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UNIVERSITÀ COMMERCIALE LUIGI BOCCONI MASTER OF SCIENCE FINANCE –QUANTITATIVE FINANCE
A new Tree-Pricing approach to overcame log normality
Giulio Laudani 1256809
14/02/2013
Anna Battauz – Relatore Fabrizio Iozzi – Controrelatore
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Ringraziamenti:
Quando si scrivono i ringraziamenti, si ha sempre un po’ lo sguardo al futuro, a
quando presi dalla nostalgia di questi giorni fantastici, si rivivono quei passati
ricordi. Questo è uno dei pochi momenti che si cristallizzano nella nostra memoria
come cesura tra due fasi: da giovane uomo a uomo maturo che con si appresta a
mettere a frutto tutte quelle conoscenze (costate anche sacrifici e dolori). Con
questo animo mi appresto a scrivere, quindi, non solo per le persone a me care, ma
anche al mio “me del futuro”.
Dopo molti sforzi e sacrifici finalmente anche questi due anni si sono conclusi.
Questa esperienza formativa mi ha permesso di avvicinarmi al mondo del lavoro
pronto a nuove sfide. In questi anni non sono mancati momenti di tensione, ma ho
avuto la fortuna di non essere mai stato solo.
Come molti miei colleghi l’università è stata anche un momento di crescita umana:
la prima volta a dover vivere soli, la prima volta a doversi gestire e prevendersi le
prime responsabilità. Ricordo come ogni cosa, anche la più piccola, sembrava essere
un evento e con quanto entusiasmo le ho vissute. Credo che in tutto ciò mi abbia
aiutato l’aver sempre avuto idee chiare sul mio futuro, di avere precisi obbiettivi.
Questo mio modo di vivere ha anche esasperato alcuni dei miei rapporti, essendo
poco incline al compromesso e tendenzialmente un idealista. Fortunatamente
l’ambiente come l’esperienza universitaria mi ha permesso di avere anche quel
pizzico di flessibilità unita a malizia, o forse di cinismo, per migliorare.
Vorrei ringraziare oggi tutte le persone con cui ho passato questi momenti e chi mi
hanno sostenuto e aiutato nei momenti più difficili. I miei genitori per i quali dedico
un ringraziamento più sentito per avermi sostenuto e avermi fatto dono sempre
della loro disponibilità e fiducia. Ai molti amici che ho conosciuto; per alcuni la
nostra amicizia inizia proprio con i cinque anni di università. Un po’ come soldati in
arme la nostra amicizia è diventata quasi una fratellanza temprata dalle comuni
difficoltà e gioie.
Qualche parola anche alla Bocconi. Credo di poter dire che in pochi amano la propria
università quanto me. Per me la Bocconi è stata un po’ come una seconda mamma,
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è il luogo dove sono diventato, quello che, probabilmente, sarò per il prossimo
futuro. Ricorderò sempre piacevolmente le ore di studio passate nei corridoi, le
interminabili chiacchierate e la paura nel ricevere i fatidici voti via SMS, vero test
per i nostri cuori.
Anche Milano riveste un ruolo importante, infatti molti dei miei ricordi più felici
rimarranno legati a questa città, ormai sentita come casa. Questa città non è stata
solo luogo di studio, ma anche di lavoro, dove le mie prime esperienze hanno avuto
modo di formarsi. Ed anche ai miei colleghi devo dei ringraziamenti, infatti grazie
anche alla loro esperienza ed insegnamenti che sono riuscito a redigere questa
stessa tesi.
LAUDANI, Giulio
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Section I:_________________________________________________ 6
1.a. Market Data Analysis: ____________________________ 10
1.b. The Derivatives chosen: ___________________________ 16
Section II: _______________________________________________ 21
1.a. The Methodologies: ______________________________ 23
1.b. The Barrier results _______________________________ 33
1.c. Stress and Non Arbitrage test: ______________________ 35
1.d. The empirical Comparison: _________________________ 38
Section III: ______________________________________________ 41
1.a. Strength & Weakness of the model: __________________ 41
1.b. Possible future evolution: _________________________ 43
Conclusions: _____________________________________________ 45
Appendix: _______________________________________________ 46
1.a. GARCH and JB test insight _________________________ 46
1.b. MatLab Code insight: _____________________________ 47
Opere citate: _____________________________________________ 51
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Abstract:
We study a new lattice method specifically tailored to overcome the log-
normality hypothesis of the underlying financial asset. Our work analyzes the
performance of this new method in both the pricing and the hedging of
different derivatives, finding higher accuracy at any level; thereby we
suggest it as a future “good” market practice. The novelty of our model is
the fitting of the pricing algorithms up to the fourth moments computed
using historical data. The chosen derivatives belong to the barrier option
family and the underlying is the S&P 500 index, since it is a very liquid and
actively traded security. We have used adaptive mesh techniques to improve
the accuracy of the pricing/hedging approximation. We have also run specific
optimization procedure to properly specify the correct time window to be
used for fitting the historical data and the model robustness under different
market conditions.
6
Section I:
This paper aims to introduce a new pricing algorithm to overcome the log-
normality hypothesis by using a tree path evolution model initially presented
in “A relaxed lattice option pricing model: implied skewness and kurtosis”
(Brorsen & Dasheng, 2009) seminal paper and later developed by the same
authors in the paper “A recombining lattice option pricing model that relaxes
the assumption of lognormality” (Dasheng & Brorsen, 2010). We want to
show i) the superiority of this approach compared to the other market most
used techniques and ii) the hedging strategy based on this approach.
We have chosen the S&P 500 index since it is a well-recognized index, in
which all components are characterized by high liquidity and they are vastly
used as underlying in the derivatives market1. To support this statement we
provide in [Tab.1] the average 30 days volume, Last Price and other Security
Information.
In [Tab.2] we provide some information on the benchmark European call and
Put option written on S&P 500 index vs. Apple US options.
1 The Chicago Board Options Exchange (CBOE) now offers Mini-SPX options, based on the
Standard & Poor's 500 Stock Index.
Tab. 1
SPX index [Blomberg Ticker]
Average Last 30 Days (M.) 555.87
Last Price 1507.84
Last Data 29/01/2013
Source: Bloomberg
7
The chosen derivative shows higher open interest than the Apple one,
although the Bid-Ask ratio2 is lower. The chart below provides us an
historical comparison from 25/10/2012 to 29/01/2013. This costant
difference is related to the higher complexity in replicating the SPX security
(500 components).
The chart shows in logartim scale3 the total number of contract written4 (call
and Put) on SPX Index and AAPL US Equity option. It is clear that the SPX
2 We present only the call Bid-ask ratio, because put ratio shows a more erratic behavior,
alight the interpretation is not in contrast with the one presented based on call ratio. 3 We have chosen to present data under logarithmic scale to reduce the scaling effect which
might reduce chart’s readability.
Tab. 2
SPX US 03/16/13 C1400 Index SPX US 03/16/13 P1400 Index
Open Interest 118,579 161,206
Last Price 109.05 4.85
Last Data 29/01/2013 29/01/2013
BidAsk Ratio 97.91% 90.20%
AAPL US 03/16/13 C450 Equity AAPL US 03/16/13 p450 Equity
Open Interest 3,936 15,707
Last Price 20.3 14.175
BidAsk Ratio 98.05% 97.56%
Source: Bloomberg
85%
90%
95%
100%
ott-12 nov-12 nov-12 nov-12 nov-12 nov-12 dic-12 dic-12 dic-12 dic-12 gen-13 gen-13 gen-13 gen-13
Bid-Ask Ratio Comparison
SPX Call Ratio Apple Call Ratio
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option volume is by far higher than any of its single components, signaling
its higher liquidity (price meaningfulness). The upwarding trend is related
with the usual option market behavior: the closer the option gets to
maturity, the more it becomes liquid (the liquidity dry down in the delivery
month) (Donna, 2001)
On top of that, Vix index5 is a vastly used security to trace volatility in the
market, the popularity of this instrument allows us to assume that S&P 500
4 Open interest provide useful information that should be considered when entering an
option position. Unlike stock trading, in which there is a fixed number of shares to be traded,
option trading can involve the creation of a new option contract when a trade is placed.
Open interest will tell you the total number of option contracts that are currently open—in
other words, contracts that have been traded but not yet liquidated by either an offsetting
trade or an exercise or assignment. Open interest also gives you key information regarding
the liquidity of an option. If there is no open interest for an option, there is no secondary
market for that option. When options have large open interest, it means they have a large
number of buyers and sellers, and an active secondary market will increase the odds of
getting option orders filled at good prices. So, all other things being equal, the bigger the
open interest, the easier it will be to trade that option at a reasonable spread between the
bid and ask. Declining open interest means that the market is liquidating and implies that
the prevailing price trend is coming to an end. A knowledge of open interest can prove
useful toward the end of major market moves. 5 VIX is a trademarked ticker symbol for the Chicago Board Options Exchange Market
Volatility Index, a popular measure of the implied volatility of S&P 500 index options. It
represents one measure of the market's expectation of stock market volatility over the next
30 day period. Prof. Menachem Brenner and Prof. Dan Galai (Brenner & Galai, 1986) first
developed the idea of a volatility index, and financial instruments based on such an index.
In 1992, the CBOE retained Prof. Robert Whaley to create a stock market volatility index
based on index option prices, which was launched in January 1993. Subsequently, the CBOE
has computed VIX on a real-time basis. The market convention on VIX is: percentage points
1
10
100
1,000
10,000
100,000
1,000,000
ott-12 nov-12 nov-12 nov-12 nov-12 nov-12 dic-12 dic-12 dic-12 dic-12 gen-13 gen-13 gen-13 gen-13
Total Open Interest
Apple Tot Open Interest SPX Total Open Interest
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index together with Vix quote have significant price and that they are less
sensitive to market rumors, typically affecting single components. However
SPX index is highly recognized as a market proxy, hence whenever new
information related to macroeconomic data or new data release are
disseminated in the market, then market players do prefer to
correct/reallocate their exposure by trading on this security, rather than
correct single security exposure in their portfolio. This market behavior
allows us to assume that all significant information is promptly incorporated
into the security price.
The chart presented above provides the SPX Index and VIX index time series
(VIX scale is on the left) between January 2000 to January 2013. We want to
point out the strong relationship between downtrend and increased market
implied volatility. Moreover peak variation/level corresponds, as we can
expect, to IT Bubble, Subprime Bubble and Sovereign Debt crisis.
The paper is divided into three sections. In Section I, we outline why the new
pricing approach is worth using, by analyzing underlying historical time
and translates, roughly, to the expected movement in the S&P 500 index over the upcoming
30-day period, which is then annualized.
0
10
20
30
40
50
60
70
80
90
0
200
400
600
800
1,000
1,200
1,400
1,600
1,800
Price Time Series
SPX Index Vix Index
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series, and what is so peculiar for barrier option: basically we address i) its
main features and ii) sensitivity on underlying distribution assumptions.
Section II is focused on the core of our dissertation, where we explain the
methodology, the pricing techniques results, the stress test performed as
well as the robustness test. Later, we analyze market condition evolution and
on other pricing techniques and we develop our hedging procedure. In
Section III, we conclude by summing up the strength of the model proposed
and its drawback, we also dedicate a sub section on possible future develops.
1.a. Market Data Analysis:
Given the S&P 500 nature, we should expect that this time series behaves
closely to Gaussian hypothesis, however this paper shows that although an
index should show less extreme and more stable time evolution, it
significantly diverges from normal assumptions.
Recent market evolution has shown the inadequacy of Gaussian normal
hypothesis on probability distribution. The 2008 financial crisis has pushed
researchers to develop proper model framework to describe data behavior.
On top of that, pricing derivatives has changed to take into account other
risk sources (such as counterpart risk, liquidity risk and so on). Those
distributions cannot be described anymore with parametric function with just
two parameters (mean and variance). Academia has started to focus on
analyzing the impact of excess Kurtosis and Skewness. Those two features
have a significant impact on risk management and on option pricing (where
the problem gets more severe the more exotic is the derivatives payoff
itself).
Return has been computed as price ratio according to the logarithm
standard, since Logarithmic returns are best used for single stock return
over time, in this case the return will only depend on the initial and last
element of the time series
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(( )
) ∏
∑ ; ∑ [ ∑ ] ∑ ( ).
We remember that the relationship between logarithmic and arithmetic
returns can be better understood by using the Taylor expansion for the log
return, which is the same of the linear if truncated at the first parameter
( ) ( ) ( ) ( ) This formula shows how the
difference between the linear and log return (for price ratio far away from 1)
will be always greater than zero, since ( ) .
Normal distribution assumption fails to capture empirical data excess kurtosis
as well as asymmetry. We have chosen the time period between January
2000 and September 2012 to show S&P 500 Gaussian hypothesis failure, by
using QQ plot, by reproducing Jarque Bera test (Jarque & Bera, 1980) results
and by plotting time dynamics of mean, variance, skewness and kurtosis.
In the chart above, we present the three QQ plots.
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In the chart above we show the standardize distribution via kernel6 plots. We
have chosen to represent the distribution in the first row using the kernel
instead of the usual histogram, since the first is a generalization of the
latest. We have chosen the kernel Gaussian function
∫ ( )
( )
√
First figure represents data standardized by sample variance
∑
, as we can see the distribution has a strong excess
kurtosis, despite the quite symmetric behavior.
According to “Value at Risk when daily changes in market variables are not
normally distributed” (Hull & White, 1998) article this behavior can be
dramatically reduced by assuming heteroskedasticity7. That’s exactly what
we have showed in second figure: those figures are based on data
6 A kernel density estimator is an empirical density “smoother” (from discrete to continuous distribution) based on the choice of the two objects: the kernel function itself ( ) and the
bandwidth parameter “h”, which is chosen according most used convention (equation
reported above) and also the standard one used in MatLab 7 Heteroskedasticity means that the error variance is correlated with the values of the
independent variables in the regression. The Breush-Pagan (1979) test is widely used in
finance research because of its generality, we didn’t run this test.
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standardized with variance computed with GARCH8 (1,1). Here in the
following we provide the equation used to fit the GARCH model, in the annex
is provided the MatLab printout.
(
) ( ) where
.
Although there is an improvement in the QQ plot figure, we notice the
mismatch on the left tail is still significant, signaling asymmetry as well as
excess kurtosis behavior. Hence also standardized time series by GARCH
(1,1) fails to respect Gaussian assumption.
We run a GARCH(1,1) assuming a t-student underlying distribution, the
results are shown in figure 3. As it can be noted there is no significant
improvements, excess kurtosis and asymmetry is still there.
Besides the graphical examination, here in the following we present our
results using the univariate Jarque Bera test (Jarque & Bera, 1980).
( )
Where we define the asymptotic distribution of the sample estimators of
skewness and kurtosis as in the following:
√ ( ) √ ( ) ( )
We have run the test on both empirical data and standardized one. As
already suggested by QQ plot figure, data fails JB test (details provided in
Annex), providing further evidence on the misrepresentation of Gaussian
assumption when trying to model data distribution.
Here in the following we provide four charts representing: mean, volatility,
skewness and kurtosis of the chosen underlying between January 2000 and
September 2012. The blue line has been drown with a time rolling range of
8 Generalized Autoregressive Conditional Heteroskedasticity (Bollerslev, 1986) (p, q) (where
p is the order of the GARCH terms and q is the order of the ARCH terms )
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233 days9, while the red one represents the whole sample size respective
centered moments10.
The above charts provide us two important information: i) time dependence
of data moments and ii) strong divergence from “long term” level at 2001 IT
bubble and at 2008 subprime bubble. By analyzing those figures we can
infer: i) the correlation between crisis/financial crush and ii) the divergence
from Gaussian assumptions. Given the material effect on pricing and
hedging, developing models, which does fit with these features, has become
a crucial task.
Here in the following we propose the same analysis based on the ratio of
price
rather than returns. We present below charts since the relaxed
model is featured to use this ratio instead of returns.
9 We didn’t directly choose 233 as time windows, it is obtained dividing the total sample size
by 20 10 We used the standard MatLab functions. We have used those values as proxy to long
term/natural levels
15
Those charts represent the un-centered moments, from whom we may
obtain the equivalent mean, variance, skewness and kurtosis given their
definition in term of moments:
( ( ))
( ( ) )
( ( ))
( ( ) )
Market players have historically recognized this potential risk; however there
wasn’t any significant case/event which might justify a framework change.
Market participants have tracked this problem by systematically overpricing
deep in/out of the money options. That’s the reason behind the constant put
option higher implied volatility than at the money option in the equity market
(the so called “Crush Phobia” represented by volatility Skew11).
We would like to propose an interesting parallel between 1987 Crush and
2008 financial crush. As before 1987 there wasn’t any evidence of Volatility
skewness, today we find ourselves in the same situation back in time, where
11 When implied volatility is plotted against strike price, the resulting graph is typically
downward sloping for equity markets, or valley-shaped for currency markets. For markets
where the graph is downward sloping, such as for equity options, the term "volatility skew"
is often used. For other markets, such as FX options or equity index options, where the
typical graph turns up at either end, the more familiar term "volatility smile" is used.
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suddenly the used models have proofed themselves to be wrong. We cannot
count any more on demand-offer driven volatility pricing and Gaussian
simplification (i.e. closed formula).
Thanks to IT development12, simulation/numerical approximation models
does not suffer any more of the usual time constrain and bias problem. Our
work tries to move one step closer to a new market standard by exploiting
the flexibility granted by lattice methods.
1.b. The Derivatives chosen:
In the late 1980’s and early 1990’s, exotic options became more visible and
popular in the over-the counter (OTC) market. Corporations, financial
institutions, fund managers, private bankers are the typical users. The most
popular group of exotic options is path-dependent options, and barrier option
is among this group. The most appreciate feature of barrier option is to offer
a cheaper protection compared to vanilla option. As an example if we take a
down-and-out barrier call option, a trader with a bull perspective view on the
market may regard the condition of the barrier being reached as quite
unlikely and be more interested in it than the regular one. Or as another
example we might consider the case of a hedger buying a barrier contract to
hedge a position with a natural barrier, e.g. the foreign currency exposure on
a deal that will take place only if the exchange rate remains above a certain
level.
The barrier options (also called trigger options) are essentially conditional
options, dependent on whether some barriers (“H”) or triggers are breached
12 A multi-core processor is a single computing component with two or more independent
actual central processing units (called "cores"), which are the units that read and execute
program instructions. The instructions are ordinary CPU instructions such as add, move
data, and branch, but the multiple cores can run multiple instructions at the same time,
increasing overall speed for programs amenable to parallel computing. In the best case, so-
called embarrassingly parallel problems may realize speedup factors near the number of
cores, or even more if the problem is split up enough to fit within each core's cache(s),
avoiding use of much slower main system memory
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within the lives of the options. Hence, the main feature is to activate a
specific event (becoming a regular option or getting a rebate), depending on
whether the barrier is breached or not during the life of the option. According
to the relative position of H and S, there are four kinds of typical barrier,
which are outlined below.
Down and Out: knock-out options with H < S.
Down and In: knock-in options with H < S.
Up and Out: knock-out options with H > S.
Up and In: knock-in options with H > S.
Barriers may be monitored continuously or discretely. Continuous monitoring
of the barrier means that the barrier is active at any time prior to maturity.
Discrete monitoring implies that the barrier is active only at discrete times,
such as daily or weekly intervals.
The importance of barrier options is growing together with the evolution and
more sophistication of market operators. Most models for pricing barrier
options assume continuous monitoring of the barrier; we have developed our
model according to this assumption, which simplify the model set up.
“Complex Barrier Option” paper (Cheuk & Vorst, 1996) shows that even
hourly versus continuous monitoring can make a significant difference in
option value. Discrete barrier options, convergence may be slow and erratic,
producing great errors even with thousands of time steps and millions of
node calculations. The reason is that the payoff of a barrier option is very
sensitive to the position of the barrier in the lattice.
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The chart above provides us with graphical example of the typical jump
around the Strike (100) for ATM option. The dashed line represents the true
probability density, while the light dash line is the discrete approximation
made by the model, where each node contribution is made by its value time
its probability. The bias arise from the difference between the hard and light
dash line, this problem get more severe the less linear is the option payoff
around critical area.
Barrier option can be priced with i) “closed form” expressed in terms of
multivariate normal probabilities, ii) approximation methods, such as
standard lattice techniques, or Monte Carlo simulation. It has been paid more
attention to analytic solutions than to lattice methods, which obtains the
correct valuation asymptotically. However, the convergence of the closed-
form solutions is slow, and the results tend to have a large bias when the
asset price is close to the barrier.
19
We have used as market proxy the Bloomberg function “OV” (option
Valuation), we provide in the following some screenshots of Bloomberg
function:
20
The model was used more as a directional control rather than try as an exact
benchmark to fit our model. The Bloomberg valuation platform does allow
user to choose among several options, note that by default Bloomberg
propose a Heston model for the volatility, while our work does assume a
deterministic and flat volatility.
21
Section II:
The need to capture this empirical features has pushed the researchers to
develop new pricing approaches, among those we want to show the
superiority of lattice model as proposed in “A relaxed lattice option pricing
model: implied skewness and kurtosis” (Brorsen & Dasheng, 2009) and “A
recombining lattice option pricing model that relaxes the assumption of
lognormality” (Brorsen & Dasheng, 2010). The idea is similar to a Taylor
expansion or the nonparametric approach using the Cornish Fisher expansion
(Cornish & Fisher, 1937) where we add the third and fourth moments used
as new parameters to better fit the empirical distribution.
The paper “Option pricing: A simplified approach” (Cox, Ross, & Rubinstein,
1979) (“CRR”) pioneered the lattice approach. They developed a discrete-
time, binomial approach to option pricing. The essence of their approach is
the construction of a binomial lattice of stock prices where the risk neutral
valuation rule is maintained. With a particular selection of binomial
parameters including probabilities and jumps, they showed that the CRR
binomial model converges to the Black-Scholes model (Black & Scholes,
1973). The CRR methodology has been extended subsequently by various
researchers. The article “Option valuation using a tree Jump process” (Boyle
P. P., 1986) took the CRR methodology one step further and proposed a
trinomial option pricing model, a three-jump model, where the stock price
can either move upwards, downwards, or stay unchanged in a given time
period. Most recent studies have been developed to handle non-normal
distribution by varying the parameters within the lattice, such as “Implied
binomial tree” paper (Rubinstein, 1994). “A modified Lattice approach to
option pricing” paper (Tian, 1993) defines binomial and trinomial models
where it has been relaxed the symmetry restriction, but it has still retained
recombination and lognormal underlying distribution assumption. (Brorsen &
22
Dasheng, 2009) and (Brorsen & Dasheng, 2010) papers have further relaxed
the restrictions, maintaining only the recombination constrain.
Their articles were tested on the Commodity market (option written on
Wheat), we want to propose their algorithm for equity derivatives and to
extend it to the Barrier option case.
Although lattice models provide powerful, intuitive and asymptotically exact
approximations there are essentially two related but distinct kinds of
approximation errors in any pricing techniques of lattice framework, which
we refer to as distribution error and nonlinearity error:
Distribution error: The true asset price distribution is unknown,
hence when we develop price model we need to assume an underlying
distribution. Usually market players have used lognormal density,
approximated by a finite set of nodes with probabilities. Even though
the mean and variance of the continuous distribution are matched by
the discrete distribution of lattice model, there is the possibility that
the model itself is biased given the wrong initial assumption on the
distribution, hence the option does not converge to the correct value.
Nonlinearity error: The finite set of nodes with probabilities used by
lattice model can be thought as a set of probability weighted average
option price over a range of the continuous price space around the
node. If the option payoff function is highly nonlinear, evaluating the
nonlinear region with only one or several nodes would give a poor
approximation to the average value over the whole interval.
We will minimize/solve the later by the adaptive mesh model with slight
computation increase, while the first issue will be solved via the intuition
initially proposed by (Brorsen & Dasheng, 2009).
23
1.a. The Methodologies:
Let and be the asset price at the current time and at one period later.
We will define the price ratio as
and we assume to take value “u” with
probability “q” and “d” with probability “ – ” in a binomial tree. In a
trinomial tree, the price ratio has three possible values “u”, “m” and “d” with
corresponding probabilities , and respectively. There are nodes
at the final step in an n-step binomial tree, while there are in the n-
step trinomial tree.
We used a backward13 way algorithm to run our initial pricing procedure on
European and American option.
In “Bumping up against Barrier option with the Binomial Method” (Boyle &
Lau, 1994) the authors show that for continuous barrier options there is a
potentially large pricing errors, resulting from a lattice, even with a large
number of steps. “Pricing Barrier option with adaptive mesh model” article
(Ahn, Figlewske, & Gao, 1999) provides an approach that greatly increases
the efficiency in lattice models, they proposed an “adaptive mesh” method
(“AMM”), to deal with ”barrier-too-close” problem in continuous barrier
options. The AMM model is very powerful in both efficiency and flexibility as
many papers have testified.
Firstly, we have developed our model testing their performance under the
plain vanilla case. Our test has been performed using both the “A
recombining lattice option pricing model that relaxes the assumption of
lognormality” (Brorsen & Dasheng, 2010) algorithms proposed: i) use
historical moments, and ii) bootstrap the implied moments by fitting the
model parameters to the market quotation. Secondly, we have used the
historical moments to run our barrier options.
13 The option values at the nodes of the final step of the lattice are calculated first. Then, the
values at the step before the final one are calculated. This procedure is continued until the
initial step, i.e. the current time.
24
1. We have chosen the “SPX US 03/16/13 C1400 Index & SPX US
03/16/13 P1400 Index14” option, since they were at the money
(“ATM”) option at inception [Strike at 1400; SPX index at 1409
@19/03/2012]
Here we provide the two Bloomberg screenshot of the call and put
European option used as reference derivatives in developing our model.
14 Note that the date convention showed in the Bloomberg ticker is the American one.
25
2. Rather than use directly the Libor15 quote (with tenor chosen according
to option maturity), we have bootstrapped from Put-Call Parity
equation the implied rate used in the market. This method allows us to
avoid producing an interest rate model.
3. We have bootstrapped16 the implied volatility surface via Black Scholes,
using put and call options with strike 1200 1300 1400 1500 1600
between March 2012 to January 2013, with maturity at 16/03/2013. In
the above chart we present the volatility surface; at the top there is
the implied call options volatility, while at the second there are the
results from put options.
15 Libor is a trimmed average of reported funding rates for unsecured borrowing deals from a
panel of large banks with the highest credit quality. It is used as risk free rate, rather than
Treasury Bills, since that market is highly influenced by central Banks and currency flotation
(Duffee, 1996). Although Libor is not a deterministic function, the source of risk coming from
the equity underlying is far higher than the one coming from money market rate, hence
assuming a deterministic dynamics won’t cause big errors/bias in pricing 16 We have minimized the difference between market price and theoretical price by fixing all
parameters besides volatility.
26
We didn’t use any smoother function and we have taken the closing options
ASK quote, hence the time evolution shows some abrupt/erratic change,
however the smile shape is clearly represented. The sharp increase in the
implied call volatility in the last days, for deep in the money options, is
caused i) the options considered were getting closer to maturity (intrinsic
value component is predominant in option price) and ii) January ending
month improvements in all markets given the higher confidence.
4. We have bootstrapped17 the implied third and fourth moments using
the relaxed lattice (Brorsen & Dasheng, 2010) trinomial model.
Plain vanilla option case:
In the n-step binomial tree case we have at the n+1 nodes of the final step
the following payoff (we denote with “c” the call and with “p” the put):
( )
( )
17 Differently form the BS case, where we have separately bootstrapped the implied volatility
(call and put; strike by strike), in this case we do have minimized the absolute difference,
hence we have considered both call and option as well as all the 5 Strike altogether. We
have forced the model to accept as input for first and second moments:
27
At the nodes n-1 for the European option the payoff is defined:
(
( ) )
(
( ) )
The American option payoff is defined as:
( | | (
( ) ) )
( | | (
( ) ) )
In the n-step trinomial case the above formula becomes:
( ( ) | | ( ) )
( ( ) | | ( ))
(
)
(
)
( ( ) | | ( ) (
) )
( ( ) | | ( ) (
) )
The relaxed tree models do not assume a specific parametric distribution for
the ratio Y, which is unknown. Instead they are based on matching the
moments of the underlying distribution. In practice, the moments could be
estimated directly from empirical data or they could be taken from an
estimated parametric distribution. The relaxed binomial model is constructed
such that the first 3 moments are matched via a discrete approximation to a
continuous via a Gaussian quadrature approach18 (DeVuyst & Preckel, 2007)
( ) ( ) ∫ ( )
Our unknown parameters are “u”, “d” and “q” for the n-step binomial model.
We will compute them by solving the approximation equation corresponding
18 K nodes allow matching the first 2k-1 moments, therefore we can match up to the third
moments for the binomial and till the fourth moments for the trinomial tree
28
to the moments up to the third, as suggested by “Discrete approximation for
probability distribution” article (Miller & Rice, 1983).
( ) ( ) ( ) ( )
The binomial case do allow for a close solution. We define a polynomial
( ) ( )( ) , where ( )
√
√
( )
and are than expressed as function of moments, given the moments.
We have taken the approximation equation and we have multiply the first by
the second by and the third by one (the case where i=0 is redundant
since it imposes that the probability sum to one). The solutions are:
( ( ) ( ) ( ( ))
)
( ) ( ( ))
( ( ) ( ) ( ( ))
)
( ) ( ( ))
The only requirement to the Y is to have finite moments and to come from a
valid probability distribution: note that the Gaussian quadrature procedure
cannot be applied directly to subjective estimates of continuous distributions
because neither the form nor the moments of these distributions are known.
Typically, the probability assessment process produces only a graph of the
cumulative distribution. However, a two-step procedure can be used to
determine a discrete approximation based solely on the information
contained in the graph. The first step uses Gaussian quadrature to determine
the moments of a continuous distribution, and the second step uses
Gaussian quadrature again to determine a discrete approximation with these
moments. Note that the time represented by each step in the lattice is the
29
time interval for the moments. For example, to represent daily step we need
to use daily data19. “A recombining lattice option pricing model that relaxes
the assumption of lognormality” (Brorsen & Dasheng, 2010) paper show that
lognormal binomial model is a special case of this framework, which
generalize the previous one.
The n-step trinomial case does not allow a closed formula solution, since the
equation system as shown below:
( )
( ) ( )
The equation20 is over-determinate and the recombine condition is not linear,
thus to avoid it we need to simultaneous solve the system by minimizing the
squared differences between the two side of the equation:
∑( ( ) )
Subject to the following constrain
Alternatively we can imply the parameters by minimizing the sum of
squared errors as suggested by ”Valuation of American call options on
dividend paying stock” paper (Whaley, 1982):
19 In cases where the central limit theorem holds, the higher moments will change
nonlinearly as time steps change. Carr and Wu (2003) find that the volatility smile does not
flatten out as maturity increases, which implies that the higher moments are stable under
addition. Wu (2006), however, finds that this is not always the case. Regardless, users of
the relaxed lattice approach must always be careful to match moments with the size of the
time step. 20 Note that the trinomial model do allow to fit up to the fifth moments, however we do not
undergo this possibility, since there is no literature on fifth moments behavior
30
∑( ( )) ∑( ( ))
where is the observed option premium for the ith call contract and
is the observed option premium for the i-th put contract. We have taken 5
contracts per option type: one ATM, two in the money (“ITM”) and two out of
the money (“OTM”).
In a binomial model, three parameters, i.e. “u”, “d”, and “q”, need to be
implied. In a trinomial model, though there are six parameters, only four of
them need to be implied, because the probability condition, ,
and the recombining condition, , have to be satisfied. Since the
lognormal distribution constraint is relaxed, the move sizes and transition
probabilities capture the information about the underlying distribution that is
contained in the option prices.
A by-product of implying the parameters by the relaxed lattice models is that
the parameters can be used directly to estimate the moments of the
unknown underlying distribution. With the binomial, the asset price ratio is
assumed to take value u with probability q and d with probability 1 - q. So its
first moment is ( ) . Similarly, we can approximate the k-th moment
of the asset price ratio based on the continuous dynamic process by the k-th
moment based on the binomial framework. The highest moment that can be
implied by a binomial model is skewness. The relaxed trinomial model yields
both skewness and kurtosis. The approximation equations based on the
trinomial framework are:
( ( )) ( ( ))
( ( ))
( ( ))
“Y” represents the ratio of the underlying asset price at the next step and the
current step. Riskless arbitrage and efficient markets were assumed and the
additional restriction of a zero mean was imposed. Note that the implied
31
moments do depend on the chosen number of steps, unless the underlying
distribution is of the stable Paretian class21.
Now we provide some numerical example:
The chart provides the binomial (red line) and trinomial (green line) relaxed
call price computed with historical moments with a rolling windows of 260
days:
( ) ( )
Here in the following we show the graph of American put option (yellow line)
and the European put option (blue line).
21 In probability theory, a random variable is said to be stable (or to have a stable
distribution) if it has the property that a linear combination of two independent copies of the
variable has the same distribution, up to location and scale parameters. “Let X1 and X2 be
independent copies of a random variable X. Then X is said to be stable if for any constants a
> 0 and b > 0 the random variable aX1 + bX2 has the same distribution as cX + d for some
constants c > 0 and d. The distribution is said to be strictly stable if this holds with d = 0”
(Nolan, 2009)
32
It is evident that the model correctly incorporates the difference between
European (lower premium) and American, note that we have increased the
risk free rate up to “9%” to increase the difference. Here in the following we
do provide the same figure with market rate (around 0.5%):
The early exercise option is not sizable at all, since the negligible time value,
and it gets smaller, the closer gets the maturity.
33
There is no difference between call option premiums since we do not assume
any dividend yield/dividend payment.
8
1.b. The Barrier results
The mean is assumed to be equal to risk free with respect to NA argument,
while the variance values are taken from implied 9022 days traded option
volatility. The skewness and the kurtosis fitting are based on historical data
with a time range of 260 days (roughly one year of daily observation).
As speaking about our study, we do not model a specific view on assessing
the proper moments/input parameters. We have just tested if the mean over
the time period chosen (90 days) is statistically different from zero, and as
excepted the t-test do not allow us to reject the zero assumption, hence we
have forced the model to factor in a zero mean.
Remember that historical and implied volatility although they are correlated
they are different, moreover there exist several implied volatilities in the
market based on different future time horizon. Here in the following we show
the historical vs. implied 30 days volatility.
22 ATM option. 90 days is the tenor chosen for our options.
34
As we have early stated, there isn’t any significant difference between the
two time series, furthermore there is no clear direction, even if we may
address the relative lag between implied and historical volatility, where the
last seem to be less reactive to market change.
Here in the following we show the results of our model with and without
adaptive mesh model for a Knock-in call option written on SPX index.
As expected the Adaptive mesh model reduce the erratic behavior and speed
up the convergence of the model.
0
10
20
30
40
50
60
70
Volatility 30d Implied vs Historical
Implied Historic
35
1.c. Stress and Non Arbitrage test:
Our paper is supported by several stress and robustness test, moreover we
have always tracked its relative performance against other pricing approach.
We have test Non-arbitrage by checking put-call parity23 condition running a
500 steps simulation.
( )
Where “K” is the strike and the sub-scrip “i” represents time. The model does
respect the condition all across the time considered as the chart above show:
there is a small divergence from the value “0” by a non-significant amount,
mainly due to simulation approximation error, note that this difference is
largely within SPX index options bid-ask spread.
23 put–call parity defines a relationship between the price of a European call option and
European put option, both with the identical strike price and expiry, namely that a portfolio
of long a call option and short a put option is equivalent to (and hence has the same value
as) a single forward contract at this strike price and expiry. This is because if the price at
expiry is above the strike price, the call will be exercised, while if it is below, the put will be
exercised, and thus in either case one unit of the asset will be purchased for the strike price,
exactly as in a forward contract.
36
Here in the following we present the price convergence (plain vanilla case) of
the lattice model as function of increased number of steps by controlling for
time elapse
We have tested only the trinomial model, since it more demanding compared
with the binomial one. Time as function of step is an exponential function,
however we address that a 5500 step trinomial tree has required around 2
seconds.
The above figure show that with more than 1000 steps the lattice model has
a small error vs. theoretical market price (117.775). 1000 steps lattice
37
model requires less than half a second, hence the model does allow a good
approximation within a suitable time window.
Now we describe the hedging procedure: we have computed option delta by
approximating the first derivative with respect underling price change with
Central Difference Estimator24 (“CDE”) as follow:
[ ( )]
( ( ) ( ))
Given this definition we will proxy this quantity with:
( ) ( )
( ) [ ( )]
( )
∑ ( )
( )
∑ ( )
The CDE is biased: ( ) ( )
24 Note that this approach is superior to the forward difference estimator (“FDE”), since the bias goes to zero faster than the FDE: ( )
38
The two charts above show the step evolution of Delta Greek letter25.
1.d. The empirical Comparison:
Here in this section we will compare the result of our model against standard
Binomial and Trinomial lognormal lattice model, together with the classic
Black and Scholes formula (European case) and Monte Carlo simulation
approach.
Since this paper is mainly oriented to market participants we have
specifically addressed the computational time burden of our new algorithms
compared to a Monte Carlo simulation with control variate26 and Antithetic27
variate variance reduction techniques, with a parametric Gaussian
distribution. We have set up a specific procedure to stress our model under
time constrain. This procedure consists on running a Monte Carlo with
100.000, 300.000 and 500.000 simulation respectively and we have saved
the time required by them. Then we have optimized the number of nodes in
our model to respect the three time constrains.
25 the Greeks are the quantities representing the sensitivities of the price of derivatives
such as options to a change in underlying parameters on which the value of an instrument
or portfolio of financial instruments is dependent. 26 Control Variate is based on using the error in the estimate of known quantities to reduce the error in the
estimate of the unknown one. We will use the combination of the known variable and the unknown one ( )
( ( )), which it will be used as estimator.
This estimator is unbiased for ( ) ( ( ( ))) ( )
So we need to choose a parameter “b” to minimize the new estimator variance to ensure ( ( )) ( ). This
method allows to reduce the variance if the control variate is correlated to the unknown, the sign do not
matter, only size the higher the better ( ) with the trivial requirement
If we joint estimate b and X we will have a bias, in fact those variables will be correlated so ( ( ( ))
( ) ( ( )) . To solve this issue we need to run two independent simulation, the first regressing Y on X to
obtain “b” ( it converges to the correct value b) and the second running the simulation for the estimator
itself
The “b” comes from ( ( ( ))) ( ) ( ( ( ))) ( )
, now we
can compute the FOC or just notice that it is a parabola so the vertex is the minimum as well. Note that
( ( )) ( )
27 Antithetic Variate consists on using for each simulation the given percentile and its opposite, so that they have
the same distribution but they are not independent, but negatively correlated. [ ] ( [ ] [ ]). It allows
doubling the sample size without doubling the time burden.
39
Here in the following we want to test our model without specifying any
models for computing input moments, we have used the historical one as
previously defined. The redline represents the relaxed model, the yellow one
is standard model together with Black and Scholes formula and the dashed
blue line is market price of the call options SPX US C1400 index:
The relaxed model diverges from the lognormal assumptions, however they
share the same time evolution. As we do expect the skewness and the
excess kurtosis do play an important role. The case of the call written on the
S&P 500 we see that our model do diverge from the Black and Scholes and
binomial/trinomial models according to the direction of underlying empirical
distribution departure from normality. Moreover, in relation with historical
market price we can see how the relaxed model does perform better. Note
that we have used historical moments as previously defined without
specifying any assumptions on them.
In the plots above we provide a graphical representation of the difference
between the historical moments used to run the above charts and the
implied moments bootstrapped from market price time series, according to
40
∑( ( )) ∑( ( ))
The first chart provides the implied parameters ( ), in order form the red
line to the blue line we have the implied parameters of order tree to 1. As we
can note there is a significant difference between the two plots.
41
Section III:
1.a. Strength & Weakness of the model:
The below chart allows us to graphically understand the difference between
the standard model (red points ‘X’) (Cox, Ross, & Rubinstein, 1979) and the
relaxed model (blue points ‘X’) (Brorsen & Dasheng, 2010). We report in
Tab3 the moments used as input to define the model parameters.
This graph provides (based on binomial case) us a clear example where by
fitting up to Skewness, the relaxed lattice tree does better fit with empirical
distribution behavior.
42
The chart above based on binomial case provides another graphical intuition
of the effect of the positive skewness, the blue points describe the dynamics
of the relaxed model. Tab4 summarizes the moments used to run the above
chart.
43
The chart above shows the tree plot based on trinomial tree (blue point
represents relaxed model, while red ones the standard lattice). To run this
plot we used the moments in Tab3, as we can see the skewness effect is
predominant on kurtosis. Note that the relaxed model value range is larger
than the standard one.
Our thesis aims to overcome the usual implied volatility surface modeling by
properly assessing asymmetry and extreme value occurrence. This relaxed
Lattice model is one step closer to define a new generalized framework by
exploiting the recent IT develops which allow exploiting simulation technique
flexibility without suffering of the usual computational time burden.
1.b. Possible future evolution:
Possible future research can accommodate this relaxed model to fit the
stochastic volatility behavior such as Heston model (Heston, 1993), where
mean and variance are allowed to change. Note that this lattice extension
would cause recombination condition to fail.
On top of this theoretic improvement there are more practical issues that can
be further analyzed. We didn’t propose a specific econometric framework to
44
compute the proper moments to be used as input in our model. There are
lots of papers on how properly forecast variance, however there is not such
agreement on higher moments. A possible evolution can be to set up an
econometric framework to properly incorporate market expectations and to
implement a cross Strike options strategy to trade on volatility surface to see
if the relaxed model grants an excess return. Note that there is a strong
evidence of non stationarity time series, this fact may seem like a
mathematical subtlety, but it is not. If we use ordinary least squares to
estimates randomly distributed independent variable, which is a lagged value
of the dependent variable, our statistical inference may be invalid. To
conduct valid statistical inference, we must make a key assumption in time-
series analysis.
Furthermore there is still room to improve the model to handle double or
more exotic barrier option payoff. In double barrier option cases there are
two areas that would generate nonlinearity error at every barrier monitored
dates. The error would be cumulative that incorrectness is even mounting
when the barrier condition is checked more frequently. On top of that the
usual approach to halve price step will quadruple the number of time steps
and would make the node value calculation become 16 times more
consuming. It is notorious how this programming structure gets unfriendly
due to overlapping nodes.
45
Conclusions:
This thesis does not only implement the new relaxed Lattice approach
(Brorsen & Dasheng, 2010) for European and American equity options but
also extend model to price barrier options. Besides, we also comprehensively
implement other competitive methods with detailed numerical results to
compare with our relaxed lattice such as the standard trinomial/binomial
tree, Monte Carlo simulation and American option via finite differences. We
have also implemented for the Barrier option case an Adaptive Mesh Model to
reduce nonlinearity error. From our research data, we numerically prove i)
the efficiency of relaxed lattice in better tailoring empirical distribution, ii)
the closer mimicking of market option prices and iii) parsimonious time
burden.
46
Appendix:
1.a. GARCH and JB test insight
The MatLab Printout showed above refers to the Gaussian GARCH(1,1). As
we can see all the model parameters are statistically different form zero (t-
test are far above the z-percentile at 95%). The parameters do respect the
condition stated in the paper to avoid that variance will explode.
The same considerations are inferred from the t-GARCH(1,1), as we can see
all variable are statistically different form zero (@95%). Note that we have
force the model to fit to the degree of freedom
( ) as computed according to the following MatLab algorithm:
47
df_init = 4; [df,qmle] = fminsearch('logL1',df_init,[],n_ldayly(1:end),sigma);
The target equation is:
function [sumloglik] = logL1(df,ret,sigma) d=df; y=ret; s=sigma;
[R,C]=size(y); logL=NaN(R,1); logL=gammaln((d+1)/2)-gammaln(d/2)-0.5*log(pi)-0.5*log(d-2)-
0.5*(1+d)*log(1+(y./s).^2./(d-2));
sumloglik=-sum(logL);
The JB tests, performed on the empirical distribution as well as on the
standardized ones, have a p-value smallest than 0.0001, signaling a strong
divergence from the Gaussian assumptions.
1.b. MatLab Code insight:
We have chosen MatLab since it is a flexible platform with a huge
programming community with lots of solutions and pre-defined functions,
which have speed up our coding preparation. Our general aim is to offer an
efficient and readable code, however we didn’t focus our work on delivering a
user friendly interface.
At first the naming convention used is to name the variables with
combination of worlds as clear as possible, divided with the Upper case, in
some routines there are some service variables which are usually named
with fuzzy or dummy name. Matrix variables are named with the suffix “m_”
while vector with “v_”.
We didn’t define any error handler since the aims of this code was to test our
statements rather than producing a re-usable/commercial code
The code is structured into three levels:
48
The main routine “OptionControl”, where all the other sub routine as
well as functions are managed. There are some control points where
the user is asked to provide some parameters on/off switch points. The
code is divided into 8 sections as showed here in the following:
disp(' Steps: ') disp('1. Data Loading ---------------------------------') disp('2. Re-elaborate Data ----------------------------') disp('3. Compute Implied Moments ----------------------') disp('4. Compute Historical Moments and Gaussian Test -') disp('5. Pricing Plain Vanilla and Check on Data ------') disp('6. Relative Difference plain vanilla case -------') disp('7. Pricing American case ------------------------') disp('8. Pricing Barrier case -------------------------')
In the sub routine “dtAnalysis” we will perform all the data analysis
and test on Gaussian distribution
In the Sub routine “SenAnalysis” we will perform some sensitivity
analysis and specific algorithmic to test time burden and efficiency
Among the many user-defined functions we will mention:
The optimization function to fit parameters in the trinomial relaxed
model, using the MatLab function (“fmincon”), which finds a
constrained minimum of a function of several variables. It attempts to
solve problems of the form:
( ) (linear constraints)
( ) ( ) (nonlinear constraints)
(bounds)
“myfun” is the target function used:
f=0; for i=0:4 f = f + (x(1)*(x(3)^i)+x(2)*(x(4)^i)+(1-x(1)-x(2))*(x(5)^i)-new(i+1))^2; end
49
where “x” is the target parameters to be optimized under the non-
linear constrains defined in “mycons”
c= [-x(1) ; -x(2); x(1)-1; x(2)-1];
ceq=[(x(3)*x(5))-(x(4)^2)];
The bootstrapping function to compute implied moments:
options = optimset('Algorithm','interior-point','Display','notify'); for t=1:2 for j=1:5 for i=1:tot_days x= fmincon(@(x)
optimp(x,m_option(i,j,t),ts_rate(i),ts_underlying(i),v_tenor(i),v_strike(j),t)
,x0,-1,0,[],[],[],[],[],options); m_vol(i,j,t) = x; clear x end end end
we have used the interior-point approach28 to constrained minimization is to
solve a sequence of approximate minimization problems. The method
consists of a self-concordant barrier function used to encode the convex set.
The function “optimp” have as target the equation:
( )
Where represents the implied volatility which is the parameter that we are
looking after.
The tree plots provided in section 3 have been drown with the user-
defined function “plottree” and “plottree_tri”
St = St*Up^max((i-j),0)*Down^esimo;
St = AssetP * Up ^ max(esimo-nSteps,0) * Mid ^ max((i-abs(j-i)),0) *
Down ^ max(i-j, 0);
Where we have one by one computed each tree arm and saved the into a
user-defined matrix, which is than plotted
28 “The interior point method was invented by John von Neumann, who suggested a new
method of linear programming, using the homogeneous linear system of Gordan (1873)
which was later popularized by Karmarkar's algorithm in 1984 for linear programming.”
50
The AMM model has been set in the following way:
We have find the minimum number of Up or Down step to breach the
barrier. We have used the function “fminsearch” targeting the absolute
difference between the underlying dynamics and the barrier level
At the critical node instead of running the usual “n” step lattice method
with defined step size “h” we plug in our Adaptive Mesh model with
step size equal to till the end (“n”)
The plug is made using a user defined function “adaptivemesh” where
we pass our parameters, as defined into the calling lattice model. The
parameters are then resized to fit new step size and to ensure the
respect the isomorphism condition. We have used the simplification:
.
51
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