The document discusses pricing the Margrabe option using Monte Carlo simulation and an explicit closed-form solution. It begins by defining the Margrabe option and explaining its use. It then presents Margrabe's closed-form solution, which prices the option as a European call using a change of numeraire approach. Next, it analyzes the option's sensitivity to various parameters. Finally, it outlines different option pricing methods and focuses on Monte Carlo simulation and the change of numeraire approach.

1. Bocconi University
Fall 2011
Applied Numerical Finance - code 20247
Monte Carlo for the Margrabe option
Giulio Laudani 1256809
Francesco Manenti 1260827
Michele Mazzoni 1262799
Luca Scollo 1244847
Gabriele Giovanni Vecchio 1259910
Part I
What is Margrabe option?
A Margrabe option, known also as exchange option, gives to the owner the
right, but not the obligation, to exchange one asset into another asset at a
speci…c maturity date T; we consider only options of the european type, for
which the buyer has the right to be paid, at the maturity date T, the di¤erence
S1 (T ) S2 (T ), i.e. it is a claim with payo¤
X (T ) = max fS1 (T ) S2 (T ) ; 0g
at time t.
The Margrabe Option can be viewed as a Spread option1 where the strike
is set equal to zero. The use of this kind of options is widespread. Due to their
1 A spread option is an option written on the di¤erence of two underlying assets, whose
values at time t we denote by S1 (t) and S2 (t). To exercise the option, the buyer must pay at
maturity a pre-speci…ed price K, known as the strike, or the exercise price of the option. In
other words, the payo¤ of a spread option at maturity T is (S1 (T ) S2 (T ) K)+ . Besides
the fact that the case K = 0 leads to a solution in closed form, it has also a practical appeal
to the market participants. Indeed, it can be viewed as an option to exchange one product
for another at no cost.
1

2. generic nature, spread options are used in various …nancial markets such as
currency and foreign exchange markets, commodity futures markets, and …xed
income markets.These options can be traded on an exchange, but the bulk of
the volume comes from over-the-counter trades.
Here we are going to describe a possible example to give a better insight.
Let us suppose that an investor wants to buy one of two stocks S1 or S2 , but
is not quite sure of his pick. Therefore he might buy an option to exchange S2
for S1 in case S1 outperforms S2 at time T . In this way his investment is the
better of the two stocks at time T . So if one would like to own either one of
the underlyings at time T , one could simply buy the …rst one and purchase the
above Margrabe option. This will guarantee that, short of the premium of the
option, we do as well as if we had bought the one ending up being the cheaper
of the two in all cases.
1 Margrabe’ closed formula
s
Following Margrabe’ work (1978), it is possible to derive the closed formula,
s
using the change of numeraire approach, taking asset S_2 as numeraire. In this
way the option to exchange asset two for asset one is a call option on asset one,
with exercise price equal to unity and interest rate equal to zero2 .
Let us suppose the capital market is perfect, of course, and let S1 and S2 be
the price of assets one and two, where and S1 and S2 are log-normal. Thus
q1 T q2 T
Sm argrab e = S1 (0) e N (d1 ) S2 (0) e N (d2 )
where
S1 (0)
ln S2 (0) + q2 q1 + 2 T
d1 = p
T
p
d2 = d1 T
2 2 2
= 1 + 2 2 1 2
2 Option sensitivity analysis
In this paragraph we are going to highlight the option sensitivity with respect to
the Margrabe option input parameter. So we created a Matlab code that allows
to …gure out how the option value changes as the option parameters change.
Therefore the program using function mesh creates the graphs, reported below.
As we can see in the …rst graph, the option is worth at least zero, and no more
than S1 , if assets one and two are worth at least zero. It’ easy to see that
s
Margrabe option is increasing with respect to S1 .
2 W.Margrabe, The value o fan option to exchange one asset for another, The Journal of
Finance March 1978, pag. 177-186
2

3. Indeed, if S1 is greater than S2 at maturity, option intrinsic value is greater
than zeros and reaches the peak as the di¤erence between assets one and two is
the highest.
Moreover, we are going to consider how option worth as asset volatility
changes. As we can see in the graph 2a Margrabe option behave in the same
way both for the …rst asset and for the second one. As the matter of fact
that, such as for a call option, as the volatility increases, the probability that
option will be in the money at maturity increases. On the other hand the graph
2b shows how option value varies as correlation between the two underlying
asset and time change. Therefore, ceteris paribus, option value increase as the
correlation between asset one and two decrease. The Margrabe option goes up
in value if security one and two are negatively correlated.
Finally we are going to show how Margrabe option price changes as contin-
uous dividend yield changes.
How we can see in the third …gure the option has negative Epsilon, price
sensitivity with respect to the dividend yield, computed for the S1 and a positive
Epsilon for S2 . So given asset one’ dividend yield equals to zero, Margrabe
s
option price goes up when asset two’ dividend yield increases.
s
3

4. 4

5. Part II
Option pricing: main issues
Generally there are several methods for pricing options; all of them are based
on evaluate an expectation. Depending on the payo¤ we can solve it by:
Explicit solution
Numerical integration (for instance Monte Carlo simulations)
Converting to a PDE and solving with boundary conditions (full blown
PDE methods or approximate solutions by simplifying the PDE)
Approximate the underlying state variable (lattice methods)
In our report we focus on two of these methods: explicit solution, introducing
the idea behind the change of numeraire approach, and Monte Carlo method
together with its feasible variance reduction techniques.
5

6. 3 The change of numeraire
Change of numeraire is a useful tool being used to price both …xed income
instruments and exotic derivatives on several underlying assets. Speci…cally this
methodology can be applied on Margrabe option, which is an option written on
two underlying assets and this approach can simplify the computation of the
price.
Numeraire is a positive traded asset, and it must be self …nancing; some
examples of numeraire are the money market account, a zero coupon bond, a
non dividend paying asset . If there are no arbitrage opportunities in the market,
then (Harrison and Kreps, 1979) relative prices (i.e. discounted numerarires)
are martingales under risk adjusted probabilities.
We recall that Riesz representation theorem allows us to make a bridge
between pricing functions and numeraires and that di¤erent numeraires imply
di¤erent probability measures and hence di¤erent expectations. If we de…ne P
as the equivalent martingale measure for numeraire pt and Q as the equivalent
martingale measure for a numeraire qt , then
P pt Q qt
vt = Et vT = Et vT
pT qT
In this way we can move from one measure to another. If P and Q are equiva-
dP
lent, then there exists a function L = dQ , called likelihood function or Radon-
Nikodym derivative, such that
dP
dPt = (t) dQt
dQ
Radon-Nikodym derivative is also the ratio of the numeraires, hence we have
Z Z Z Z
pt pt dP pt pT q t qt
vt = vT dPt = vT (t) dQt = vT dQt = vT dQt
pT pT dQ pT pt q T qT
so to change measure it is necessary to multiply by the Radon-Nikodym deriva-
tive. In pricing put or call european options the numeraire is the money market
account but if we want to price an option on two underlying assets like the Mar-
grabe one, we may choose as numeraire one of the underlying assets in order to
have a nice distribution under the new measure. We set as discounted numeraire
e (t)eq2 t
N (t) = S2B(t) , we change from the risk neutral measure to the new QN and
after some algebra we get
" #
h i S1 (t)
+
Q rT + QN q2 T
price = E e (S1 (t) S2 (t)) = S2 (0) E e 1
S2 (t)
S1
Applying Ito formula to compute d S2 (t) and then Girsanov theorem to have
the distribution under the new measure we …nally get
q1 T q2 T
price = S1 (0) e N (d1 ) S2 (0) e N (d2 )
that is the closed formula for pricing exchange options.
6

7. 4 Monte Carlo simulation and variance reduc-
tion techniques
Instead of using only a closed formula to price this kind of option, we develop
also a Monte Carlo simulation. Let
n
1X
b n := f (Xi )
n i=1
be the Monte Carlo estimate of E [f (X)]. This estimate is consistent by the
Strong Law of Large Numebers and it is also unbiased. Generally Monte Carlo
1
has a rate of convergence O n 2 , while the error in a trapezoidal rule for d
2
dimensional integrals is O n d . Hence Monte Carlo methods are attractive in
evaluating integrals when d > 4. Margrabe option pricing requires to evaluate
a two dimensional integral since its price can be viewed as
h i ZZ
+ +
E e rT (S1 (T ) S2 (T )) = e rT (s1 s2 ) 'T (s1 ; s2 ) ds1 ds2
where ' (s1 ; s2 ) is the joint density between the two underlying assets.You can
notice that d = 2 and for this option Monte Carlo is not a so attractive method.
However it is interesting to compare the price got from the closed formula with
the one given by MC estimate and analyze the con…dence interval, the absolute
error and the convergence.
To simulate the …nal payo¤ of the exchange option we have to simulate two
correlated Wiener processes, thus we …rst simulate two independent standard
normal variates and and then we use
"1 = Z1
p
"2 = Z1 + 1 2Z
2
to drive path. Moreover we de…ne the absolute error as jb j and the con…dence
b
interval as b z pn .In the next paragraphs we will describe the main steps
n
2
for writing a VBA and a Matlab code that compute these quantities and then
we will show our numerical and graphical results. To improve the Monte Carlo
estimante we introduced also two variance reduction techniques:
antithetic variate technique
control variate technique
The …rst one exploits the negative correlation between pair of replications
because it reduces the variance of the estimation if the correlation of the pairs
is negative; this method can be applied if we have a monotone function f of a
random variable Z. In our case f N (0; 1), then we sample Z1 and we set
Z2 = Z1 , obtaining negative dependence. The estimate will be
n
1X1
b AV = [f (Zi ) + f ( Zi )]
n i=1 2
7

8. The second technique is the control variate and it consists of exploiting the
information about the error in the estimate of known quantities to reduce the
error of the unkonwn quantities. Suppose we have the expected value of another
function g of Z. We set f as the discounted payo¤ of the Margrabe option and
the control variate g as the di¤erence of the two underlying assets at maturity.
Thus the control variate estimate is
n
1X
b CV = f (Zi ) + b [g (Zi ) E [g (Z)]]
n i=1
where b is chosen in such a way that it minimizes the variance of the estimator.
It can be shown that the optimal parameter is given by
Cov (f (Z) ; g (Z))
b =
V ar (g (Z))
algorithm we …rst run a fast Monte Carlo simulation to compute the covari-
ance and the variance in order to get the value of b and then a Monte Carlo
simulation to compute the price of the option.
Part III
The VBA code
5 Program functionality
Our assignment is divided into two parts: one entirely dedicated to the user
usage and the second used only for tracking purpose and for simplifying the
production of some speci…c plots3 . This decision is motivated by our acknowl-
edge of the VBA superiority regarding the User-friendly interface and, on the
other hand, of the higher power and of the higher sophistication of MatLab.
Our general aim is to o¤er a good Interface to price Option (speci…cally it
is provided a pricing program for Margrabe option) and a plot generator to
track down the di¤erent computation e¢ ciency techniques provided within the
program itself. The Excel …le “Margrabe_project” is divided into three sheet:
The …rst one is the Main, where there are the Print out of the Pricing
procedure (in the ListBox), the two plots and the Starting Button.
The second sheet, named DataSet, is a service sheet used to save the input
inserted in the main UserMask to allow recalling them. It was initially
used to print the result of each simulation, but in the released version this
functionality has been disabled for saving time.
3 Those plots has been used mainly for the presentation, even if they will o¤er an useful
prospective to discuss about some interesting topic such as the dependency of the pricing
formula to change of main input
8

9. The last sheet, named ListTypesOption, is the one used to insert the
possible option type available when we call the main UserMask (both in
the ComboBox and in the page name within the multipage control). This
functionality allow to easily add new pricing formula section, however
those sections must be …ll with controls in the code, in fact the program
will automatically add just a Label warning the user.
The …rst part has been produced relying on VBA and it consists on a Main
Mask (called “Insert Option Elements” where is requested to provide the main
),
input4 by the user, and a Second Mask (called “Settings”and available from the
…rst one), where the user is allowed to change both some accessory variables5
and to choose the reduction techniques within the available one6 . Both the
masks are available from the Button “Option Price Manual “in the “Foglio1”
sheet. The user can decide two di¤erent outcomes:
The Standard by clicking on the button “Pricing” it consist on printing
,
out into the ListBox a summary of what has been asked, if there has been
any problem or wired/fuzzy outcome and the …nal output. We have chosen
the ListBox instead of the Textbox since is it is easier in coding appending
new line of info, depending of the pricing. The print out is divided into
three/four section:
1. The …rst will communicate if something went wrong or not and then
will show the closed formula price and Monte Carlo one and their
absolute di¤erence
2. The second will provide some info on the Simulation
3. The third will provide the IC of the simulation
4. The fourth section is activate only if the program has revealed some
fuzzy input, meaning numbers that have overcome the error checking,
but might be too big.
The Plot generator, by clicking on the button “Plot” that will produce
,
two di¤erent plot in the “Main” Sheet on the right to the ListBox, We
have decided to print them in the sheet instead of creating a new sheet for
each graph since we have wanted to present all the outcome in one single
page, than the user can resize the plots:
1. The …rst represents the price level of the Closed Formula, the Anti-
thetic, the Standard and the Control variate Technique
2. The second one represents the IC of all the pricing formula provided,
we have decided to color the pair of line with the some color to ease
the readability.
4 Those inputs must be express with an annual frequency
5 The risk free rate used and the number of iteration, as well as the level of con…dence
6 The program o¤ers the Antithetic and the Control Variate methods
9

10. 6 Convention used and Internal set up
In this section we will go through the code and we will o¤er a detail justi…cation
and explanation. At …rst the naming convention used is to name the variables
with combination of words as clear as possible, divided with the Upper case;
in some routines there are some service variables which are usually named with
fuzzy or dummy name. For further details we remand to the comments on the
code, where each functionality has been described7 . There are some functions
that are not needed any more in the released version, but we have decided to
leave them to allow the professor to track how we have built the program. These
functions are mainly in the module “Output_Sheet” which has been used to
,
print in the spreadsheet the simulation results.
The VBA code is divided into di¤erent modules; when describes their fea-
tures I’ group them into categories and I’ de…ne them in the subsequent
ll ll
subtitle:
6.1 Error handler
As a preliminary introduction we will spend some words on the general frame-
work: since the program is structured as a cascade we have preferred to put
all the error check at the highest level, which is when the user is inserting the
required input. This procedure has been done in the “ErrorControl” module,
where there are four routines, two for each mask.
Those routine share the same structure the …rst dealt with the “type” of
input insert, meaning will check if the input is a number while the second will
check that the input is inside an acceptable range. Those routines will handle
the correction procedure as well. Note that we have created a sort of loop by
calling each of the routines by themselves to ensure that the program is able to
solve any problems till its …nal solution.
The …rst task is performed with the trick: to force the program to convert
the input into a number together with the function “on error goto” In case of
.
error we have used a dummy variable
“count”to control where the error has happened, so that to properly inform
the user what went wrong and how the program is expecting to receive data.
The second task is structured as the …rst one. Note that there are other error
handlers in some secondary routine to check for any residual problem. This two
steps approach should grant the maximum consistency to the program.
6.2 Output Coding
This section is about the ListBox …ll-in and the Sheet anchorage. The ListBox
module is divided into two routines, the …rst is the main one and it dealt with
the …rst three outputs as described in the introduction, while the second dealt
7 As …nal remark the code has been written with the 2010 Excel version, we have checked
for compatibility issues with older version, however for any function/runtime problem please
contact: giulio.laudani@fastwebnet.it
10

11. with the accessory forth point. We have tried to use String variable as clearer
as possible. In fact the Variable Output has been …lled in separately from the
ListBox …ll-in loop. Besides this main string we have created two other of them
“AddIn” and “MyString” which are used to de…ne the clock functionality and
,
the variance reduction technique used.
The second routine dealt with any possible fuzzy inputs meaning those inputs
that are in the acceptance region, but too big relative to the other variable,
suggesting a possible typo. In this version we have decided to control just for the
time and volatility variable, but it might be extended with more sophisticated
algorithms. Note that in this case the output variable instead of being pre
determinate, it is …lled up case by case and contextually used to …ll-in the
ListBox.
6.3 Plot coding
In this module there are all the routines needed to transform array into series8
and the setup of the two plots, meaning the placement in the spreadsheet and
other secondary features.
There are two plot generators, very similar in the structure, although the
…rst one (“PlotCV” is more sophisticated9 in one way, it has a max and min in
)
the “y” axis speci…cally de…ned to increase the series visibility. As mentioned
in the comment in the VBA code the need of two functions to convert the
array into series depends on the di¤erent format of the incoming input, one is
a Double while the other is Variant. This di¤erence is due to the di¤erent need
for storing information from the function “PricingForPlot” and “ICForPlot” ,
we will discuss about this later on.
The last routine has been registered, that’ why it is longer than needed, we
s
haven’ modi…ed it since we want to show that some task are better and quickly
t
performed without spending tons of time in coding, what matter is just the …nal
result. Furthermore we have disabled some of the functionality since they are
not compatible with older VBA version.
6.4 Pricing formula
In this module there are both the pricing formula need to produce the ListBox
Outcome and the Plot one. The …rst task is quite straight forward:
We have use the “select case” structure to manage the possible outcome
and the Monte Carlo simulation tools are the one presented in the class
or in the lab.
We have initialized the simulation with the function Randomize
8 Since
the Excel plot function need those kind of format to work
9 The"PlotCV” has two variables more than the “PlotIC”, that are the max and min
computed form the original array containing the series
11

12. We have used the Static10 function Clock to track the simulation duration.
The function precision is up to seconds
The payo¤ for each cycle has been saved in an array “payo¤()” that is
re-sized at inception
Note that the function “MC_MargrabeOption” is set as variant since we
need to pass more than one set of results, speci…cally two: the price and
the variance needed to compute the IC in later stage.
The second task is achieved using two routine: the …rst to compute the price
and equivalent variance, while the second will compute the IC using the results
of the …rst function. It is important to point out that the function output is not
a matrix, but a sort of box with two di¤erent elements: the …rst is the matrix
with all the price level, the other the equivalent variance matrix11 .
6.5 Mask Feature
We create two speci…c modules one for each user mask to ease the readability
of the program. In those modules we have inserted all the anchorage and open
features, meaning the default value, if any, and the style for each Textbox.
However most of the mask functions have been inserted inside the mask itself.
Both the UserMask, as mentioned early, have lots of code to deal with some
speci…c event, such as the selection of a Textbox to grant to erase all the numbers
already …lled-in to ease their change or to deal with the exit event to ensure
that the input format is always the correct one. Some further details on the
main User mask:
The Multipage control is linked with the Combobox, in fact it is possible
to change the page both by clicking in the ComboBox menu or on the
Multipage Page controls
We have created a speci…c program to add to any new pages, which are
not de…ned by the programmer, a new Label warning the user.
Just few world on the Checkbox in the “UserMask” which is the main one.
,
This command allows to price the option with inverted Stock. We have thought
that it may be useful to add this functionality.
1 0 The type static was needed to save the starting point and make the di¤erence with the
ending value when call back
1 1 We pointed out this di¤erence because when we will call the resulting matrix we need to
use the notation “answer(0) (. . . )” or “answer(1)(. . . )” depending on which elements we want
to use and within that element which speci…c value we need
12

13. Part IV
Graphs and results
We performed a Monte Carlo (MC) simulation of the price of the Margrabe
option and compared it to the closed formula price. We repeated the process
inputting an increasing number of simulations parameter (n) and obtained the
error of our MC simulation as a function of n. We did this for a standard Monte
Carlo simulation, for a MC simulation with Antithetic Variates and for a MC
Simulation with Control Variates.
As you can see from …gure [MC001], the Control Variate MC has the lowest
level of standard error at all levels of n. Also, after roughly 200 thousand simu-
lations the error stabilizes around its “structural”value. As the theory suggests,
the Antithetic Variates and the standard MC simulations are respectively the
second and third best techniques. Note how the result is neatly plotted in the
graph, with standard MC error dominating Antithetic Variates at all levels of
n.
Effects of variance reduction tecniques
0.7
0.6 Standard MC
Control Variates MC
0.5 Antithetic Variates MC
0.4
Sam ple variance
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Number of simulations 5
x 10
In …gure [MC002], we plotted the actual closed formula price and the sim-
ulated price as a function of n. As previously pointed out, the error clearly
stabilizes around two hundred thousand simulations.
13

14. Confidence Interval at 95% converges to closed formula price
12
11.5
11
10.5
P rice
10
9.5
9
8.5
8
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Number of simulations x 10
5
Part V
References
Arbitrage theory in continuous time; Bjork T.; Oxford University Press,
New York, 3rd ed., 2009
Interest rate modelling; James J., Webber N.; Wiley, Chichester, 2000
Martingales and arbitrage in multiperiod securities markets; Harrison
J.M., Kreps D.M., Journal of Economic Theory, 1979
Monte Carlo methods in …nancial engineering; Glasserman P.; Springer,
New York, 2003
Numerical methods in …nance and economics; Brandimarte P.; Wiley,
Hoboken, 2006
Options, futures and other derivatives; Hull J.C.; Pearson International,
Upper Suddle River, 2006
The value of an option to exchange one asset for another; Margrabe W.;
The Journal of Finance, vol XXXIII, March 1978
14