Modeling of Oil Prices - Ke Du, Eckhard Platen and Renata Rendek

 
Modeling of Oil Prices - Ke Du, Eckhard Platen and Renata Rendek
QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE F

INANCE RESEARCH CENTRE

               QUANTITATIVE FINANCE RESEARCH CENTRE

           Research Paper 321                                      December 2012

                         Modeling of Oil Prices
                         Ke Du, Eckhard Platen and Renata Rendek

             ISSN 1441-8010                                www.qfrc.uts.edu.au
Modeling of Oil Prices - Ke Du, Eckhard Platen and Renata Rendek
Modeling of Oil Prices

       Ke Du 1 , Eckhard Platen                 2
                                                    and Renata Rendek 3

                                   December 21, 2012

 Abstract: The paper derives a parsimonious two-component affine diffusion
 model with one driving Brownian motion to capture the dynamics of oil prices.
 It can be observed that the oil price behaves in some sense similarly to the US
 dollar. However, there are also clear differences. To identify these the paper
 studies the empirical features of an extremely well diversified world stock in-
 dex, which is a proxy of the numéraire portfolio, in the denomination of the oil
 price. Using a diversified index in oil price denomination allows us to disen-
 tangle the factors driving the oil price. The paper reveals that the volatility of
 the numéraire portfolio denominated in crude oil, increases at major oil price
 upward moves. Furthermore, the log-returns of the index in oil price denom-
 ination appear to follow a Student-t distribution. These and other stylized
 empirical properties lead to the proposed tractable diffusion model, which has
 the normalized numéraire portfolio and market activity as components. An
 almost exact simulation technique is described, which illustrates the charac-
 teristics of the proposed model and confirms that it matches well the observed
 stylized empirical facts.

JEL Classification: G10, C10, C15
1991 Mathematics Subject Classification: 62P05, 62P20, 62G05, 62-07, 68U20

Key words and phrases: commodities, oil price, numéraire portfolio, market ac-
tivity, square root processes, benchmark approach.

   1
     University of Technology Sydney, Finance Discipline Group, PO Box 123, Broadway, NSW,
2007, Australia.
   2
     University of Technology Sydney, Finance Discipline Group and School of Mathematical
Sciences, PO Box 123, Broadway, NSW, 2007, Australia, Email: Eckhard.Platen@uts.edu.au,
Phone: +61295147759, Fax: +61295147711.
   3
     University of Technology Sydney, School of Mathematical Sciences, PO Box 123, Broadway,
NSW, 2007, Australia, Email: Renata.Rendek@uts.edu.au, Phone: +61295147781.
1     Introduction
Motivated by the fact that oil prices behave in some sense similarly to the US
dollar because oil is traded in this currency, and since the log-returns of the world
stock index in oil price denomination appear to follow a Student-t distribution,
we model oil prices by a similar methodology as was proposed in Platen & Rendek
(2012a) for currencies. This methodology employs a well diversified stock index
as proxy of the numéraire portfolio (NP). The latter equals the growth optimal
portfolio, which maximizes expected logarithmic utility from terminal wealth.
The two components of the proposed new model are the normalized approximate
NP denominated in units of oil, and the inverse of the respective market activity.
Both quantities are modeled as square root processes, where the first one is moving
slower than the second one. They are both driven only by one Brownian motion,
modeling the nondiversifiable uncertainty of the market with respect to oil price
denomination. It turns out that the crucial difference to the model proposed in
Platen & Rendek (2012a), is the correlation between the normalized NP in oil
denomination and its market activity. It is observed that in contrast to the NP
in currency denomination, the NP in oil denomination has positive correlation
with its market activity. This property can be interpreted as an anti-leverage
effect, generating higher market activity and volatility when the NP in oil price
denomination increases, that is, it declines substantially relative to the NP. This
type of change in market activity makes economic sense because a lower oil price
is likely to trigger increased economic activity, including trading activity. A
diversified index in currency denomination behaves differently. Here we have
the leverage effect where the market activity increases when the NP in currency
denomination decreases.
As it turned out during the investigation, in order to capture over long time
periods realistically the oil price evolution, the model needs to be formulated
in a general financial modeling framework, which goes beyond the classical no-
arbitrage paradigm. By interpreting a well diversified world stock index as NP
the fitting of the proposed parsimonious model can be accomplished such that it
captures well reality, in particular, the long term dynamics of the price of the oil
price relative to the NP which is approximated by a well diversified stock index.
The benchmark approach, see Platen & Heath (2010) and Platen (2011) provides
the mathematical framework for the modeling. It generalizes the classical no-
arbitrage modeling and pricing framework towards a much richer modeling world.
In particular, pricing is performed under the real world probability measure with
the NP as numéraire. The central building block of the benchmark approach is its
benchmark, the NP, which is also the growth optimal portfolio, see Long (1990)
and Kelly (1956). This portfolio is employed as the fundamental unit of value in
the analysis, which is significantly different to the classical approach, where one
uses typically the savings account as denominator. Using an approximate NP in
oil price denomination allows us to disentangle the factors driving the oil price

                                         2
and those driving the currency. This is important for the statistical analysis as
well as the modeling.
Since, oil prices have grown considerably over the last decades, it is important to
approximate closely the NP, which is in many ways the ”best” performing port-
folio. The Naive Diversification Theorem in Platen & Rendek (2012b) states that
the equi-weighted index (EWI) approximates well the NP of a given investment
universe when the number of constituents is large and the given market is well
securitized. The latter property essentially means that the risk factors driving the
underlying risky securities are sufficiently different. The EWI used in this paper
is an extremely well diversified index constructed in Platen & Rendek (2012c),
where the details for its construction can be found.
The paper is organized as follows: Section 2 describes the object of study, that
is, the discounted equi-weighted index denominated in the oil price. Section 3
extracts a list of stylized empirical facts for the observed dynamics. Section 4
proposes a parsimonious, tractable model for these dynamics involving the power
of a time transformed affine diffusion. It also discusses the volatility and market
activity dynamics arising from the proposed model. Section 5 describes a robust
step-by-step methodology for fitting the proposed model and visualizes volatility
and market activity as they emerge under the model. Section 6 describes for
the model an almost exact simulation method, which allows us to confirm that
the empirical properties of the model match the list of stylized empirical facts of
Section 3. Finally, Section 7 summarizes the model described in Platen & Rendek
(2012a) for the denomination of the numéraire portfolio in currency denomination.
This allows us to model the oil price in currency denomination.

2    Approximate Numéraire Portfolio in Oil Price
     Denomination
The aim of this section is to introduce the object of study, which is the discounted
NP denominated in units of oil. It is important to approximate well the dynamics
of the NP denominations. The oil price, which has grown dramatically over the
years, will use the ratio of the denomination of the NP in domestic currency over
the denomination of the NP in oil. The current paper focuses on the modeling of
the dynamics of the NP in oil denomination. For the denomination of the NP in
currency denomination we will refer to Platen & Rendek (2012a).

Numéraire Portfolio

The numéraire portfolio (NP) is a strictly positive portfolio which when used as
benchmark turns all benchmarked nonnegative portfolios into supermartingales,
see Platen & Heath (2010). Denote by St an approximate value of the NP in

                                         3
units of the domestic currency (say US dollar) at time t ≥ 0. Following Platen &
Rendek (2012b), we use an equi-weighted index (EWI) as approximate NP.
The EWI considered in this paper is identical to the one calculated in Platen
& Rendek (2012c). It was built from almost 10, 000 stocks, whose total return
prices were obtained from Thomson Reuters Datastream. The EWI was built
in three stages: first, country subsector equi-weighted indices were constructed;
second, from these constituents country equi-weighted indices were built; and
third, finally the EWI was calculated by equal value weighting the country equi-
weighted indices. We always took 40 basis points proportional transaction costs
into account. In Platen & Rendek (2012b, 2012c) a description of the method
for building such equi-weighted indices is described in detail and a Naive Di-
versification Theorem is proved that gives the theoretical reasoning behind the
approximation of the NP.

         18

         16

         14

         12

                            ln(EWI)
         10

                                             ln(MCI)
          8

          6

          4
          1973   1978    1983         1988   1993      1998   2003   2008   2013

Figure 2.1: Logarithms of the MCI and the EWI under 40 bp transaction costs.

In Fig. 2.1 we display the logarithm of the EWI in US dollar denomination to-
gether with the market capitalization weighted index (MCI). The MCI displayed
in this figure is the global Datastream index (with mnemonics TOTMKWD) used
also in Platen & Rendek (2012a). It fluctuates and performs very similarly to
the FTSE all-cap index and the MSCI total return world index. The EWI is
clearly growing on average faster than the MCI. This is mainly due to its better
diversification resulting from the construction methodology used.

Numéraire Portfolio Denominated in Units of Oil

The oil price itself in US dollar denomination is driven by the uncertainty of two
major securities. These are the commodity oil and the currency US dollar. One
needs to disentangle their combined influence on the oil price, which is usually

                                             4
30000

       25000

       20000

       15000

       10000

        5000

           0
         04/1985     04/1990     04/1995               04/2000   04/2005   04/2010

                     Figure 2.2: The discounted EWI of oil.

given in US dollar denomination. We do this by involving a well-diversified global
index, the EWI, which we interpret here also as proxy of the NP. In Platen &
Rendek (2012a) the dynamics of an EWI in US dollar denomination has been an-
alyzed and modeled. Similarly we analyze and model in this paper the dynamics
of an EWI in oil price denomination. In some sense we obtain a least disturbed
observation of the dynamics of the commodity oil, when we denominate the ex-
tremely well diversified global index, the EWI and proxy of the NP, in units of
oil.
The NP in oil denomination St at time t can be expressed by the ratio
                                               Ct
                                    St =          ,                                  (2.1)
                                               Xt
where Xt is the oil spot price in US dollar at time t ≥ 0 and Ct is the US dollar
denomination of the NP at time t.

Oil Savings Account

The next step is to construct the oil savings account
                                           Rt
                                                   rs1 ds
                                  Bt1 = e      0            ,                        (2.2)
where the convenience yield rt1 for oil is approximated by the expression
                                            
                             1       1       Ft
                           rt = − ln             + rt0 .                             (2.3)
                                    ∆        Xt
                                                    3
Here Ft is the three months oil futures price, ∆ = 12 and rt0 is the US dollar
interest rate at time t ≥ 0, see e.g. Du & Platen (2012) for details on this
approximation.

                                           5
Discounted Numéraire Portfolio for Oil

The object of our study is now the, by the oil savings account (2.2), discounted
NP. That is,
                                         St
                                   S̄t = 1                                  (2.4)
                                        Bt
for t ≥ 0. Fig.2.2 plots the oil discounted NP for the period from 02/04/1985 until
18/03/2010. We note an approximately exponential increase of the oil discounted
NP. In the next section we will apply some standard statistical methods in order
to identify stylized empirical properties of the oil discounted NP.

3    Empirical Observations
Platen & Rendek (2012a) observed seven stylized empirical facts pertaining to
diversified world stock indices in currency denomination. Below we check whether
similar or different properties emerge for the oil discounted NP, that is, the EWI
denominated in an oil savings account.

(i) Uncorrelated Returns
     Fig. 3.1 displays the autocorrelation function for the log-returns of the oil
     discounted EWI with 95% confidence bounds. Similarly, to the log-returns
     of the index in currency denominations, the autocorrelation of log-returns
     of the oil discounted EWI is close to zero.

                                                          Sample Autocorrelation Function

                                 0.8
       Sample Autocorrelation

                                 0.6

                                 0.4

                                 0.2

                                  0

                                −0.2
                                       0   10   20   30       40        50        60        70   80   90   100
                                                                       Lag

Figure 3.1: Autocorrelation function for log-returns of the oil discounted EWI.

                                                                      6
(ii) Correlated Absolute Returns
     Fig. 3.2 plots the autocorrelation function of the absolute log-returns of the
     oil discounted EWI. Even for large lags the autocorrelation is non-negligible
     and does not seem to show an exponential decline.

                                                         Sample Autocorrelation Function

                                0.8
      Sample Autocorrelation

                                0.6

                                0.4

                                0.2

                                 0

                               −0.2
                                      0   10   20   30       40        50        60        70   80   90   100
                                                                      Lag

Figure 3.2: Autocorrelation function for the absolute log-returns of the oil dis-
counted EWI.

(iii) Student-t Distributed Returns
     Fig. 3.3 displays the log-histogram of normalized log-returns of the oil dis-
     counted EWI with the logarithm of the Student-t density with 3.13 degrees
     of freedom, see last column in Table 3.1 for the estimated degrees of free-
     dom. Visually the fit seems to be very good. In order to further quantify
     the fit of the Student-t distribution we perform a log-likelihood ratio test in
     the family of the symmetric generalized hyperbolic (SGH) distributions, see
     Rao (1973) and Platen & Rendek (2008). Table 3.1 reports the test statis-
     tics calculated for four special cases of the SGH distribution. These are: the
     Student-t distribution, the normal inverse Gaussian (NIG) distribution, the
     hyperbolic distribution and the variance gamma (VG) distribution. The
     test statistics are here distributed according to the chi-square distribution
     with one degree of freedom. Therefore, the hypothesis of the Student-t dis-
     tribution being the best candidate distribution in the family of the SGH
     distributions cannot be rejected at the 99.9% level of significance, since
     0.00000002 < χ20.001,1 ≈ 0.000002.

(iv) Volatility Clustering
     Fig. 3.4 illustrates the estimated annualized volatility Vti of the oil dis-
     counted EWI. The squared volatility Vt2i at time ti is obtained from squared
     log-returns via exponential smoothing. For the discretization time ti = ∆i,

                                                                     7
0
            10

              −1
            10

                                                              log−empirical density
              −2
            10

              −3
            10

                                                                                 log−Student−t density
              −4
            10

              −5
            10

              −6
            10
              −10          −5         0                   5              10                15              20

Figure 3.3: Logarithms of empirical density of normalized log-returns of the oil
discounted EWI and Student-t density with 3.13 degrees of freedom.

Table 3.1: Log-Maximum likelihood test statistic for the log-returns of the oil
discounted EWI.

Commodity               Student-t              NIG               Hyperbolic                         VG              df.

Crude Oil             0.00000002          61.168568              182.120161                181.189575              3.13

            2.5

                 2

            1.5

                 1

            0.5

              0
            04/1985         04/1990        04/1995             04/2000                04/2005            04/2010

  Figure 3.4: Estimated volatility from log-returns of the oil discounted EWI.

       for i ∈ {0, 1, 2, . . . }, the exponential smoothing is applied to squared log-

                                                      8
1

         0.5

          0

        −0.5

         −1

        −1.5

         −2

        −2.5
         04/1985     04/1990      04/1995       04/2000    04/2005    04/2010

Figure 3.5: Logarithms of normalized EWI for oil (upper graph) and its volatility
(lower graph).

         3.5

          3

         2.5

          2

         1.5

          1

         0.5

          0

        −0.5
         04/1985     04/1990      04/1995       04/2000    04/2005    04/2010

Figure 3.6: Quadratic covariation between the logarithms of normalized EWI for
oil and its volatility.

     returns Rt2i in the following way:
                                      √              √
                            Vt2i+1 = α ∆Rt2i + (1 − α ∆)Vt2i ,                  (3.1)

     for i ∈ {0, 1, 2, . . . }. Here the smoothing parameter λ is assumed to equal
     α = 0.92. This choice works well and has been used in Platen & Rendek
     (2012a).
     The volatility of the oil discounted EWI in Fig 3.4 exhibits periods of low
     volatility and periods of high volatility. It can be conjectured that such
     volatility is potentially a stationary stochastic process.

                                            9
(v) Long Term Exponential Growth
     Fig. 5.1 illustrates the logarithm of the oil discounted EWI with a trend
     line fitted by linear regression. The logarithm of the oil discounted EWI
     exhibits consistent long term linear growth, which in turn results in the
     long term exponential growth for the oil discounted EWI.

(vi) Anti-Leverage Effect
     A leverage effect is typically observed for the currency discounted world
     stock index, see Platen & Rendek (2012a). This empirical fact is, however,
     not observed for the oil discounted EWI and its normalized version, shown
     in Fig. 5.2, where its average long term growth is taken out by dividing with
     a respective exponential function of time. In Fig. 3.5 we plot the logarithms
     of the normalized EWI for oil and its volatility. When the normalized
     EWI for oil moves upwards, in general, the volatility increases and vice
     versa. This implies an anti-leverage effect for the oil discounted EWI and
     its volatility. In fact, the covariation function between the normalized EWI
     for oil and its volatility, displayed in Fig. 3.6, indicates a mostly positive
     correlation between the increments for the normalized EWI for oil and its
     volatility.

(vii) Extreme Volatility at Major Commodity Discounted Index Moves
     Extreme volatility at major index downturns was observed in Platen &
     Rendek (2012a) for the discounted world stock index in currency denom-
     inations. Fig. 3.5 and Fig. 3.6, however, indicate that for the normalized
     EWI for oil the volatility increases when the index moves strongly up and
     the increase is more substantial compared to the ”normal” moves of the
     index.

4    Modeling of Oil Prices
This section derives a parsimonious two-component model for the oil discounted
EWI. It follows to some extent the methodology described in Platen & Rendek
(2012a) with some important changes in the design of the dependencies in the
two-component model.

Discounted Index

The discounted index S̄t , which is the oil discounted index introduced in Section
2, is expressed by the product

                                 S̄t = Aτt (Yτt )q                            (4.1)

                                        10
for t ≥ 0, see Platen & Rendek (2012a). An exponential function Aτt of a given
τ -time, the market activity time (to be specified below), models the long term
average growth of the discounted index as

                                 Aτt = A exp{aτt }                                   (4.2)

for t ≥ 0.
We use in (4.2) the initial parameter A > 0 and the long term average net growth
rate a ∈ ℜ with respect to market activity time.

Normalized Index

As a consequence of equation (4.1), the ratio (Yτt )q = AS̄τt denotes the normalized
                                                             t
index, that is the normalized index for oil, at time t. This normalized index is
assumed to form an ergodic diffusion process evolving according to τ -time. We
assume that it satisfies the SDE

                     
                                    δ
                                               ! q1     
                     δ 1        Γ   2
                                        +q                         p
              dYτ =  −                 δ
                                                      Yτ  dτ +       Yτ dW (τ ),   (4.3)
                     4 2            Γ    2

for τ ≥ 0 with Y0 > 0. Only the two parameters δ > 2 and q > 0 enter the SDE
(4.3) together with its initial value Y0 > 0.

Market Activity Time

We model the market activity time τt via the ordinary differential equation

                                        dτt = Mt dt                                  (4.4)

for t ≥ 0 with τ0 ≥ 0. Here we call the derivative of τ -time with respect to
calendar time t the market activity dτ
                                     dt
                                       t
                                         = Mt at time t ≥ 0. In Platen & Rendek
(2012a) market activity has been modeled by the inverse of a square root process.
Similarly, but different, the process M1 = { M1t , t ≥ 0} is assumed to be a fast
moving square root process in t-time with the dynamics
                                                 r
                         1        ν       1             γ
                     d        =     γ−ǫ       dt −        dWt ,             (4.5)
                        Mt        4      Mt            Mt

for t ≥ 0 with M0 > 0, where γ > 0, ν > 2 and ǫ > 0. Note the negative sign
in front of the diffusion term which indicates the main difference of the model to
the one in Platen & Rendek (2012a).
The Brownian motion W (τ ), which models in market activity time the long
term nondiversifiable uncertainty with respect to oil denomination, is driving

                                              11
the normalized index Yτ . This process is linked to the standard Brownian motion
W = {Wt , t ≥ 0} in t-time through the market activity M in the following way:
                                   r
                                     dτt         p
                        dW (τt ) =       dWt = Mt dWt                      (4.6)
                                     dt
for t ≥ 0 with W0 = 0. The Brownian motion W = {Wt , t ≥ 0} in (4.6) is identical
to the one introduced in the equation (4.5). The above setup produces a two-
component model with only one source of uncertainty. Note that the increments
of the inverse of market activity are positively correlated to the increments of the
normalized index.

Expected Rate of Return and Volatility

By application of the Itô formula one obtains from (4.1), (4.2), (4.3), (4.4) and
(4.6) for the discounted index S̄t the stochastic differential equation (SDE)
                              dS̄t = S̄t (µt dt + σt dWt )                       (4.7)
for t ≥ 0, with initial value S̄0 = A0 (Y0 )q and expected rate of return
            
                             δ
                                     ! q1                          
                         Γ     +  q
                                                            
               a     q       2              δ    1             1 
      µt =       −            δ
                                          +    q + q(q − 1)            Mt .      (4.8)
              Mt 2          Γ 2              4    2           Mt Yτt

The volatility with respect to t-time emerges in the form
                                          s
                                            Mt
                                   σt = q       .                                (4.9)
                                            Yτt

Benchmark Approach

Due to the SDE (4.7) and the Itô formula, the dynamics for the benchmarked
                                   B1
savings account B̂t1 = (S̄t )−1 = Stt , which is the inverse of the oil discounted NP,
is characterized by the SDE
                       dB̂t1 = B̂t1 −µt + σt2 dt − σt dWt ,
                                                            
                                                                                 (4.10)
for t ≥ 0, see (4.8) and (4.9). It follows if for all t ≥ 0 one has
                                       σt2 ≤ µt                                 (4.11)
then the benchmarked savings account B̂t forms an (A, P )-super-
martingale. This is the key property needed to accommodate the model under
the benchmark approach, see Platen & Heath (2010).
To guarantee almost surely in the proposed model the inequality (4.11), one has
by (4.8) and (4.9) to satisfy the following two conditions:

                                          12
Assumption 4.1 First, the dimension δ of the square root process Y needs to
satisfy the equality
                             δ = 2(q + 1).                           (4.12)

Assumption 4.2 The long term average net growth rate a with respect to τ -time
has to satisfy the inequality
                                                    q1
                               q       Γ (2q + 1)
                                                           ≤ a.                (4.13)
                               2       Γ (q + 1)

When equality holds in (4.13) for the proposed model the benchmarked savings
account is a local martingale as assumed in the version of the benchmark ap-
proach formulated in Platen & Heath (2010). For a more general version of the
benchmark approach, where there is no equality in (4.13), we refer to Platen
(2011) and Platen & Rendek (2012a).

5     Model Fitting
Let us now describe the model fitting procedure to the oil discounted EWI. In
the simplified version of the model we assume q = 1 in (4.1), therefore δ = 4
in (4.3) and ν = 4 in (4.5). The main reason for this assumption is the fact
that it is empirically extremely difficult to give a sufficiently precise estimate for
the degrees of freedom of the observed Student-t distributed log-returns, see also
(iii) in Section 6. On the other hand, we may employ arguments from Platen &
Rendek (2012a), which suggest theoretically for currency denominated log-returns
a Student-t distribution with four degrees of freedom. The data indicate with the
estimated 3.13 degrees of freedom for the index log-returns that four degrees of
freedom would work well for a model and would make it very tractable.

Step 1: Normalization of Index

By the fact that Mt has an inverse gamma density with ν degrees of freedom the
mean of Mt is explicitly known. By the ergodic theorem this mean amounts to

                             1 t           4 ǫ
                              Z
                         lim     Ms ds =          P-a.s.                  (5.1)
                        t→∞ t 0          ν −2γ

Therefore, it is possible to approximate (4.2) by the following expression
                                         n 4aǫ      o
                             Aτt ≈ A exp           t ,                          (5.2)
                                          γ(ν − 2)
for t ≥ 0.

                                             13
11

          10

           9

           8

           7                           0.21 t +5.41

           6

           5

           4
         04/1985      04/1990     04/1995             04/2000   04/2005   04/2010

        Figure 5.1: Logarithm of the oil discounted EWI and linear fit.

           3

         2.5

           2

         1.5

           1

         0.5

           0
         04/1985      04/1990     04/1995             04/2000   04/2005   04/2010

                       Figure 5.2: Normalized EWI for oil.

Therefore, since a line can be fitted to the logarithm of the discounted EWI of
                                                                       4aǫ
oil, see Fig. 5.1, it is straightforward to calculate A = 223.32 and γ(ν−2) ≈ 0.21.
Fig. 5.2 illustrates the normalized EWI of oil obtained as the ratio of the oil
discounted EWI over the function in (5.2).

Step 2: Observing Market Activity

By (4.3), (4.4) and an application
                               p of the Itô formula, one obtains as time derivative
of the quadratic variation for Yτt the expression
                                √
                              d[ Y ]τt   1 dτt    Mt
                                       =        =     ,                        (5.3)
                                 dt      4 dt      4
which is proportional to market activity. The estimation of the trajectory of the
market activity process M is performed using daily observations. First, the ”raw”

                                            14
6

           5

           4

           3

           2

           1

            0
          04/1985       04/1990          04/1995        04/2000   04/2005    04/2010

                                  Figure 5.3: Market activity.
                          √
                        d[ Y ]τt
time derivative Qt =       dt
                                    at the ith observation time t = ti is estimated from
the finite difference                      √            √
                                          [ Y ]τti+1 − [ Y ]τti
                                   Q̂ti =                                           (5.4)
                                                ti+1 − ti
for i ∈ {0, 1, . . . }. Second, exponential smoothing is applied to the observed finite
differences according to the recursive standard moving average formula
                                p                       p
                      Q̃ti+1 = α ti+1 − ti Q̂ti + (1 − α ti+1 − ti )Q̃ti ,        (5.5)

i ∈ {0, 1, . . . }, with weight parameter α > 0.
Fig. 5.3 displays the resulting trajectory  √ of Mt for daily observations, when in-
terpreting this value as estimate of 4 dtd [ Y ]τt , for t ≥ 0. Here an initial value of
M0 ≈ 0.21 emerged and the time average of the trajectory of (Mt )−1 amounted
to 11.98.

Step 4: Parameter γ

Fig. 5.4 plots the quadratic variation of the square root of the estimated process
1
M
  . Our estimate for the slope equals here 10.94. Since under the proposed model
            hq i
                1
we have dtd     M
                    = 41 γ, we obtain γ ≈ 43.76.
                    t

Step 5: Parameters ν and ǫ

Fig. 5.5 displays the histogram of market activity with inverse gamma fit with
ν = 2.80 degrees of freedom. Since, we consider a simplified version of the model
the same degrees of freedom for δ = 4 and ν = 4, we obtain from the average
                               4ǫ
value of the market activity γ(ν−2) ≈ 0.21 the estimate for ǫ = 4.57.

                                                   15
300

         250

         200

         150

         100

          50

           0
         04/1985            04/1990       04/1995        04/2000       04/2005         04/2010

                                                                                 1
                   Figure 5.4: Quadratic variation of the square root of         M
                                                                                   .

           5

         4.5

           4

         3.5

           3

         2.5

           2

         1.5

           1

         0.5

           0
               0            1         2              3             4         5           6

       Figure 5.5: Histogram of market activity with inverse gamma fit.

Step 6: Long Term Average Net Growth Rate

Finally, we obtain the long term average net growth rate a ≈ 1, since the average
                                  4ǫ
value of the market activity is γ(ν−2) ≈ 0.21. This indicates, the condition (4.13)
is approximately satisfied as an equality. Therefore, the benchmark approach can
be applied, as described in Platen & Heath (2010) and Platen & Rendek (2012a).

6    Simulation Study
The aim of this section is to describe an almost exact simulation method for the
model introduced in Section 4. As indicated before, both of the processes M1 and
Y are square root processes of dimension δ = ν = 4 in the stylized version of
the model, which we propose. The transition density of the square root process

                                                    16
is the non-central chi-square density, therefore, the simulation can be considered
to be almost exact when sampling from this transition density. More precisely,
it is exact for the process M1 and almost exact for Y . The following four steps
describe the simulation of the normalized index and its volatility:

                                    1
1. Simulation of the Process        M

          16

          14

          12

          10

           8

           6

           4

           2

           0
               0    5      10       15      20      25      30      35      40

                         Figure 6.1: Simulated path of M.

First, we describe the simulation of the inverse M1 of the market activity process.
It is described by the SDE (4.5) and is a square root process of dimension ν = 4.
This process can be sampled exactly due to its non-central chi-square transition
density of dimension ν = 4. That is, we have
                                         s                                  !2 
                   −ǫ(ti+1 −ti )                             )
    1       γ(1 − e              ) 2         4ǫe−ǫ(ti+1 −ti          1
         =                         χ3,i +                        )
                                                                         − Zi  , (6.1)
  Mti+1             4ǫ                      γ(1 − e−ǫ(ti+1 −t  i   ) Mti

for ti = ∆i, i ∈ {0, 1, . . . }; see also Broadie & Kaya (2006) and Platen & Rendek
(2012a). Here Zi is an independent standard Gaussian distributed random vari-
able and χ23,i is an independent chi-square distributed random variable with three
degrees of freedom. Then the right hand side of (6.1) becomes a non-central chi-
square distributed random variable with the requested non-centrality and four
degrees of freedom.
Fig. 6.1 plots the simulated path of the market activity M. The market activity
displayed in this figure has more pronounced spikes compared to the estimated
market activity in Fig. 5.3. We will see later that when the market activity is
estimated from the path of the simulated index it resembles closely the historically
observed path in Fig. 5.3.

                                          17
2. Calculation of τ -Time

The next step of the simulation generates the market activity time, the τ -time.
By (4.4) one aims for the increment
                                 Z ti+1
                   τti+1 − τti =        Ms ds ≈ Mti (ti+1 − ti ),          (6.2)
                                           ti

i ∈ {0, 1, . . . }.
Fig. 6.2 plots the simulated τ -time, which is the market activity time obtained
from the path of the simulated market activity in Fig. 6.1 with the use of the
approximation (6.2).

3. Calculation of the Y Process

              9

              8

              7

              6

              5

              4

              3

              2

              1

              0
                  0         5      10      15     20      25     30      35      40

                      Figure 6.2: Simulated τ -time, the market activity time.

The simulation of the Y process is very similar to the simulation of the square
root process M1 . Both processes are square root processes of dimension four and
both are driven by the same source of uncertainty. We, therefore, employ in
each time step the same Gaussian random variable Zi and the same chi-square
distributed random variable χ23,i , as in (6.1), to obtain the new value of the Y
process,
                                          s                             2 
                   −(τti+1 −τti )                 −(τti+1 −τti )
              1−e                 χ23,i +     4e
     Yτti+1 =                                                    Yτt + Zi   , (6.3)
                     4                        1 − e−(τti+1 −τti ) i

for ti = ∆i, i ∈ {0, 1, . . . }. Note that the difference τti+1 − τti was approximated
by using in (6.2) the market activity of the previous step.

                                                18
2.5

          2

         1.5

          1

         0.5

          0
               0   5       10      15     20      25      30        35    40

         Figure 6.3: Simulated trajectory of the normalized index Yτt .

Fig. 6.3 displays the simulated trajectory of the normalized index Y obtained by
the formula (6.3). This trajectory resembles the normalized EWI for oil displayed
in Fig. 5.2.
By analyzing the increments of the two processes M1 and Y for vanishing time
step size, one can show with arguments as employed in Diop (2003) and Alfonsi
(2005) that the pair of the simulated solutions (6.1) and (6.3) converges weakly
to the solution of the two dimensional SDE given by equations (4.5) and (4.3).
Note that in a weak sense the simulation of M1 can be interpreted as being exact
and that of Yτ as being almost exact.

4. Calculating the Volatility Process

         4.5

          4

         3.5

          3

         2.5

          2

         1.5

          1

         0.5

          0
               0   5       10      15     20      25      30        35    40

                   Figure 6.4: Simulated volatility of the index.

The volatility process at time ti is calculated under the stylized model with q = 1

                                        19
4

          3.5

           3

          2.5

           2

          1.5

           1

          0.5

           0
                0   5      10      15       20          25   30    35      40

         Figure 6.5: Estimated market activity of the simulated index.

         140

         120

         100

          80

          60

          40

          20

           0
                0   5      10      15       20          25   30    35      40

Figure 6.6: Quadratic variation of the square root of the inverse of estimated
market activity.

as                                          s
                                                 Mti
                                    σti =                                       (6.4)
                                                 Yτti
for i ∈ {0, 1, 2, . . . }, see (4.9). The simulated volatility, obtained by (6.4) from
the trajectory of the simulated market activity, displayed in Fig. 6.1, and the
trajectory of the simulated normalized index, plotted in Fig. 6.3, is illustrated in
Fig. 6.4. It again exhibits more pronounced spikes when compared to the esti-
mated volatility of the oil discounted EWI in Fig.3.4. These spikes are practically
removed when estimating from the simulated trajectory. Fig. 6.5 plots the esti-
mated market activity of the simulated index. Note that smoothing removes most
spikes of the simulated market activity in Fig. 6.1. Moreover, the quadratic vari-
ation of the square root of the inverse of the estimated market activity is more in
line with the quadratic variation of the inverse of market activity obtained from
the normalized EWI for oil, see Fig. 6.6 and Fig. 5.4.

                                         20
Empirical Properties of the Proposed Model

Let us now check the seven empirical stylized facts described in Section 3. The
estimation methods of Section 3 are now applied to the simulated trajectory of
the index.

(i) Uncorrelated Returns

                                                         Sample Autocorrelation Function

                                0.8
      Sample Autocorrelation

                                0.6

                                0.4

                                0.2

                                 0

                               −0.2
                                      0   10   20   30       40        50        60        70   80   90   100
                                                                      Lag

  Figure 6.7: Autocorrelation function for log-returns of the simulated index.

                                                         Sample Autocorrelation Function

                                0.8
      Sample Autocorrelation

                                0.6

                                0.4

                                0.2

                                 0

                               −0.2
                                      0   10   20   30       40        50        60        70   80   90   100
                                                                      Lag

Figure 6.8: Autocorrelation function for absolute log-returns of the simulated
index.

     Fig.6.7 displays the autocorrelation function for log-returns of the simulated
     index. Similarly as in Fig. 3.1, the autocorrelation function decreases fast

                                                                    21
to zero and stays at zero for large lags. In fact, it is located between the
     95% confidence bounds.
(ii) Correlated Absolute Returns
     Fig. 6.8 plots the autocorrelation function for the absolute log-returns of
     the simulated index. Such autocorrelation of absolute log-returns does not
     decrease to zero. It is located outside the 95% confidence bounds even for
     large lags. This is in line with the autocorrelation function of the absolute
     log-returns of the oil discounted EWI displayed in Fig. 3.2.
(iii) Student-t Distributed Returns

          0
        10

          −1
        10

                                                       log−empirical density

          −2
        10

                                                              log−Student−t density

          −3
        10

          −4
        10
          −10   −8    −6     −4    −2     0     2      4          6            8      10

Figure 6.9: Logarithms of the empirical distribution of the normalized log-returns
of the simulated index and Student-t density with four degrees of freedom.

     Fig. 6.9 illustrates the logarithms of the empirical distribution of the nor-
     malized log-returns of the simulated index and Student-t density with four
     degrees of freedom. As expected from the design of the model in Section
     4 the distribution of log-returns of the simulated index is Student-t with
     four degrees of freedom. Note that the estimated degrees of freedom may
     vary significantly for the simulated trajectories, as was illustrated in Platen
     & Rendek (2012a). Such deviations can be easily as big as one degree of
     freedom. This is also one of the reasons why we fixed the parameters δ and
     ν to four in the proposed stylized version of the model.
(iv) Volatility Clustering
     As expected from the model design, the estimated volatility of the simulated
     index, plotted in Fig. 6.10, has periods of higher and periods of lower values.
     The estimated squared volatility was obtained by exponential smoothing
     (3.1) with α = 0.92 applied to the squared log-returns of the simulated
     index.

                                        22
1.6

        1.4

        1.2

         1

        0.8

        0.6

        0.4

        0.2

         0
              0         5       10     15      20     25            30   35   40

                  Figure 6.10: Estimated volatility of the simulated index.

(v) Long Term Exponential Growth

        15

        14

        13
                                                           0.21 t+5.07
        12

        11

        10

         9

         8

         7

         6

         5
              0         5       10     15      20     25            30   35   40

              Figure 6.11: Logarithm of simulated index with linear fit.

    Given the simulated normalized index in Fig. 6.3 it is straightforward to
    calculate the index values by multiplication of the normalized index with
    the exponential function given in (5.2). The logarithm of the simulated
    index is displayed in Fig. 6.11 with the least squares linear fit. The model
    clearly recovers the long term exponential growth of the EWI for oil.

(vi) Anti-Leverage Effect
    It has been noticed in Section 3 that the normalized EWI for oil is mostly
    positively correlated to its volatility. When sudden upward moves in the

                                             23
1.5

          1

         0.5

          0

        −0.5

         −1

        −1.5

         −2
               0   5     10      15      20      25     30      35      40

Figure 6.12: Logarithms of simulated normalized index (upper graph) and its
estimated volatility (lower graph).

         0.6

         0.5

         0.4

         0.3

         0.2

         0.1

          0

        −0.1
               0   5     10      15      20      25     30      35      40

Figure 6.13: Quadratic covariation between the logarithms of simulated normal-
ized index and its estimated volatility.

     simulated normalized index for oil are observed, the volatility spikes up.
     This means that the market activity increases when the prices of oil are
     low relative to the NP. This anti-leverage effect for the EWI of oil is also
     recovered by the model in Section 4. The logarithms of simulated nor-
     malized index and its estimated volatility are illustrated in Fig. 6.12. The
     positive correlation is here clearly noticeable. Such positive correlation is
     even clearer when comparing the simulated market activity in Fig. 6.1 and
     the simulated normalized index in Fig. 6.3.
     Additionally, Fig. 6.13 plots the quadratic covariation between the loga-

                                       24
rithms of simulated index and its estimated volatility. It resembles the
      corresponding quadratic covariation for the normalized EWI for oil and its
      estimated volatility in Fig. 3.6.
(vii) Extreme Volatility at Major Index Moves
      Finally, the model produces extreme volatility at major index upward moves.
      This was already visible in Fig. 6.12. During sudden upward moves in the
      index the volatility jumps up. This models the fact that the market is
      more active when the normalized EWI for oil moves strongly upward. This
      corresponds usually with a strong downward move of the oil price.

In summary, one can say that the proposed model captures well all seven styl-
ized empirical facts listed in Section 3 and cannot be easily falsified on these
grounds, see Popper (1934). The paper has shown that it is possible to identify
a parsimonious model for a diversified equity index denominated in oil. It has
only one driving Brownian motion, three initial parameters and three structural
parameters.

7     Modeling the Spot Price of Oil
We can model the oil denominated NP in the way as proposed in this paper.
On the other hand, we can model the currency denominated NP, as described in
Platen & Rendek (2012a). Therefore, it is possible to express the oil spot price
dynamics by the SDEs derived for these quantities.
By (2.1) we can express the spot price of oil as the ratio of domestic currency
denominated (US dollar) NP, Ct , over the oil denominated NP, St , that is
                                                  Ct
                                       Xt =          ,                      (7.1)
                                                  St
for t ≥ 0.
We model the currency denominated NP as in Platen & Rendek (2012a), and use
an analogous notation to the oil denomination. Therefore we set
                                      Ct = C̄t Bt0 ,                        (7.2)
where                                        nZ           t            o
                               Bt0   = exp                    rs0 ds        (7.3)
                                                      0
for t ≥ 0. Here rt0 is the short rate of the domestic currency.
The discounted NP in the currency denomination is modeled as in Platen &
Rendek (2012a), and resembles the model described in this paper. The domestic
savings account discounted NP C̄t at time t is equal to
                                     C̄t = A0τ 0 Yτ00 ,                     (7.4)
                                                  t           t

                                             25
with
                               A0τ 0 = A0 exp{a0 τt0 }                         (7.5)
                                  t

for t ≥ 0. Additionally, the normalized NP Yτ00 in τ 0 -time can be expressed as a
square root process of dimension four as follows:
                                              q
                     dYτ00 = (1 − Yτ00 )dτ 0 + Yτ00 dW 0 (τ 0 ).             (7.6)

For simplicity, W 0 is assumed to be an independent Brownian motion in τ 0 -time.
The τ 0 -time is given by an ordinary differential equation involving the currency
market activity M 0 . That is,
                                   dτt0 = Mt0 dt,                             (7.7)
where the inverse of currency market activity satisfies the SDE
                                                    s
                                                      γ0
                           
                         1         0    0  1
                    d         = (γ   − ǫ      )dt +       dWt0                 (7.8)
                        Mt0               Mt0         Mt0

for t ≥ 0. Here the Brownian motion Wt0 in t-time is related to the Brownian
motion W 0 (τt0 ) in τ 0 -time by relation
                                        r
                                          dτ 0       p
                          dW 0 (τt0 ) =        dWt0 = Mt0 dWt0 .        (7.9)
                                           dt
It is imprtant to note the difference in the sign in front of the diffusion term of
the SDE (7.8), which is opposite to the one in the SDE (4.5) for the commodity
oil. The fit of the model to the US dollar denomination of the discounted EWI
provided the parameters: A0 = 2922.08, Y00 = 0.76, M00 = 0.044, γ 0 = 511.33,
ǫ0 = 11.31 and a0 = 6.31.
In this manner we have constructed an oil spot price model, which separately
models the movements of the oil price relative to the index and the currency
relative to the same index. This disentangles the impact of the two main factors
that drive the oil price in US dollar denomination. One notes also the influence of
the oil convenience yield and the US interest rate on the long term evolution of the
oil price under the proposed model. This model permits a more realistic pricing
of oil derivatives then previous models, in particular for long dated derivatives,
as explained in Du & Platen (2012).

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                                         26
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                                      27
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