The parametric approach to estimate the risk neutral density function directly starts with the assumption that the risk neutral distribution of ST belongs to a parametric family W . Rr/ of one-dimensional continuous distributions.
For any parameter vector# 2 and every strike priceK we may calculate the hypothetical prices for the callC.K; #/;the put P .K; #/both with expirationT;
and the forwardFby
C.Kj#/DerT Z1
K
.xK/ q.xj/ dx (10.12)
P .Kj#/DerT ZK 0
.Kx/ q.xj/ dx (10.13)
F# DerT Z1
0
x q.xj#/ dx (10.14)
Therein,q.j#/denotes any probability density function of the distribution fromW
associated with#:
The estimation of the risk neutral density function reduces to the estimation of the distribution parameter vector#:The most common approach is based onS0; observed pricesY1; : : : ; Ynfor calls with strikesK1; : : : ; Km;andYQ1; : : : ;YQm with strikesKQ1; : : : ;KQn:Both, calls and puts with expirationT:The parameter vector# is estimated by minimizing the sum of the squared differences between the observed call, put and forward price and the hypothetical ones. More precisely, the estimation involves the solution of the following minimization problem
min Xm iD1
fYiC.Kij#/g2 C Xn jD1
˚YQiP .KQij#/2
(10.15) C erTS0F#
2
s.t.#2:
The crucial step to implement this parametric approach is to find a proper statistical family W as a model for the risk neutral distribution. Usually, either a very general class a distribution is selected or mixtures of log-normal distributions are utilized. As general classes we shall discuss the benchmark case of log-normal distributions and the generalized Gamma distributions. Let us start with assumption of log-normal distributions.
10.4.1 Estimation Using Log-Normal Distributions
Closely related to the Black Scholes model the log-normal distributions are sometimes used for the risk neutral distribution, indicated as a benchmark case (cf. e.g.Jondeau and Rockinger(2000)). Recall that a probability density function fLN.;/of a log-normal distribution with parameters2Rand > 0is given by
fLN.;/.x/defD 8ˆ
<
ˆ: p 1
2x e fln.x/g
2
22 W x > 0
0 W otherwise
:
For fixed 2 and different 1; 2 the respective probability density functions fLN.1;/andfLN.2;/are linked by
fLN.2;/De.12/fLN.1;/
˚x e.12/
: (10.16)
Then applying the change of variables theorem for integration we obtain the following relationships between the call and put prices
C.Kj2; /De.21/C˚
K e.12/j1;
(10.17) P .Kj2; /De.21/P˚
K e.12/j1;
: (10.18)
The equations suggest to express prices C.Kj; / andP .Kj; / in terms of Black Scholes formulas, noticing thatCBS.K; /DC
Kj
r2
2
TCln.S0/;
andPBS.K; / D P
Kj
r2 2
TCln.S0/;
holds for any strikeK: For 2Rand > 0we obtain
CBS.Kj; /defDC.Kj; / (10.19)
De
8<
: 0
@r2 2
1 ATCln.S0/
9=
;CBS 8ˆ ˆ<
ˆˆ :K e
0
@r2 2
1
ATCln.S0/
; 9>
>=
>>
;
PBS.Kj; /defDC.Kj; / (10.20)
De
8<
: 0
@r2 2
1 ATCln.S0/
9=
;PBS 8ˆ ˆ<
ˆˆ :K e
0
@r2 2
1
ATCln.S0/
; 9>
>=
>>
;: With a slight abuse of convention we shall callCBS.Kj; /andPBS.Kj; / Black Scholes call and put prices too.
Next we want to introduce the approach to substitute log-normal distributions for the risk neutral distributions by mixtures of them.
10.4.2 Estimation Using Log-Normal Mixtures
The use of log-normal mixtures to model the risk neutral distribution of ST
was initiated by Ritchey (1990) and became further popular even in financial industries by the studiesBahra(1997),Melick and Thomas(1997) andS¨oderlind and Swensson (1997). The idea is to model the risk neutral density function as a weighted sum of probability density functions of possibly different log-normal distribution. Namely, we set
q.xj1; : : : ; k; 1; : : : ; k; 1; : : : ; k/defD Xk iD1
ifLN.i;i/.x/;
wherefLN.i;i/denotes a probability density function of the log-normal distribution with parametersi 2 Ras well asi > 0;and nonnegative weights1; : : : ; k
summing up to1:
This approach might be motivated w.r.t. two aspects. Firstly such density functions are flexible enough to model a great variety of potential shapes for the risk neutral density function. Secondly, we may compute easily the hypothetical call and put prices in terms of respective Black-Scholes formulas by
C.Kj1; : : : ; k; 1; : : : ; k; 1; : : : ; k/D Xk iD1
iCBS.Kji; i/(10.21)
P .Kj1; : : : ; k; 1; : : : ; k; 1; : : : ; k/D Xk iD1
iPBS.Kji; i/:(10.22)
Additionally, drawing on well known formulas for the expectations of log-normal distributions, we obtain
F1;:::;k;1;:::;k;1;:::;k D Xk iD1
ie.iC22irT /
Recalling that the parameter estimation is based on observations ofmcall andnput prices we have to take into account the problem of overfitting. More precisely, the number3k1of parameters should not exceedmCn;the number of observations.
Furthermore in order to reduce the numerical complexity of the minimization problem underlying the estimation it is often suggested to restrict estimation to the choice ofk2 f2; 3g:
Empirical evidence (cf. e.g. Corrado and Su (1997); Savickas (2002, 2005)) shows that the implied skewness of the underlying used in options is often negative, in contrary to the skewness of log-normal distributions. In order to take into account negative skewness Savickas proposed to use Weibull distributions (cf. Savickas (2002,2005)). InFabozzi et al.(2009) this suggestion has been extended to the fam- ily of generalized gamma distributions that will be considered in the next subsection.
10.4.3 Estimation Using Generalized Gamma Distributions
According to#defD.˛; ˇ; k/2defD0;1Œ3a respective probability density function fG.j˛; ˇ; ı/is given by
fG.j˛; ˇ; k/D 8<
: 1 .k/
ˇ
˛ x
˛ ˇk1
exp
x
˛ ˇ
W x > 0
0 W otherwise
;
where denotes the Gamma function. The corresponding cumulative distribution functionG.j˛; ˇ; k/is given by
G.xj˛; ˇ; k/defDI
k;
x
˛ ˇ
; whereI denotes the incomplete gamma function defined as
I .k; y/defD 1 .k/
Zy
0
xk1exdx:
It is known thatkD1leads to a Weibull distribution, whenˇD1we get a gamma distribution, when ˇ D k D 1 we obtain an exponential distribution and when k ! 1 we arrive a log-normal distribution. Explicit formulas for the respective hypothetical pricesC.Kj˛; ˇ; k/; P .Kj˛; ˇ; k/andF˛;ˇ;k, the moment generating function, have been derived inFabozzi et al. (2009) (pp. 58, 70). They read as follows.
F˛;ˇ;kD˛ .kC ˇ1/
.k/ (10.23)
C.Kj˛; ˇ; k/DerTF˛;ˇ;k erTK (10.24)
"
F˛;ˇ;kI (
k 1 ˇ;
K
˛ ˇ)
C K I (
k;
K
˛ ˇ) #
P .Kj˛; ˇ; k/DerT
"
K I (
k;
K
˛ ˇ)
F˛;ˇ;kI (
kC 1 ˇ;
K
˛ ˇ) #
: (10.25) A different class of methods to estimate the risk neutral density start with a parametric model of the whole stock price process which determines in an analytic way the risk neutral distribution. Then the risk neutral density will be estimated indirectly via calibration of the stock price process.