of the SPD 177

An alternative speciп¬Ѓcation assumes that the call option function is homoge-
neous of degree one in St and K (as in the Black-Scholes formula) so that:

C(St , K, П„, rt,П„ , Оґt,П„ ) = KC(St /K, П„, rt,П„ , Оґt,П„ ). (8.8)

Combining the assumptions of (8.7) and (8.8) the call pricing function can be
further reduced to a function of three variables ( FK , П„, rt,П„ ).
t,П„

Another approach is to use a semiparametric speciп¬Ѓcation based on the Black-
Scholes implied volatility. Here, the implied volatility Пѓ is modelled as a non-
parametric function, Пѓ(Ft,П„ , K, П„ ):

C(St , K, П„, rt,П„ , Оґt,П„ ) = CBS (Ft,П„ , K, П„, rt,П„ ; Пѓ(Ft,П„ , K, П„ )). (8.9)

Empirically the implied volatility function mostly depends on two parameters:
the time to maturity П„ and the moneyness M = K/Ft,П„ . Almost equivalently,
Лњ Лњ
one can set M = St /K where St = St в€’ D and D is the present value of
the dividends to be paid before the expiration. Actually, in the case of a
dividend yield Оґt , we have D = St (1 в€’ eв€’Оґt ). If the dividends are discrete, then
Dti eв€’rt,П„i where ti is the dividend payment date of the ith dividend
D=
ti в‰¤t+П„
and П„i is its maturity.
Therefore, the dimension of the implied volatility function can be reduced to
Пѓ(K/Ft,П„ , П„ ). In this case the call option function is:

C(St , K, П„, rt,П„ , Оґt,П„ ) = CBS (Ft,П„ , K, П„, rt,П„ ; Пѓ(K/Ft,П„ , П„ )). (8.10)

Л†
в€‚ Ct (В·)
Л† Л†
Once a smooth estimate of Пѓ (В·) is obtained, estimates of Ct (В·), в€†t =
Л† в€‚St ,
Л† Л†
в€‚ 2 Ct (В·) в€‚ 2 Ct (В·)
Л†
Л† and ftв€— = ert,П„ П„
О“t = , can be calculated.
2 в€‚K 2
в€‚St

8.3.2 Further dimension reduction

The previous section proposed a semiparametric estimator of the call pricing
function and the necessary steps to recover the SPD. In this section the di-
mension is reduced further using the suggestion of Rookley (1997). Rookley
178 8 Estimating State-Price Densities with Nonparametric Regression

uses intraday data for one maturity and estimates an implied volatility surface
where the dimension are the intraday time and the moneyness of the options.
Here, a slightly diп¬Ђerent method is used which relies on all settlement prices
of options of one trading day for diп¬Ђerent maturities to estimate the implied
volatility surface Пѓ(K/Ft,П„ , П„ ). In the second step, these estimates are used for
a given time to maturity which may not necessarily correspond to the maturity
of a series of options. This method allows one to compare the SPD at diп¬Ђerent
dates because of the п¬Ѓxed maturity provided by the п¬Ѓrst step. This is interesting
if one wants to study the dynamics and the stability of these densities.
Fixing the maturity also allows us to eliminate П„ from the speciп¬Ѓcation of the
implied volatility function. In the following part, for convenience, the deп¬Ѓnition
Лњ
of the moneyness is M = St /K and we denote by Пѓ the implied volatility. The
notation в€‚f (xв€‚xi n ) denotes the partial derivative of f with respect to xi and
1 ,...,x

df (x)
the total derivative of f with respect to x.
dx

Moreover, we use the following rescaled call option function:
Cit
cit = ,
Лњ
St
Лњ
St
Mit = .
Ki

where Cit is the price of the ith option at time t and Ki is its strike price.
The rescaled call option function can be expressed as:

eв€’rП„ О¦(d2 )
= c(Mit ; Пѓ(Mit )) = О¦(d1 ) в€’
cit ,
Mit
log(Mit ) + rt + 1 Пѓ(Mit )2 П„
2
в€љ
d1 = ,
Пѓ(Mit ) П„
в€љ
= d1 в€’ Пѓ(Mit ) П„ .
d2

The standard risk measures are then the following partial derivatives (for no-
tational convenience subscripts are dropped):
в€‚C в€‚C Лњ в€‚c ,
в€†= = = c(M, Пѓ(M )) + S
Лњ Лњ
в€‚S в€‚S в€‚S

в€‚2C в€‚2C 2
в€‚в€† в€‚c Лњв€‚ c.
О“= = = =2 +S
Лњ Лњ Лњ
в€‚S 2
в€‚S в€‚S2 в€‚S2
в€‚S
8.3 Semiparametric estimation of the SPD 179

where
в€‚c dc в€‚M dc 1
= = ,
Лњ Лњ
dM в€‚ S dM K
в€‚S
2
d2 c
в€‚2c 1
= .
Лњ dM 2 K
в€‚S2
The SPD is then the second derivative of the call option function with respect
to the strike price:
2 2
rП„ в€‚ C rП„ Лњ в€‚ c
в€—
f (В·) = e =e S . (8.13)
в€‚K 2 в€‚K 2
The conversion is needed because c(В·) is being estimated not C(В·). The analyt-
ical expression of (8.13) depends on:
2
d2 c
в€‚2c M dc M
= +2
в€‚K 2 dM 2 dM K 2
K

dc
The functional form of is:
dM

dc dd1 О¦ (d2 ) dd2 О¦(d2 )
в€’ eв€’rП„ + eв€’rП„
= О¦ (d1 ) , (8.14)
M2
dM dM M dM
d2 c
while is:
dM 2

2
d2 c d2 d1 dd1
в€’ d1
= О¦ (d1 )
dM 2 dM 2 dM
2
eв€’rП„ О¦ (d2 ) d2 d2 2 dd2 dd2
в€’ в€’ в€’ d2
2
M dM M dM dM
2eв€’rП„ О¦(d2 )
в€’ (8.15)
M3
180 8 Estimating State-Price Densities with Nonparametric Regression

The quantities in (8.14) and (8.15) are a function of the following п¬Ѓrst deriva-
tives:
dd1 в€‚d1 в€‚d1 в€‚Пѓ
= + ,
dM в€‚M в€‚Пѓ в€‚M
dd2 в€‚d2 в€‚d2 в€‚Пѓ
= + ,
dM в€‚M в€‚Пѓ в€‚M
в€‚d1 в€‚d2 1
в€љ,
= =
в€‚M в€‚M MПѓ П„
в€љ
в€‚d1 log(M ) + rП„ П„
в€љ
в€’
= + ,
Пѓ2 П„
в€‚Пѓ 2
в€љ
в€‚d2 log(M ) + rП„ П„
в€љ
в€’ в€’
= .
Пѓ2 П„
в€‚Пѓ 2

For the remainder of this chapter, we deп¬Ѓne:

V = Пѓ(M ),
в€‚Пѓ(M )
V = ,
в€‚M
в€‚ 2 Пѓ(M )
V = . (8.16)
в€‚M 2

The quantities in (8.14) and (8.15) also depend on the following second deriva-
tive functions:
в€љ
d2 d1 1 1 V П„ log(M ) + rП„
в€љ в€љ
=в€’ в€’
+ +V
dM 2 Пѓ2 П„
MПѓ П„ M Пѓ 2
log(M ) + rП„ 1
в€љ в€љ,
в€’
+ V 2V (8.17)
3П„ M Пѓ2 П„
Пѓ

в€љ
d2 d2 1 1 V П„ log(M ) + rП„
в€љ в€љ
=в€’ в€’V
+ +
dM 2 Пѓ2 П„
MПѓ П„ M Пѓ 2
log(M ) + rП„ 1
в€љ в€љ.
в€’
+ V 2V (8.18)
Пѓ3 П„ M Пѓ2 П„

Local polynomial estimation is used to estimate the implied volatility smile and
its п¬Ѓrst two derivatives in (8.16). A brief explanation will be described now.
8.3 Semiparametric estimation of the SPD 181

8.3.3 Local Polynomial Estimation

Consider the following data generating process for the implied volatilities:

Пѓ = g(M, П„ ) + Пѓ в€— (M, П„ )Оµ,

where E(Оµ) = 0, Var(Оµ) = 1. M, П„ and Оµ are independent and Пѓ в€— (m0 , П„0 ) is
the conditional variance of Пѓ given M = m0 , П„ = П„0 . Assuming that all third
derivatives of g exist, one may perform a Taylor expansion for the function g
in a neighborhood of (m0 , П„0 ):

1 в€‚2g
в€‚g
(m в€’ m0 )2
g(m, П„ ) в‰€ g(m0 , П„0 ) (m в€’ m0 ) +
+
2 в€‚M 2
в€‚M m0 ,П„0 m0 ,П„0
2
в€‚g 1в€‚ g
(П„ в€’ П„0 )2
(П„ в€’ П„0 ) +
+
2 в€‚П„ 2
в€‚П„ m0 ,П„0 m0 ,П„0
2
1 в€‚g
(m в€’ m0 )(П„ в€’ П„0 ).
+ (8.19)
2 в€‚M в€‚П„ m0 ,П„0

This expansion suggests an approximation by local polynomial п¬Ѓtting, Fan
and Gijbels (1996). Hence, to estimate the implied volatility at the target
point (m0 , П„0 ) from observations Пѓj (j = 1, . . . , n), we minimize the following
expression:

n
Пѓj в€’ ОІ0 + ОІ1 (Mj в€’ m0 ) + ОІ2 (Mj в€’ m0 )2 + ОІ3 (П„j в€’ П„0 )
j=1
(8.20)
2
2
+ОІ4 (П„j в€’ П„0 ) + ОІ5 (Mj в€’ m0 )(П„j в€’ П„0 ) KhM ,hП„ (Mj в€’ m0 , П„j в€’ П„0 )

where n is the number of observations (options), hM and hП„ are the bandwidth
controlling the neighborhood in each directions and KhM ,hП„ is the resulting
kernel function weighting all observation points. This kernel function may be
a product of two univariate kernel functions.
For convenience use the following matrix deп¬Ѓnitions:
(M1 в€’ m0 )2 (П„1 в€’ П„0 )2
M1 в€’ m0 П„1 в€’ П„0 (M1 в€’ m0 )(П„1 в€’ П„0 )
пЈ« пЈ¶
1
(M2 в€’ m0 )2 (П„2 в€’ П„0 )2
M2 в€’ m0 П„2 в€’ П„0 (M2 в€’ m0 )(П„2 в€’ П„0 ) пЈ·
пЈ¬1
X = пЈ¬. пЈ·,
пЈ¬ пЈ·
. . . . .
пЈ­. . . . . .
. . . . . . пЈё
(Mn в€’ m0 )2 (П„n в€’ П„0 )2
Mn в€’ m0 П„n в€’ П„0 (Mn в€’ m0 )(П„n в€’ П„0 )
1
182 8 Estimating State-Price Densities with Nonparametric Regression

пЈ« пЈ¶ пЈ« пЈ¶
Пѓ1 ОІ0
Пѓ = пЈ­ . пЈё, ОІ = пЈ­ . пЈё.
W = diag{KhM ,hП„ (Mj в€’ m0 , П„j в€’ П„0 )}
. .
and
пЈ¬пЈ· пЈ¬пЈ·
. .
Пѓn ОІ5

Hence, the weighted least squares problem (8.20) can be written as

min (Пѓ в€’ XОІ) W (Пѓ в€’ XОІ) . (8.21)
ОІ

and the solution is given by
в€’1
Л†
ОІ = X WX X W Пѓ. (8.22)

A nice feature of the local polynomial method is that it provides the estimated
implied volatility and its п¬Ѓrst two derivatives in one step. Indeed, one has from
(8.19) and (8.20):

в€‚g Л†
= ОІ1 ,
в€‚M m0 ,П„0

в€‚2g Л†
= 2ОІ2 .
в€‚M 2 m0 ,П„0

One of the concerns regarding this estimation method is the dependence on the
bandwidth which governs how much weight the kernel function should place
on an observed point for the estimation at a target point. Moreover, as the
call options are not always symmetrically and equally distributed around the
ATM point, the choice of the bandwidth is a key issue, especially for estimation
at the border of the implied volatility surface. The bandwidth can be chosen
global or locally dependent on (M, П„ ). There are methods providing вЂќoptimalвЂќ
bandwidths which rely on plug-in rules or on data-based selectors.
In the case of the volatility surface, it is vital to determine one bandwidth for the
maturity and one for the moneyness directions. An algorithm called Empirical-
Bias Bandwidth Selector (EBBS) for п¬Ѓnding local bandwidths is suggested by
Ruppert (1997) and Ruppert, Wand, Holst and HВЁssler (1997). The basic idea
o
of this method is to minimize the estimate of the local mean square error at
each target point, without relying on asymptotic result. The variance and the
bias term are in this algorithm estimated empirically.
8.4 An Example: Application to DAX data 183

Using the local polynomial estimations, the empirical SPD can be calculated
with the following quantlet:

lpspd = spdbl(m,sigma,sigma1,sigma2,s,r,tau)
estimates the semi-parametric SPD.

The arguments for this quantlet are the moneyness (m), V (sigma), V (sigma1),
V (sigma2), underlying price (s) corrected for future dividends, risk-free in-
terest rate (r), and the time to maturity (tau). The output consist of the local
polynomial SPD (lpspd.fstar), в€† (lpspd.delta), and the О“ (lpspd.gamma)
of the call-options.

8.4 An Example: Application to DAX data
This section describes how to estimate the Black-Scholes and local polynomial
SPD using options data on the German DAX index.

8.4.1 Data

The dataset was taken from the п¬Ѓnancial database MD*BASE located at CASE
(Center for Applied Statistics and Economics) at Humboldt-UniversitВЁt zu
a
Berlin. Since MD*BASE is a proprietary database, only a limited dataset
is provided for demonstration purposes.
This database is п¬Ѓlled with options and futures data provided by Eurex. Daily
series of 1, 3, 6 and 12 months DM-LIBOR rates taken from the T homson
F inancial Datastream serve as riskless interest rates. The DAX 30 futures
and options settlement data of January 1997 (21 trading days) were used in this
study. Daily settlement prices for each option contract are extracted along with
contract type, maturity and strike. For the futures, the daily settlement prices,
maturities and volumes are the relevant information. To compute the interest
rates corresponding to the option maturities a linear interpolation between the
available rates was used.
The DAX is a performance index which means that dividends are reinvested.
However, assuming no dividend yields when inverting the Black-Scholes for-
mula results in diп¬Ђerent volatilities for pairs of puts and calls contrary to the
184 8 Estimating State-Price Densities with Nonparametric Regression

no-arbitrage assumption contained in the Put-Call parity. This remark can
be explained by the fact that until January 2002 domestic investors have an
advantage as they may receive a portion or all of the dividend taxes back de-
pending on their tax status. Dividend tax means here the corporate income
tax for distributed gains from the gross dividend.
Since the dividends are rebated to domestic investors the DAX should fall by
an amount contained between 0 and these dividend taxes. Indeed, the value of
this fall depends on the level of these taxes which may be equal to zero and on
the weights of domestic and foreign investors trading the DAX. These dividend
taxes have the same eп¬Ђects as ordinary dividends and should therefore be used
for computing the implied volatilities and the future price implicit in the Black
Scholes formula.
Hafner and Wallmeier (2001) suggest a method in order to get around this
problem which consists in computing dividends implied by the Put-Call parity.
Indeed, combining the futures pricing formula

Ft,П„F = St ert,П„F П„F в€’ Dt,П„f

and the Put-Call parity

Ct в€’ Pt = St в€’ Dt,П„O в€’ Keв€’rt,П„o П„o

we obtain:

Ct в€’ Pt = Ft,П„F eв€’rt,П„F + Dt,П„F ,П„O в€’ Keв€’rt,П„O П„O (8.23)

where П„O is the maturity of the options, П„F is the maturity of the nearest
forward whose volume is positive and Dt,П„F ,П„O = Dt,П„F в€’ Dt,П„O is the diп¬Ђerence
between the present values of the dividends.
Using (8.23), implied dividends were computed for each pair of put and call
with the same strike. Theoretically, for a given time to maturity there must
be only one value for these implied dividends. For each maturity the average
of these implied dividends was used to compute the corrected price. Using this
method implied volatilities are more reliable as the systematic вЂњgapвЂќ between
put and call volatilities disappears. The only uncertainty at this stage is due
to the interpolated rates for the maturity П„O .
The dataset consists of one п¬Ѓle XFGData9701 with 11 columns.
8.4 An Example: Application to DAX data 185

1 Day
2 Month
3 Year
4 Type of option (1 for calls, 0 for puts)
5 Time to maturity (in calendar days)
6 Strike prices
7 Option prices
8 Corrected spot price (implied dividends taken into account)
9 Risk-free interest rate
10 Implied volatility
11 Non-corrected spot price

issuing the following command:

Next extract all call options on January 3, 1997 with the paf command:

data=paf(data,(data[,1]==3)&&(data[,4]==1))

8.4.2 SPD, delta and gamma

This section provides an example using XploRe to calculate the semiparametric
SPD using DAX index options data. It is assumed that the quantlib finance
XFGSPDonematurity.xpl plots the SPD of the series of options closest to
maturity. This п¬Ѓrst example only uses smoothing method in one dimension.
XFGSPDoneday.xpl calculates and plots the local polynomial SPD for Jan-
uary 10, 1997 for diп¬Ђerent times to maturity (П„ = 0.125, 0.25, 0.375). After
loading the data, the implied volatility is estimated using the volsurf quantlet,
while the п¬Ѓrst and second derivatives are estimated using lpderxest quantlet.
In this example the grid size is 0.01. The bandwidth is chosen arbitrarily at 0.15
and 0.125 for the moneyness and maturity directions respectively. The criteria
used is a visual inspection of the п¬Ѓrst and second derivatives to ensure that
they are continuous and smooth. Next the quantlet spdbl is used to calculate
the SPD which is п¬Ѓnally displayed in Figure 8.1.
186 8 Estimating State-Price Densities with Nonparametric Regression

This п¬Ѓgure shows the expected eп¬Ђect of time to maturity on the SPD, which
is a loss of kurtosis. The x-axis represents the terminal prices ST . The local
polynomial SPD displays a negative skew compared to a theoretical Black-
Scholes SPD. The major reason for the diп¬Ђerence is the measure of implied
volatility. Using the local polynomial estimators one captures the eп¬Ђect of the
вЂњvolatility smileвЂќ and its eп¬Ђects on the higher moments such as skewness and
kurtosis. This result is similar to what AВЁ
Д±t-Sahalia and Lo (1998) and Rookley
(1997) found in their study.

Semi-parametric SPD: 10-01-1997
5 10 15 20 25
Density*E-4

0

2500 3000 3500
Stock price at expiry
Figure 8.1. Local Polynomial SPD for П„ = 0.125 (blue,п¬Ѓlled), П„ = 0.25
(black,dashed) and П„ = 0.375 (red,dotted).
XFGSPDoneday.xpl

Figure 8.2 and Figure 8.3 show Delta and Gamma for the full range of strikes
and for three diп¬Ђerent maturities. This method allows the user to get in one
step both greeks in one estimation for all strikes and maturities.
A natural question that may arise is how do the SPDs evolve over time. In
this section an illustrative example is used to show the dynamics of the SPD
over the month of January 1997. XFGSPDonemonth.xpl estimates and plots
the SPD for each trading day in January 1997. The x-axis is the moneyness,
y-axis is the trading day, and the z-axis is the SPD. Figure 8.4 shows the local
polynomial SPD for the three п¬Ѓrst weeks of January, 1997.
8.4 An Example: Application to DAX data 187

Semi-parametric Delta: 10-01-1997
1
0.5
Delta

0

2500 3000 3500
Strike prices
Figure 8.2. Local Polynomial Delta for П„ = 0.125 (blue,п¬Ѓlled), П„ = 0.25
(black,dashed) and П„ = 0.375 (red,dotted).

Semi-parametric Gamma: 10-01-1997
5 10 15 20 25
Gamma*E-4

0

2500 3000 3500
Strike prices
Figure 8.3. Local Polynomial Gamma for П„ = 0.125 (blue,п¬Ѓlled), П„ =
0.25 (black,dashed) and П„ = 0.375 (red,dotted).
XFGSPDoneday.xpl

8.4.3 Bootstrap conп¬Ѓdence bands

RookleyвЂ™s method serves to estimate the SPD, where V , V and V from (8.16)
are computed via local polynomials. The method is now applied to estimate
a SPD whose maturity is equal to the maturity of a series of options. In this
case, the nonparametric regression is a univariate one.
188 8 Estimating State-Price Densities with Nonparametric Regression

Local-Polynomial SPD: 01-1997, tau=0.250

5.97
24
23
4.48
22
21
20
3.00
17
16
15
14
1.51
13
10
9
8
7
6 1.20
1.10
1.00
0.90

Figure 8.4. Three weeks State-Price Densities on a moneyness scale.
XFGSPDonemonth.xpl

With a polynomial of order p = 2 and a bandwidth h = nв€’1/9 , it can be
shown that
Л†в€—
E|fn в€’ f в€— |2 = O nв€’4/9 ,

because
Л† = O nв€’8/9 ,
E|Vn в€’ V |2
Л† = O nв€’4/9 ,
E|Vn в€’ V |2
Л† = O nв€’4/9 .
E|Vn в€’ V |2
8.4 An Example: Application to DAX data 189

This result can be obtained using some theorems related to local polynomial
estimation, for example in Fan and Gijbels (1996), if some boundary conditions
are satisп¬Ѓed.
Л†в€— Л†в€—
An asymptotic approximation of fn is complicated by the fact that fn is a
non linear function of V , V and V . Analytical conп¬Ѓdence intervals can be
obtained using delta methods proposed by AВЁ Д±t-Sahalia (1996). However, an
alternative method is to use the bootstrap to construct conп¬Ѓdence bands. The
idea for estimating the bootstrap bands is to approximate the distribution of
Л†
sup |f в€— (k) в€’ f в€— (k)|.
k

The following procedure illustrates how to construct bootstrap conп¬Ѓdence
bands for local polynomial SPD estimation.

1. Collect daily option prices from MD*BASE, only choose those options
with the same expiration date, for example, those with time to maturity
49 days on Jan 3, 1997.
2. Use the local polynomial estimation method to obtain the empirical SPD.
Notice that when П„ is п¬Ѓxed the forward price F is also п¬Ѓxed. So that the
implied volatility function Пѓ(K/F ) can be considered as a п¬Ѓxed design
situation, where K is the strike price.
3. Obtain the conп¬Ѓdence band using the wild bootstrap method. The wild
bootstrap method entails:
вЂў Suppose that the regression model for the implied volatility function
Пѓ(K/F ) is:
Ki
+ Оµi , i = 1, В· В· В· , n.
Yi = Пѓ
F
вЂў Choose a bandwidth g which is larger than the optimal h in or-
der to have oversmoothing. Estimate the implied volatility function
Пѓ(K/F ) nonparametrically and then calculate the residual errors:

Ki
Оµi = Yi в€’ Пѓh
Лњ Л† .
F

вЂў Replicate B times the series of the {Лњi } with wild bootstrap ob-
Оµ
в€—,j
taining {Оµi } for j = 1, В· В· В· , B, HВЁrdle (1990), and build B new
a
190 8 Estimating State-Price Densities with Nonparametric Regression

bootstrapped samples:

Ki
Yiв€—,j = Пѓg + Оµв€—,j .
Л† i
F

вЂў Estimate the SPD f в€—,j using bootstrap samples, RookleyвЂ™s method
and the bandwidth h, and build the statistics
Л†
Tf = sup |f в€—,j (z) в€’ f в€— (z)|.
в€—
z

Л† Л†
вЂў Form the (1 в€’ О±) bands [f в€— (z) в€’ tf в€— ,1в€’О± , f в€— (z) + tf в€— ,1в€’О± ],
в€—
where tf в€— ,1в€’О± denotes the empirical (1 в€’ О±)-quantile of Tf .

Two SPDs (Jan 3 and Jan 31, 1997) whose times to maturity are 49 days
were estimated and are plotted in Figure (8.5). The bootstrap conп¬Ѓdence
band corresponding to the п¬Ѓrst SPD (Jan 3) is also visible on the chart. In
Figure (8.6), the SPDs are displayed on a moneyness metric. It seems that the
diп¬Ђerences between the SPDs can be eliminated by switching to the moneyness
metric. Indeed, as can be extracted from Figure 8.6, both SPDs lie within
the 95 percent conп¬Ѓdence bands. The number of bootstrap samples is set to
B = 100. The local polynomial estimation was done on standardized data, h
is then set to 0.75 for both plots and g is equal to 1.1 times h. Notice that
greater values of g are tried and the conclusion is that the conп¬Ѓdence bands
are stable to an increase of g.

8.4.4 Comparison to Implied Binomial Trees

In Chapter 7, the Implied Binomial Trees (IBT) are discussed. This method is a
close approach to estimate the SPD. It also recovers the SPD nonparametrically
from market option prices and uses the Black Scholes formula to establish the
relationship between the option prices and implied volatilities as in RookelyвЂ™s
method. In Chapter 7, the Black Scholes formula is only used for Barle and
Cakici IBT procedure, but the CRR binomial tree method used by Derman
and Kani (1994) has no large diп¬Ђerence with it in nature. However, IBT and
nonparametric regression methods have some diп¬Ђerences caused by diп¬Ђerent
modelling strategies.
The IBT method might be less data-intensive than the nonparametric regres-
sion method. By construction, it only requires one cross section of prices. In the
8.4 An Example: Application to DAX data 191

SPDs and bootstrap CB, tau= 49 days
10 15 20 25 30
density*E-4
5
0

2400 2600 2800 3000
Stock price at expiry S(T)

Figure 8.5. SPD estimation and bootstrap conп¬Ѓdence band.
XFGSPDcb.xpl

SPDs and bootstrap CB, tau= 49 days
8
6
density
4
2
0

0.8 0.9 1 1.1
S(T)/F(t,T)

Figure 8.6. SPD estimation and bootstrap conп¬Ѓdence band (moneyness
metric).
XFGSPDcb2.xpl

earlier application with DAX data, option prices are used with diп¬Ђerent times
to maturity for one day to estimate the implied volatility surface п¬Ѓrst in order
192 8 Estimating State-Price Densities with Nonparametric Regression

SPD estimations: 19970103, tau= 77 days

0.2
0.15
density*E-2

0.1
0.05
0

2000 2500 3000 3500
stock price

Figure 8.7. Comparison of diп¬Ђerent SPD estimations, by RookleyвЂ™s
method (blue) and IBT (black, thin).
XFGSPDcom.xpl

to construct the tree using the relation formula between option prices and risk-
neutral probabilities. The precision of the SPD estimation using IBT is heavily
aп¬Ђected by the quality of the implied volatility surface and the choice of the
levels of the implied tree. Furthermore, from the IBT method only risk-neutral
probabilities are obtained. They can be considered as a discrete estimation of
the SPD. However, the IBT method is not only useful for estimating SPD, but
also for giving a discrete approximation of the underlying process.
The greatest diп¬Ђerence between IBTs and nonparametric regression is the re-
quirement of smoothness. The precision of RookleyвЂ™s SPD estimation is highly
dependent on the selected bandwidth. Even if very limited option prices are
given, a part of the SPD estimation still can be obtained using nonparametric
regression, while the IBT construction has to be given up if no further struc-
ture is invoked on the volatility surface. RookleyвЂ™s method has on п¬Ѓrst sight
no obvious diп¬Ђerence with AВЁ Д±t-SahaliaвЂ™s method theoretically, AВЁД±t-Sahalia and
Lo (1998). But investigating the convergence rate of the SPD estimation using
AВЁД±t-SahaliaвЂ™s method allows one to conduct statistical inference such as test of
the stability of the SPD and tests of risk neutrality.
8.4 An Example: Application to DAX data 193

The quantlet XFGSPDcom.xpl shows a comparison of the SPD estimates by
IBT and RookleyвЂ™s methods. The diп¬Ђerences between these two SPD estimates
may be due to the selection of the bandwidths in RookleyвЂ™s method, the choice
of steps in the construction of the IBT and the use of DAX implied dividends
in RookleyвЂ™s method. Figure 8.7 shows the implied binomial trees and the local
polynomial SPDs for January, 3 1997.
Both densities seems to be quiet diп¬Ђerent. Indeed, the IBTs SPD shows a fatter
left tail than the RookleyвЂ™s one and the RookleyвЂ™s SPD shows a larger kurtosis.
To test which of both densities is more reliable, a cross-validation procedure is
performed. The idea of this test is to compare the theoretical prices based on
(8.1) with those observed on the market. However, as the whole tails are not
available for the RookleyвЂ™s SPD, the test is done on butterп¬‚y spreads deп¬Ѓned in
Section 8.2 since their prices should not be inп¬‚uenced by the tails of the SPDs.
For cross-validation, we remove the three calls used to calculate the observed
butterп¬‚y prices from the sample before estimating the SPD. Moreover, since
the largest diп¬Ђerence between both SPDs is observed at the ATM point (see
Figure 8.7), the test is applied only to the two butterп¬‚y spreads whose centers
surround the ATM point. The width 2в€†K of the butterп¬‚y spread is set to 200.
This procedure is done for the 21 days of January 1997. Figure 8.8 displays
the results in term of relative pricing error E:
Pobserved в€’ PSP D
E=
Pobserved
where Pobserved is the observed price of the butterп¬‚y spread and PSP D is the
price computed using the SPD estimate and (8.1). It seems that both SPDs
have a too small kurtosis since the observed prices of butterп¬‚y spreads are
larger than those of both SPDs in most of the cases. However, RookleyвЂ™s SPD
is in mean nearer to the observed price than the IBTвЂ™s one.

Bibliography
AВЁ
Д±t-Sahalia, Y. (1996). The Delta method for Nonparametric Kernel Function-
als, mimeo.
AВЁ
Д±t-Sahalia, Y. and Lo, A. W. (1998). Nonparametric estimation of state-price
densities implicit in п¬Ѓnancial asset prices, Journal of Finance 53: 499вЂ“547.
194 8 Estimating State-Price Densities with Nonparametric Regression

20
Pricing error in %

10
0
-20 -10

5 10 15 20 25 30
Date

20
Pricing error in %

10
0
-10

5 10 15 20 25 30
Date

Figure 8.8. The upper graph display the relative pricing errors for the
butterп¬‚y spread centered on the nearest strike on the left side of the
ATM point. The second graph corresponds to the butterп¬‚y spread
centered on the nearest strike on the right side of the ATM point. The
black lines represent the IBTвЂ™s pricing errors and the blue the RookleyвЂ™s
errors.

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196 8 Estimating State-Price Densities with Nonparametric Regression

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9 Trading on Deviations of Implied
and Historical Densities
Oliver Jim Blaskowitz and Peter Schmidt

9.1 Introduction
In recent years a number of methods have been developed to infer implied
state price densities (SPD) from cross sectional option prices, Chapter 7 and
8. Instead of comparing this density to a historical density extracted from the
observed time series of the underlying asset prices, i.e. a risk neutral density to
an actual density, AitвЂ“Sahalia, Wang and Yared (2000) propose to compare two
risk neutral densities, one obtained from cross sectional S&P 500 option data
and the other from the S&P 500 index time series. Furthermore, they propose
trading strategies designed to exploit diп¬Ђerences in skewness and kurtosis of
DAX index. While the option implied SPD is estimated by means of the Barle
and Cakici, Barle and Cakici (1998), implied binomial tree version, the time
series density is inferred from the time series of the DAX index by applying a
method used by AitвЂ“Sahalia, Wang and Yared (2000). Based on the comparison
of both SPDs the performance of skewness and kurtosis trades is investigated.
We use options data included in MD*BASE. This is a database located at
CASE (Center for Applied Statistics and Economics) of HumboldtвЂ“UniversitВЁta
zu Berlin. The time period is limited to data of the period between 01/01/97
and 12/31/99 for which MD*BASE contains daily closing prices of the DAX
index, EUREX DAX option settlement prices and annual interest rates which
are adjusted to the time to maturity of the above mentioned EUREX DAX
options.
While Section 9.2 applies the Barle and Cakici implied binomial tree algorithm
198 9 Trading on Deviations of Implied and Historical Densities

which estimates the option implied SPD using a two week cross section of DAX
index options, Section 9.3 explains and applies the method to estimate DAX
time series SPD from 3 months of historical index prices. Following, in Section
9.4 we compare the conditional skewness and kurtosis of both densities. Section
9.5 and 9.6 complete the chapter with the investigation of 4 trading strategies
and Section 9.7 completes with some critical remarks.

9.2 Estimation of the Option Implied SPD
BarleвЂ“CakiciвЂ™s modiп¬Ѓcation of DermanвЂ“KaniвЂ™s Implied Binomial Tree (IBT)
yields a proxy for the option implied SPD, f в€— , see Chapter 7. XploRe provides
quantlets computing DermanвЂ“KaniвЂ™s and BarleвЂ“CakiciвЂ™s IBTвЂ™s. Since the latter
proved to be slightly more robust than the former, Jackwerth (1999), we decide
to use BarleвЂ“CakiciвЂ™s IBT to compute the option implied SPD. In the following
subsection, we follow closely the notation used in Chapter 7. That is, N denotes
the number of evenly spaced time steps of length в€†t in which the tree is divided
into (so we have N + 1 levels). Fn,i = erв€†t sn,i is the forward price of the
underlying, sn,i , at node i at, level n. Each level n corresponds to time tn =
nв€†t.

9.2.1 Application to DAX Data

Using the DAX index data from MD*BASE, we estimate the 3 month option
implied IBT SPD f в€— by means of the XploRe quantlets IBTbc and volsurf and
a two week cross section of DAX index option prices for 30 periods beginning
in April 1997 and ending in September 1999. We measure time to maturity
(TTM) in days and annualize it using the factor 360, giving the annualized
time to maturity П„ = TTM/360. For each period, we assume a п¬‚at yield curve.
We extract from MD*BASE the maturity consistent interest rate.
We describe the procedure in more detail for the п¬Ѓrst period. First of all, we
estimate the implied volatility surface given the two week cross section of DAX
option data and utilizing the XploRe quantlet volsurf which computes the 3
dimensional implied volatility surface (implied volatility over time to maturity
and moneyness) using a kernel smoothing procedure. Friday, April 18, 1997
is the 3rd Friday of April 1997. On Monday, April 21, 1997, we estimate the
volatility surface, using two weeks of option data from Monday, April 7, 1997,
to Friday, April 18, 1997. Following, we start the IBT computation using the
9.2 Estimation of the Option Implied SPD 199

DAX price of this Monday, April 21, 1997. The volatility surface is estimated
for the moneyness interval [0.8, 1.2] and the time to maturity interval [0.0, 1.0].
Following, the XploRe quantlet IBTbc takes the volatility surface as input and
computes the IBT using Barle and CakiciвЂ™s method. Note that the observed
smile enters the IBT via the analytical BlackвЂ“Scholes pricing formula for a call
C(Fn,i , tn+1 ) and for a put P (Fn,i , tn+1 ) which are functions of St1 = s1,1 ,
K = Fn,i , r, tn+1 and Пѓimpl (Fn,i , tn+1 ). We note, it may happen that at the
edge of the tree option prices, with associated strike prices Fn,i and node prices
sn+1,i+1 , have to be computed for which the moneyness ratio sn+1,i+1 /Fn,i is
outside the intverall [0.8, 1.2] on which the volatility surface has been estimated.
In these cases, we use the volatility at the edge of the surface. Note, as well,
that the mean of the IBT SPD is equal to the futures price by construction of
the IBT.
Finally, we transform the SPD over sN +1,i into a SPD over logвЂ“returns uN +1,i =
ln(sN +1,i /s1,1 ) as follows:

sN +1,i x
P(sN +1,i = x) = P ln = ln = P uN +1,i = u
s1,1 s1,1

where u = ln(x/s1,1 ). That is, sN +1,i has the same probability as uN +1,i . See
Figure 9.1 for the SPD computed with parameters N = 10 time steps and
interest rate r = 3.23.
A crucial aspect using binomial trees is the choice of the number of time steps
N in which the time interval [t, T ] is divided. In general one can state, the more
time steps are used the better is the discrete approximation of the continuous
diп¬Ђusion process and of the SPD. Unfortunately, the bigger N , the more node
prices sn,i possibly have to be overridden in the IBT framework. Thereby we are
eп¬Ђectively losing the information about the smile at the corresponding nodes.
Therefore, we computed IBTвЂ™s for diп¬Ђerent numbers of time steps. We found
no hint for convergence of the variables of interest, skewness and kurtosis. Since
both variables seemed to п¬‚uctuate around a mean, we compute IBTвЂ™s with time
steps 10, 20, . . . , 100 and consider the average of these ten values for skewness
and kurtosis as the option implied SPD skewness and kurtosis.
Applying this procedure for all 30 periods, beginning in April 1997 and ending
in September 1999, we calculate the time series of skewness and kurtosis of the
3 month implied SPD f в€— shown in Figures 9.3 and 9.4. We see that the implied
SPD is clearly negatively skewed for all periods but one. In September 1999 it
is slightly positively skewed. The pattern is similar for the kurtosis of f в€— which
is leptokurtic in all but one period. In October 1998 the density is platykurtic.
200 9 Trading on Deviations of Implied and Historical Densities

IBT SPD

0.2
0.15
SPD

0.1
0.05
0

-0.3 -0.2 -0.1 0 0.1 0.2
LogReturn
Figure 9.1. Option implied SPD estimated on April 21, 1997, by an
IBT with N = 10 time steps, S0 = 3328.41, r = 3.23 and П„ = 88/360.

9.3 Estimation of the Historical SPD
While the previous section was dedicated to п¬Ѓnding a proxy for f в€— used by
investors to price options, this section approximates the historical underlyingsвЂ™
density g в€— for date t = T using all the information available at date t = 0. Of
course, if the process governing the underlying asset dynamics were common
knowledge and if agents had perfect foresight, then by no arbitrage arguments
both SPDs should be equal. Following AitвЂ“Sahalia, Wang and Yared (2000),
the density extracted from the observed underlyingsвЂ™ data is not comparable to
the density implied by observed option data without assumptions on investorвЂ™s
preferences. As in HВЁrdle and Tsybakov (1995), they apply an estimation
a
method which uses the observed asset prices to infer indirectly the time series
SPD. First, we will explain the estimation method for the underlyingsвЂ™ SPD.
Second, we apply it to DAX data.
9.3 Estimation of the Historical SPD 201

9.3.1 The Estimation Method

Assuming the underlying S to follow an Л† diп¬Ђusion process driven by a Brow-
Ito
nian motion W :

dSt = Вµ(St )dt + Пѓ(St )dWt . (9.1)

AitвЂ“Sahalia, Wang and Yared (2000) rely on GirsanovвЂ™s characterization of the
change of measure from the actual density to the SPD. It says the diп¬Ђusion
function of the assetвЂ™s dynamics is identical under both the risk neutral and
the actual measure and only the drift function needs to be adjusted, leading to
the risk neutral asset dynamics:
в€—
(rt,П„ в€’ Оґt,П„ )St dt + Пѓ(St )dWtв€— .
в€— в€—
dSt = (9.2)

Let gt (St , ST , П„, rt,П„ , Оґt,П„ ) denote the conditional density of ST given St gen-
в€—
erated by the dynamics deп¬Ѓned in equation (9.1) and gt (St , ST , П„, rt,П„ , Оґt,П„ )
denote the conditional density generated by equation (9.2) then f в€— can only be
compared to the risk neutral density g в€— and not to g.
A crucial feature of this method is that the diп¬Ђusion functions are identical
under both the actual and the risk neutral dynamics (which follows from Gir-
sanovвЂ™s theorem). Therefore, it is not necessary to observe the risk neutral path
of the DAX index {St }. The function Пѓ(вЂў) is estimated using N в€— observed
в€—

index values {St } and applying FlorensвЂ“ZmirouвЂ™s (1993) (FZ) nonparametric
version of the minimum contrast estimators:
N в€— в€’1 в€’S
KF Z ( SiF Z )N в€— {S(i+1)/N в€— в€’ Si/N в€— }2
i=1 h
ПѓF Z (S)
Л† = , (9.3)
Nв€— в€’S
KF Z ( SiF Z )
i=1 h

where KF Z (вЂў) is a kernel function and hF Z is a bandwidth parameter such
that:

(N в€— hF Z )в€’1 ln(N в€— ) N в€— h4 Z
в†’ в†’
0 and 0
F

as N в€— в†’ в€ћ. Without imposing restrictions on the drift function ПѓF Z is an
Л†
unbiased estimator of Пѓ in the model speciп¬Ѓed in equation (9.2). Since the DAX
index is a performance index (Оґt,П„ = 0), the risk neutral drift rate of equation
(9.2) is equal to rt,П„ .
Once Пѓ(вЂў) is estimated, the time series SPD g в€— can be computed by Monte
Carlo integration. Applying the Milstein scheme (Kloeden, Platen and Schurz
202 9 Trading on Deviations of Implied and Historical Densities

(1994)), we simulate M = 10, 000 paths of the diп¬Ђusion process:
в€—
= rt,П„ St dt + ПѓF Z (St )dWtв€—
в€— в€—
dSt Л† (9.4)
for a time period of 3 months, starting value St=0 equal to the DAX index value
at the beginning of the period, collect the endpoints at T of these simulated
paths {ST,m : m = 1, . . . , M } and annualize the index logвЂ“returns. Then
g в€— is obtained by means of a nonparametric kernel density estimation of the
continuously compounded logвЂ“returns u:
M
um в€’ u
1
в€—
pt (u)
Л† = KM C (9.5)
M hM C hM C
m=1

where um is the logвЂ“return at the end of the mth path and KM C (вЂў) is a kernel
function and hM C is a bandwidth parameter. The equation:

log(S/St )
pв€— (u)du
P(ST в‰¤ S) P(u в‰¤ log(S/St ))
= = t
в€’в€ћ

with u = ln(ST /St ) relates this density estimator to the SPD g в€— :

pв€— (log(S/St ))
в€— в€‚
в‰¤ S)
gt (S) = в€‚S P(ST = .
t
S

в€љ
This method results in a nonparametric estimator g в€— which is N в€— вЂ“consistent
Л†
as M в†’ в€ћ even though ПѓF Z converges at a slower rate (AitвЂ“Sahalia, Wang
Л†
and Yared (2000)).
In the absence of arbitrage, the futures price is the expected future value of the
spot price under the risk neutral measure. Therefore the time series distribu-
tion is translated such that its mean matches the implied future price. Then the
bandwidth hM C is chosen to best match the variance of the IBT implied distri-
bution. In order to avoid overвЂ“ or undersmoothing of g в€— , hM C is constrained
to be within 0.5 to 5 times the optimal bandwidth implied by SilvermanвЂ™s rule
of thumb. This procedure allows us to focus the density comparison on the
skewness and kurtosis of the two densities.

9.3.2 Application to DAX Data

Using the DAX index data from MD*BASE we estimate the diп¬Ђusion function
Пѓ 2 (вЂў) from equation (9.2) by means of past index prices and simulate (forward)
M = 10, 000 paths to obtain the time series density, g в€— .
9.3 Estimation of the Historical SPD 203

Time Series Density
3
2
Density

1

-0.3 -0.2 -0.1 0 0.1 0.2 0.3
LogReturns
Figure 9.2. Mean and variance adjusted estimated time series density
on Friday, April 18, 1997. Simulated with M = 10, 000 paths, S0 =
3328.41, r = 3.23 and П„ = 88/360.

To be more precise, we explain the methodology for the п¬Ѓrst period in more
detail. First, note that Friday, April 18, 1997, is the 3rd Friday of April 1997.
Thus, on Monday, April 21, 1997, we use 3 months of DAX index prices from
Monday, January 20, 1997, to Friday, April 18, 1997, to estimate Пѓ 2 . Following,
on the same Monday, we start the 3 months вЂ˜forwardвЂ™ Monte Carlo simulation.
The bandwidth hF Z is determined by Cross Validation applying the XploRe
quantlet regxbwcrit which determines the optimal bandwidth from a range
of bandwidths by using the resubstitution estimator with the penalty function
вЂ™Generalized Cross ValidationвЂ™.
Knowing the diп¬Ђusion function it is now possible to Monte Carlo simulate the
index evolution. The Milstein scheme applied to equation (9.2) is given by:
Si/N в€—в€— = S(iв€’1)/N в€—в€— + rS(iв€’1)/N в€—в€— в€†t + Пѓ(S(iв€’1)/N в€—в€— )в€†Wi/N в€—в€— +
1 в€‚Пѓ
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