SlideShare une entreprise Scribd logo
1  sur  35
Télécharger pour lire hors ligne
University	
  Cattolica	
  del	
  Sacro	
  Cuore	
  di	
  Milano	
  
	
  
	
  
Faculty	
  of	
  Banking,	
  Financial	
  and	
  Insurance	
  Sciences	
  
	
  
The	
  Decline	
  of	
  Saras	
  S.p.a.	
  
	
  
	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  
	
  
	
  	
   	
   	
   	
   	
   	
   Benedetti	
  Kevin	
  
	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  Matr.	
  4008804	
  
	
  
Course:	
  Applied	
  Statistics	
  for	
  Finance	
  
Prof.	
  Iacus	
  
Prof.	
  Zappa	
  
	
  
Accademic	
  year	
  2011-­‐2012	
  
	
  
	
  
INDEX
Chapter 1 – Preliminary Stock Analysis
1.1 Company Characteristics
1.2 Change Point Analysis
Chapter 2 – Option Valuation
2.1 Valuation of Financial Options: introduction
2.2 The Black & Scholes Model
2.2.1 Comments on B&S Model
2.3 The Monte Carlo Method
Chapter 3 – Lévy Process
3.1 Fast Fourier Transform
3.2 Monte Carlo Approach
Chapter 4 – Greeks Analysis
4.1 Greeks
4.2 Conclusions
Chapter 1 - Preliminary Stock Analysis – Saras S.p.a.
1.1 – Abstract
1.2 – Company Characteristics
The Saras Group, whose operations were started by Angelo Moratti in 1962, has approximately
2,200 employees and total revenues of about 11.0 billion Euros as of 31st December 2011. The
Group is active in the energy sector, and is a leading Italian and European crude oil refiner. It sells
and distributes petroleum products in the domestic and international markets, directly and through
the subsidiaries Saras Energia S.A. (in Spain) and Arcola Petrolifera S.p.A. (in Italy). The Group
also operates in the electric power production and sale, through the subsidiaries Sarlux S.r.l. and
Sardeolica S.r.l.. In addition, the Group provides industrial engineering and scientific research
services to the oil, energy and environment sectors through the subsidiary Sartec S.p.A.. Finally, in
July 2011, the Group created a new subsidiary called Sargas S.r.l., which operates in the fields of
exploration and development, as well as transport, storage, purchase and sale of gaseous
hydrocarbons.
Here are the market performance of the stock integrated with an important indicator that are the
volumes:
As we can notice from this picture the stock registered a small decrease in the stock price in the
very first part of the graph but it went up again jast before july 2008. From this point the stock went
down rapidly and it never stopped. Even today the trand of the stock in quite negative. We have to
1
2
3
4
5
SRS.MI [2007-05-21/2012-05-18]
Last 0.74
Volume (millions):
2,239,500
0
10
20
30
40
50
Mag 21
2007
Lug 01
2008
Lug 01
2009
Lug 01
2010
Lug 01
2011
Mag 18
2012
say that in the period considered for the analysis the whole world had to face the financial crisis
started in 2008 and the bad situation of this company doesn’t surprise. Deslpite the negative trend
on January 2012 we can observe a n incredibly high value of volumes: during this period, in fact,
there was the probability for the company to be delisted.
1.3 – Change Point Analysis
Given the decline of the stock and the the pattern of the prices during this period of crisis we
observe a “roller coaster” graph. Now we are going through an analysis that could help us to
explain this performance in order to catch the points where a turnaround has been registered.
In this paragraph I want to use an important tool in volatility analysis: the Change Point Analysis.
Considering a process: X = {Xt, 0 ≤ t ≤ T} à dXt = b(Xt)dt + √θσ(Xt)dBt and X0 = x0, 0<θ1,
θ2<∞{Bt, t ≥ 0} à Bt is a Brownian motion and the coefficients are defined and known.
The aim of this analysis is to find a point called τ0(tau0) associated to a parameter called θ(theta).
R-software compute for us this kind of operation and it gives us two values of theta: θ1, the
volatility just before the change point, and θ2, the volatility immediately after the change point.
In our case we are going to analyze a one-year period – from May 20, 2011 to May 20, 2012 – in
order to observe the reaction of the market in a tough span of time: in fact from May 20, 2011 there
were rumors of a probability of delisting, denied on December 1st, 2011.This announcement could
have brought some “good news” fo investors in a definitely not easy period. Therefore we are going
to expect a change point on around this date: from high range of volatility to a more attenuate one.
 
The picture above confirms our expectations: the value τ0 – the change point – is set on January 10,
2012. In fact after the announcements, where we can observe a steep rise in stock price, we find few
days more of high volatility and then, after January 10 a reduction in volatility.
Analitically speaking here are the R results:
τ0 = 2012-01-10
θ1 = 0.0444403
θ2 = 0.02976848
∆2-1=0.02976848 – 0.0444403 = -0.01467182
0.8
1.0
1.2
1.4
1.6
1.8
S [2011-05-20/2012-05-18]
0.8
1.0
1.2
1.4
1.6
1.8
Last 0.74
Bollinger Bands (20,2) [Upper/Lower]: 0.991/0.730
Mag 20
2011
Ago 01
2011
Ott 03
2011
Dic 01
2011
Feb 01
2012
Apr 02
2012
Chapter 2 – Option Valuation
2.1 – Valuation of Financial Options Introduction
A financial option contract gives its owner thr right (but not the obligation) to purchase or sell an
asset at fixed price at some future date. Two distinct kinds of option contracts exist: call options and
put options. A call option gives the owner the right to buy the asset; a put option gives the owner
the right to sell the asset. The most commonly encountered option contracts are options on shares of
stock: a stock option gives the holder the option to buy or sell a share of stock on or before a given
date for a given price.
When a holder of an option enforces the agreement and buys or sells a share of stock at the agreed-
upon price, he is exercising the option. The price at which the holder buys or sells the share of
stock when the option is exercised is called the strike price.
There are two kinds of options. American options, the most common kind, allow their holders to
exercise the option on any date up to and including a final date called the expiration date.
European option allow their holders to exercise the option only on the expiration date.
The price of an European Option derive basically from the difference between the reference price,
the strike price(K), and the value of the underlying asset (S) plus a premium based on the remaining
time until the expiration date of the option
C = max(S – K, 0)
P = max(K –S, 0)
As we can easily deduce from these equations the value of a call option can’t be negative because if
the value drops below zero the owner doesn’t exercise the option at the expiration date. It can be
usefull having a representation of how these options work. Here is the graph of a call option and a
put option with the same strike price:
	
  	
  
0.0 0.5 1.0 1.5 2.0
0.00.20.40.60.81.0
Payoff Functions
x
f(x)
Call
Put
Nowadays the value of an option is calculated relying on several mathematics models that help us
in predicting the value of an option changes related to changing in the underlying conditions.
In evaluating options analysts should keep clearly in their mind some main conditions: first of all
they surely have to consider the market price of the underlying security, the strike price of the
option and the relationship that stands between them because, as we know, the option price changes
a lot depending on whether the option is in the money or out of the money. Then they have to focus
on the cost of holding the underlying security, the expiration date and the expected volatility of the
underlying security’s price with respect to the life of the option.
2.2 The Black & Scholes Model
Most of the models that, today, are used by analysts all over the world have a common root: the
model devoloped by Fisher Black and Myron Scholes (1973) called the “Black&Scholes Model”
that allows, taking in consideration some assumptions, the pricing of European Call and Put option
using a simple formula.
The assumptions mentioned above are:	
  
1)	
  The	
  stock	
  pays	
  no	
  dividends	
  during	
  the	
  option's	
  life	
  
Most companies pay dividends to their share holders, so this might seem a serious limitation to the
model considering the observation that higher dividend yields elicit lower call premiums. A
common way of adjusting the model for this situation is to subtract the discounted value of a future
dividend from the stock price.
2)	
  European	
  exercise	
  terms	
  are	
  used	
  
European exercise terms dictate that the option can only be exercised on the expiration date.
American exercise term allow the option to be exercised at any time during the life of the option,
making american options more valuable due to their greater flexibility. This limitation is not a
major concern because very few calls are ever exercised before the last few days of their life. This
is true because when you exercise a call early, you forfeit the remaining time value on the call and
collect the intrinsic value. Towards the end of the life of a call, the remaining time value is very
small, but the intrinsic value is the same.
3)	
  Markets	
  are	
  efficient	
  
This assumption suggests that people cannot consistently predict the direction of the market or an
individual stock. The market operates continuously with share prices following a continuous Itô
process. To understand what a continuous Itô process is, you must first know that a Markov process
is "one where the observation in time period t depends only on the preceding observation." An Itô
process is simply a Markov process in continuous time. If you were to draw a continuous process
you would do so without picking the pen up from the piece of paper.
	
  
4)	
  No	
  commissions	
  are	
  charged	
  
Usually market participants do have to pay a commission to buy or sell options. Even floor traders
pay some kind of fee, but it is usually very small. The fees that Individual investor's pay is more
substantial and can often distort the output of the model.
5)	
  Interest	
  rates	
  remain	
  constant	
  and	
  known	
  
The Black and Scholes model uses the risk-free rate to represent this constant and known rate. In
reality there is no such thing as the risk-free rate, but the discount rate on U.S. Government
Treasury Bills with 30 days left until maturity is usually used to represent it. During periods of
rapidly changing interest rates, these 30 day rates are often subject to change, thereby violating one
of the assumptions of the model.
6)	
  Returns	
  are	
  lognormally	
  distributed	
  
This assumption suggests, returns on the underlying stock are normally distributed, which is
reasonable for most assets that offer options.
7) the stock price follows a geometric Brownian motion with constant drift and volatility.
This part will be widley covered in the “comments on B&S” paragraph
	
  
Suppose that all of the assumptions above are verified: in this case we can plainly calculate the
price of an option using the exact formula of B&S.
Here are the Call and the Put formula, respectively:
Call	
  price:	
  	
   	
  
Put	
  price:	
  	
  
	
  	
  	
   	
  	
  	
  	
  	
  	
  	
   	
  
Where:	
  
	
  
	
  
	
  
	
  
	
  
	
  
and:	
  
	
  
	
  
What we are going to do now is try to compute the price of a Call and a Put option on “Saras S.p.a”
using the Black&Scholes formula in order to verify if the results of the model are plausible with
respect to the real market value of the same options.
First of all we look on a financial web site (Yahoo.fianance, Google Finance etc.) focusing on data
we need to proceed in our calculations. We have to remember that the expiration date is going to be
expressed as effective trading days (252). Here are the parameters:
	
  
S0= 0.75 K=0.72 T=20/252 r=0.005 σ=?
	
  
At this point of our calculations, unfortunately, we have a missed value: the volatility.
This parameter is not directly observable on the market so we have to find it out by ourselves and
using R it is quite immediate to obtain that the historical volatility, in the period from 2011-01-05 to
2012-01-05, is equal to 0.54220241
.
SARAS S.p.a
MKT “C” B&S”C” MKT”P” B&S”P”
0.039 0.06149544 0.029 0.03120970
Looking at the results we can easily deduce that both Call and Put options prices of Saras S.p.a.
computed using Black&Scholes formula are higher than the market prices of the same options.
This could be caused by the value of the volatility used in the computation: in fact there is the
possibility that our value is higher than the one used by analysts in the market.
These results tell us that the future expected volatility is lower than the historical volatility of the
past year (the one we used) and this reveals that there is a lower chance for the option to be in the
money at the maturity date reflecting the possibility of lower oprion prices.
Now our goal is to understand if our expectations about the value of the historical volatility are
confirmed and see how much is the difference between the historical volatility and the implied one.
R software gives us a function that can do this for us and the results are:
	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  
1
I wanted to be sure about the historical volatility so I checked it out using the Call-Parity equation
and it has been confirmed.
Implied Volatility for a CALL Historical Volatility
σ = 0.2468025 σ = 0.5422024
Implied Volatility for a PUT
σ = 0.5443167
	
  
A we can see from these results the implied volatility used by analysts to price the options are
different from the historical in the CALL option case and this is reflected in the difference between
the market price and the one computed using the B&S formula.
On the other hand the implied volatility used in pricing the PUT is extremely similar to the
historical one; in fact the market price is 0.029 against the B&S one of 0.031.
	
  
	
  
	
  
Looking at this graph, that represents the trend of the Saras stock prices over the last year, we can
see some indicators tant could help us with the volatility pattern.
In order to do so some two technical tools have been included in this chart and they will surely help
our comprehension of what is the overall situation. These tools are: Bollinger Bands (BBands) and
the Average True Range(ATR).
These indicators, combined together, are very common in fincance in order to predict the inversion
of trend of a security (a stock in our case).
0.8
1.0
1.2
1.4
1.6
1.8
S [2011-05-05/2012-05-04]
Last 0.88
Bollinger Bands (20,2) [Upper/Lower]: 0.967/0.846
0.04
0.05
0.06
0.07
0.08
0.09
0.10
Mag 05
2011
Lug 01
2011
Set 01
2011
Nov 01
2011
Gen 02
2012
Mar 01
2012
Mag 02
2012
The Average True Range has the aim of calculate prices volatility as the breadth of their
fluctuations. This mechanism is based on the idea that, during a turnaround, the volatility assume
extreme values (high or low).
On the other hand we have the Bollinger Bands that are based on the volatility calculation as well
but there is a superior band and an inferior one so that we can have an idea of what the volatility
range is. From an operating point of view the Bands give us signals of buying or selling when these
conditions occure:
-­‐ when the price graph goes out of the upper band and it goes in again, this is a selling sign;
this could represent an increase in price followed by an adjustment.
-­‐ When the same thing happens with respect to the lower band, we have a buying sign; this
means that the price has gone down very quickly up to the point of turnaround.
Concerning Saras S.p.a. we can see some of these situations. For example we see that on August
2011 the graph crossed the lower band meaning a strong decrease in price in fact in this period we
registered the peak of the Euro debt crisis. We can see that even from the volatility graph that on
that date had a steep rise. The price went down up to the point of turnaround, crossing again the
lower band, giving a signal of strong buying of the stock.
Conversely, on December 2011, we can see the same process that led to a steep rise in the stock
price in according to the volatility graph that registered a strong increase. In this case the stock price
went up and, just after having crossed the upper line, went down meaning that we were in front of a
turnaround and people were selling their shares.
Looking at the very last part of the chart we can see that, in these days, the stock is having a
negative trend and this is the reason why the implied volatility of the put option is higher than the
call one telling us that the market expectations for the near future is still a down-trend.
2.2.1 – Comments
The Black&Scholes formula, the one we used before to price Saras S.p.a. options, doesn’t match
very well with the real world we live in.
That’s why, as we have seen in the previous paragrph, is based on some assumptions that are
impossible to notice in real circumstances: in particular what we are going to prove now is the
soundness of the assumption that says stock price follows a geometric Brownian motion with
constant drift and constant volatility. In order to understand better what we are going to talk about
let’s see, first, what a geormetric Brownian motion actually is.
One of the most important assumption of the B&S model is that stock prices follow a Normal
distributed process and this processi s known as the geometric Brownian motion. It defined as
follows:
dSt = µSt dt + σSt dWt
where:
σ is the volatility and it is assumed constant.
µ is the expected return.
Wt is Wiener Process that is the stochastic component of the process. In order to compute our
demonstration we are not going to use the returns of the stock. Instead, we are going to use the
logarithm of the returns (more reliable) given by the ratio: returns in time t over returns in time t-1.
Here is the expression:
log.returns = returnst / returnst-1
	
  
Gen 02
2007
Gen 02
2008
Gen 02
2009
Gen 04
2010
Gen 03
2011
Gen 02
2012
-0.100.000.100.20
log.returns
 
	
  
	
  
	
  
	
  
	
  
-0.10 -0.05 0.00 0.05 0.10 0.15 0.20
05101520
density.default(x = log.returns, na.rm = TRUE)
N = 1364 Bandwidth = 0.004264
Density
-3 -2 -1 0 1 2 3
-0.100.000.100.20
Normal Q-Q Plot
Theoretical Quantiles
SampleQuantiles
These graphs are a very clear demonstration of what we were looking for. The density function of
our company are not properly ditributed and we are going to comment some indicators that tell us
why: first of all, looking at the second graph(the density function), we can easily see that the tails of
the function are not linear adn well defined as farther we move from central values. This fact
underline the fact that the price does not follow a geometric Brownian motion so that the
assumption does not hold. Finally, the third picture, should give us the explanation of why our
prices does not follow a Normally distributed function. If we look at it we can see a straight line
indicating the path that our prices should follow to be Normally distributed and the path they
actually follow. Concerning central values they seem to be as the ideal funtion wants them to be but
if we move from central value we notice that they go astray.
2.2.2 – Volatility Smiles
As we wrote before, B&S formula assesses that prices follow a normally distributed trend with
constant volatiltiy. In this part we are to verify this last statement.
Here are some options (call and put) with the same expiration date:
	
  
Saras S.p.a.
Expiration June 15, 2012
CALL PUT
Mkt Strike(K) Mkt
0.0435 0.74 0.03
0.0415 0.76 0.037
0.032 0.78 0.047
0.0205 0.8 0.051
0.008 0.85 0.086
0.003 0.9
0.0005 0.95
0.0005 1
0.0005 1.05
For semplicity we show and comment only the graph related to Call Option.
 
	
  
As we can immediatly see the stock prices are not characterized by constant volatility and, instead,
it changes as the strike price changes.
The volatility seems to follow a particular path that, as analysts call it, can be similar to a “smile”.
This kind of graph is called “Volatility Smile”. The reason of this trend is related to the fact that
values are high when the option is deeply in, or conversely out, of the money and decrease when the
option is near to the “at the money point”.
	
  
2.3 – The Monte Carlo Method
As we have already mentioned, the Black&Scholes model is used all over the world for pricing
options but our paper shows that the assumptions it is based on are quite unrealistic and through our
calculations we disproved them.
Now we are going to challenge another method, the Monte Carlo Method, based on a simulation: a
random generation of thousand of combination of prices . The successive step is going to be the
calculation of the payoff of the option for each simulation; the discounted results will be the price of
the option we are looking for.
0.75 0.80 0.85 0.90 0.95 1.00 1.05
0.400.450.500.55
Volatility smile SARAS S.P.A
K
smile
!^
In this part of the paper we want to compare two different prices of the same option: we have
already calculated one of these two prices that is generated by B&S model. The second one is going
to be generated by the Monte Carlo method. Then we are going to discuss our results.
As Monte Carlo simulation is based on repeated price generation we are goint to take different
scenarios characterised by different number of simulation (1000, 10000, 100000, 1000000) and
what we expect is that the higher are repetitions the more precise would be our price according to
the market one.
SARAS S.p.a.
CALL M PUT
0.05976969 1000 0.03035832
0.06110445 10000 0.03120731
0.06154876 100000 0.0311551
0.06154721 1000000 0.03115751
As we expected the accuracy of prices with respect to ones calculated by B&S is as higher as the
number of the simulations increase.
This is the main characteristic of the Monte Carlo simulation based on the “large number law” and
it is plain also using a function of R software called “speed of convergence” that show us
graphically the characteristic we’ve mentioned above.
To simplify our calculations, our analysis on convergency is going to be done only on one option:
the characteristics of this option are S0=0.75 and the strike price(K)= 0.72. The interest rate is, as
for other operations, is the interest rate of deutsch bank.
 
Chapter 3 – Lévy Process
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a
stochastic process that starts at 0, admits càdlàag(continue à droite, limitée à gauche) modification
and has "stationary independent increments" — this phrase will be explained below. It is a
stochastic analog of independent and identically distributed random variables, and the most well
known examples are the Wiener process and the Poisson process.
It is defined as follows:
A stochastic process X = {Xt : t ≥ 0} is said to be a Lèvy process if,
Speed of Convergency - Saras S.p.a.
MC replications
MCprice
10 100 200 500 1000
1) X0 = 0 almost surely
2) Indipendent increments: For any 0 ≤ t1 < t2 < … < tn < ∞, Xt2 – Xt1, Xt3 – Xt2, … , Xtn – Xtn-1
are indipendent.
3) Stationary increments: for any s < t, Xt – Xs is equal in distribution to Xt-s
4) t -> Xt is almost surely right continuous with left limits.
In order to better understand what is wrong in the B&S Model we are going to discuss and represent
graphically the problem. First we have to say that distribustions are not normal:
	
  
	
  
	
  
Looking at the graph it is evident that the distribution of the logarithm of the returns is definitely
different from a normal distributed function (as we have already computed for the B&S Model).
Our goal is, now, find a new model which should not be based on geometric Brownian motion like
the B&S model. Well, a solution to our problem could be represented by Lévy proccesses.
Using R-software we are able to compute some Lévy processes and to plot them giving us a sketch
of what could be a solution to our problem:
-0.1 0.0 0.1 0.2
02468101214
density.default(x = Ret.Saras)
N = 253 Bandwidth = 0.008547
Density
 	
  
In the above we can see:
Picture a) the NORMAL parameter estimation.
MEAN:-0.003379408 SD:0.036455639
Picture b) the NORMAL INVERSE GAUSSIAN parameter estimation
ALPHA:28.375193345 BETA:1.564509140
DELTA:0.036703176MU:-0.005406176
Picture c) the HYPERBOLIC parameter estimation
ALPHA:43.912065188 BETA:1.505010969
DELTA:0.016400880MU:-0.005299175
Picture d) the GENERALIZED HYPERBOLIC parameter estimation
ALPHA:3.813755900 BETA:2.074031129
DELTA:0.057400423MU:-0.006110047
-0.10 -0.05 0.00 0.05 0.10
-2-1012
x
logf(x)
NORMAL: Parameter Estimation
-0.10 0.00 0.05 0.10 0.15 0.20
-4-202
x
logf(x)
NIG Parameter Estimation
-0.10 0.00 0.05 0.10 0.15 0.20
-6-4-202
logf(x)
HYP Parameter Estimation
-0.10 0.00 0.05 0.10 0.15 0.20
-0.50.51.5
logf(x)
GH Parameter Estimation
LAMBDA:-2.229897013
Once we have introduced what the Lévy process is we have to make another important assumption:
it says that Lévy markets, even if we condider simpliest one, are not complete.
Now we can proceed in pricing an option and, in this particular case, this operation can follow two
ways:
1) the Fast Fourier Transform
2) the Monte Carlo Approach
In order to simplify my processes we chose to proceed just for the first method giving a sketch of
theory for the second one.
3.1 – Fast Fourier Transform
A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier
transform (DFT) and its inverse. There are many distinct FFT algorithms involving a wide range of
mathematics, from simple complex-number arithmetic to group theory and number theory. A DFT
decomposes a sequence of values into components of different frequencies. This operation is useful
in many fields but computing it directly from the definition is often too slow to be practical. An
FFT is a way to compute the same result more quickly: computing a DFT of N points in the naive
way, using the definition, takes O(N2
) arithmetical operations, while an FFT can compute the same
result in only O(N log N) operations. For example, corcerning the gepmetric Brownian motion, the
characteristic function of Zt is: φ(u)= exp( iu(µ – 1/2 σ2
) - σ2
u2
/2) where φ is known, the proce of an
option can be approximated to this equation:
CT(k) ≈ e-α
k
/π ∑e-iv
j
k
ψT(vj)η, where vj = η(j-1) and k = logK.
The constant alpha is considered as the dampening factor and it is usally equal to one. This model
can we easily used on R and the results we obtained are:
B&S Price FFT Price ∆
0.06149541 - 0.05932576 = 0.00216965
3.2 – The Monte Carlo Approach
The first step we have to take is identifying the distribution of the returns. Then we just have to
simulate the patterns of the stochastic process and apply the payoff function to the final value.
ST = S0eZ
T
Chapter 4 – Greeks Analysis and Conclusions
4.1 – Greeks
In mathematical finance, 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. The name is used because the most
common of these parameters are often denoted by Greek letters. Collectively these have also been
called the risk sensitivities, risk measures or hedge parameters.
The Greeks are vital tools in risk management. Each Greek measures the sensitivity of the value of
a portfolio to a small change in a given underlying parameter, so that component risks may be
treated in isolation, and the portfolio rebalanced accordingly to achieve a desired exposure.
The Greeks in the Black–Scholes model are relatively easy to calculate, a desirable property of
financial models, and are very useful for derivatives traders, especially those who seek to hedge
their portfolios from adverse changes in market conditions. For this reason, those Greeks which are
particularly useful for hedging delta, theta, and vega are well-defined for measuring changes in
Price, Time and Volatility. Although rho is a primary input into the Black–Scholes model, the
overall impact on the value of an option corresponding to changes in the risk-free interest rate is
generally insignificant and therefore higher-order derivatives involving the risk-free interest rate are
not common.
The most common of the Greeks are the first order derivatives: Delta, Vega, Theta and Rho as
well as Gamma, a second-order derivative of the value function. The remaining sensitivities in this
list are common enough that they have common names, but this list is by no means exhaustive.
FIRST ORDER GREEKS
Delta
Delta, , measures the rate of change of option value with respect to changes in the underlying
asset's price. Delta is the first derivative of the value of the option with respect to the underlying
instrument's price .
For a vanilla option, delta will be a number between 0.0 and 1.0 for a long call (and/or short put)
and 0.0 and −1.0 for a long put (and/or short call) – depending on price, a call option behaves as if
one owns 1 share of the underlying stock (if deep in the money), or owns nothing (if far out of the
money), or something in between, and conversely for a put option.
Vega
Vega measures sensitivity to volatility. Vega is the derivative of the option value with respect to the
volatility of the underlying asset.
Vega can be an important Greek to monitor for an option trader, especially in volatile markets,
since the value of some option strategies can be particularly sensitive to changes in volatility. The
value of an option straddle, for example, is extremely dependent on changes to volatility.
Theta
Theta, θ, measures the sensitivity of the value of the derivative to the passage of time: the "time
decay.”
The mathematical result of the formula for theta is expressed in value per year. By convention, it is
usual to divide the result by the number of days in a year, to arrive at the amount of money per
share of the underlying that the option loses in one day. Theta is almost always negative for long
calls and puts and positive for short calls and puts. An exception is a deep in-the-money European
put. The total theta for a portfolio of options can be determined by summing the thetas for each
individual position.
The value of an option can be analysed into two parts: the intrinsic value and the time value. The
intrinsic value is the amount of money you would gain if you exercised the option immediately,
while the time value is the value of having the option of waiting longer before deciding to exercise.
Rho
Rho, , measures sensitivity to the interest rate: it is the derivative of the option value with respect
to the risk free interest rate (for the relevant outstanding term).
Except under extreme circumstances, the value of an option is less sensitive to changes in the risk
free interest rate than to changes in other parameters. For this reason, rho is the least used of the
first-order Greeks.
Rho is typically expressed as the amount of money, per share of the underlying, that the value of the
option will gain or lose as the risk free interest rate rises or falls by 1.0% per annum (100 basis
points).
Lambda
Lambda, , omega, , or elasticity is the percentage change in option value per percentage
change in the underlying price, a measure of leverage, sometimes called gearing.
SECOND ORDER GREEKS
Gamma
Gamma, , measures the rate of change in the delta with respect to changes in the underlying price.
Gamma is the second derivative of the value function with respect to the underlying price. All long
options have positive gamma and all short options have negative gamma. Gamma is greatest
approximately at-the-money (ATM) and diminishes the further out you go either in-the-money
(ITM) or out-of-the-money (OTM).
When a trader seeks to establish an effective delta-hedge for a portfolio, the trader may also seek to
neutralize the portfolio's gamma, as this will ensure that the hedge will be effective over a wider
range of underlying price movements. However, in neutralizing the gamma of a portfolio, alpha (the
return in excess of the risk-free rate) is reduced.
Thanks to R, calculating the coefficients of Greeks on Saras Call options is quite easy and using a
simple function here are the results we obtained:
SARAS S.p.a
As the Underlying Stock Price Changes - Delta and Gamma
Delta measures the sensitivity of an option's theoretical value to a change in the price of the
underlying asset. It is normally represented as a number between -1 and 1, and it indicates how
much the value of an option should change when the price of the underlying stock rises by one euro.
As an alternative convention, the delta can also be shown as a value between -100 and +100 to
show the total euro sensitivity on the value 1 option, which comprises of 100 shares of the
underlying.
Call options have positive deltas and put options have negative deltas. At-the-money options
generally have deltas around 50. Deep-in-the-money options might have a delta of 80 or higher,
while out-of-the-money options have deltas as small as 20 or less.
In our case if we multiply the value of Delta we obtain: 0.6354121*100=63.54121.
Delta 0.6354121
Gamma 3.279769
Theta 0.06149541
Vega 0.07938833
Lambda 7.749506
Rho 0.03294156
The value can be considered standing around 50 so we are describing an option that stands just
beyond the “at the money” zone but, at the same time, it is not in “in the money” zone.
Furthermore, if the underlying price changes by 1 euro the related variation in the option proce is
going to be almost 0.64.
Another thing we are interested in is how delta may change as the stock proce moves: Gamma
measures the rate of change in the delta for each one-point increase in the underlying asset. It is a
valuable tool in helping you forecast changes in the delta of an option or an overall position.
Gamma will be larger for the at-the-money options, and gets progressively lower for both the in-
and out-of-the-money options. Unlike delta, gamma is always positive for both calls and puts.
In our case this indicator helps us to better understand what kind of option we are dealing with in
fact as we said above lower values of gamma indicates a “in the money” option (almost 3.28).
Besides, it tells us that for one point our delta is going to change by 3.28.
Changes in Volatility and the Passage of Time - Theta and Vega
Theta is a measure of the time decay of an option, the euro amount that an option will lose each day
due to the passage of time. For at-the-money options, theta increases as an option approaches the
expiration date. For in- and out-of-the-money options, theta decreases as an option approaches
expiration.
Theta is one of the most important concepts for a beginning option trader to understand, because it
explains the effect of time on the premium of the options that have been purchased or sold. The
further out in time you go, the smaller the time decay will be for an option. If you want to own an
option, it is advantageous to purchase longer-term contracts. If you want a strategy that profits from
time decay, then you will want to short the shorter-term options, so that the loss in value due to time
happens quickly.
In our case theta is low (almost 0.06) indicating that our option approaches the expiration and it
actually does because the expiration of the option we considered in the calculation expires within a
month(June 15, 2012). Since the expiration date is not far this indicator tells us that the amount
money we are going to loose up to the expiration is small.
Vega measures the sensitivity of the price of an option to changes in volatility. A change in
volatility will affect both calls and puts the same way. An increase in volatility will increase the
prices of all the options on an asset, and a decrease in volatility causes all the options to decrease in
value.
However, each individual option has its own vega and will react to volatility changes a bit
differently. The impact of volatility changes is greater for at-the-money options than it is for the in-
or out-of-the-money options. While vega affects calls and puts similarly, it does seem to affect calls
more than puts. Perhaps because of the anticipation of market growth over time.
Since we are analyzing an “in the money” option even vega has a low value indicating a small
impact of volatility.
Changes with respect to interest rate - Rho
Since it measures the sensitivity with respect to the interest rate we are going to multiply by 100 the
value we obatained on R: 0.03294156*100=3,294156. This value in the gain of our option related to
a variation of 1.0% of the interest rate.
Elasticity - Lambda
Finally we are going to discuss the only second order greek that gives us a measure of leverage. In
our case it is 7.749506 and this is the percentage variation (almost 7.75%) in our option per
percentage change in the underlying asset value.
4.2 - Conclusions
To be honest working on this paper actually helped me to better understand the reality of the
financial market but, first of all, the reality of an incredibly important company as Saras within the
financial market. My technical skills were low at the beginning of the process but this lack helped
me consolidating my theory and my practical skills. Concerning the results obtained I have to say
that I am satisfied. According to these results the company is facing a tough crisis period and the
stock prices reflect it. If we look at the options value, the price of a put option still worths more than
a call indicating the trend is not going to have turnaround. The differences in prices is not very
consistent due to the different methods we have implented: for example Lévy process should have
given a price improving the Black & Scholes formula based on the geometric Brownian motion but
the difference between the two prices is very low. Afterall, going through my analysis I found one
of the most powerfull company in the field of refining facing the difficulties in a very bad way.
Besides, the problems are not coming only from financial markets in facts the company is working
to fix the problem of its employers’ death while on working . The overall situation is very clear and
it tells that this company is going down, the stock price registered a decreasing trend and I don’t
think is going to stop.
SCRIPT
## Chapter 1 ##
library(quantmod)
get.symbols(“SRS.MI”, from=”2007-05-20”, to=”2012-05-20”)
chartSeries(“SRS.MI”,
theme="white", TA=c(addVo(), addBBands()))
##Change Point##
library (tseries)
S <- get.hist.quote("SRS.MI", start = "2011-05-01", end="2012-04-30")
chartSeries(S, TA = c(addVo(), addBBands()), theme="white")
S <- S$Close
require(sde)
cpoint(S)
addVLine = function(dtlist) plot(addTA(xts(rep(TRUE, + + NROW (dtlist)),
dtlist), on = 1, col = "red"))
addVLine(cpoint(S)$tau0)
##Chapter 2##
##Option Valuation##
f.call <- function(X) sapply(X, function(X) max(X-K,0)))
f.put <- function(X) saplly(X, function(X) max(c(K-X,0)))
K<-1
curve(f.call, 0,2,main=”Payoff Functions”,col=”Blue”, Ity=4,
lwd=4,ylab=expression(f(x)))
curve(f.put,0,2,col=”red”,add=TRUE,Ity=4,lwd=4)
temp<-legend(“top”,legend=c(“Call”,”Put”),Ity=1,
lwd=5,col=c(“blue”,”red”)
##Black&Scholes Model##
require(fImport)
S<-yahooSeries(“SRS.MI”, from=”2011-05-20”, to=”2012-05-12”)
Head(S)
Close<-S[, “SRS.MI.Close”]
Require(quantmod)
chartSeries(Close, theme=”white”)
X<-returns(Close)
Delta<-1/252
sigma.hat<-sqrt(var(X)/Delta)[1,1]
sigma.hat
##Call Option (Saras)##
call.price<-function(x=1, t=0, T=1, r=1, sigma=1,K=1){
d2<-(log(X/K) + (r-0.5*sigma^2)*(T-t)/sigma*sqrt(T-t))
d1<-d2 + sigma*sqrt(T-t)
x*pnorm(d1)-K*exp(-r*(T-t))*pnorm(d2)}
S0<-0.75
K<-0.72
T<-20/252
r<-0.005
sigma<-0.5422024
C<-call.price(x=S0, t=0, T=T, r=r, K=K, sigma=sigma)
C
## Put Option (Saras)##
put.price<-function(x=1, t=0, T=1, r=1, sigma=1,K=1){
d2<-(log(X/K) + (r-0.5*sigma^2)*(T-t))/(sigma*sqrt(T-t))
d1<-d2 + sigma*sqrt(T-t)
K*exp(-r*(T-t))*pnorm(-d2) –x*pnorm(-d1)}
S0<-0.75
K<-0.72
T<-20/252
r<-0.005
sigma<-0.5422024
p<-put.price(x=S0, t=0, T=T, r=r, K=K, sigma=sigma)
p
##Put-Call Parity##
C – S0 + K * exp(-r * T)
##Implied Volatility##
#call
p<-0.039
Delta<-1/252
T<-20*Delta
S0<-0.75
K<-0.72
r<-0.005
sigma.imp<-GBSVolatility(p,”c”,S=S0, X=K, Time=T, r=r, b=r)
sigma.imp
#put
p<-0.029
sigma.imp<-GBSVolatility(p, “c”, S=S0, X=K, Time=T, r = r, b = r)
sigma.imp
##Plot B&S##
library(tseries)
S <- get.hist.quote(“SRS.MI”, start=”2011-05-20”, end=”2012-05-20”)
chartSeries(S,TA=c(addVo(),addBBands(), addATR()), theme=”white”)
##geometric Brwnian motion##
getSymbols(“SRS.MI”)
plot(SRS.MI)
str(SRS.MI)
S<-SRS.MI[, “SRS.MI.Close”]
X<-diff(log(S))
plot(X)
plot(density(X, na.rm=TRUE))
qqnorm(na.omit(X))
qqline(na.omit(X))
##Log-Returns##
getSymbols(“SRS.MI”)
plot(SRS.MI)
str(SRS.MI)
##Volatility Smile##
S<- (SRS.MI[, “SRS.MI.Close”]
X<-returns(Close)
Delta<-1/252
sigma.hat<-sqrt(var(X)/Delta)
sigma.hat
Pt<-c(0.0435, 0.0415, 0.032, 0.0205, 0.008, 0.003, 0.0005, 0.0005)
K<-c(0.74, 0.76, 0.78, 0.8, 0.85, 0.9, 0.95, 1, 1.05)
S0<-0.75
nP<-length(Pt)
T<-20*Delta
R<-0.005
smile<- sapply(1:nP, function(i) GBSVolatiltiy(Pt[i], “c”, S=S0, X=K[i],
Time=T, r=r, b=r))
vals<-c(smile, sigma.hat)
plot(K, smile, type =”l”, lty=3, col=”orange”)
axis(2, sigma.hat, expression(hat(sigma)), col=”blue”)
	
  
##MONTE CARLO Method##
S0<-0.75
K<-0.72
r<-0.005
Delta<-1/252
T<-20*Delta
sigma<-0.5422024
MCPrice<-function(x=1, t=0, T=1, r=1, sigma=1, M=1000, f){
h<-function(m){
u<- rnorm(m/2)
tmp<-c(x*exp((r-0.5*sigma^2)*(T-t)+sigma*sqrt(T-t)*u), x*exp((r-
0.5*sigma^2)*(T-t)+sigma*sqrt(T-t)*(-u)))
mean(sapply(tmp, function(xx) f(xx)))
}
p<-h(M)
p*exp(-r*(T-t))
}
##CALL##
f<-function(x) max(0, x-K)
set.seed(123)
M<-1000
MCPrice(x=S0, t=0, T=T, r=r, sigma=sigma.hat, M=M, f=f)
M<-10000
MCPrice(x=S0, t=0, T=T, r=r, sigma=sigma.hat, M=M, f=f)
M<-100000
MCPrice(x=S0, t=0, T=T, r=r, sigma=sigma.hat, M=M, f=f)
M<-1000000
MCPrice(x=S0, t=0, T=T, r=r, sigma=sigma.hat, M=M, f=f)
	
  
##PUT##
f<- function(x) max(0,K-x)
M<-1000
MCPrice(x=S0, t=0, T=T, r=r, sigma=sigma.hat, M=M, f=f)
M<-10000
MCPrice(x=S0, t=0, T=T, r=r, sigma=sigma.hat, M=M, f=f)
M<-100000
MCPrice(x=S0, t=0, T=T, r=r, sigma=sigma.hat, M=M, f=f)
M<- 1000000
MCPrice(x=S0, t=0, T=T, r=r, sigma=sigma.hat, M=M, f=f)
	
  
##Speed of Convergency##
S0<-0.75
K<- 0.72
r<-0.005
Delta<- 1/252
sigma.hat<-0.5422024
f<- function(x) max(0,x-K)
set.seed(123)
m<-c(10, 50, 100, 150, 200, 250,500, 1000)
p1<-NULL
err<-NULL
nM<-length(m)
repl<-100
mat<-matrix(, repl, nM)
for(k in 1:nM){
tmp<-numeric(repl)
fot(i in 1:repl) tmp[i] <- MCPrice(x=S0, t=0, T=T, r=r, sigma, M=m[k],
f=f)
mat[, k] <- tmp
p1<-c(p1, mean(tmp))
err<-c(err, sd(tmp))
}
colnames(mat)<-m
p0<-GBSOption(TypeFlag=”c”, S=S0, X=K, Time=T, r=r, b=r, sigma=sigma)
minP<-min(p1-err)
maxP<-max(p1 + err)
plot(m, p1, type=”n”, ylim=c(minP, maxP), axes=F, ylab=”MC price”, xlab0
“MC replications”, main= “Speed of Convergency – Saras S.p.a.”)
lines(m, p1 + err, col= “blue”)
lines(m, p1-err, col= “blue”)
axis(2, p0, “B&S price”)
axis(1,m)
boxplot(mat, add=TRUE, at=m, boxwex=15, col= “orange”, axes=F)
points(m, p1, col= “blue”, lwd=3, lty=3)
	
  
##Greeks	
  Analysis##	
  
require(fOptions)
r<- 0.005
T<-20/252
K<-0.72
sigma<- 0.5422024
S0<- 0.75
GBSCharacteristics(TypeFlag= “c”, S=S0, X=K, Time=T, r=r, b=r,
sigma=sigma)
	
  
	
  
##Lévy process##
require(fImport)
data <- yahooSeries("ENI.MI", from="2009-01-01", to="2009-12-31")
S <- data[, "ENI.MI.Close"] > X <- returns(S) > require(quantmod) >
lineChart(S,layout=NULL,theme="white")
lineChart(X,layout=NULL,theme="white")
sigma.hat <- sqrt( var(X)/deltat(S) )
alpha.hat <- mean(X)/deltat(S)
mu.hat <- alpha.hat + 0.5 * sigma.hat^2
plot(density(X))
d <- function(x) dnorm(x, mean=mu.hat, sd=sigma.hat)
curve(d, -.2, .2, col="red",add=TRUE, n=500)
par(mfrow=c(2,2))
par(mar=c(3,3,3,1))
grid <- NULL
library(fBasics)
nFit(X)
nigFit(X,trace=FALSE)
hypFit(X,trace=FALSE)
ghFit(X,trace=FALSE)
##FFT##
FFTcall.price <- function(phi, S0, K, r, T, alpha = 1, N = 2^12, eta =
0.25) {
m <- r - log(phi(-(0+1i)))
phi.tilde <- function(u) (phi(u) * exp((0+1i) * u * m))^T
psi <- function(v) exp(-r * T) * phi.tilde((v - (alpha +
1) * (0+1i)))/(alpha^2 + alpha - v^2 + (0+1i) * (2 *
alpha + 1) * v)
lambda <- (2 * pi)/(N * eta)
b <- 1/2 * N * lambda
ku <- -b + lambda * (0:(N - 1))
v <- eta * (0:(N - 1))
tmp <- exp((0+1i) * b * v) * psi(v) * eta * (3 + (-1)^(1:N) –
((1:N) - 1 == 0))/3
ft <- fft(tmp)
res <- exp(-alpha * ku) * ft/pi
inter <- spline(ku, Re(res), xout = log(K/S0))
return(inter$y * S0)
}
phiBS <- function(u) exp((0+1i) * u * (mu - 0.5 * sigma^2) - 0.5 *
sigma^2 * u^2)
S0<- 0.75
K<-0.72
R<-0.005
T<-20/252
Sigma<-0.5422024
mu <- 1
require(fOptions)
GBSOption(TypeFlag = "c", S = S0, X = K, Time = T, r = r, b = r, sigma =
sigma)
FFTcall.price(phiBS, S0 = S0, K = K, r = r, T = T)
}

Contenu connexe

Similaire à Benedetti Kevin George 4008804

Statistical Arbitrage Pairs Trading, Long-Short Strategy
Statistical Arbitrage Pairs Trading, Long-Short StrategyStatistical Arbitrage Pairs Trading, Long-Short Strategy
Statistical Arbitrage Pairs Trading, Long-Short Strategy
z-score
 
Option Pricing Models Lecture NotesThis week’s assignment is .docx
Option Pricing Models Lecture NotesThis week’s assignment is .docxOption Pricing Models Lecture NotesThis week’s assignment is .docx
Option Pricing Models Lecture NotesThis week’s assignment is .docx
hopeaustin33688
 
Ahmed_Exp22_PPT_Ch03_CumulativeAssessment_IT Careers.pptxIT .docx
Ahmed_Exp22_PPT_Ch03_CumulativeAssessment_IT Careers.pptxIT .docxAhmed_Exp22_PPT_Ch03_CumulativeAssessment_IT Careers.pptxIT .docx
Ahmed_Exp22_PPT_Ch03_CumulativeAssessment_IT Careers.pptxIT .docx
robert345678
 
Investigation of Frequent Batch Auctions using Agent Based Model
Investigation of Frequent Batch Auctions using Agent Based ModelInvestigation of Frequent Batch Auctions using Agent Based Model
Investigation of Frequent Batch Auctions using Agent Based Model
Takanobu Mizuta
 
The Validity of Company Valuation Using Dis.docx
The Validity of Company Valuation  Using Dis.docxThe Validity of Company Valuation  Using Dis.docx
The Validity of Company Valuation Using Dis.docx
christalgrieg
 

Similaire à Benedetti Kevin George 4008804 (20)

Solution to Black-Scholes P.D.E. via Finite Difference Methods (MatLab)
Solution to Black-Scholes P.D.E. via Finite Difference Methods (MatLab)Solution to Black-Scholes P.D.E. via Finite Difference Methods (MatLab)
Solution to Black-Scholes P.D.E. via Finite Difference Methods (MatLab)
 
Statistical Arbitrage Pairs Trading, Long-Short Strategy
Statistical Arbitrage Pairs Trading, Long-Short StrategyStatistical Arbitrage Pairs Trading, Long-Short Strategy
Statistical Arbitrage Pairs Trading, Long-Short Strategy
 
White paper risk management in exotic derivatives trading - ch cie gra
White paper   risk management in exotic derivatives trading - ch cie graWhite paper   risk management in exotic derivatives trading - ch cie gra
White paper risk management in exotic derivatives trading - ch cie gra
 
Risk management in exotic derivatives trading
Risk management in exotic derivatives tradingRisk management in exotic derivatives trading
Risk management in exotic derivatives trading
 
Damien lamberton, bernard lapeyre, nicolas rabeau
Damien lamberton, bernard lapeyre, nicolas rabeauDamien lamberton, bernard lapeyre, nicolas rabeau
Damien lamberton, bernard lapeyre, nicolas rabeau
 
Option Pricing Models Lecture NotesThis week’s assignment is .docx
Option Pricing Models Lecture NotesThis week’s assignment is .docxOption Pricing Models Lecture NotesThis week’s assignment is .docx
Option Pricing Models Lecture NotesThis week’s assignment is .docx
 
Ahmed_Exp22_PPT_Ch03_CumulativeAssessment_IT Careers.pptxIT .docx
Ahmed_Exp22_PPT_Ch03_CumulativeAssessment_IT Careers.pptxIT .docxAhmed_Exp22_PPT_Ch03_CumulativeAssessment_IT Careers.pptxIT .docx
Ahmed_Exp22_PPT_Ch03_CumulativeAssessment_IT Careers.pptxIT .docx
 
The Analysis of the Impact of Capital Mobility on Bubbly Episodes Creation in...
The Analysis of the Impact of Capital Mobility on Bubbly Episodes Creation in...The Analysis of the Impact of Capital Mobility on Bubbly Episodes Creation in...
The Analysis of the Impact of Capital Mobility on Bubbly Episodes Creation in...
 
Sovereign credit risk, liquidity, and the ecb intervention: deus ex machina? ...
Sovereign credit risk, liquidity, and the ecb intervention: deus ex machina? ...Sovereign credit risk, liquidity, and the ecb intervention: deus ex machina? ...
Sovereign credit risk, liquidity, and the ecb intervention: deus ex machina? ...
 
Fair valuation of participating life insurance contracts with jump risk
Fair valuation of participating life insurance contracts with jump riskFair valuation of participating life insurance contracts with jump risk
Fair valuation of participating life insurance contracts with jump risk
 
Evaluation of options portfolios for exchange rate hedges
Evaluation of options portfolios for exchange rate hedgesEvaluation of options portfolios for exchange rate hedges
Evaluation of options portfolios for exchange rate hedges
 
Statistics&Business Project
Statistics&Business ProjectStatistics&Business Project
Statistics&Business Project
 
Pres fibe2015-pbs-org
Pres fibe2015-pbs-orgPres fibe2015-pbs-org
Pres fibe2015-pbs-org
 
Pres-Fibe2015-pbs-Org
Pres-Fibe2015-pbs-OrgPres-Fibe2015-pbs-Org
Pres-Fibe2015-pbs-Org
 
Business
BusinessBusiness
Business
 
Investigation of Frequent Batch Auctions using Agent Based Model
Investigation of Frequent Batch Auctions using Agent Based ModelInvestigation of Frequent Batch Auctions using Agent Based Model
Investigation of Frequent Batch Auctions using Agent Based Model
 
2008_BNP_Deriatives 101.pdf
2008_BNP_Deriatives 101.pdf2008_BNP_Deriatives 101.pdf
2008_BNP_Deriatives 101.pdf
 
thesis
thesisthesis
thesis
 
Science in the City 2022 Prof J Jamesv[2022]
Science in the City 2022 Prof J Jamesv[2022]Science in the City 2022 Prof J Jamesv[2022]
Science in the City 2022 Prof J Jamesv[2022]
 
The Validity of Company Valuation Using Dis.docx
The Validity of Company Valuation  Using Dis.docxThe Validity of Company Valuation  Using Dis.docx
The Validity of Company Valuation Using Dis.docx
 

Benedetti Kevin George 4008804

  • 1. University  Cattolica  del  Sacro  Cuore  di  Milano       Faculty  of  Banking,  Financial  and  Insurance  Sciences     The  Decline  of  Saras  S.p.a.                                                                                                                 Benedetti  Kevin                                                                                                                                                Matr.  4008804     Course:  Applied  Statistics  for  Finance   Prof.  Iacus   Prof.  Zappa     Accademic  year  2011-­‐2012      
  • 2. INDEX Chapter 1 – Preliminary Stock Analysis 1.1 Company Characteristics 1.2 Change Point Analysis Chapter 2 – Option Valuation 2.1 Valuation of Financial Options: introduction 2.2 The Black & Scholes Model 2.2.1 Comments on B&S Model 2.3 The Monte Carlo Method Chapter 3 – Lévy Process 3.1 Fast Fourier Transform 3.2 Monte Carlo Approach Chapter 4 – Greeks Analysis 4.1 Greeks 4.2 Conclusions
  • 3. Chapter 1 - Preliminary Stock Analysis – Saras S.p.a. 1.1 – Abstract 1.2 – Company Characteristics The Saras Group, whose operations were started by Angelo Moratti in 1962, has approximately 2,200 employees and total revenues of about 11.0 billion Euros as of 31st December 2011. The Group is active in the energy sector, and is a leading Italian and European crude oil refiner. It sells and distributes petroleum products in the domestic and international markets, directly and through the subsidiaries Saras Energia S.A. (in Spain) and Arcola Petrolifera S.p.A. (in Italy). The Group also operates in the electric power production and sale, through the subsidiaries Sarlux S.r.l. and Sardeolica S.r.l.. In addition, the Group provides industrial engineering and scientific research services to the oil, energy and environment sectors through the subsidiary Sartec S.p.A.. Finally, in July 2011, the Group created a new subsidiary called Sargas S.r.l., which operates in the fields of exploration and development, as well as transport, storage, purchase and sale of gaseous hydrocarbons. Here are the market performance of the stock integrated with an important indicator that are the volumes: As we can notice from this picture the stock registered a small decrease in the stock price in the very first part of the graph but it went up again jast before july 2008. From this point the stock went down rapidly and it never stopped. Even today the trand of the stock in quite negative. We have to 1 2 3 4 5 SRS.MI [2007-05-21/2012-05-18] Last 0.74 Volume (millions): 2,239,500 0 10 20 30 40 50 Mag 21 2007 Lug 01 2008 Lug 01 2009 Lug 01 2010 Lug 01 2011 Mag 18 2012
  • 4. say that in the period considered for the analysis the whole world had to face the financial crisis started in 2008 and the bad situation of this company doesn’t surprise. Deslpite the negative trend on January 2012 we can observe a n incredibly high value of volumes: during this period, in fact, there was the probability for the company to be delisted. 1.3 – Change Point Analysis Given the decline of the stock and the the pattern of the prices during this period of crisis we observe a “roller coaster” graph. Now we are going through an analysis that could help us to explain this performance in order to catch the points where a turnaround has been registered. In this paragraph I want to use an important tool in volatility analysis: the Change Point Analysis. Considering a process: X = {Xt, 0 ≤ t ≤ T} à dXt = b(Xt)dt + √θσ(Xt)dBt and X0 = x0, 0<θ1, θ2<∞{Bt, t ≥ 0} à Bt is a Brownian motion and the coefficients are defined and known. The aim of this analysis is to find a point called τ0(tau0) associated to a parameter called θ(theta). R-software compute for us this kind of operation and it gives us two values of theta: θ1, the volatility just before the change point, and θ2, the volatility immediately after the change point. In our case we are going to analyze a one-year period – from May 20, 2011 to May 20, 2012 – in order to observe the reaction of the market in a tough span of time: in fact from May 20, 2011 there were rumors of a probability of delisting, denied on December 1st, 2011.This announcement could have brought some “good news” fo investors in a definitely not easy period. Therefore we are going to expect a change point on around this date: from high range of volatility to a more attenuate one.
  • 5.   The picture above confirms our expectations: the value τ0 – the change point – is set on January 10, 2012. In fact after the announcements, where we can observe a steep rise in stock price, we find few days more of high volatility and then, after January 10 a reduction in volatility. Analitically speaking here are the R results: τ0 = 2012-01-10 θ1 = 0.0444403 θ2 = 0.02976848 ∆2-1=0.02976848 – 0.0444403 = -0.01467182 0.8 1.0 1.2 1.4 1.6 1.8 S [2011-05-20/2012-05-18] 0.8 1.0 1.2 1.4 1.6 1.8 Last 0.74 Bollinger Bands (20,2) [Upper/Lower]: 0.991/0.730 Mag 20 2011 Ago 01 2011 Ott 03 2011 Dic 01 2011 Feb 01 2012 Apr 02 2012
  • 6. Chapter 2 – Option Valuation 2.1 – Valuation of Financial Options Introduction A financial option contract gives its owner thr right (but not the obligation) to purchase or sell an asset at fixed price at some future date. Two distinct kinds of option contracts exist: call options and put options. A call option gives the owner the right to buy the asset; a put option gives the owner the right to sell the asset. The most commonly encountered option contracts are options on shares of stock: a stock option gives the holder the option to buy or sell a share of stock on or before a given date for a given price. When a holder of an option enforces the agreement and buys or sells a share of stock at the agreed- upon price, he is exercising the option. The price at which the holder buys or sells the share of stock when the option is exercised is called the strike price. There are two kinds of options. American options, the most common kind, allow their holders to exercise the option on any date up to and including a final date called the expiration date. European option allow their holders to exercise the option only on the expiration date. The price of an European Option derive basically from the difference between the reference price, the strike price(K), and the value of the underlying asset (S) plus a premium based on the remaining time until the expiration date of the option C = max(S – K, 0) P = max(K –S, 0) As we can easily deduce from these equations the value of a call option can’t be negative because if the value drops below zero the owner doesn’t exercise the option at the expiration date. It can be usefull having a representation of how these options work. Here is the graph of a call option and a put option with the same strike price:     0.0 0.5 1.0 1.5 2.0 0.00.20.40.60.81.0 Payoff Functions x f(x) Call Put
  • 7. Nowadays the value of an option is calculated relying on several mathematics models that help us in predicting the value of an option changes related to changing in the underlying conditions. In evaluating options analysts should keep clearly in their mind some main conditions: first of all they surely have to consider the market price of the underlying security, the strike price of the option and the relationship that stands between them because, as we know, the option price changes a lot depending on whether the option is in the money or out of the money. Then they have to focus on the cost of holding the underlying security, the expiration date and the expected volatility of the underlying security’s price with respect to the life of the option. 2.2 The Black & Scholes Model Most of the models that, today, are used by analysts all over the world have a common root: the model devoloped by Fisher Black and Myron Scholes (1973) called the “Black&Scholes Model” that allows, taking in consideration some assumptions, the pricing of European Call and Put option using a simple formula. The assumptions mentioned above are:   1)  The  stock  pays  no  dividends  during  the  option's  life   Most companies pay dividends to their share holders, so this might seem a serious limitation to the model considering the observation that higher dividend yields elicit lower call premiums. A common way of adjusting the model for this situation is to subtract the discounted value of a future dividend from the stock price. 2)  European  exercise  terms  are  used   European exercise terms dictate that the option can only be exercised on the expiration date. American exercise term allow the option to be exercised at any time during the life of the option, making american options more valuable due to their greater flexibility. This limitation is not a major concern because very few calls are ever exercised before the last few days of their life. This is true because when you exercise a call early, you forfeit the remaining time value on the call and collect the intrinsic value. Towards the end of the life of a call, the remaining time value is very small, but the intrinsic value is the same. 3)  Markets  are  efficient   This assumption suggests that people cannot consistently predict the direction of the market or an individual stock. The market operates continuously with share prices following a continuous Itô process. To understand what a continuous Itô process is, you must first know that a Markov process is "one where the observation in time period t depends only on the preceding observation." An Itô process is simply a Markov process in continuous time. If you were to draw a continuous process you would do so without picking the pen up from the piece of paper.  
  • 8. 4)  No  commissions  are  charged   Usually market participants do have to pay a commission to buy or sell options. Even floor traders pay some kind of fee, but it is usually very small. The fees that Individual investor's pay is more substantial and can often distort the output of the model. 5)  Interest  rates  remain  constant  and  known   The Black and Scholes model uses the risk-free rate to represent this constant and known rate. In reality there is no such thing as the risk-free rate, but the discount rate on U.S. Government Treasury Bills with 30 days left until maturity is usually used to represent it. During periods of rapidly changing interest rates, these 30 day rates are often subject to change, thereby violating one of the assumptions of the model. 6)  Returns  are  lognormally  distributed   This assumption suggests, returns on the underlying stock are normally distributed, which is reasonable for most assets that offer options. 7) the stock price follows a geometric Brownian motion with constant drift and volatility. This part will be widley covered in the “comments on B&S” paragraph   Suppose that all of the assumptions above are verified: in this case we can plainly calculate the price of an option using the exact formula of B&S. Here are the Call and the Put formula, respectively: Call  price:       Put  price:                           Where:              
  • 9. and:       What we are going to do now is try to compute the price of a Call and a Put option on “Saras S.p.a” using the Black&Scholes formula in order to verify if the results of the model are plausible with respect to the real market value of the same options. First of all we look on a financial web site (Yahoo.fianance, Google Finance etc.) focusing on data we need to proceed in our calculations. We have to remember that the expiration date is going to be expressed as effective trading days (252). Here are the parameters:   S0= 0.75 K=0.72 T=20/252 r=0.005 σ=?   At this point of our calculations, unfortunately, we have a missed value: the volatility. This parameter is not directly observable on the market so we have to find it out by ourselves and using R it is quite immediate to obtain that the historical volatility, in the period from 2011-01-05 to 2012-01-05, is equal to 0.54220241 . SARAS S.p.a MKT “C” B&S”C” MKT”P” B&S”P” 0.039 0.06149544 0.029 0.03120970 Looking at the results we can easily deduce that both Call and Put options prices of Saras S.p.a. computed using Black&Scholes formula are higher than the market prices of the same options. This could be caused by the value of the volatility used in the computation: in fact there is the possibility that our value is higher than the one used by analysts in the market. These results tell us that the future expected volatility is lower than the historical volatility of the past year (the one we used) and this reveals that there is a lower chance for the option to be in the money at the maturity date reflecting the possibility of lower oprion prices. Now our goal is to understand if our expectations about the value of the historical volatility are confirmed and see how much is the difference between the historical volatility and the implied one. R software gives us a function that can do this for us and the results are:                                                                                                                           1 I wanted to be sure about the historical volatility so I checked it out using the Call-Parity equation and it has been confirmed.
  • 10. Implied Volatility for a CALL Historical Volatility σ = 0.2468025 σ = 0.5422024 Implied Volatility for a PUT σ = 0.5443167   A we can see from these results the implied volatility used by analysts to price the options are different from the historical in the CALL option case and this is reflected in the difference between the market price and the one computed using the B&S formula. On the other hand the implied volatility used in pricing the PUT is extremely similar to the historical one; in fact the market price is 0.029 against the B&S one of 0.031.       Looking at this graph, that represents the trend of the Saras stock prices over the last year, we can see some indicators tant could help us with the volatility pattern. In order to do so some two technical tools have been included in this chart and they will surely help our comprehension of what is the overall situation. These tools are: Bollinger Bands (BBands) and the Average True Range(ATR). These indicators, combined together, are very common in fincance in order to predict the inversion of trend of a security (a stock in our case). 0.8 1.0 1.2 1.4 1.6 1.8 S [2011-05-05/2012-05-04] Last 0.88 Bollinger Bands (20,2) [Upper/Lower]: 0.967/0.846 0.04 0.05 0.06 0.07 0.08 0.09 0.10 Mag 05 2011 Lug 01 2011 Set 01 2011 Nov 01 2011 Gen 02 2012 Mar 01 2012 Mag 02 2012
  • 11. The Average True Range has the aim of calculate prices volatility as the breadth of their fluctuations. This mechanism is based on the idea that, during a turnaround, the volatility assume extreme values (high or low). On the other hand we have the Bollinger Bands that are based on the volatility calculation as well but there is a superior band and an inferior one so that we can have an idea of what the volatility range is. From an operating point of view the Bands give us signals of buying or selling when these conditions occure: -­‐ when the price graph goes out of the upper band and it goes in again, this is a selling sign; this could represent an increase in price followed by an adjustment. -­‐ When the same thing happens with respect to the lower band, we have a buying sign; this means that the price has gone down very quickly up to the point of turnaround. Concerning Saras S.p.a. we can see some of these situations. For example we see that on August 2011 the graph crossed the lower band meaning a strong decrease in price in fact in this period we registered the peak of the Euro debt crisis. We can see that even from the volatility graph that on that date had a steep rise. The price went down up to the point of turnaround, crossing again the lower band, giving a signal of strong buying of the stock. Conversely, on December 2011, we can see the same process that led to a steep rise in the stock price in according to the volatility graph that registered a strong increase. In this case the stock price went up and, just after having crossed the upper line, went down meaning that we were in front of a turnaround and people were selling their shares. Looking at the very last part of the chart we can see that, in these days, the stock is having a negative trend and this is the reason why the implied volatility of the put option is higher than the call one telling us that the market expectations for the near future is still a down-trend. 2.2.1 – Comments The Black&Scholes formula, the one we used before to price Saras S.p.a. options, doesn’t match very well with the real world we live in. That’s why, as we have seen in the previous paragrph, is based on some assumptions that are impossible to notice in real circumstances: in particular what we are going to prove now is the soundness of the assumption that says stock price follows a geometric Brownian motion with constant drift and constant volatility. In order to understand better what we are going to talk about let’s see, first, what a geormetric Brownian motion actually is. One of the most important assumption of the B&S model is that stock prices follow a Normal distributed process and this processi s known as the geometric Brownian motion. It defined as follows: dSt = µSt dt + σSt dWt where: σ is the volatility and it is assumed constant.
  • 12. µ is the expected return. Wt is Wiener Process that is the stochastic component of the process. In order to compute our demonstration we are not going to use the returns of the stock. Instead, we are going to use the logarithm of the returns (more reliable) given by the ratio: returns in time t over returns in time t-1. Here is the expression: log.returns = returnst / returnst-1   Gen 02 2007 Gen 02 2008 Gen 02 2009 Gen 04 2010 Gen 03 2011 Gen 02 2012 -0.100.000.100.20 log.returns
  • 13.             -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 05101520 density.default(x = log.returns, na.rm = TRUE) N = 1364 Bandwidth = 0.004264 Density -3 -2 -1 0 1 2 3 -0.100.000.100.20 Normal Q-Q Plot Theoretical Quantiles SampleQuantiles
  • 14. These graphs are a very clear demonstration of what we were looking for. The density function of our company are not properly ditributed and we are going to comment some indicators that tell us why: first of all, looking at the second graph(the density function), we can easily see that the tails of the function are not linear adn well defined as farther we move from central values. This fact underline the fact that the price does not follow a geometric Brownian motion so that the assumption does not hold. Finally, the third picture, should give us the explanation of why our prices does not follow a Normally distributed function. If we look at it we can see a straight line indicating the path that our prices should follow to be Normally distributed and the path they actually follow. Concerning central values they seem to be as the ideal funtion wants them to be but if we move from central value we notice that they go astray. 2.2.2 – Volatility Smiles As we wrote before, B&S formula assesses that prices follow a normally distributed trend with constant volatiltiy. In this part we are to verify this last statement. Here are some options (call and put) with the same expiration date:   Saras S.p.a. Expiration June 15, 2012 CALL PUT Mkt Strike(K) Mkt 0.0435 0.74 0.03 0.0415 0.76 0.037 0.032 0.78 0.047 0.0205 0.8 0.051 0.008 0.85 0.086 0.003 0.9 0.0005 0.95 0.0005 1 0.0005 1.05 For semplicity we show and comment only the graph related to Call Option.
  • 15.     As we can immediatly see the stock prices are not characterized by constant volatility and, instead, it changes as the strike price changes. The volatility seems to follow a particular path that, as analysts call it, can be similar to a “smile”. This kind of graph is called “Volatility Smile”. The reason of this trend is related to the fact that values are high when the option is deeply in, or conversely out, of the money and decrease when the option is near to the “at the money point”.   2.3 – The Monte Carlo Method As we have already mentioned, the Black&Scholes model is used all over the world for pricing options but our paper shows that the assumptions it is based on are quite unrealistic and through our calculations we disproved them. Now we are going to challenge another method, the Monte Carlo Method, based on a simulation: a random generation of thousand of combination of prices . The successive step is going to be the calculation of the payoff of the option for each simulation; the discounted results will be the price of the option we are looking for. 0.75 0.80 0.85 0.90 0.95 1.00 1.05 0.400.450.500.55 Volatility smile SARAS S.P.A K smile !^
  • 16. In this part of the paper we want to compare two different prices of the same option: we have already calculated one of these two prices that is generated by B&S model. The second one is going to be generated by the Monte Carlo method. Then we are going to discuss our results. As Monte Carlo simulation is based on repeated price generation we are goint to take different scenarios characterised by different number of simulation (1000, 10000, 100000, 1000000) and what we expect is that the higher are repetitions the more precise would be our price according to the market one. SARAS S.p.a. CALL M PUT 0.05976969 1000 0.03035832 0.06110445 10000 0.03120731 0.06154876 100000 0.0311551 0.06154721 1000000 0.03115751 As we expected the accuracy of prices with respect to ones calculated by B&S is as higher as the number of the simulations increase. This is the main characteristic of the Monte Carlo simulation based on the “large number law” and it is plain also using a function of R software called “speed of convergence” that show us graphically the characteristic we’ve mentioned above. To simplify our calculations, our analysis on convergency is going to be done only on one option: the characteristics of this option are S0=0.75 and the strike price(K)= 0.72. The interest rate is, as for other operations, is the interest rate of deutsch bank.
  • 17.   Chapter 3 – Lévy Process In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process that starts at 0, admits càdlàag(continue à droite, limitée à gauche) modification and has "stationary independent increments" — this phrase will be explained below. It is a stochastic analog of independent and identically distributed random variables, and the most well known examples are the Wiener process and the Poisson process. It is defined as follows: A stochastic process X = {Xt : t ≥ 0} is said to be a Lèvy process if, Speed of Convergency - Saras S.p.a. MC replications MCprice 10 100 200 500 1000
  • 18. 1) X0 = 0 almost surely 2) Indipendent increments: For any 0 ≤ t1 < t2 < … < tn < ∞, Xt2 – Xt1, Xt3 – Xt2, … , Xtn – Xtn-1 are indipendent. 3) Stationary increments: for any s < t, Xt – Xs is equal in distribution to Xt-s 4) t -> Xt is almost surely right continuous with left limits. In order to better understand what is wrong in the B&S Model we are going to discuss and represent graphically the problem. First we have to say that distribustions are not normal:       Looking at the graph it is evident that the distribution of the logarithm of the returns is definitely different from a normal distributed function (as we have already computed for the B&S Model). Our goal is, now, find a new model which should not be based on geometric Brownian motion like the B&S model. Well, a solution to our problem could be represented by Lévy proccesses. Using R-software we are able to compute some Lévy processes and to plot them giving us a sketch of what could be a solution to our problem: -0.1 0.0 0.1 0.2 02468101214 density.default(x = Ret.Saras) N = 253 Bandwidth = 0.008547 Density
  • 19.     In the above we can see: Picture a) the NORMAL parameter estimation. MEAN:-0.003379408 SD:0.036455639 Picture b) the NORMAL INVERSE GAUSSIAN parameter estimation ALPHA:28.375193345 BETA:1.564509140 DELTA:0.036703176MU:-0.005406176 Picture c) the HYPERBOLIC parameter estimation ALPHA:43.912065188 BETA:1.505010969 DELTA:0.016400880MU:-0.005299175 Picture d) the GENERALIZED HYPERBOLIC parameter estimation ALPHA:3.813755900 BETA:2.074031129 DELTA:0.057400423MU:-0.006110047 -0.10 -0.05 0.00 0.05 0.10 -2-1012 x logf(x) NORMAL: Parameter Estimation -0.10 0.00 0.05 0.10 0.15 0.20 -4-202 x logf(x) NIG Parameter Estimation -0.10 0.00 0.05 0.10 0.15 0.20 -6-4-202 logf(x) HYP Parameter Estimation -0.10 0.00 0.05 0.10 0.15 0.20 -0.50.51.5 logf(x) GH Parameter Estimation
  • 20. LAMBDA:-2.229897013 Once we have introduced what the Lévy process is we have to make another important assumption: it says that Lévy markets, even if we condider simpliest one, are not complete. Now we can proceed in pricing an option and, in this particular case, this operation can follow two ways: 1) the Fast Fourier Transform 2) the Monte Carlo Approach In order to simplify my processes we chose to proceed just for the first method giving a sketch of theory for the second one. 3.1 – Fast Fourier Transform A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse. There are many distinct FFT algorithms involving a wide range of mathematics, from simple complex-number arithmetic to group theory and number theory. A DFT decomposes a sequence of values into components of different frequencies. This operation is useful in many fields but computing it directly from the definition is often too slow to be practical. An FFT is a way to compute the same result more quickly: computing a DFT of N points in the naive way, using the definition, takes O(N2 ) arithmetical operations, while an FFT can compute the same result in only O(N log N) operations. For example, corcerning the gepmetric Brownian motion, the characteristic function of Zt is: φ(u)= exp( iu(µ – 1/2 σ2 ) - σ2 u2 /2) where φ is known, the proce of an option can be approximated to this equation: CT(k) ≈ e-α k /π ∑e-iv j k ψT(vj)η, where vj = η(j-1) and k = logK. The constant alpha is considered as the dampening factor and it is usally equal to one. This model can we easily used on R and the results we obtained are: B&S Price FFT Price ∆ 0.06149541 - 0.05932576 = 0.00216965 3.2 – The Monte Carlo Approach The first step we have to take is identifying the distribution of the returns. Then we just have to simulate the patterns of the stochastic process and apply the payoff function to the final value. ST = S0eZ T Chapter 4 – Greeks Analysis and Conclusions 4.1 – Greeks In mathematical finance, 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
  • 21. instrument or portfolio of financial instruments is dependent. The name is used because the most common of these parameters are often denoted by Greek letters. Collectively these have also been called the risk sensitivities, risk measures or hedge parameters. The Greeks are vital tools in risk management. Each Greek measures the sensitivity of the value of a portfolio to a small change in a given underlying parameter, so that component risks may be treated in isolation, and the portfolio rebalanced accordingly to achieve a desired exposure. The Greeks in the Black–Scholes model are relatively easy to calculate, a desirable property of financial models, and are very useful for derivatives traders, especially those who seek to hedge their portfolios from adverse changes in market conditions. For this reason, those Greeks which are particularly useful for hedging delta, theta, and vega are well-defined for measuring changes in Price, Time and Volatility. Although rho is a primary input into the Black–Scholes model, the overall impact on the value of an option corresponding to changes in the risk-free interest rate is generally insignificant and therefore higher-order derivatives involving the risk-free interest rate are not common. The most common of the Greeks are the first order derivatives: Delta, Vega, Theta and Rho as well as Gamma, a second-order derivative of the value function. The remaining sensitivities in this list are common enough that they have common names, but this list is by no means exhaustive. FIRST ORDER GREEKS Delta Delta, , measures the rate of change of option value with respect to changes in the underlying asset's price. Delta is the first derivative of the value of the option with respect to the underlying instrument's price . For a vanilla option, delta will be a number between 0.0 and 1.0 for a long call (and/or short put) and 0.0 and −1.0 for a long put (and/or short call) – depending on price, a call option behaves as if one owns 1 share of the underlying stock (if deep in the money), or owns nothing (if far out of the money), or something in between, and conversely for a put option. Vega
  • 22. Vega measures sensitivity to volatility. Vega is the derivative of the option value with respect to the volatility of the underlying asset. Vega can be an important Greek to monitor for an option trader, especially in volatile markets, since the value of some option strategies can be particularly sensitive to changes in volatility. The value of an option straddle, for example, is extremely dependent on changes to volatility. Theta Theta, θ, measures the sensitivity of the value of the derivative to the passage of time: the "time decay.” The mathematical result of the formula for theta is expressed in value per year. By convention, it is usual to divide the result by the number of days in a year, to arrive at the amount of money per share of the underlying that the option loses in one day. Theta is almost always negative for long calls and puts and positive for short calls and puts. An exception is a deep in-the-money European put. The total theta for a portfolio of options can be determined by summing the thetas for each individual position. The value of an option can be analysed into two parts: the intrinsic value and the time value. The intrinsic value is the amount of money you would gain if you exercised the option immediately, while the time value is the value of having the option of waiting longer before deciding to exercise. Rho Rho, , measures sensitivity to the interest rate: it is the derivative of the option value with respect to the risk free interest rate (for the relevant outstanding term). Except under extreme circumstances, the value of an option is less sensitive to changes in the risk free interest rate than to changes in other parameters. For this reason, rho is the least used of the first-order Greeks. Rho is typically expressed as the amount of money, per share of the underlying, that the value of the option will gain or lose as the risk free interest rate rises or falls by 1.0% per annum (100 basis points). Lambda
  • 23. Lambda, , omega, , or elasticity is the percentage change in option value per percentage change in the underlying price, a measure of leverage, sometimes called gearing. SECOND ORDER GREEKS Gamma Gamma, , measures the rate of change in the delta with respect to changes in the underlying price. Gamma is the second derivative of the value function with respect to the underlying price. All long options have positive gamma and all short options have negative gamma. Gamma is greatest approximately at-the-money (ATM) and diminishes the further out you go either in-the-money (ITM) or out-of-the-money (OTM). When a trader seeks to establish an effective delta-hedge for a portfolio, the trader may also seek to neutralize the portfolio's gamma, as this will ensure that the hedge will be effective over a wider range of underlying price movements. However, in neutralizing the gamma of a portfolio, alpha (the return in excess of the risk-free rate) is reduced. Thanks to R, calculating the coefficients of Greeks on Saras Call options is quite easy and using a simple function here are the results we obtained: SARAS S.p.a As the Underlying Stock Price Changes - Delta and Gamma Delta measures the sensitivity of an option's theoretical value to a change in the price of the underlying asset. It is normally represented as a number between -1 and 1, and it indicates how much the value of an option should change when the price of the underlying stock rises by one euro. As an alternative convention, the delta can also be shown as a value between -100 and +100 to show the total euro sensitivity on the value 1 option, which comprises of 100 shares of the underlying. Call options have positive deltas and put options have negative deltas. At-the-money options generally have deltas around 50. Deep-in-the-money options might have a delta of 80 or higher, while out-of-the-money options have deltas as small as 20 or less. In our case if we multiply the value of Delta we obtain: 0.6354121*100=63.54121. Delta 0.6354121 Gamma 3.279769 Theta 0.06149541 Vega 0.07938833 Lambda 7.749506 Rho 0.03294156
  • 24. The value can be considered standing around 50 so we are describing an option that stands just beyond the “at the money” zone but, at the same time, it is not in “in the money” zone. Furthermore, if the underlying price changes by 1 euro the related variation in the option proce is going to be almost 0.64. Another thing we are interested in is how delta may change as the stock proce moves: Gamma measures the rate of change in the delta for each one-point increase in the underlying asset. It is a valuable tool in helping you forecast changes in the delta of an option or an overall position. Gamma will be larger for the at-the-money options, and gets progressively lower for both the in- and out-of-the-money options. Unlike delta, gamma is always positive for both calls and puts. In our case this indicator helps us to better understand what kind of option we are dealing with in fact as we said above lower values of gamma indicates a “in the money” option (almost 3.28). Besides, it tells us that for one point our delta is going to change by 3.28. Changes in Volatility and the Passage of Time - Theta and Vega Theta is a measure of the time decay of an option, the euro amount that an option will lose each day due to the passage of time. For at-the-money options, theta increases as an option approaches the expiration date. For in- and out-of-the-money options, theta decreases as an option approaches expiration. Theta is one of the most important concepts for a beginning option trader to understand, because it explains the effect of time on the premium of the options that have been purchased or sold. The further out in time you go, the smaller the time decay will be for an option. If you want to own an option, it is advantageous to purchase longer-term contracts. If you want a strategy that profits from time decay, then you will want to short the shorter-term options, so that the loss in value due to time happens quickly. In our case theta is low (almost 0.06) indicating that our option approaches the expiration and it actually does because the expiration of the option we considered in the calculation expires within a month(June 15, 2012). Since the expiration date is not far this indicator tells us that the amount money we are going to loose up to the expiration is small. Vega measures the sensitivity of the price of an option to changes in volatility. A change in volatility will affect both calls and puts the same way. An increase in volatility will increase the prices of all the options on an asset, and a decrease in volatility causes all the options to decrease in value. However, each individual option has its own vega and will react to volatility changes a bit differently. The impact of volatility changes is greater for at-the-money options than it is for the in- or out-of-the-money options. While vega affects calls and puts similarly, it does seem to affect calls more than puts. Perhaps because of the anticipation of market growth over time. Since we are analyzing an “in the money” option even vega has a low value indicating a small impact of volatility.
  • 25. Changes with respect to interest rate - Rho Since it measures the sensitivity with respect to the interest rate we are going to multiply by 100 the value we obatained on R: 0.03294156*100=3,294156. This value in the gain of our option related to a variation of 1.0% of the interest rate. Elasticity - Lambda Finally we are going to discuss the only second order greek that gives us a measure of leverage. In our case it is 7.749506 and this is the percentage variation (almost 7.75%) in our option per percentage change in the underlying asset value. 4.2 - Conclusions To be honest working on this paper actually helped me to better understand the reality of the financial market but, first of all, the reality of an incredibly important company as Saras within the financial market. My technical skills were low at the beginning of the process but this lack helped me consolidating my theory and my practical skills. Concerning the results obtained I have to say that I am satisfied. According to these results the company is facing a tough crisis period and the stock prices reflect it. If we look at the options value, the price of a put option still worths more than a call indicating the trend is not going to have turnaround. The differences in prices is not very consistent due to the different methods we have implented: for example Lévy process should have given a price improving the Black & Scholes formula based on the geometric Brownian motion but the difference between the two prices is very low. Afterall, going through my analysis I found one of the most powerfull company in the field of refining facing the difficulties in a very bad way. Besides, the problems are not coming only from financial markets in facts the company is working to fix the problem of its employers’ death while on working . The overall situation is very clear and it tells that this company is going down, the stock price registered a decreasing trend and I don’t think is going to stop.
  • 26. SCRIPT ## Chapter 1 ## library(quantmod) get.symbols(“SRS.MI”, from=”2007-05-20”, to=”2012-05-20”) chartSeries(“SRS.MI”, theme="white", TA=c(addVo(), addBBands())) ##Change Point## library (tseries) S <- get.hist.quote("SRS.MI", start = "2011-05-01", end="2012-04-30") chartSeries(S, TA = c(addVo(), addBBands()), theme="white") S <- S$Close require(sde) cpoint(S) addVLine = function(dtlist) plot(addTA(xts(rep(TRUE, + + NROW (dtlist)), dtlist), on = 1, col = "red")) addVLine(cpoint(S)$tau0) ##Chapter 2## ##Option Valuation## f.call <- function(X) sapply(X, function(X) max(X-K,0))) f.put <- function(X) saplly(X, function(X) max(c(K-X,0))) K<-1 curve(f.call, 0,2,main=”Payoff Functions”,col=”Blue”, Ity=4, lwd=4,ylab=expression(f(x))) curve(f.put,0,2,col=”red”,add=TRUE,Ity=4,lwd=4) temp<-legend(“top”,legend=c(“Call”,”Put”),Ity=1, lwd=5,col=c(“blue”,”red”) ##Black&Scholes Model## require(fImport) S<-yahooSeries(“SRS.MI”, from=”2011-05-20”, to=”2012-05-12”) Head(S)
  • 27. Close<-S[, “SRS.MI.Close”] Require(quantmod) chartSeries(Close, theme=”white”) X<-returns(Close) Delta<-1/252 sigma.hat<-sqrt(var(X)/Delta)[1,1] sigma.hat ##Call Option (Saras)## call.price<-function(x=1, t=0, T=1, r=1, sigma=1,K=1){ d2<-(log(X/K) + (r-0.5*sigma^2)*(T-t)/sigma*sqrt(T-t)) d1<-d2 + sigma*sqrt(T-t) x*pnorm(d1)-K*exp(-r*(T-t))*pnorm(d2)} S0<-0.75 K<-0.72 T<-20/252 r<-0.005 sigma<-0.5422024 C<-call.price(x=S0, t=0, T=T, r=r, K=K, sigma=sigma) C ## Put Option (Saras)## put.price<-function(x=1, t=0, T=1, r=1, sigma=1,K=1){ d2<-(log(X/K) + (r-0.5*sigma^2)*(T-t))/(sigma*sqrt(T-t)) d1<-d2 + sigma*sqrt(T-t) K*exp(-r*(T-t))*pnorm(-d2) –x*pnorm(-d1)} S0<-0.75 K<-0.72 T<-20/252
  • 28. r<-0.005 sigma<-0.5422024 p<-put.price(x=S0, t=0, T=T, r=r, K=K, sigma=sigma) p ##Put-Call Parity## C – S0 + K * exp(-r * T) ##Implied Volatility## #call p<-0.039 Delta<-1/252 T<-20*Delta S0<-0.75 K<-0.72 r<-0.005 sigma.imp<-GBSVolatility(p,”c”,S=S0, X=K, Time=T, r=r, b=r) sigma.imp #put p<-0.029 sigma.imp<-GBSVolatility(p, “c”, S=S0, X=K, Time=T, r = r, b = r) sigma.imp ##Plot B&S## library(tseries) S <- get.hist.quote(“SRS.MI”, start=”2011-05-20”, end=”2012-05-20”) chartSeries(S,TA=c(addVo(),addBBands(), addATR()), theme=”white”)
  • 29. ##geometric Brwnian motion## getSymbols(“SRS.MI”) plot(SRS.MI) str(SRS.MI) S<-SRS.MI[, “SRS.MI.Close”] X<-diff(log(S)) plot(X) plot(density(X, na.rm=TRUE)) qqnorm(na.omit(X)) qqline(na.omit(X)) ##Log-Returns## getSymbols(“SRS.MI”) plot(SRS.MI) str(SRS.MI) ##Volatility Smile## S<- (SRS.MI[, “SRS.MI.Close”] X<-returns(Close) Delta<-1/252 sigma.hat<-sqrt(var(X)/Delta) sigma.hat Pt<-c(0.0435, 0.0415, 0.032, 0.0205, 0.008, 0.003, 0.0005, 0.0005) K<-c(0.74, 0.76, 0.78, 0.8, 0.85, 0.9, 0.95, 1, 1.05)
  • 30. S0<-0.75 nP<-length(Pt) T<-20*Delta R<-0.005 smile<- sapply(1:nP, function(i) GBSVolatiltiy(Pt[i], “c”, S=S0, X=K[i], Time=T, r=r, b=r)) vals<-c(smile, sigma.hat) plot(K, smile, type =”l”, lty=3, col=”orange”) axis(2, sigma.hat, expression(hat(sigma)), col=”blue”)   ##MONTE CARLO Method## S0<-0.75 K<-0.72 r<-0.005 Delta<-1/252 T<-20*Delta sigma<-0.5422024 MCPrice<-function(x=1, t=0, T=1, r=1, sigma=1, M=1000, f){ h<-function(m){ u<- rnorm(m/2) tmp<-c(x*exp((r-0.5*sigma^2)*(T-t)+sigma*sqrt(T-t)*u), x*exp((r- 0.5*sigma^2)*(T-t)+sigma*sqrt(T-t)*(-u))) mean(sapply(tmp, function(xx) f(xx))) } p<-h(M) p*exp(-r*(T-t))
  • 31. } ##CALL## f<-function(x) max(0, x-K) set.seed(123) M<-1000 MCPrice(x=S0, t=0, T=T, r=r, sigma=sigma.hat, M=M, f=f) M<-10000 MCPrice(x=S0, t=0, T=T, r=r, sigma=sigma.hat, M=M, f=f) M<-100000 MCPrice(x=S0, t=0, T=T, r=r, sigma=sigma.hat, M=M, f=f) M<-1000000 MCPrice(x=S0, t=0, T=T, r=r, sigma=sigma.hat, M=M, f=f)   ##PUT## f<- function(x) max(0,K-x) M<-1000 MCPrice(x=S0, t=0, T=T, r=r, sigma=sigma.hat, M=M, f=f) M<-10000 MCPrice(x=S0, t=0, T=T, r=r, sigma=sigma.hat, M=M, f=f) M<-100000 MCPrice(x=S0, t=0, T=T, r=r, sigma=sigma.hat, M=M, f=f) M<- 1000000 MCPrice(x=S0, t=0, T=T, r=r, sigma=sigma.hat, M=M, f=f)   ##Speed of Convergency##
  • 32. S0<-0.75 K<- 0.72 r<-0.005 Delta<- 1/252 sigma.hat<-0.5422024 f<- function(x) max(0,x-K) set.seed(123) m<-c(10, 50, 100, 150, 200, 250,500, 1000) p1<-NULL err<-NULL nM<-length(m) repl<-100 mat<-matrix(, repl, nM) for(k in 1:nM){ tmp<-numeric(repl) fot(i in 1:repl) tmp[i] <- MCPrice(x=S0, t=0, T=T, r=r, sigma, M=m[k], f=f) mat[, k] <- tmp p1<-c(p1, mean(tmp)) err<-c(err, sd(tmp)) } colnames(mat)<-m p0<-GBSOption(TypeFlag=”c”, S=S0, X=K, Time=T, r=r, b=r, sigma=sigma) minP<-min(p1-err) maxP<-max(p1 + err) plot(m, p1, type=”n”, ylim=c(minP, maxP), axes=F, ylab=”MC price”, xlab0 “MC replications”, main= “Speed of Convergency – Saras S.p.a.”)
  • 33. lines(m, p1 + err, col= “blue”) lines(m, p1-err, col= “blue”) axis(2, p0, “B&S price”) axis(1,m) boxplot(mat, add=TRUE, at=m, boxwex=15, col= “orange”, axes=F) points(m, p1, col= “blue”, lwd=3, lty=3)   ##Greeks  Analysis##   require(fOptions) r<- 0.005 T<-20/252 K<-0.72 sigma<- 0.5422024 S0<- 0.75 GBSCharacteristics(TypeFlag= “c”, S=S0, X=K, Time=T, r=r, b=r, sigma=sigma)     ##Lévy process## require(fImport) data <- yahooSeries("ENI.MI", from="2009-01-01", to="2009-12-31") S <- data[, "ENI.MI.Close"] > X <- returns(S) > require(quantmod) > lineChart(S,layout=NULL,theme="white") lineChart(X,layout=NULL,theme="white") sigma.hat <- sqrt( var(X)/deltat(S) ) alpha.hat <- mean(X)/deltat(S) mu.hat <- alpha.hat + 0.5 * sigma.hat^2
  • 34. plot(density(X)) d <- function(x) dnorm(x, mean=mu.hat, sd=sigma.hat) curve(d, -.2, .2, col="red",add=TRUE, n=500) par(mfrow=c(2,2)) par(mar=c(3,3,3,1)) grid <- NULL library(fBasics) nFit(X) nigFit(X,trace=FALSE) hypFit(X,trace=FALSE) ghFit(X,trace=FALSE) ##FFT## FFTcall.price <- function(phi, S0, K, r, T, alpha = 1, N = 2^12, eta = 0.25) { m <- r - log(phi(-(0+1i))) phi.tilde <- function(u) (phi(u) * exp((0+1i) * u * m))^T psi <- function(v) exp(-r * T) * phi.tilde((v - (alpha + 1) * (0+1i)))/(alpha^2 + alpha - v^2 + (0+1i) * (2 * alpha + 1) * v) lambda <- (2 * pi)/(N * eta) b <- 1/2 * N * lambda ku <- -b + lambda * (0:(N - 1)) v <- eta * (0:(N - 1)) tmp <- exp((0+1i) * b * v) * psi(v) * eta * (3 + (-1)^(1:N) – ((1:N) - 1 == 0))/3 ft <- fft(tmp)
  • 35. res <- exp(-alpha * ku) * ft/pi inter <- spline(ku, Re(res), xout = log(K/S0)) return(inter$y * S0) } phiBS <- function(u) exp((0+1i) * u * (mu - 0.5 * sigma^2) - 0.5 * sigma^2 * u^2) S0<- 0.75 K<-0.72 R<-0.005 T<-20/252 Sigma<-0.5422024 mu <- 1 require(fOptions) GBSOption(TypeFlag = "c", S = S0, X = K, Time = T, r = r, b = r, sigma = sigma) FFTcall.price(phiBS, S0 = S0, K = K, r = r, T = T) }