Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Term structure of implied risk neutral densities
Enrique ter Horst
CESA and IESA
Joint work with Roberto Casarin (Università Ca’ Foscari) and Fabrizio
Leisen (Kent University) and German Molina (Idalion Hedge Fund)
CESA and IESA
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
• Modelling data naturally defined on bounded interval.
The analysis of time series data defined on a bounded interval
(such as rates or proportions) has been a challenging issue for
many years and still represents an open issue.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
• Modelling data naturally defined on bounded interval.
The analysis of time series data defined on a bounded interval
(such as rates or proportions) has been a challenging issue for
many years and still represents an open issue.
• Focus on Beta Autoregressive (BAR) time series models.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
• Modelling data naturally defined on bounded interval.
The analysis of time series data defined on a bounded interval
(such as rates or proportions) has been a challenging issue for
many years and still represents an open issue.
• Focus on Beta Autoregressive (BAR) time series models.
• Bayesian inference for BAR models. We follow a Markov-chain
Monte Carlo (MCMC) approach.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
• Modelling data naturally defined on bounded interval.
The analysis of time series data defined on a bounded interval
(such as rates or proportions) has been a challenging issue for
many years and still represents an open issue.
• Focus on Beta Autoregressive (BAR) time series models.
• Bayesian inference for BAR models. We follow a Markov-chain
Monte Carlo (MCMC) approach.
• Selection of the lag structure for the BAR model (model
selection). We follow a Reversible-Jump MCMC approach.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
• The dynamics of the unemployment rate is one of the most
studied problem in time series analysis (Bean(1994) and Nickel
(1997)).
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
• The dynamics of the unemployment rate is one of the most
studied problem in time series analysis (Bean(1994) and Nickel
(1997)).
• Modeling and forecasting the unemployment rate still represent
challenging issues in econometrics (Neftci (1984), Montgomery et
al. (1998) Koop and Potter (1999)). The unemployment rate is
usually characterized by relatively short periods of rapid economic
contraction and by relatively long periods of slow expansion.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
• The dynamics of the unemployment rate is one of the most
studied problem in time series analysis (Bean(1994) and Nickel
(1997)).
• Modeling and forecasting the unemployment rate still represent
challenging issues in econometrics (Neftci (1984), Montgomery et
al. (1998) Koop and Potter (1999)). The unemployment rate is
usually characterized by relatively short periods of rapid economic
contraction and by relatively long periods of slow expansion.
• We focus on another fundamental feature: it is naturally defined
on a bounded interval, let us say the (0, 1) interval. Data
transformation (Wallis (1987)). Beta autoregressive (Rocha and
Cribari-Neto (2009)).
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
0.12
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
0.00
0.04
0.08
US Unemployment Rate
EU Unemployment Rate
0
100
200
300
400
500
time
US and Euro Area unemployment rates, monthly frequency.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
0.12
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
0.00
0.04
0.08
US Unemployment Rate
EU Unemployment Rate
0
100
200
300
400
500
time
US and Euro Area unemployment rates, monthly frequency.
• US: Feb 1971 to Dec 2009, 467 observations, seasonally adjusted.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
0.12
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
0.00
0.04
0.08
US Unemployment Rate
EU Unemployment Rate
0
100
200
300
400
500
time
US and Euro Area unemployment rates, monthly frequency.
• US: Feb 1971 to Dec 2009, 467 observations, seasonally adjusted.
• Euro Area: Jan 1995 to Dec 2009, 180 observations, seasonally
adjusted, aggregated for 13 countries.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
• The capacity utilization deals with all production factors (i.e.
labor force and stock of capital) of an economic system or sector.
Definition and statistical measures: Klein and Su (1979).
Economic theory: Burnside and Eichenbaum (1996) and Cooley et
al. (1995), Macroeconometrics: Klein and Su (1979))
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
• The capacity utilization deals with all production factors (i.e.
labor force and stock of capital) of an economic system or sector.
Definition and statistical measures: Klein and Su (1979).
Economic theory: Burnside and Eichenbaum (1996) and Cooley et
al. (1995), Macroeconometrics: Klein and Su (1979))
• A decreasing capacity utilization is interpreted as slowdown of
the aggregated demand and consequently a reduction of the
inflation level. An increase of the capacity utilization reveals an
expansion of the level of economic activity.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
• The capacity utilization deals with all production factors (i.e.
labor force and stock of capital) of an economic system or sector.
Definition and statistical measures: Klein and Su (1979).
Economic theory: Burnside and Eichenbaum (1996) and Cooley et
al. (1995), Macroeconometrics: Klein and Su (1979))
• A decreasing capacity utilization is interpreted as slowdown of
the aggregated demand and consequently a reduction of the
inflation level. An increase of the capacity utilization reveals an
expansion of the level of economic activity.
• The series exhibits a negative trend. A deterministic trend could
be naturally included in the Beta regression, however we focus on
the autoregressive components (two steps procedure).
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
0.2
0.4
0.6
0.8
1.0
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
0.0
US Capacity Utilizatoin Rate
Detrended US Capacity Utilizatoin Rate
0
100
200
300
400
500
time
US capacity utilization rate (solid line), and the detrended capacity
rate (dashed line)
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
0.2
0.4
0.6
0.8
1.0
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
0.0
US Capacity Utilizatoin Rate
Detrended US Capacity Utilizatoin Rate
0
100
200
300
400
500
time
US capacity utilization rate (solid line), and the detrended capacity
rate (dashed line)
• Jan 1967 to May 2010, all industry sectors, seasonally adjusted.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
Option-based probability assessments are expressed as CDFs
{Gi }N
i=1 , of random quantities which are realized outcomes of the
expiration date price of the underlying futures contract, {YTi }N
i=1
R yT
and Gi ≡ 0 i fRN (x)dx where fRN (x) is the risk-neutral density
estimated at time 0 to forecast for time Ti .
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
Option-based probability assessments are expressed as CDFs
{Gi }N
i=1 , of random quantities which are realized outcomes of the
expiration date price of the underlying futures contract, {YTi }N
i=1
R yT
and Gi ≡ 0 i fRN (x)dx where fRN (x) is the risk-neutral density
estimated at time 0 to forecast for time Ti .
We say that the {Gi (yTi )}N
i=1 are calibrated if the proportion of
times Gi (yTi ) is less than or equal to any given value u on the
interval [0, 1] is u.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
Option-based probability assessments are expressed as CDFs
{Gi }N
i=1 , of random quantities which are realized outcomes of the
expiration date price of the underlying futures contract, {YTi }N
i=1
R yT
and Gi ≡ 0 i fRN (x)dx where fRN (x) is the risk-neutral density
estimated at time 0 to forecast for time Ti .
We say that the {Gi (yTi )}N
i=1 are calibrated if the proportion of
times Gi (yTi ) is less than or equal to any given value u on the
interval [0, 1] is u.
More formally, let U be a random variable on the interval [0, 1]
with CDF C(U), defined by Ui = Gi (YTi ). The process used to
generate the Gi is calibrated if U is uniformly distributed on the
interval [0, 1], i.e., if C(u)=u for all u in [0, 1].
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
If the option-based assessment process is well calibrated or
forecasts well the underlying process, the ui , calculated in this
manner from a sample of independent assessments will be
uniformly distributed.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
If the option-based assessment process is well calibrated or
forecasts well the underlying process, the ui , calculated in this
manner from a sample of independent assessments will be
uniformly distributed.
The function C, the CDF of U, is called a calibration function.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
If the option-based assessment process is well calibrated or
forecasts well the underlying process, the ui , calculated in this
manner from a sample of independent assessments will be
uniformly distributed.
The function C, the CDF of U, is called a calibration function.
The calibration function provides a means of transforming a
noncalibrated assessment into a calibrated one. The distribution
defined by the transformation F (YTi ) = C (G (YTi ))will always be a
calibrated distribution because, for any Z,
C (G (yTi )) = Prob(G (YTi ) 5 C (G (yTi ))) = Prob(YTi 5 yTi )
When the Gi are calibrated, Fi = Gi . When the Gi , are not
calibrated, the calibration function can provide insights into the
nature of the problems in assessment process and their resolution.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
de Vincent-Humphreys and Noss (2012)
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
de Vincent-Humphreys and Noss (2012)
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
de Vincent-Humphreys and Noss (2012)
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
de Vincent-Humphreys and Noss (2012)
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
Fackler and King (1990), which forms the foundation for this
paper, use q-q plots to illustrate that option-implied RNDs provide
a poor probabilistic description of agricultural commodity price
out-turns.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
Fackler and King (1990), which forms the foundation for this
paper, use q-q plots to illustrate that option-implied RNDs provide
a poor probabilistic description of agricultural commodity price
out-turns.
Liu et al (2007) also employ q-q plots to compare the PIT to a
uniform distribution, which as previously explained is equivalent to
checking whether the empirical distribution function of the PIT lies
on the gradient-one line.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
Fackler and King (1990), which forms the foundation for this
paper, use q-q plots to illustrate that option-implied RNDs provide
a poor probabilistic description of agricultural commodity price
out-turns.
Liu et al (2007) also employ q-q plots to compare the PIT to a
uniform distribution, which as previously explained is equivalent to
checking whether the empirical distribution function of the PIT lies
on the gradient-one line.
Fackler and King (1990) find that the CDF of a beta distribution
can be used to calibrate the RND to observed outcomes, removing
the systematic difference between the two. They find the CDF of
the beta distribution offers a parsimonious yet flexible calibration
function. Although it depends on only two parameters, it nests
many simple forms of transformation such as a mean shift,
mean-preserving changes in variance, and changes involving mean,
variance and skewness.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
The calibration approach estimates real-world probabilities as some
function of the risk-neutral probabilities. That function is called
the calibration function. Equation (1) expresses this general
relationship in terms of cumulative real-world and risk-neutral
cumulative probability density functions F P and F Q :
F P (x) = C (F Q (x))
(1)
C (), must be restricted to functions which are themselves
cumulative probability functions (example the Beta distribution), ie
have a range [0,1] and are nondecreasing. It can be recast in terms
of probability densities:
f P (x) = C 0 (F Q (x))f Q (x)
Enrique ter Horst
(2)
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
The RNDs are estimated using a refinement of the non-parametric
method set out in Panigirtzoglou and Skiadopoulos (2004). That
method exploits the result of Breeden and Litzenberger (1978),
which relates the RND to the curvature of the call price function.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
The RNDs are estimated using a refinement of the non-parametric
method set out in Panigirtzoglou and Skiadopoulos (2004). That
method exploits the result of Breeden and Litzenberger (1978),
which relates the RND to the curvature of the call price function.
The procedure for estimating RNDs consists of five stages. First,
the observed option prices are filtered to remove those prices that
violate the no-arbitrage restrictions of monotonicity and convexity.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
The RNDs are estimated using a refinement of the non-parametric
method set out in Panigirtzoglou and Skiadopoulos (2004). That
method exploits the result of Breeden and Litzenberger (1978),
which relates the RND to the curvature of the call price function.
The procedure for estimating RNDs consists of five stages. First,
the observed option prices are filtered to remove those prices that
violate the no-arbitrage restrictions of monotonicity and convexity.
The data are then translated from (strike, price) into (delta,
implied volatility) using the Black-Scholes formulae, as in Malz
(1997). This does not assume that the Black-Scholes
option-pricing paradigm holds true. Rather, this transformation
should be viewed as purely for numerical convenience,
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
The RNDs are estimated using a refinement of the non-parametric
method set out in Panigirtzoglou and Skiadopoulos (2004). That
method exploits the result of Breeden and Litzenberger (1978),
which relates the RND to the curvature of the call price function.
The procedure for estimating RNDs consists of five stages. First,
the observed option prices are filtered to remove those prices that
violate the no-arbitrage restrictions of monotonicity and convexity.
The data are then translated from (strike, price) into (delta,
implied volatility) using the Black-Scholes formulae, as in Malz
(1997). This does not assume that the Black-Scholes
option-pricing paradigm holds true. Rather, this transformation
should be viewed as purely for numerical convenience,
which facilitates the third stage: fitting a natural smoothing spline
through the transformed data. The spline is evaluated at 1,000
delta values and transformed back into (strike, call price) space
then second derivate computed.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
de Vincent-Humphreys and Noss (2012)
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
de Vincent-Humphreys and Noss (2012)
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
Estimating the parameters p and q of the Beta calibration function
is done from observing independent PIT using MLE. This means
that expirations τi ≡ T − ti cannot have any time intersections.
Otherwise the PITs are not anymore independent and one has
dependent PITs.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
Motivations Unemployment Rate
Motivations Capacity Utilization Rate
Calibration of Option-Based Probability Assessments
One week
One month
1
1
0.9
0.9
0.8
0.8
0.7
0.7
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0.2
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Two months
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Three months
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Six months
1
0.9
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0
0
Enrique ter Horst
100
200
300
400
500
600
700
800
900
CESA
Term
structure
Joint
of impliedwork
risk neutral
with
densities
Roberto Casa
Figure : PIT time series
of and
theIESA
EUR/USD.
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
• Underlying Let xt,τi , i = 1, . . . , M, t = 1, . . . , T , be a set of
underlying realized forward levels, available at time t for the
Q
P (x) denote the
different maturities τ1 , . . . , τM . Let Ft,τ
(x) and Ft,τ
i
i
risk neutral and the physical cumulative density functions (cdf),
Q
P (x) their probability density
respectively and ft,τ
(x) and ft,τ
i
i
functions (pdf).
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
• Underlying Let xt,τi , i = 1, . . . , M, t = 1, . . . , T , be a set of
underlying realized forward levels, available at time t for the
Q
P (x) denote the
different maturities τ1 , . . . , τM . Let Ft,τ
(x) and Ft,τ
i
i
risk neutral and the physical cumulative density functions (cdf),
Q
P (x) their probability density
respectively and ft,τ
(x) and ft,τ
i
i
functions (pdf).
• Joint Calibration Then assume the following joint deformation
Q
Q
model FtP (xt,τ1 , . . . , xt,τM ) = Ct (Ft,τ
(xt,τ1 ), . . . , Ft,τ
(xt,τM ))
1
M
M
where Ct : [0, 1] → [0, 1], t = 1, . . . , T , is a sequence of
deformation functions. Or:
P
ft,τ
(xt,τ1 , . . . , xt,τM )
=
Q
Q
ct (Ft,τ
(xt,τ1 ), . . . , Ft,τ
(xt,τM ))
1
M
M
Y
Q
ft,τ
(xt,τj )
j
j=1
where ct is the mixed partial derivative of Ct with respect all the
arguments
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Q
• Beta Markov Random Field Let yjt = Ft,τ
(xt,τj ), j = 1, . . . , M,
j
then in order to model the dependence of the prediction densities
at different dates, our modelling assumption is a beta dynamic
Markov random field (β-MRF).
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References
Q
• Beta Markov Random Field Let yjt = Ft,τ
(xt,τj ), j = 1, . . . , M,
j
then in order to model the dependence of the prediction densities
at different dates, our modelling assumption is a beta dynamic
Markov random field (β-MRF).
• Local Neighbourhood Let E = [0, 1] be the phase space and
S = {1, . . . , M} the finite set of sites (see Bremaud (1999), ch. 7)
corresponding to the different maturities, then our β-MRF is
defined by the following local specification:
ct (y1t , . . . , yMt ) =
M
1 Y
cjt (yjt |yN(j) )
Zt
(3)
j=1
where yN(j) = {ykt , k ∈ N(j) ⊂ S} with N(j) a member of the
neighbourhood system N, cjt represents the j-th components of
the joint calibration function ct and Zt is a normalization function
which may depend on the parameter of the calibration model and
Enriquemay
ter Horst
CESA
Termand
structure
IESA Joint
of impliedwork
risk neutral
with
densities
Roberto Casa
be not known.
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
• First Model Modelling the full dependence structure between
densities at the different maturities and allowing for time-change
could lead to over-fitting problems. Therefore, we consider two
parsimonious beta MRF models by assuming a time-invariant
topology (S, N):
∅
if j = 1
N(j) =
{j − 1} if j 6= 1
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References
• First Model Modelling the full dependence structure between
densities at the different maturities and allowing for time-change
could lead to over-fitting problems. Therefore, we consider two
parsimonious beta MRF models by assuming a time-invariant
topology (S, N):
∅
if j = 1
N(j) =
{j − 1} if j 6= 1
• Second Model and the second one is a proximity model

if j = 1
 {2}
{j − 1, j + 1} if j 6= 1, M
N(j) =

{M − 1}
if j = M
connecting each density with the two adjacent densities in terms of
maturity.
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• Following the standard practice in calibration literature (e.g., see
Fackler and King (1990)) we assume that the j-th component of
the joint calibration function is the probability density function of a
beta distribution.
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• Following the standard practice in calibration literature (e.g., see
Fackler and King (1990)) we assume that the j-th component of
the joint calibration function is the probability density function of a
beta distribution.
• In order to account for possible time dependence in the PITs we
let the parameter of the beta calibration function of the density at
maturity τj to depend on the past values of the PITs for the same
maturity.
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• Following the standard practice in calibration literature (e.g., see
Fackler and King (1990)) we assume that the j-th component of
the joint calibration function is the probability density function of a
beta distribution.
• In order to account for possible time dependence in the PITs we
let the parameter of the beta calibration function of the density at
maturity τj to depend on the past values of the PITs for the same
maturity.
• We use the re-parametrization of beta pdf used in Bayesian
mixture models (e.g., see Robert and Rousseau (2002) and
Bouguila et al. (2006)) and Bayesian beta autoregressive processes
(e.g., see Casarin et al. (2012))
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•
µ γjt −1
cjt (yjt |yN(j) ) = Bjt yjt jt
Enrique ter Horst
(1 − yjt )(1−µjt )γjt −1
(4)
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•
µ γjt −1
cjt (yjt |yN(j) ) = Bjt yjt jt
with
Bjt =
Enrique ter Horst
(1 − yjt )(1−µjt )γjt −1
(4)
Γ(µjt )
Γ(µjt ψjt )Γ((1 − µjt )ψjt )
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•
µ γjt −1
cjt (yjt |yN(j) ) = Bjt yjt jt
with
Bjt =
(1 − yjt )(1−µjt )γjt −1
(4)
Γ(µjt )
Γ(µjt ψjt )Γ((1 − µjt )ψjt )
•

µjt
=
γjt
= γj
ϕ α0j +
p
X
k=1

αkj yt−k,j +
X
βkj yt,k 
(5)
k∈N(j)
(6)
with ϕ : R 7→ [0, 1] a twice differentiable strictly monotonic link
function. We assume a logistic function.
Enrique ter Horst
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A dynamic calibration model
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References
•The extraction of parametric and nonparametric risk-neutral
densities has been important not only for traders in order to use
this density to price other more exotic derivatives but for central
bankers as well and policy makers Aı̈t-Sahalia and Duarte (2003);
Rouah and Vainberg (2007).
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
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Motivation and Previous Works
A dynamic calibration model
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Conclusions and Future Work
Bayesian Inference
References
•The extraction of parametric and nonparametric risk-neutral
densities has been important not only for traders in order to use
this density to price other more exotic derivatives but for central
bankers as well and policy makers Aı̈t-Sahalia and Duarte (2003);
Rouah and Vainberg (2007).
• Recently a great deal of interest has grown in predicting the both
the nonparametric risk-neutral and its physical counterpart
simultaneously for the 3-month Euribor interest rate using the beta
calibration function as provided by Vesela and Gutiérrez (2013).
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
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Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
•The extraction of parametric and nonparametric risk-neutral
densities has been important not only for traders in order to use
this density to price other more exotic derivatives but for central
bankers as well and policy makers Aı̈t-Sahalia and Duarte (2003);
Rouah and Vainberg (2007).
• Recently a great deal of interest has grown in predicting the both
the nonparametric risk-neutral and its physical counterpart
simultaneously for the 3-month Euribor interest rate using the beta
calibration function as provided by Vesela and Gutiérrez (2013).
• These constant maturity risk-neutral densities are interpolated in
practice from fixed expiration densities as done in Vergote and
Gutiérrez (2012).
Enrique ter Horst
Joint work with Roberto Casa
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A dynamic calibration model
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References
•As an application of our estimation algorithm, we shall consider a
geometric Brownian motion process, St , t ∈ [0, T ], to model the
price of the underlying as in Black and Scholes (1973) and Merton
(1973), i.e.
Z t
Z t
St = S0 +
Su µdu +
Su σdW (u)
(7)
0
0
where Wt , t ∈ [0, T ], is a Wiener process.
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•As an application of our estimation algorithm, we shall consider a
geometric Brownian motion process, St , t ∈ [0, T ], to model the
price of the underlying as in Black and Scholes (1973) and Merton
(1973), i.e.
Z t
Z t
St = S0 +
Su µdu +
Su σdW (u)
(7)
0
0
where Wt , t ∈ [0, T ], is a Wiener process.
• We simulate price sample paths under the physical measure for
3, 6 and 12 months for a time interval of T = 2 years, µ = 0.20,
r = 0.05, σ = 0.15, τ1 = 0.25, τ2 = 0.5, and τ3 = 1, j = 1, 2, 3.
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•In our simulation exercise we assume that a year has 252 trading
days (prices) and that 3 (6 and 12) months correspond to 63 (126
and 252) trading days respectively.
Enrique ter Horst
Joint work with Roberto Casa
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Termand
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A dynamic calibration model
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Bayesian Inference
References
•In our simulation exercise we assume that a year has 252 trading
days (prices) and that 3 (6 and 12) months correspond to 63 (126
and 252) trading days respectively.
• In our calibration exercise we also assume that in the evaluation
of the risk-neutral densities the wrong value of the parameter σ is
used. We consider two settings, σ = 0.2 (volatility overestimation)
and σ = 0.1 (volatility underestimation).
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σ = 0.2
σ = 0.1
1
1
NC
C
NC
C
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0
0
0.2
0.2
0.4
0.6
0.8
1
1
0
0
0.8
0.6
0.6
0.4
0.4
0.2
0.6
0.8
1
0.4
0.6
0.8
1
0.4
0.6
0.8
1
0.2
0.2
0.4
0.6
0.8
1
1
0
0
0.2
1
NC
C
NC
C
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0
0
0.4
NC
C
0.8
0
0
0.2
1
NC
C
0.2
0.2
0.4
0.6
0.8
1
0
0
0.2
Figure : Non calibrated and calibrated risk neutral distribution for
different maturities (rows) and volatility levels (columns).
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References
σ = 0.2
σ = 0.1
5
8
4
6
3
4
2
2
1
0
60
80
100
120
140
160
5
0
60
80
100
120
140
160
80
100
120
140
160
80
100
120
140
160
8
4
6
3
4
2
2
1
0
60
80
100
120
140
160
5
0
60
8
4
6
3
4
2
2
1
0
60
80
100
120
140
160
0
60
Figure : Non calibrated (dashed line) and calibrated (solid line) risk
neutral distribution and price level (vertical dotted line) at last point of
the sample, i.e. t = 504, for different maturities (rows) and different
volatility levels (columns).
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References
σ = 0.2
1
0.8
NC
C
U
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
0.4
0.6
0.8
1
0.4
0.6
0.8
1
1
0.8
NC
C
U
0.6
0.4
0.2
0
0
0.2
1
0.8
NC
C
U
0.6
0.4
0.2
0
0
0.2
Figure : Non calibrated (solid line), perfectly calibrated (dotted line), and
β-MRF calibrated (gray dashed line) risk neutral distributions for
different maturities (rows). In each plot, gray areas represent the 95%
HPD region.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
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Motivation and Previous Works
A dynamic calibration model
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Bayesian Inference
References
• This paper provides an extension in the context of random fields
of beta models existing in the literature and builds upon the
Bayesian autoregressive model by Casarin et al. (2012).
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
• This paper provides an extension in the context of random fields
of beta models existing in the literature and builds upon the
Bayesian autoregressive model by Casarin et al. (2012).
• We have provided a new modelling framework using both the
derivative and spot markets for the term structure of the implied
probability, which accounts for the possible dependence between
pits at different maturities and different dates for a given maturity,
therefore allowing borrowing of information between the different
tenors for both the risk-neutral and the physical measures.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
• This paper provides an extension in the context of random fields
of beta models existing in the literature and builds upon the
Bayesian autoregressive model by Casarin et al. (2012).
• We have provided a new modelling framework using both the
derivative and spot markets for the term structure of the implied
probability, which accounts for the possible dependence between
pits at different maturities and different dates for a given maturity,
therefore allowing borrowing of information between the different
tenors for both the risk-neutral and the physical measures.
• We also provide a proper inferential Bayesian framework that
allows us to include parameter uncertainty in the density
calibration functions, and therefore in the physical densities.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
• This paper provides an extension in the context of random fields
of beta models existing in the literature and builds upon the
Bayesian autoregressive model by Casarin et al. (2012).
• We have provided a new modelling framework using both the
derivative and spot markets for the term structure of the implied
probability, which accounts for the possible dependence between
pits at different maturities and different dates for a given maturity,
therefore allowing borrowing of information between the different
tenors for both the risk-neutral and the physical measures.
• We also provide a proper inferential Bayesian framework that
allows us to include parameter uncertainty in the density
calibration functions, and therefore in the physical densities.
• Comments are welcomed!
Enrique ter Horst
Joint work with Roberto Casa
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Termand
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A dynamic calibration model
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References
Let xt = (xt,τ1 , . . . , xt,τM ) be a set of observations for different
maturities, and xp+1:T = (xp+1 , . . . , xT ), then the likelihood of the
model writes as
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Let xt = (xt,τ1 , . . . , xt,τM ) be a set of observations for different
maturities, and xp+1:T = (xp+1 , . . . , xT ), then the likelihood of the
model writes as
L(xp+1:T |θ) =
T
Y
P
ft,τ
(xt,τ1 , . . . , xt,τM )
t=p+1
=
T
Y
t=p+1
Enrique ter Horst
1
Zt
M
Y
µjt γj −1
Q
Bjt (µjt γj , (1 − µjt )γj ) Ft,τ
(x
)
t,τj
j
j=1
(1−µjt )γj −1
Q
Q
(xt,τj )
1 − Ft,τ
(x
)
ft,τ
t,τj
j
j
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Let xt = (xt,τ1 , . . . , xt,τM ) be a set of observations for different
maturities, and xp+1:T = (xp+1 , . . . , xT ), then the likelihood of the
model writes as
L(xp+1:T |θ) =
T
Y
P
ft,τ
(xt,τ1 , . . . , xt,τM )
t=p+1
=
T
Y
t=p+1
1
Zt
M
Y
µjt γj −1
Q
Bjt (µjt γj , (1 − µjt )γj ) Ft,τ
(x
)
t,τj
j
j=1
(1−µjt )γj −1
Q
Q
(xt,τj )
1 − Ft,τ
(x
)
ft,τ
t,τj
j
j
Note that this is a pseudo-likelihood, since we assume that the p
initial values of the β-MRF are known.
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We assume the following hierarchical specification of the prior
distribution
i.i.d.
N (αj , sj2 ), k = 0, . . . , p, j = 1, . . . , M
(8)
i.i.d.
N (βj , gj2 ), k = 1, . . . , mj , j = 1, . . . , M
(9)
i.i.d.
IG(ξ/2, ξ/2), j = 1, . . . , M
(10)
i.i.d.
∼
N (α, s 2 ), j = 1, . . . , M
(11)
i.i.d.
∼
N (β, g 2 ), j = 1, . . . , M
(12)
α
∼
(13)
β
∼
N (a, s02 )
N (b, g02 )
αkj
βkj
γj
αj
βj
∼
∼
∼
(14)
where mj = Card(N(j)) is the number of elements of N(j).
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After some tedious algebra, the joint posterior distribution can be
written as

T
T
M
X
X
X
π(θ|xp+1:T ) ∝ exp −
log Zt −
log Bjt (15)
t=p+1
+
T
M X
X
t=p+1 j=1

Q
Ajt µjt + (1 − Ft,τ
(xt,τj )) exp(σj ) f (θ)
j
t=p+1 j=1
where
Bjt = Bjt (µjt exp(σj ), (1 − µjt ) exp(σj ))
and
Q
Q
Ajt = log(Ft,τ
(xt,τj )/(1 − Ft,τ
(xt,τj )))
j
j
Enrique ter Horst
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References
• A major problem with this model is that the normalizing
constants Zt , t = p + 1, . . . , T , in the likelihood function and in
the posterior distribution are unknown and possibly depend on the
parameters. Thus, samples from π(θ|xp+1:T ) cannot be easily
obtained with standard MCMC procedures.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
• A major problem with this model is that the normalizing
constants Zt , t = p + 1, . . . , T , in the likelihood function and in
the posterior distribution are unknown and possibly depend on the
parameters. Thus, samples from π(θ|xp+1:T ) cannot be easily
obtained with standard MCMC procedures.
• For instance, the standard MH algorithm cannot be directly
applied because the acceptance probability involves ratios of
unknown normalizing constants. In the last two decades, various
approximation methods have been proposed in order to circumvent
the problem of intractable normalizing constants.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
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IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
• A major problem with this model is that the normalizing
constants Zt , t = p + 1, . . . , T , in the likelihood function and in
the posterior distribution are unknown and possibly depend on the
parameters. Thus, samples from π(θ|xp+1:T ) cannot be easily
obtained with standard MCMC procedures.
• For instance, the standard MH algorithm cannot be directly
applied because the acceptance probability involves ratios of
unknown normalizing constants. In the last two decades, various
approximation methods have been proposed in order to circumvent
the problem of intractable normalizing constants.
• We define the parameter vector θ = (θ 1 , . . . , θ M , α, β) where
θ j = (αj , β j , σj , αj , βj ), αj = (α0j , α1j , . . . , αpj ) and
β j = (β1j , . . . , βmj j ) for j = 1, . . . , M.
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References
• Recently Møller et al. (2006) proposed an auxiliary variable
MCMC algorithm, which is a feasible simulation procedure for
many models with intractable normalizing constant. The Møller
et al. (2006)’s single auxiliary variable method has been succesfully
improved by Murray et al. (2006). They propose the exchange
algorithm, which removes the need to estimate the parameter
before sampling begins, and has higher acceptance probability than
Møller et al. (2006) ’s algorithm.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
• Recently Møller et al. (2006) proposed an auxiliary variable
MCMC algorithm, which is a feasible simulation procedure for
many models with intractable normalizing constant. The Møller
et al. (2006)’s single auxiliary variable method has been succesfully
improved by Murray et al. (2006). They propose the exchange
algorithm, which removes the need to estimate the parameter
before sampling begins, and has higher acceptance probability than
Møller et al. (2006) ’s algorithm.
• Unfortunately both the single auxiliary variable and the
exchange algorithms require exact sampling of the auxiliary
variable from its conditional distribution, which can be
computational expensive for many statistical models.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
An exact simulation algorithm for our beta MRF model is not
available, thus in this paper we follow and alternative route and
apply the double MH algorithm proposed by Liang (2010). The
double MH avoids the exact simulation step by applying an
internal MH step to generate the auxiliary variable.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
The Double MH Sampler of Liang (2010)
Assume the current value of the MH chain is θ (t) = θ, then the
double MH sampler iterates over the following steps:
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
The Double MH Sampler of Liang (2010)
Assume the current value of the MH chain is θ (t) = θ, then the
double MH sampler iterates over the following steps:
• (a) Simulate a new sample θ 0 from π(θ) using a MH algorithm
starting with θ.
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
The Double MH Sampler of Liang (2010)
Assume the current value of the MH chain is θ (t) = θ, then the
double MH sampler iterates over the following steps:
• (a) Simulate a new sample θ 0 from π(θ) using a MH algorithm
starting with θ.
• (b) Generate the auxiliary variable zp+1:T ∼ Pθn0 (zp+1:T |xp+1:T )
and accept it with probability min{1, ρ(θ, θ 0 , zp+1:T |xp+1:T )}
where ρ(θ, θ 0 , zp+1:T |xp+1:T ) is equal to:
ρ(θ, θ 0 , zp+1:T |xp+1:T ) =
Enrique ter Horst
L(zp+1:T |θ) L(xp+1:T |θ 0 )
L(xp+1:T |θ) L(zp+1:T |θ 0 )
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
The Double MH Sampler of Liang (2010)
Assume the current value of the MH chain is θ (t) = θ, then the
double MH sampler iterates over the following steps:
• (a) Simulate a new sample θ 0 from π(θ) using a MH algorithm
starting with θ.
• (b) Generate the auxiliary variable zp+1:T ∼ Pθn0 (zp+1:T |xp+1:T )
and accept it with probability min{1, ρ(θ, θ 0 , zp+1:T |xp+1:T )}
where ρ(θ, θ 0 , zp+1:T |xp+1:T ) is equal to:
ρ(θ, θ 0 , zp+1:T |xp+1:T ) =
L(zp+1:T |θ) L(xp+1:T |θ 0 )
L(xp+1:T |θ) L(zp+1:T |θ 0 )
• (c) Set θ (t+1) = θ 0 if the auxiliary variable is accepted and
θ t+1 = θ otherwise. Two MH in (b): one for auxiliary variable and
one for accepting θ 0 .
Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
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Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
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Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
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Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities
Motivation and Previous Works
A dynamic calibration model
Simulation and Application
Conclusions and Future Work
Bayesian Inference
References
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Enrique ter Horst
Joint work with Roberto Casa
CESA
Termand
structure
IESA of implied risk neutral densities