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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 1, Eckhard Platen 2 and Renata Rendek 3
December 21, 2012
Abstract: The paper derives a parsimonious two-component affine diffusionmodel 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 USdollar. However, there are also clear differences. To identify these the paperstudies the empirical features of an extremely well diversified world stock in-dex, which is a proxy of the numeraire portfolio, in the denomination of the oilprice. 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 ofthe numeraire portfolio denominated in crude oil, increases at major oil priceupward moves. Furthermore, the log-returns of the index in oil price denom-ination appear to follow a Student-t distribution. These and other stylizedempirical properties lead to the proposed tractable diffusion model, which hasthe normalized numeraire portfolio and market activity as components. Analmost exact simulation technique is described, which illustrates the charac-teristics of the proposed model and confirms that it matches well the observedstylized empirical facts.
JEL Classification: G10, C10, C151991 Mathematics Subject Classification: 62P05, 62P20, 62G05, 62-07, 68U20
Key words and phrases: commodities, oil price, numeraire portfolio, market ac-tivity, square root processes, benchmark approach.
1University of Technology Sydney, Finance Discipline Group, PO Box 123, Broadway, NSW,
2007, Australia.2University of Technology Sydney, Finance Discipline Group and School of Mathematical
Sciences, PO Box 123, Broadway, NSW, 2007, Australia, Email: [email protected],
Phone: +61295147759, Fax: +61295147711.3University of Technology Sydney, School of Mathematical Sciences, PO Box 123, Broadway,
NSW, 2007, Australia, Email: [email protected], Phone: +61295147781.
1 Introduction
Motivated by the fact that oil prices behave in some sense similarly to the USdollar because oil is traded in this currency, and since the log-returns of the worldstock 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 indexas proxy of the numeraire portfolio (NP). The latter equals the growth optimalportfolio, which maximizes expected logarithmic utility from terminal wealth.The two components of the proposed new model are the normalized approximateNP 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 movingslower than the second one. They are both driven only by one Brownian motion,modeling the nondiversifiable uncertainty of the market with respect to oil pricedenomination. It turns out that the crucial difference to the model proposed inPlaten & Rendek (2012a), is the correlation between the normalized NP in oildenomination and its market activity. It is observed that in contrast to the NPin currency denomination, the NP in oil denomination has positive correlationwith its market activity. This property can be interpreted as an anti-leverageeffect, generating higher market activity and volatility when the NP in oil pricedenomination increases, that is, it declines substantially relative to the NP. Thistype of change in market activity makes economic sense because a lower oil priceis likely to trigger increased economic activity, including trading activity. Adiversified index in currency denomination behaves differently. Here we havethe leverage effect where the market activity increases when the NP in currencydenomination decreases.
As it turned out during the investigation, in order to capture over long timeperiods realistically the oil price evolution, the model needs to be formulatedin a general financial modeling framework, which goes beyond the classical no-arbitrage paradigm. By interpreting a well diversified world stock index as NPthe fitting of the proposed parsimonious model can be accomplished such that itcaptures well reality, in particular, the long term dynamics of the price of the oilprice relative to the NP which is approximated by a well diversified stock index.
The benchmark approach, see Platen & Heath (2010) and Platen (2011) providesthe 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 withthe NP as numeraire. The central building block of the benchmark approach is itsbenchmark, 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 inthe analysis, which is significantly different to the classical approach, where oneuses typically the savings account as denominator. Using an approximate NP inoil 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 aswell as the modeling.
Since, oil prices have grown considerably over the last decades, it is important toapproximate closely the NP, which is in many ways the ”best” performing port-folio. The Naive Diversification Theorem in Platen & Rendek (2012b) states thatthe equi-weighted index (EWI) approximates well the NP of a given investmentuniverse when the number of constituents is large and the given market is wellsecuritized. The latter property essentially means that the risk factors driving theunderlying risky securities are sufficiently different. The EWI used in this paperis 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, thatis, the discounted equi-weighted index denominated in the oil price. Section 3extracts a list of stylized empirical facts for the observed dynamics. Section 4proposes a parsimonious, tractable model for these dynamics involving the powerof a time transformed affine diffusion. It also discusses the volatility and marketactivity dynamics arising from the proposed model. Section 5 describes a robuststep-by-step methodology for fitting the proposed model and visualizes volatilityand market activity as they emerge under the model. Section 6 describes forthe model an almost exact simulation method, which allows us to confirm thatthe empirical properties of the model match the list of stylized empirical facts ofSection 3. Finally, Section 7 summarizes the model described in Platen & Rendek(2012a) for the denomination of the numeraire portfolio in currency denomination.This allows us to model the oil price in currency denomination.
2 Approximate Numeraire Portfolio in Oil Price
Denomination
The aim of this section is to introduce the object of study, which is the discountedNP denominated in units of oil. It is important to approximate well the dynamicsof the NP denominations. The oil price, which has grown dramatically over theyears, will use the ratio of the denomination of the NP in domestic currency overthe denomination of the NP in oil. The current paper focuses on the modeling ofthe dynamics of the NP in oil denomination. For the denomination of the NP incurrency denomination we will refer to Platen & Rendek (2012a).
Numeraire Portfolio
The numeraire portfolio (NP) is a strictly positive portfolio which when used asbenchmark 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 returnprices were obtained from Thomson Reuters Datastream. The EWI was builtin three stages: first, country subsector equi-weighted indices were constructed;second, from these constituents country equi-weighted indices were built; andthird, finally the EWI was calculated by equal value weighting the country equi-weighted indices. We always took 40 basis points proportional transaction costsinto account. In Platen & Rendek (2012b, 2012c) a description of the methodfor building such equi-weighted indices is described in detail and a Naive Di-versification Theorem is proved that gives the theoretical reasoning behind theapproximation of the NP.
1973 1978 1983 1988 1993 1998 2003 2008 20134
6
8
10
12
14
16
18
ln(EWI)
ln(MCI)
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 displayedin this figure is the global Datastream index (with mnemonics TOTMKWD) usedalso in Platen & Rendek (2012a). It fluctuates and performs very similarly tothe FTSE all-cap index and the MSCI total return world index. The EWI isclearly growing on average faster than the MCI. This is mainly due to its betterdiversification resulting from the construction methodology used.
Numeraire Portfolio Denominated in Units of Oil
The oil price itself in US dollar denomination is driven by the uncertainty of twomajor securities. These are the commodity oil and the currency US dollar. Oneneeds to disentangle their combined influence on the oil price, which is usually
4
04/1985 04/1990 04/1995 04/2000 04/2005 04/20100
5000
10000
15000
20000
25000
30000
Figure 2.2: The discounted EWI of oil.
given in US dollar denomination. We do this by involving a well-diversified globalindex, 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 dynamicsof an EWI in oil price denomination. In some sense we obtain a least disturbedobservation 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 ofoil.
The NP in oil denomination St at time t can be expressed by the ratio
St =Ct
Xt
, (2.1)
where Xt is the oil spot price in US dollar at time t ≥ 0 and Ct is the US dollardenomination of the NP at time t.
Oil Savings Account
The next step is to construct the oil savings account
B1t = e
∫ t
0r1sds, (2.2)
where the convenience yield r1t for oil is approximated by the expression
r1t = − 1
∆ln
(
Ft
Xt
)
+ r0t . (2.3)
Here Ft is the three months oil futures price, ∆ = 312
and r0t is the US dollarinterest rate at time t ≥ 0, see e.g. Du & Platen (2012) for details on thisapproximation.
5
Discounted Numeraire Portfolio for Oil
The object of our study is now the, by the oil savings account (2.2), discountedNP. That is,
St =St
B1t
(2.4)
for t ≥ 0. Fig.2.2 plots the oil discounted NP for the period from 02/04/1985 until18/03/2010. We note an approximately exponential increase of the oil discountedNP. In the next section we will apply some standard statistical methods in orderto identify stylized empirical properties of the oil discounted NP.
3 Empirical Observations
Platen & Rendek (2012a) observed seven stylized empirical facts pertaining todiversified world stock indices in currency denomination. Below we check whethersimilar or different properties emerge for the oil discounted NP, that is, the EWIdenominated in an oil savings account.
(i) Uncorrelated Returns
Fig. 3.1 displays the autocorrelation function for the log-returns of the oildiscounted EWI with 95% confidence bounds. Similarly, to the log-returnsof the index in currency denominations, the autocorrelation of log-returnsof the oil discounted EWI is close to zero.
0 10 20 30 40 50 60 70 80 90 100−0.2
0
0.2
0.4
0.6
0.8
Sample Autocorrelation Function
Lag
Sam
ple
Aut
ocor
rela
tion
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 theoil discounted EWI. Even for large lags the autocorrelation is non-negligibleand does not seem to show an exponential decline.
0 10 20 30 40 50 60 70 80 90 100−0.2
0
0.2
0.4
0.6
0.8
Sample Autocorrelation Function
Lag
Sam
ple
Aut
ocor
rela
tion
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 degreesof 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 quantifythe fit of the Student-t distribution we perform a log-likelihood ratio test inthe family of the symmetric generalized hyperbolic (SGH) distributions, seeRao (1973) and Platen & Rendek (2008). Table 3.1 reports the test statis-tics calculated for four special cases of the SGH distribution. These are: theStudent-t distribution, the normal inverse Gaussian (NIG) distribution, thehyperbolic distribution and the variance gamma (VG) distribution. Thetest statistics are here distributed according to the chi-square distributionwith one degree of freedom. Therefore, the hypothesis of the Student-t dis-tribution being the best candidate distribution in the family of the SGHdistributions cannot be rejected at the 99.9% level of significance, since0.00000002 < χ2
0.001,1 ≈ 0.000002.
(iv) Volatility Clustering
Fig. 3.4 illustrates the estimated annualized volatility Vti of the oil dis-counted EWI. The squared volatility V 2
tiat time ti is obtained from squared
log-returns via exponential smoothing. For the discretization time ti = ∆i,
7
−10 −5 0 5 10 15 2010
−6
10−5
10−4
10−3
10−2
10−1
100
log−empirical density
log−Student−t density
Figure 3.3: Logarithms of empirical density of normalized log-returns of the oildiscounted 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 oildiscounted EWI.
Commodity Student-t NIG Hyperbolic VG df.
Crude Oil 0.00000002 61.168568 182.120161 181.189575 3.13
04/1985 04/1990 04/1995 04/2000 04/2005 04/20100
0.5
1
1.5
2
2.5
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
04/1985 04/1990 04/1995 04/2000 04/2005 04/2010−2.5
−2
−1.5
−1
−0.5
0
0.5
1
Figure 3.5: Logarithms of normalized EWI for oil (upper graph) and its volatility(lower graph).
04/1985 04/1990 04/1995 04/2000 04/2005 04/2010−0.5
0
0.5
1
1.5
2
2.5
3
3.5
Figure 3.6: Quadratic covariation between the logarithms of normalized EWI foroil and its volatility.
returns R2tiin the following way:
V 2ti+1
= α√∆R2
ti+ (1− α
√∆)V 2
ti, (3.1)
for i ∈ {0, 1, 2, . . . }. Here the smoothing parameter λ is assumed to equalα = 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 lowvolatility and periods of high volatility. It can be conjectured that suchvolatility 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 trendline fitted by linear regression. The logarithm of the oil discounted EWIexhibits consistent long term linear growth, which in turn results in thelong term exponential growth for the oil discounted EWI.
(vi) Anti-Leverage Effect
A leverage effect is typically observed for the currency discounted worldstock index, see Platen & Rendek (2012a). This empirical fact is, however,not observed for the oil discounted EWI and its normalized version, shownin Fig.5.2, where its average long term growth is taken out by dividing witha respective exponential function of time. In Fig.3.5 we plot the logarithmsof the normalized EWI for oil and its volatility. When the normalizedEWI for oil moves upwards, in general, the volatility increases and viceversa. This implies an anti-leverage effect for the oil discounted EWI andits volatility. In fact, the covariation function between the normalized EWIfor oil and its volatility, displayed in Fig. 3.6, indicates a mostly positivecorrelation between the increments for the normalized EWI for oil and itsvolatility.
(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 normalizedEWI for oil the volatility increases when the index moves strongly up andthe increase is more substantial compared to the ”normal” moves of theindex.
4 Modeling of Oil Prices
This section derives a parsimonious two-component model for the oil discountedEWI. It follows to some extent the methodology described in Platen & Rendek(2012a) with some important changes in the design of the dependencies in thetwo-component model.
Discounted Index
The discounted index St, which is the oil discounted index introduced in Section2, is expressed by the product
St = Aτt(Yτt)q (4.1)
10
for t ≥ 0, see Platen & Rendek (2012a). An exponential function Aτt of a givenτ -time, the market activity time (to be specified below), models the long termaverage 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 growthrate a ∈ ℜ with respect to market activity time.
Normalized Index
As a consequence of equation (4.1), the ratio (Yτt)q = St
Aτt
denotes the normalized
index, that is the normalized index for oil, at time t. This normalized index isassumed to form an ergodic diffusion process evolving according to τ -time. Weassume that it satisfies the SDE
dYτ =
δ
4− 1
2
(
Γ(
δ2+ q)
Γ(
δ2
)
) 1
q
Yτ
dτ +√
Yτ dW (τ), (4.3)
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 = Mtdt (4.4)
for t ≥ 0 with τ0 ≥ 0. Here we call the derivative of τ -time with respect tocalendar time t the market activity dτt
dt= 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 1
M= { 1
Mt, t ≥ 0} is assumed to be a fast
moving square root process in t-time with the dynamics
d
(
1
Mt
)
=
(
ν
4γ − ǫ
1
Mt
)
dt−√
γ
Mt
dWt, (4.5)
for t ≥ 0 with M0 > 0, where γ > 0, ν > 2 and ǫ > 0. Note the negative signin front of the diffusion term which indicates the main difference of the model tothe one in Platen & Rendek (2012a).
The Brownian motion W (τ), which models in market activity time the longterm nondiversifiable uncertainty with respect to oil denomination, is driving
11
the normalized index Yτ . This process is linked to the standard Brownian motionW = {Wt, t ≥ 0} in t-time through the market activity M in the following way:
dW (τt) =
√
dτt
dtdWt =
√
MtdWt (4.6)
for t ≥ 0 withW0 = 0. The Brownian motionW = {Wt, t ≥ 0} in (4.6) is identicalto 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 incrementsof the inverse of market activity are positively correlated to the increments of thenormalized index.
Expected Rate of Return and Volatility
By application of the Ito formula one obtains from (4.1), (4.2), (4.3), (4.4) and(4.6) for the discounted index St the stochastic differential equation (SDE)
dSt = St (µtdt+ σtdWt) (4.7)
for t ≥ 0, with initial value S0 = A0(Y0)q and expected rate of return
µt =
a
Mt
− q
2
(
Γ(
δ2+ q)
Γ(
δ2
)
)1
q
+
(
δ
4q +
1
2q(q − 1)
)
1
MtYτt
Mt. (4.8)
The volatility with respect to t-time emerges in the form
σt = q
√
Mt
Yτt
. (4.9)
Benchmark Approach
Due to the SDE (4.7) and the Ito formula, the dynamics for the benchmarked
savings account B1t = (St)
−1 =B1
t
St, which is the inverse of the oil discounted NP,
is characterized by the SDE
dB1t = B1
t
((
−µt + σ2t
)
dt− σtdWt
)
, (4.10)
for t ≥ 0, see (4.8) and (4.9). It follows if for all t ≥ 0 one has
σ2t ≤ µt (4.11)
then the benchmarked savings account Bt forms an (A, P )-super-martingale. This is the key property needed to accommodate the model underthe benchmark approach, see Platen & Heath (2010).
To guarantee almost surely in the proposed model the inequality (4.11), one hasby (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 tosatisfy the equality
δ = 2(q + 1). (4.12)
Assumption 4.2 The long term average net growth rate a with respect to τ -timehas to satisfy the inequality
q
2
(
Γ (2q + 1)
Γ (q + 1)
)1
q
≤ a. (4.13)
When equality holds in (4.13) for the proposed model the benchmarked savingsaccount 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 thebenchmark 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. Inthe simplified version of the model we assume q = 1 in (4.1), therefore δ = 4in (4.3) and ν = 4 in (4.5). The main reason for this assumption is the factthat it is empirically extremely difficult to give a sufficiently precise estimate forthe 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-returnsa Student-t distribution with four degrees of freedom. The data indicate with theestimated 3.13 degrees of freedom for the index log-returns that four degrees offreedom 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 themean of Mt is explicitly known. By the ergodic theorem this mean amounts to
limt→∞
1
t
∫ t
0
Msds =4
ν − 2
ǫ
γP-a.s. (5.1)
Therefore, it is possible to approximate (4.2) by the following expression
Aτt ≈ A exp{ 4aǫ
γ(ν − 2)t}
, (5.2)
for t ≥ 0.
13
04/1985 04/1990 04/1995 04/2000 04/2005 04/20104
5
6
7
8
9
10
11
0.21 t +5.41
Figure 5.1: Logarithm of the oil discounted EWI and linear fit.
04/1985 04/1990 04/1995 04/2000 04/2005 04/20100
0.5
1
1.5
2
2.5
3
Figure 5.2: Normalized EWI for oil.
Therefore, since a line can be fitted to the logarithm of the discounted EWI ofoil, see Fig. 5.1, it is straightforward to calculate A = 223.32 and 4aǫ
γ(ν−2)≈ 0.21.
Fig. 5.2 illustrates the normalized EWI of oil obtained as the ratio of the oildiscounted EWI over the function in (5.2).
Step 2: Observing Market Activity
By (4.3), (4.4) and an application of the Ito formula, one obtains as time derivativeof the quadratic variation for
√
Yτt the expression
d[√Y ]τtdt
=1
4
dτt
dt=
Mt
4, (5.3)
which is proportional to market activity. The estimation of the trajectory of themarket activity process M is performed using daily observations. First, the ”raw”
14
04/1985 04/1990 04/1995 04/2000 04/2005 04/20100
1
2
3
4
5
6
Figure 5.3: Market activity.
time derivative Qt =d[√Y ]τtdt
at the ith observation time t = ti is estimated fromthe finite difference
Qti =[√Y ]τti+1
− [√Y ]τti
ti+1 − ti(5.4)
for i ∈ {0, 1, . . . }. Second, exponential smoothing is applied to the observed finitedifferences according to the recursive standard moving average formula
Qti+1= α
√
ti+1 − tiQti + (1− α√
ti+1 − ti)Qti , (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 d
dt[√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 process1M. Our estimate for the slope equals here 10.94. Since under the proposed model
we have ddt
[√
1M
]
t= 1
4γ, we obtain γ ≈ 43.76.
Step 5: Parameters ν and ǫ
Fig. 5.5 displays the histogram of market activity with inverse gamma fit withν = 2.80 degrees of freedom. Since, we consider a simplified version of the modelthe same degrees of freedom for δ = 4 and ν = 4, we obtain from the averagevalue of the market activity 4ǫ
γ(ν−2)≈ 0.21 the estimate for ǫ = 4.57.
15
04/1985 04/1990 04/1995 04/2000 04/2005 04/20100
50
100
150
200
250
300
Figure 5.4: Quadratic variation of the square root of 1M.
0 1 2 3 4 5 60
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
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 averagevalue of the market activity is 4ǫ
γ(ν−2)≈ 0.21. This indicates, the condition (4.13)
is approximately satisfied as an equality. Therefore, the benchmark approach canbe 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 themodel introduced in Section 4. As indicated before, both of the processes 1
Mand
Y are square root processes of dimension δ = ν = 4 in the stylized version ofthe 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 consideredto be almost exact when sampling from this transition density. More precisely,it is exact for the process 1
Mand almost exact for Y . The following four steps
describe the simulation of the normalized index and its volatility:
1. Simulation of the Process 1M
0 5 10 15 20 25 30 35 400
2
4
6
8
10
12
14
16
Figure 6.1: Simulated path of M .
First, we describe the simulation of the inverse 1M
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 transitiondensity of dimension ν = 4. That is, we have
1
Mti+1
=γ(1− e−ǫ(ti+1−ti))
4ǫ
χ23,i +
(√
4ǫe−ǫ(ti+1−ti)
γ(1− e−ǫ(ti+1−ti))
1
Mti
− Zi
)2
, (6.1)
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 χ2
3,i is an independent chi-square distributed random variable with threedegrees 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 fourdegrees of freedom.
Fig. 6.1 plots the simulated path of the market activity M . The market activitydisplayed in this figure has more pronounced spikes compared to the estimatedmarket activity in Fig. 5.3. We will see later that when the market activity isestimated from the path of the simulated index it resembles closely the historicallyobserved 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
τti+1− τti =
∫ ti+1
ti
Msds ≈ Mti(ti+1 − ti), (6.2)
i ∈ {0, 1, . . . }.Fig. 6.2 plots the simulated τ -time, which is the market activity time obtainedfrom the path of the simulated market activity in Fig. 6.1 with the use of theapproximation (6.2).
3. Calculation of the Y Process
0 5 10 15 20 25 30 35 400
1
2
3
4
5
6
7
8
9
Figure 6.2: Simulated τ -time, the market activity time.
The simulation of the Y process is very similar to the simulation of the squareroot process 1
M. Both processes are square root processes of dimension four and
both are driven by the same source of uncertainty. We, therefore, employ ineach time step the same Gaussian random variable Zi and the same chi-squaredistributed random variable χ2
3,i, as in (6.1), to obtain the new value of the Y
process,
Yτti+1=
1− e−(τti+1−τti )
4
χ23,i +
√
4e−(τti+1−τti )
1− e−(τti+1−τti )
Yτti+ Zi
2
, (6.3)
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
0 5 10 15 20 25 30 35 400
0.5
1
1.5
2
2.5
Figure 6.3: Simulated trajectory of the normalized index Yτt .
Fig. 6.3 displays the simulated trajectory of the normalized index Y obtained bythe formula (6.3). This trajectory resembles the normalized EWI for oil displayedin Fig. 5.2.
By analyzing the increments of the two processes 1M
and Y for vanishing timestep 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 weaklyto the solution of the two dimensional SDE given by equations (4.5) and (4.3).Note that in a weak sense the simulation of 1
Mcan be interpreted as being exact
and that of Yτ as being almost exact.
4. Calculating the Volatility Process
0 5 10 15 20 25 30 35 400
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Figure 6.4: Simulated volatility of the index.
The volatility process at time ti is calculated under the stylized model with q = 1
19
0 5 10 15 20 25 30 35 400
0.5
1
1.5
2
2.5
3
3.5
4
Figure 6.5: Estimated market activity of the simulated index.
0 5 10 15 20 25 30 35 400
20
40
60
80
100
120
140
Figure 6.6: Quadratic variation of the square root of the inverse of estimatedmarket activity.
as
σti =
√
Mti
Yτti
(6.4)
for i ∈ {0, 1, 2, . . .}, see (4.9). The simulated volatility, obtained by (6.4) fromthe trajectory of the simulated market activity, displayed in Fig. 6.1, and thetrajectory of the simulated normalized index, plotted in Fig. 6.3, is illustrated inFig. 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 practicallyremoved when estimating from the simulated trajectory. Fig. 6.5 plots the esti-mated market activity of the simulated index. Note that smoothing removes mostspikes 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 inline with the quadratic variation of the inverse of market activity obtained fromthe 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. Theestimation methods of Section 3 are now applied to the simulated trajectory ofthe index.
(i) Uncorrelated Returns
0 10 20 30 40 50 60 70 80 90 100−0.2
0
0.2
0.4
0.6
0.8
Sample Autocorrelation Function
Lag
Sam
ple
Aut
ocor
rela
tion
Figure 6.7: Autocorrelation function for log-returns of the simulated index.
0 10 20 30 40 50 60 70 80 90 100−0.2
0
0.2
0.4
0.6
0.8
Sample Autocorrelation Function
Lag
Sam
ple
Aut
ocor
rela
tion
Figure 6.8: Autocorrelation function for absolute log-returns of the simulatedindex.
Fig.6.7 displays the autocorrelation function for log-returns of the simulatedindex. 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 the95% confidence bounds.
(ii) Correlated Absolute Returns
Fig. 6.8 plots the autocorrelation function for the absolute log-returns ofthe simulated index. Such autocorrelation of absolute log-returns does notdecrease to zero. It is located outside the 95% confidence bounds even forlarge lags. This is in line with the autocorrelation function of the absolutelog-returns of the oil discounted EWI displayed in Fig. 3.2.
(iii) Student-t Distributed Returns
−10 −8 −6 −4 −2 0 2 4 6 8 1010
−4
10−3
10−2
10−1
100
log−empirical density
log−Student−t density
Figure 6.9: Logarithms of the empirical distribution of the normalized log-returnsof 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 fourdegrees of freedom. As expected from the design of the model in Section4 the distribution of log-returns of the simulated index is Student-t withfour degrees of freedom. Note that the estimated degrees of freedom mayvary significantly for the simulated trajectories, as was illustrated in Platen& Rendek (2012a). Such deviations can be easily as big as one degree offreedom. This is also one of the reasons why we fixed the parameters δ andν to four in the proposed stylized version of the model.
(iv) Volatility Clustering
As expected from the model design, the estimated volatility of the simulatedindex, 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 simulatedindex.
22
0 5 10 15 20 25 30 35 400
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Figure 6.10: Estimated volatility of the simulated index.
(v) Long Term Exponential Growth
0 5 10 15 20 25 30 35 405
6
7
8
9
10
11
12
13
14
15
0.21 t+5.07
Figure 6.11: Logarithm of simulated index with linear fit.
Given the simulated normalized index in Fig. 6.3 it is straightforward tocalculate the index values by multiplication of the normalized index withthe exponential function given in (5.2). The logarithm of the simulatedindex is displayed in Fig. 6.11 with the least squares linear fit. The modelclearly 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 mostlypositively correlated to its volatility. When sudden upward moves in the
23
0 5 10 15 20 25 30 35 40−2
−1.5
−1
−0.5
0
0.5
1
1.5
Figure 6.12: Logarithms of simulated normalized index (upper graph) and itsestimated volatility (lower graph).
0 5 10 15 20 25 30 35 40−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
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 arelow relative to the NP. This anti-leverage effect for the EWI of oil is alsorecovered by the model in Section 4. The logarithms of simulated nor-malized index and its estimated volatility are illustrated in Fig. 6.12. Thepositive correlation is here clearly noticeable. Such positive correlation iseven clearer when comparing the simulated market activity in Fig. 6.1 andthe 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 thecorresponding quadratic covariation for the normalized EWI for oil and itsestimated 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 theindex the volatility jumps up. This models the fact that the market ismore active when the normalized EWI for oil moves strongly upward. Thiscorresponds 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 thesegrounds, see Popper (1934). The paper has shown that it is possible to identifya parsimonious model for a diversified equity index denominated in oil. It hasonly one driving Brownian motion, three initial parameters and three structuralparameters.
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 inPlaten & Rendek (2012a). Therefore, it is possible to express the oil spot pricedynamics by the SDEs derived for these quantities.
By (2.1) we can express the spot price of oil as the ratio of domestic currencydenominated (US dollar) NP, Ct, over the oil denominated NP, St, that is
Xt =Ct
St
, (7.1)
for t ≥ 0.
We model the currency denominated NP as in Platen & Rendek (2012a), and usean analogous notation to the oil denomination. Therefore we set
Ct = CtB0t , (7.2)
where
B0t = exp
{
∫ t
0
r0sds}
(7.3)
for t ≥ 0. Here r0t 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 domesticsavings account discounted NP Ct at time t is equal to
Ct = A0τ0tY 0τ0t, (7.4)
25
withA0
τ0t= A0 exp{a0τ 0t } (7.5)
for t ≥ 0. Additionally, the normalized NP Y 0τ0
in τ 0-time can be expressed as asquare root process of dimension four as follows:
dY 0τ0 = (1− Y 0
τ0)dτ0 +
√
Y 0τ0
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 currencymarket activity M0. That is,
dτ 0t = M0t dt, (7.7)
where the inverse of currency market activity satisfies the SDE
d
(
1
M0t
)
= (γ0 − ǫ01
M0t
)dt+
√
γ0
M0t
dW 0t (7.8)
for t ≥ 0. Here the Brownian motion W 0t in t-time is related to the Brownian
motion W 0(τ 0t ) in τ 0-time by relation
dW 0(τ 0t ) =
√
dτ 0
dtdW 0
t =√
M0t dW
0t . (7.9)
It is imprtant to note the difference in the sign in front of the diffusion term ofthe SDE (7.8), which is opposite to the one in the SDE (4.5) for the commodityoil. The fit of the model to the US dollar denomination of the discounted EWIprovided the parameters: A0 = 2922.08, Y 0
0 = 0.76, M00 = 0.044, γ0 = 511.33,
ǫ0 = 11.31 and a0 = 6.31.
In this manner we have constructed an oil spot price model, which separatelymodels the movements of the oil price relative to the index and the currencyrelative to the same index. This disentangles the impact of the two main factorsthat drive the oil price in US dollar denomination. One notes also the influence ofthe oil convenience yield and the US interest rate on the long term evolution of theoil price under the proposed model. This model permits a more realistic pricingof oil derivatives then previous models, in particular for long dated derivatives,as explained in Du & Platen (2012).
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