Numerical Mo dels for Pricing Barrier Options
Oskar Milton
th March
Contents
Abstract
Variable sp eci cation
Intro duction
Brie y ab out options
What is a barrier option
Up and out call
Up and in call
Down and out call
Down and in call
Up and out put
Up and in put
Down and out put
Down and in put
Double barrier option
Delimitations
Standard option contract
Black Scholes analytic formula and theory
Binomial Tree Mo del
Cox Ross Rubinstein Binomial Mo del
Trigeorgis Mo del
Calculating a barrier option s value using Trigeorgis Mo del
Howtocho ose only the b est p oints
Computing hedge sensitivities
Trinomial Tree Mo del
Implementing a stretch parameter
Testing the adjusted trinomial mo del
Computing hedge sensitivities
Finite Di erence Mo del
Explicit approach
Implementation
Stability and convergence
Implicit Cranc Nicolson
Construction of the grid
Boundary conditions
Implicit not fully centered
Comparison b etween the Finite Di erence Mo dels
Monte Carlo Simulation
Comparison
Evaluation of the b est suited numerical mo del for pricing the standard contract
Evaluation of the b est suited numerical mo del for pricing contracts with asset
price very close to the barrier
Evaluation of the b est suited numerical mo del for pricing contracts with asset
price far away from the barrier
Computing Hedge Sensitivities
Delta
Gamma
Omega
Theta
Vega v
Rho
Including Discrete Dividends
Conclutions and Results
Di culties in the evaluation pro cess
Notes ab out the Finite Di erence mo del
Further Development
App endix Ab out the evaluation to ol built in Java
How to add new features to the program
The develop ers environment
Bibliography
Abstract
A Barrier options is a typ of option that is path dep endent That is the payo is dep endent
of the motion of the underlying asset from the day the contract is written until it expires In
this working pap er I am evaluating the b est suited numerical mo del for pricing this typ e of
contract Of the four di erenttyp es of mo dels I have found one that is very fast and robust
There are some earlier works in this eld but non of them has came up with a mo del as
fast as the one presented here The mo del is a Finite Di erence Mo del with Cranc Nicolson
the results is accurate due to earlier works but here is the results presented with a higher
precision this is not scienti cally proved and shall just b e used as a help to understand the
very fast convergence of this mo del
Variable sp eci cation
Symbol Description Co de notation
S Current price of the underlying asset S
S t or S Continuous price of the underlying asset at
anytime t
S Price of the underlying asset at p osition i j St i j
i j
in a tree or a grid
Standard deviation of return volatility ofas sig S
set price S
r Continuously comp ounded interest rate r
K Strike or exercise price of a contingent claim K
Drift of asset price S mu
t Time t
T Maturitydateofcontingent claim T
q Continuous dividend yield on an asset q
c K T Europ ean call price with strike price K and
time to maturityT
p K T Europ ean put price with strike price K and
time to maturityT
C t s or C The contingent claim price at time t and asset
price s for some conract
C The contingent claim price at p osition i j in C i i
i j
a tree or a grid
y The natural logarithm of the asset price S y
Risk neutral drift of y nu
u Size of prop ortional upward move of sto chastic u
variable or subscript indicating upward move
of a sto chastic variable
d Size of prop ortional downward move of d
sto chastic variable or subscript indicating
downward moveofastochastic variable
Symbol Description Co de notation
p Probabilityofupward move in a tree pu
u
p Probability of straight forward move in a tree pm
m
p Probabilityofdownward move in a tree pd
d
i Time step index in a tree or a grid i
j Asset price state index in a tree or a grid j
N Numb er of discretizations i e the number of Ni
i
time steps b etween current date and maturity
date
N Number of discretizations of the asset price Nj
j
between the lowest and the highest no de in
the tree or the grid
M Number of simulations for the MonteCarlo M
simulation mo del
B Barrier level B
X Cash rebate paied when barrier kno cks out Xrebate
r ebate
D Dividend paymentattimestepi Di
i
i
D Discounted cumulativevalue of all future div Dsum i
sum
idend payments at time step i including pay
ment at time step i
Hedge sensitivities
The rate of change of the value of an option delta
with resp ect to changes in the sto ck price
The rate of change of the delta with resp ect gamma
to changes in the sto ck price
The rate of change of the gamma with resp ect omega
to changes in the sto ck price
The rate of change of the value of an option theta
with resp ect to time
The rate of change of the value of an option rho
with resp ect to the risk free rate of interest
v The rate of change of the value of an option vega
with resp ect to volatility
For futher reading ab out the hedge sensitivities go to section
Intro duction
I ve b een commissioned by ABN Amro Software AB to evaluate di erentnumerical mo dels
for pricing barrier options
This pro ject is my thesis for MSc at the Royal Institute of Technology KTH in Sto ck
holm
The mo dels that are handled are the following
The Binomial Tree Mo del
The Trinomial Tree Mo del
The Finite Di erence Mo del
Monte Carlo Simulation
When comparing di erent mo dels for pricing derivatives it is normally the tree or grid size
i e the numb er of discretizations that is considered When a mo del is used in a transaction
system where derivative traders need to recalculate their p ortfolio several times p er hour it is
not the numb er of discretisations that is imp ortant but the calculation time These big sys
tems are time critical b ecause of the heavy load and must therefore use mo dels that give the
best value at the lowest amount of time In the comparison later on in this pap er I will show
b oth the numb er of discretizations and the calculation time for the di erent mo dels When I
say that one mo del is faster than an other it is normally with resp ect to the calculation time
that I have made the comparison
To get a clear idea of the behaviour of the di erent mo dels the diagrams are displayed
with option value on the vertical axis and computer p ower or real calculation time on the
horizontal axis The computer p ower is a relative factor that indicates how much time a
calculation of an option s price takes the real computer time on the other hand is the true
calculation time measured by the computer and is displayed in milliseconds If the computer
power or calculation time needed for one mo del is twice as high as for an other mo del for
the same accuracy in the option s price we come to the conclusion that the second mo del is
twice as fast and twice as go o d as the rst one
A computer program has been develop ed to simplify the evaluation pro cess and the com
parison This to ol makes it p ossible to draw graphs from several di erent mo dels in the
same diagram and makes it easy to compare the convergence of the di erent mo dels All the
diagrams shown in the pap er comes from this to ol It also allows making diagrams for hedge
parameters such as delta gamma omega etc The hedge parameters tell us much ab out the
mo del and it s accuracy near the barrier
The to ol is constructed so that implementation and testing of di erentderivatives are made
easy and fast This was already from the start a pronounced goal b ecause ABN Amro Soft
ware AB wants to implement several new derivatives in their system All these derivatives
need to be evaluated and tested A great thing ab out the to ol is that anyone including
those with no programming exp erience can use the to ol and get the maximum of help in
their evaluation pro cess from it In the end of this pap er I will showyou how the evaluation
pro cess is to b e done and how to use the to ol to make it fast The to ol b elongs to ABN Amro
Software AB and can not b e distributed with this pap er
Brie y ab out options
Options are traded all around the world in bigger and bigger quantities but what is an option
and why trade with them All options havean underlying asset As an example we have
the sto ck option that has a sto ck as its underlying asset Commo dities as metals or corn are
other underlying assets that an option can b e based on Often when dealing with options we
hear words like put and call or Europ ean and American but what do they really mean
If we start with the call when you by a call option contract you have bought the right
to buy a certain amount of the underlying asset to a xed price also called the strike price
with the notation K sp eci ed in the contract The put on the other hand gives you the right
to sell a certain amount of the underlying asset for the strike price K sp eci ed in the option
contract In b oth cases the one that mustsellorby the asset when the contract is exercised
is the one who originally wrote the option contract
If the option is a Europ ean style option it can only b e exercised at the maturityday For the
American style option contract it can b e exercised at any time during the options life i e
from the dayitwas written until maturity
These typ es are the most common but there are plenty of di erent typ es on the market
more or less traded Next we are going to lo ok at a typ e called barrier option it is yet not
traded in very big quantities
What is a barrier option
A barrier option is a path dep endent option with some kind of restriction of its validity That
is if the barrier is crossed the option will either b e valid and activeorinvalid and void The
barrier is a xed value due to the sto ck price e g if the current sto ck price is SEK the
barrier can b e placed at SEK the barrier will b e crossed if the sto ck price rises over SEK
The option can start it s life inactive and b ecome active when if the asset price crosses
the barrier i e kno ck in or start active and b ecome void when if it crosses the barrier i e
kno ck out The barrier can b e placed ab oveorbelow the currentstock price The option
can b e either a call or a put
If an option starts active as for kno ck out barrier options and when the option go es void
a rebate is often paid to the owner of the option contract
When combining these di erent typ es we will get eight di erent single barrier options de
scrib ed b elow
Up and out call
The up and out call starts active and stays so until the barrier is crossed or the expiry date
is reached If the barrier is crossed the rebate will b e paid If the expiry date is reached and
the sto ck price is ab ove the strike price a sum equal S K will b e paid else the value of
T
the option is If it s an American option the option can b e exercised at any time t as long
as it is active ThesumpaidwillthenbeS K Figure shows two di erent paths and
t
for an Up and Out call option Path crosses the barrier marked B as so on as it do es
it gets void and the option s price equals zero even if it gets back b elow the barrier b efore
maturityday Path that never crosses the barrier will get a payo at maturityday that is
greater than zero b ecause the asset price is b etween the strike price K and the barrier B
Up and in call
The up and in call starts inactive and stays so until the barrier is crossed If the expiry date
is reached and the barrier hasn t b een crossed the option will b e void and no money will b e
paid If the barrier is crossed the option b ecomes active but will not b ecome inactiveifit
Figure Up and Out call example The asset price following path will kno ck out when it
crosses the barrier the option b ecomes void and gets a value equal to zero If the asset price
is following path the option gets a value equal to S K
T
crosses the barrier again Therefore the value of the option is equal S K for a Europ ean
T
option and S K for an American option if it is exercised b efore maturity date If the
t
option b ecomes void no rebate will b e paid
Down and out call
The down and out call has the same behaviour as the up and out call with the di erence
that this instrument s barrier is placed b elow the sto ck price at time In this case the sto ck
price has to cross the barrier to b ecome void that is the sto ck price has to go b elow the
barrier to inactivate the instrument The sum paid is the same as for a down and out call if
the barrier is crossed a rebate will b e paid If the expiry date is reached and the sto ck price
is ab ove the strike price a sum equal S K will b e paid else the value of the option is
T
If it s an American option the option can b e executed at any time t as long as it is active
Then the sum paid will b e S K
t
Note this is the option that is going to be the standard example option all tests and
evaluations are done due to this barrier typ e See section
Down and in call
The down and in call has the same b ehaviour as the up and in call with the di erence that
this instrument s barrier is placed below the sto ck price at time In this case the sto ck
price has to cross the barrier to b ecome active that is the sto ck price has to go b elow the
barrier to activate the instrument When the barrier once is crossed it can never be void
The sum paid is the same as for the up and in call the value of the option is equal S K
T
for a Europ ean option and S K for an American option if it is exercised at time t If the
t
option b ecomes void no rebate will b e paid
Up and out put
The up and out put starts activejustlike the up and out call option if the option s payo
shall b e greater than zero the asset price has to lay under the strike price and it shall never
during its life have raised ab ove the barrier Here if the barrier never has b een crossed the
payo is K S for a Europ ean option and K S for a American option if it is exercised
T t
in advance
Up and in put
The Up and in put starts inactive and gets valid when crossing the barrier just like the Up
and in call option Here the option pays o if the asset price lay under the strike price and
if it at least one time during its life has b een ab ove the barrier As for the Up and out put
if the barier never has b een crossed the payo is K S for a Europ ean option and K S
T t
for a American option if it is exercised in advance
Down and out put
The Down and out put kno cks out and gets void liketheDown and out call if the barrier is
crossed Therefore the payo will b e greater than zero only if the asset price lay b elow the
strike price and if it never crosses the barrier during its life As for the Up and out put if
the barier never has b een crossed the payo is K S for a Europ ean option and K S for
T t
a American option if it is exercised in advance
Down and in put
The Down and in put kno cks in and gets valid and active when it crosses the barrier from
ab ove It pays o if the asset priveislower than the strike price and if the barrier has b een
crossed at least once As for the Up and out put if the barrier never has b een crossed the
payo is K S for a Europ ean option and K S for a American option if it is exercised
T t
in advance
Double barrier option
The two barrier option also referred to as corridor option can b e of twotyp es double kno ck
in or double kno ck out The rst starts inactive and b ecomes active when a barrier is crossed
from the outside the other starts active with the current asset price somewhere b etween the
barriers it gets void if any of the barriers are crossed This typeofoptionisnotgoingtobe
evaluated in this pap er this is a limitation and is discussed next
Delimitations
A barrier option can have one or more barriers as discussed ab ove In addition the barrier
can b e non constant In this pap er only single constant barrier options will b e discussed and
evaluated In section Further Development I brie y discuss how other contracts can
b e treated
Standard option contract
To make the comparison between di erent mo dels easy we will use one standard option
contract in the examples throughout this pap er It will b e a single barrier option of the typ e
Down and out call with the following data
S
K
B
r
T
No dividend is payed during the options life The de nition of all variables can b e found in
section
Black Scholes analytic formula and theory
In Fischer Black and Myron Scholes received the Nob el price for their option pricing
formula They have revolutionized the world with their formula that makes it p ossible to
calculate the option value for some options in an analytic way Before presenting the simple
analytic formulas I will show you the Black Scholes di erential equation from which the
formulas derive
C C C
S r S r C
t
S S
It is outside the framework of this pap er to derive this formula but more details can b e found
in Hull section This di erential equation is also the basis for the nite di erence
mo del see section
For the Black Scholes pricing formulas that will be presented below there are some as
sumptions that restrict the use They are as follows
The sto ckpays no dividends during the option s life
Europ ean exercise terms are used
An arbitrage free world is assumed
Interest rates and volatility remain constant and known during the option s life
The underlying asset price follows a geometric Brownian motion i e returns are log
normally distributed
The formula and all its variables is describ ed b elow Where
rT
cK T SN d Ke N d
and
rT
pK T SN d Ke N d
Here
S
log r T
K
p
d
T
and
p
T d d
The variables used ab ove
c Theoretical Call Premium
p Theoretical Put Premium
S Current sto ck price
T Time until expiration
K Option striking price
R Risk free interest rate
N Cumulative standard normal distribution
Standard deviation of sto ck returns Volatility
In order to understand the formula itself equation wedivideitinto two parts The rst
part SN d derives the exp ected b ene t from acquiring a sto ck outright This is found
bymultiplying sto ck price S bythechange in the call premium with resp ect to a change
rt
in the underlying sto ck price N d The second part of the mo del Ke N d gives the
presentvalue of paying the exercise price on the expiration day The fair market value of the
call option is then calculated by taking the di erence b etween these two parts
The assumptions mentioned ab ove make formula and applicable on a limited number
of derivatives The assumption that says that no dividends can b e paid during the option s
life restrict the use dramatically most companies pay dividend to their share holders and
when a dividend is paid the call option value for example gets lower Continuous dividend
paymentisanapproximate way to deal with discrete dividend payment Rob ert Merton came
up with a solution for this in The assumption that Europ ean exercise terms are used
means that the option can only b e exercised on the maturitydayincontrary to American
exercise terms that means that the option can b e exercised at any time during the option s life
In addition to the assumptions ab ovethe Black Scholes formula is not applicable on path
dep endent options such as barrier options Wethenhave to nd di erent mo dels that relax
the assumptions of no dividends only Europ ean exercise terms and not path dep endentop
tions For numerical mo dels in discrete time and asset price are all these assumptions relaxed
which gets us to the following chapters where di erentnumerical mo dels will b e examined
and evaluated
Binomial Tree Mo del
The binomial tree mo del is asimpleandpowerful mo del for pricing standard options such
as Europ ean and American style options For these the mo del is widely used The results
retrieved with around time steps is often a go o d approximation to the accurate theoreti
cally value and the calculation time required is very short Two di erent binomial mo dels will
be presented here the rst one prop osed byCox Ross and Rubinstein see is the most
intuitive one and is easy to understand The second mo del was prop osed byTrigeorgis
see page and is an even b etter approximation to the theoretically correct option
price I will present this mo del as an educational example to make it easier to understand
the log transformed calculation metho d also used for the trinomial and the nite di erence
mo dels presented later in this pap er
These mo dels are also applicable on options with discrete and continuous dividends Fur
thermore it is very simple to implement early exercise conditions as needed for American
option pricing Options with path dep endency as barrier options can also b e treated
When using these mo dels with the barrier condition attached to the mo del for pricing stan
dard options we will get a very unstable b ehaviour esp ecially near the barrier I will show
away to decrease this instability as suggested byBoyle and Lau which is applicable on
the mo del byCox Ross Rubinstein Moreover I will showhow to implement early exercise
condition and discrete dividends within the binomial tree framework Finally I discuss hedge
sensitivities such as delta gamma and omega
Cox Ross Rubinstein Binomial Mo del
The Cox Ross Rubinstein binomial mo del was presented in a working pap er called
Option Pricing A Simpli ed Approach Their theory has laid ground to the binomial
mo del that I will present b elow
To b egin with the binomial tree has equal time steps t througout the whole tree so are
also the up ratio u the down ratios d and the transition probability p For simplifying
the notation we decide up on a numb ering system i j as in gure Eachvariable has it s
index derived from this numb ering system
We will only lo ok at recombining trees for this mo del that is one up movefollowed by a
down move will result in the same no de as one down movefollowed by one up move ud du
Figure shows a one step binomial mo del of the asset price also called the jump pro cess
Cox Ross and Rubinstein have chosen the multiplicative jumpsizes u and d to have the
relationship
u
d
which satis es the jump condition mentioned ab ove They also show that u must satisfy the
condition
p
t
u e
Since we are working in a risk neutral and arbitrage free world see page and the
exp ected return from a sto ck is equal the risk free interest rate r Then for one time step
t the exp ected value for the option given the values one time step ahead is
r t
S e pS pS
i j i j i j
or
r t
e pu pd
Figure Numb er system for the Binomial Tree
Figure Jump pro cess for the asset price for the binomial tree
whichgives the explicit expression of the transition probability
r t
e d
p
u d
Trigeorgis Mo del
Trigeorgis has develop ed a mo del where the asset price is log transformed This means that
he assumes that it is the logarithm of the asset price that evolves as a brownian motion with
constant drift The logarithm of the asset price is normally distributed with constant mean
and variance The numb ering system will b e the same in this mo del but we ll declare the up
and down transition probabilities as p and p where p p An up move takes you to
u d d u
y y and a down movetakes you to y y here wewillcho ose equal jump size and
u d
therefore y y Figure shows a one step binomial mo del of the logarithm of the
d u
asset
Figure Jump pro cess with equal jump sizes for the logarithm of the asset price
Applying It s formula the continuous time risk neutral pro cess for y lnS t can be
shown to b e
dy dt dz
where
r
The mean and variance of the continuous time pro cess can b e approximated with the bino
mial pro cess for y this leads us to equation
E y p y p y t
u u d d
and
E y p y p y t t
u d
u d
The assumption that wehave equal jump size gives us y y y This applied to
u u
equation and gives
p y p y t
u d
and
p y p y t t
u d
which takes us to the goal
p
y t t
u
y y
d u
t
p
u
y
p p
d u
Next I m going to showhow to use this mo del for pricing barrier options
Calculating a barrier option s value using Trigeorgis Mo del
The following pro cedure to calculate an option s price is allmost the same as for the Cox
Ross Rubinstein Mo del to get a more exact description for this mo del I direct you to the
litterature in the eld Note the calculations are done for a Down and out call option as for
the rest of the examples in this pap er see section
We have all the information we need to build the tree at maturity date we start with
computing the sto ck price at the lowest p osition
N y
i
S Se
N
y
by adding e to the asset price We can then compute all other values from S to S
N N N
i i i
at the no de ab oveas
y
S S e
N j N j
Knowing all the sto ck prices at maturitydatewe can now get the call option s price as the
maximum value of S K and for all j as
N j
i
C max S K
N j N j
i
Wenow do the calculation of the option s price in an iterativeway from maturityday back
to current time We compute the option prices at time t knowing the prices at t t as
C p C p C
i j u i j d i j
After iterating N times we obtain the option s price C at current time If we during the
i
iteration want to apply the early exercise condition as for American style options wehave
to calculate the sto ck price spread at time t from the spread at time t t This can b e
done in a very simple way
S
i j
S for j i
i j
y
e
The early exercise condition can then b e added where the option s price is set to
C maxC S K
i j i j i j
We can nally add the barrier condition to the mo del This condition is also very simple
to implement The only check that has to b e done is if the barrier has b een crossed at any
no de if it has we set the option s value at that no de to zero
When we use the metho d ab ove for implementing the barrier condition we will get an error
The calculated value will always b e ab ove the theoretically correct value for a Down and out
call option Figure b elowshows how the barrier is represented in the mo del You can see
that the no de just below the true barrier lies where the mo del will represent an imaginary
barrier The distance b etween the true barrier and the imaginary one is the source of error
that will result in a error in the option price The closer the barrier lies the current asset
price the lower will the option price b e As wehave seen the imaginary barrier will always
laybelow the true one and therefore the error will always b e p ositive
Figure Error generated by the barrier
While changing the numb er of time steps the distance b etween the true barrier and the imag
inary barrier will change It will not always get smaller in a p erio dical manner the imaginary
barrier will make a jump so that the distance grows to a maximum This o ccurs when a no de
passes the true barrier from b elow when passed the imaginary barrier will lay on a new
no de b elow the true barrier The uctuation in the option price is presented in Figure
Figure The sawto oth pattern for the binomial tree metho d Option value is presented on
the vertical axis and the p ower rate on the horizontal axis
How this b ehaviour can b e eliminated bycho osing only the b est p oints I will showyou next
How to cho ose only the b est p oints
The way the b est points is chosen for a binomial tree mo del was suggested by Phelim P
Boyle and Sok Ho on Lau in and their working pap er was published in the Journal
of Derivatives The mo del is based on the binomial mo del sugested by Cox Ross and
Rubinstein earlier describ ed in this chapter Their metho d makes the option price converge
much faster than for a standard binomial tree mo del I will show why this mo del gives a
b etter approximation of the option price Lo oking att Figure we can se that the option
price converge towards a value that is close to the lowest value in the graph Weknowthat
the error Figure is at a minimum while the barrier lies just ab ovea layer of horizontal
no des What Boyle and Lau did were that they chose the numb er of time steps in a waythat
makes this criteria ful lled They came up with aformula that cho ose the numb er of time
steps for us as
m T
F m m
S
log
B
For some m the largest integer that is smaller than F m represents the numb er of time steps
for the p oint at the b ottom b etween twosawteeth see gure For a sp eci c m knowing the
F m wecancho ose the b est numb er of time steps N as the largest integer that is smaller
than F m
Here the barrier B will lie between the asset price after m down jumps and the asset
price after m down jumps this gives the following relationship
m m
Sd B Sd
The factor d is the multiplicativedown move as de ned byCox Ross and Rub enstein pre
sented earlier For di erentvalues of m we will get di erentnumb ers of time steps
When I implemented this function I wanted to calculate the option s price at a sp eci c
power rate I then calculated the m value from the number of time steps where I got the
suitable p ower rate see secion In general terms we assume that we wantto calculate
the value of a barrier option with a sp eci c numb er of time steps N we then calculate m
from
s
S
N log
B
m
T
Nowwe calculate m as the smallest integer value bigger than m Then weknow that the
numb er of time steps given by equation gives the b est p oint closest to but greater than
N thenumb er of time steps requested
I ve made a calculation using this metho d and found that the result is very good Figure
shows the values calculated in the same interval as for the standard Cox Ross Rubinstein
mo del Please make a comparison with gure but don not forget that the scale on the y
axes are di erent
As you have seen this mo del has a disadvantage the value of the option do es not converge
strictly this makes it hard to knowwhenyou have reached a correct value E g often you
want to double the time step to se how the value converge but this isn t p ossible to do for
this mo del Here you have to lo ok at a much bigger range of di erent values for di erent
numb er of time steps to get a picture of the convergence
Computing hedge sensitivities
Here I will presenthow to compute the hedge sensitivities for a binomial tree metho d Those
hedge parameters that do not di ers from the standard way of calculation will not b e dis
cussed here They can b e found under section where you also can nd de nitions of all
the hedge sensitivities mentioned in this pap er
Delta Gamma and Omega is calculated from the binomial tree We can also calculate
Figure The numb er of discretizations is chose in a way that minimize the error Option
value is presented on the vertical axis and the p ower rate on the horizontal axis
Theta from the tree in a simple way To b egin with Delta is calculated as
C C C
S S
S
To b e ab el to calculate the Delta you must have more values than the one we already have
at currenttime j for this we need asset values and option values two time steps forward
at time j Note These can b e saved during the iterative calculation pro cess as earlier
describ ed The same holds for the Gamma this parameter is calculated as
C C C C
C
S S S S
S S
S
The Omega is then calculated as follows
C C C C C C C C
S S S S S S S S
C
S S S S
S S
S
The last hedge parameter that can b e obtained directly from the tree is Theta This is the
rateofchange of the option price with time It is calculated as
C C C
t t
In the next section we re going to lo ok at a di erent typ e of mo del called Trinomial tree
mo del
Trinomial Tree Mo del
Here we assume that the asset price follows a geometric Brownian motion GBM presented
by the sto chastic di erential equation SDE b elow
dS Sdt Sdz
It is more convenient to work with the log transformed asset price y t lnS t also
describ ed for the Trigeorgis binomial tree mo del This transform equation and gives us
dy dt dz
where
Before continuing the calculation we will present the mo del and its jump probabilities Going
from one time step t to t t the log transformed asset price y can change to either y y
with probability p y y with probability p or stay the same with probability p The
u d m
mo del is illustrated b elow in gure
Figure The jump pro cess for the natural logarithm of the asset price for the trinomial tree
mo del
I have also de ned a numb ering notation system which reminds of the numb ering system
for the binomial mo del Normally thenumb ering system for the trinomial mo del has a cen
ter with a constant log transformed asset price at index it is then growing upwards with
p ositive indexes and downwards with negative indexes When implementing the mo del in
a computer calculation function using vectors for storing the values this numb ering system
is disturbing b ecause most programming languages use vectors with non negative indexes
Therefore wecho ose the lowest p osition at each time step in the tree to have index Figure
presents this numb ering notation system
Each p osition in the tree has an index i representing the time step and an index j representing
the log transformed asset price index The index i go es from to N at maturity date j go es
from at the lowest p osition in the tree up to N at the highest p osition
We can now continue the calculation of the jump probabilities and the jump size The
relationship b etween the parameters of the continuous time pro cess and the trinomial pro
cess is obtained by equating the mean and variance over the time interval t and requiring
that the probabilities sum to one this gives
E y p y p p y t
u m d
Figure The Trinomial mo del s numb ering system
E y p y p p y t t
u m d
and
p p p
u m d
When wesolve the equations and with resp ect to the probabilities we obtain
t t
p
u
t
y
y
t t
p
m
t
y
y
and
t t
p
d
t
y
y
To determine the range of asset price values in the grid we consider what Clewlow and
Strickland page says A reasonable range of asset price values at maturitydateof
the option is three standard deviations on either side of the mean So knowing the standard
deviation we can calculate the y as
p
T
y
N
i
We can now start to initialize the tree variables starting with S at maturity We rst set
N j
i
the asset price at the lowest p osition in the tree at maturitydateto
N y
i
S S t e
N
i
Then we can calculate all the asset prices ab ove in an iterativeway as
y
S S e
N j N j
i i
The option prices at maturitydaycannow b e calculated from the following expression
C max S K
N j N j
i i
Wenow start the iterative pro cess to calculate the option prices throughout the tree for each
time step backto we will calculate the option prices In general terms at time step i we
will calculate the prices from j to j i as
r t
C e p C p C p C
i j u i j m i j d i j
r t
This is the discounted exp ectation where e is the discount factor When we reach the
current time step at i we can get the option value at C If wewanttomake the mo del
valid for American style options wehave to include an exercise condition at every time step
this condition is the same as when we calculate the option prices at maturity written in a
more general wayweget
C maxC S K
i j i j i j
Here you can see that we need to know the asset price in every no de throughout the tree we
can nd these values one time step ahead in the tree where
S S
i j i j
see gur It is still p ossible to keep all asset prices in one singel vector but using equation
requires that we start at the b ottom j working upwards to j i so that we don t
overwrite the data Finally we can implement the barrier condition by adding an extra check
to see if the barrier is crossed This is done for every no de if an asset price is below the
barrier that is for a Down and out call option the value of the option will b e set to zero
in that no de That is if
S B
i j
then
C
i j
As for the standard binomial tree mo del we get a very alternating b ehaviour when the current
asset price lies near the barrier Peter Ritchken found a solution for this problem in
He de ne a stretch parameter that adjust the lattice so that a horizontal row of no des
always will lie on the barrier Then the error as we discussed in section will b e limited
In next section we discus how to implement the stretch parameter
Implementing a stretch parameter
Implementing a stretch parameter is an e cient way of adjusting the standard trinomial
mo del it is well known and often used for pricing barrier options Mayb e that is b ecause
the pro cedure is quite easy to understand and not to hard to implement I will start with
de ning the stretch parameter in general terms by lo oking at a sp eci c case with a Down
and out option First we de ne a variable n thatisthenumber of down moves that leads
to the horizontal layer of no des just ab ove the barrier The exact or decimal number of down
moves to the barrier is de ned as and is calculated as
S
ln
B
p
t
Nowwe can calculate n as the largest integer value that is smaller than and then de ne
as
when
n
Furthermore we rede ne the jump probabilities and the jumpsize adjusted to the The
jump size y as de ned in the standard mo del is chosen as
p
y t
and the jump sizes also de ned ab ove eq is chosen as
p
t
p
u
p
m
and
p
t
p
d
Notice that if the lattice collapses to the usual binomial lattice As I mentioned
this mo del is valid in general and not only for the Down and out barrier option For an
Up and out barrier option the will takeavalue based on the number of up moves b efore
crossing the barrier instead of the number of down moves
Testing the adjusted trinomial mo del
The convergence is the most imp ortant measurement of the options b ehaviour This is easiest
presented in a graph with the option s price on the vertical axis and the power rate i e
growing number of N on the horizontal axis see section In gure this is displayed
i
for the adjusted trinomial mo del where the current asset price lies near the barrier The
standard case parameters are used as describ ed in the intro duction of this pap er
Figure The adjusted trinomial tree mo del The option s value is represented on the
vertical axis and the p ower rate on the horizontal axis
This mo del do es not converge strictly you can clearly see the singularities where the price
jumps To see a comparison with other mo dels you should lo ok at section where the
adjusted binomial tree mo del this adjusted trinomial mo del and the nite di erence mo del
are compared
Computing hedge sensitivities
Here I will presenthow to compute the hedge sensitivities for a trinomial tree mo del Those
hedge parameters that do not di ers from the standard way of calculation will not b e dis
cussed here they can be found under chapter where you also can nd de nitions of all
the other hedge sensitivities mentioned in this pap er
The way of calculating the hedges do es not di er much from the way they are calculated
from the binomial tree mo del Here we can calculate the Delta one time step from current
time as
C C C
S S
S
To b e able to calculate the Delta wemust havemorevalues than the one we already haveat
currenttime j for this we need asset values and option values one time step forward at
time j The same go es for the Gamma but here we need option values two time steps
forward at time j The Gamma is calculated as
C C C C
C
S S S S
S S
S
The Omega is evaluated at the third time step from currenttime j it is calculated as
C C C C C C C C
S S S S S S S S
C
S S S S
S S
S
The last hedge parameter that can b e obtained directly from the tree is the Theta This is
the rate of change of the option price with resp ect to time It is calculated as
C C C
t t
In the following section we will lo ok at the Finite Di erence mo del
Finite Di erence Mo del
I will present for you a robust and fast nite di erence mo del that converges to a correct
value much faster than earlier published metho ds do see the titles b elow The metho d is
robust b ecause its stability dep ends on neither the numb er of time steps nor it s number of
price discretizations b etween the current price of the underlying asset and the barrier This
metho d converges fast when the price of the underlying asset is close to the barrier That is
b ecause the no des of the grid hit the barrier and the current price for the underlying asset
exactly Furthermore there is always at least one step of discretization b etween the current
price of the underlying asset and the barrier I will compare this metho d with some earlier
published work on the eld
On pricing barrier options byPeter Ritchken Trinomial Tree Mo del
An explicit nite di erence approach to the pricing of barrier options byP Boyle
and Y Tian
Application of Finite Di erence Metho d for Pricing Barrier Options by G Io e and
M Io e
When comparing di erent mo dels for pricing derivatives it s normally the number of dis
cretizations that is considered but di erent numerical mo dels use di erent typ es of dis
cretizations Therefore we will mainly consider the calculation time as the dimensioning
factor in this section
What we are going to do is to solve the Black Scholes Partial Di erential Equation PDE
That is done using numerical approximations of the derivatives We rewrite the Black Scholes
PDE equation presented in section
C C C
r S S
r C
t
S S
Before continuing wehave to do some changes in the PDE b ecause wewant to log transform
the asset price That is
y tlog S t
Boyle and Tian show how to write the log transformed Partial Di erential Equation
PDE where the outcome is
C C C
r r C
t y y
The derivatives can be de ned in several di erent ways here we ll lo ok at three di erent
approximations rst an explicit mo del and then two di erent implicit mo dels
Explicit approach
The mo del I will present here was prop osed by P Boyle and Y Tian in Their
mo del which they refer to as Mo di ed Explicit Finite Di erence MEFD makes the op
tion price converge to a theoretically correct value much faster than earlier prop osed mo dels
such as the binomial tree mo del or the trinomial tree mo del Their main idea is to
put a horizontal line of no des on the barrier The drawback is that most of the time there
will b e no no de that is laying on the current asset price implying that no no de will p ossess
the right option value They prop ose a wayofinterp olate the option value at current asset
price by quadratic interp olation using the closest no des ab ove and b elow the current asset
price To get these values at current time t we also need to make a calculation one time
step back in time t thismakes the implementation a bit harder but more ab out that later
This mo del is explicit which means that byknowing the option values at time step i
makes it p ossible to calculate the value of the option at the current time step i This is il
lustrated in gure where the option value C kan b e calculated knowing C C
i j i j i j
and C
i j
Figure An explicit nite di erence mo del
The evaluation p ointis i t S where we denote
i j
C C i t S C
i j i j
How are the derivatives approximated in that p oint We use numerical approximation where
the derivatives is calculated as
C C C
i j i j
t t
C C C
i j i j
y y
and
C C C C
i j i j i j
y y
We can nowputthederivatives into the PDE eq and then we get an explicit expression
of the option price C at a no de i j as
i j
C C C C C C C
i j i j i j i j i j i j i j
r rC
i j
t y y
If we rewrite this equation we can see the explicit b ehaviour
C p C p C p C
i j u i j m i j d i j
where p p p are given by
u m d
p t
u
y y
p t r t
m
y
p t
d
y y
and
r
We now have all the formulas for starting the implementation pro cess In the following
section I will showhow this mo del can b e implemented and how the quadratic interp olation
is done
Implementation
As b efore we will start the calculation by de ning a grid which has a asset price spread of
p
T on either side of the current asset price as describ ed in section We will have N time
i
steps b etween current time and maturitytimeand N discretizations of the asset price
i
Note that the grid is built from one step back in time t Starting at the lowest no des j
the grid is growing upwards to the highest layer of no des at index j N N Now
j i
wecho ose the barrier to lay on a horizontal line of no des with j N Because there are
d
no no des laying on the current asset price wehaveto ndthelayer of no des laying closest
De ning y as the log transformed current asset price Boyle and Tian prop ose a way of
nding the number of no des j between the barrier and the no des closest to the current
asset price as
y y
d
j h i
y
where h i returns the integer p ortion of the argument WeknowthatthereareN layers
i
of no des below this p oint the barrier will then layon a layer with index j N j
i
b ecause the layer of no des laying closest to the current asset price has index j N The
i
grid is illustrated in gure To calculate the asset prices at di erent no des going from
Figure An explicit nite di erence mo del where y lies on the barrier and y is the layer
d k
of no des that is closest to the current asset price The upp ermost layer of no des layaty
u
j up to j N we start with calculating the lowest asset price as
j
y N j y
i
d
S e
From this p osition we can easily calculate the rest of the asset prices simply by adding a
y
factor e to the asset price one level b elow in an iterativeway this can b e done using the
following expression
y
S e S
j j
Nowwe can compute the option values at maturity date by adding the exercise condition if
the barrier is crossed then
C max S K
N j j
i
else
C
N j
i
In general
p
y t
but the has to b e chosen in a way so that the mo del is stable see section Here we
cho ose
p
which according to Boyle and Tian gives b etter accuracy of the option price then other
As I mentioned b efore the time interval go es from t to t and do es not start at current
T
time t as normally This makes the implementation of the mo del confusing b ecause the
arrays as I discussed in section starts at index in most programming languages When
the calculations at maturitydayis done wehave N iterations left to do starting at t
i T
stepping backuntil we reaches t Within this iterative pro cess we will calculate the option
values with the barrier condition added as follows if the barrier is crossed then
C p C p C p C
i j u i j m i j d i j
else
C
i j
Then we add the exercise condition if it is an American style option as
C maxC
i j i j
When writing the co de only one array is needed for storing data and not a matrix as ab ove
b ecause the mo del is explicit and the old values can b e overwritten
As you have seen we do not need to calculate the values at t it is just the grid that
is built that waysowe get three valuesatcurrent time t The option values lo cated in the
array C while nished the iteration pro cess is used to calculate the option value at current
asset price Boyle and Tian describ e the basic principle of the quadratic interp olation as
follows Imagine a real valued function g x Suppose that we wish to obtain the value of the
function at x The functional form of g may not be known but the values of the function at
x x and x are known and denotedbyg x g x and g x respectively Without loss
of generality let x x x If x x x the value of g x may be approximated by
x x x x x x x x x x x x
g x g x g x g x
x x x x x x x x x x x x
Ihave written a simple function for this quadratic interp olation in which I put the output
values from the iterative calculation pro cess with Boyle and Tian notation it gives
x log S g C
N N
i i
x log S g C
N N
i i
g C x log S
N N
i i
x log S t
Here the current asset price S is closest to S as de ned in the b eginning of this section
N
i
In this mo del we have a one dimensional freedom as shown ab ove this parameter can
be chosen in a waysothatboth the current asset price and the barrier is hit exactly bya
layer of no des In this case the mo del has collapsed to the trinomial tree mo del see gure
whichgives a faster convergence
If wehavetwo barriers the parameter can b e chosen so that b oth barriers but not the current
asset price are hit exactly Without any tests I b elieve that this mo del will b e b oth accurate
and fast for pricing this typ e of contract Furthermore the implicit mo del presented later
will mayb e deliver an even faster convergence if this interp olation metho d is used More
ab out this in section Further Development Nowwe are going to lo ok at the stability
and convergence conditions for the explicit mo del
Stability and convergence
The stability is imp ortant to study in order to knowinwhich cases the mo del is applicable
As you have already seen the trinomial tree mo del can also b e derived from the PDE eq
and not only as we did in section This makes the stability conditions derived b elowvalid
for the trinomial tree as well
We want the stability conditions for the transformed PDE eq but those are the
same as for the homogeneous equation b elow
C C C
r
t y y
If we apply the discrete derivatives eq as b efore we will get the following discrete
partial di erential equation
C C C C C C C
i j i j i j i j i j i j i j
r
t y y
If we rewrite this equation we can see the explicit behaviour as earlier but with di erent
jump probabilities
C p C p C p C
i j i j i j i j
Here p p and p are given by
p t
y y
p t
y
and
p t
y y
If wenow add the dep endency condition b etween y and t eq we get the following
three equations
p
t
p
p
and
p
t
p
A su cient condition for stabilityisthatp p and p is non negative see Boyle and Tian
The can not b e negative b ecause the steps of discretization can not b e negative We
start with lo oking at p that restrict the stabilityas
The other conditions p and p are satis ed if the restrictions b elow are ful lled
p
j j t
For the trinomial mo del prop osed by Ritchken wehave another condition for the that
makes b oth the barrier and the current asset price hit by a no de se section Then the
stability condition makes it di cult to cho ose the b est suited time steps and jump sizes For
the implicit mo dels b elow we ll havenosuch problems They are allways stable and therefore
we can cho ose time steps and jump sizes indep endently this gives this mo del an advantage
to the explicit b ecause it is easier to adjust these parameters for di erent contracts i e
contracts where the current asset price is close to the barrier
Implicit Cranc Nicolson
This metho d is a so called fully centered metho d which means that it replaces the space and
time derivatives with nite di erences centered at an imaginary time step at i The
dep endencies for this mo del is describ ed in gure
Figure An implicit nite di erence mo del using Cranc Nicolson
We see here that the evaluation point p osition is i t S which means that we
i j
approximate the continuous option value in that p oint to the following value
C C
i j i j
C C i t S
i j
The partial derivatives are approximated with Crank Nicolson and are describ ed in discrete
terms in the grid as follows
C C C
i j i j
t t
C C C C C
i j i j i j i j
y y
and
C C C C C C C
i j i j i j i j i j i j
y y
The PDE eq can now b e presented in a discrete way
C C C C C C
i j i j i j i j i j i j
r
t y
C C C C C C C C
i j i j i j i j i j i j i j i j
r
y
Rewritten as
p C p C p C p C p C p C
u i j m i j d i j u i j m i j d i j
Were the p p and p are given by
u m d
p t
u
y y
r t
p t
m
x
and
t p
d
y y
Construction of the grid
The main idea with this metho d is to place b oth the barrier and the sp ot price on the grid s
no des This is always p ossible while dealing with single barrier options as in this pap er To
determine the range of asset price values in the grid we consider what Clewlow and Strick
land page says A reasonable range of asset price values at maturity date of the
option is three standard deviations either side of the mean Tomake this mo del converge
to the sixth decimal wemust cho ose an even bigger range than prop osed ab ove for the stan
dard option contract a range of standard deviations ab ove the current asset price is enough
To understand the complex grid presented below b etter we lo ok at gure where we
can se all the variables and their p osition in the grid
The lowest no des in the grid is placed on the barrier that is b ecause we know that all
option values b elow the barrier is zero for a Down and out call option We then decide
how many no des wewantbetween the barrier and the sp ot price If the mo del shall b e valid
the minimum amount of discretizations k is equal to one b ecause wemust do at least one
calculation in this area When we have determined the number of no des between the two
critical p oints in the grid we are going to decide up on how many steps of discretization we
will haveabove the no de laying on the sp ot price
We start with determine the y as
y y
d
y
k
Where
y log S
and
y log B
d
will lie next The log transformed asset price at the upp ermost horizontal layer of no des y
N
j
ab ove y which is calculated as b elow
u
p
y log T S
u
Figure The grid used for the Cranc Nicolson implicit mo del Here y lies on the barrier
d
and y lies on the current asset price The upp ermost layer of no des y lies next ab ove y
k N u
j
When having the discretisation step of the sto ck price we can determine howmany no des
N that are required to get the needed price spread N mustbeaninteger value so wehave
j j
to cho ose the upp ermost no de to lie ab ove the upp er limit of the price spread Whichgives
y y y y
u d u d
i h i N hk
j
y y y
d
Where h i is the op erator that returns the upp er integer value of the argument Then the
price spread of the underlying asset in the mo del go es from the barrier B at p osition up
to S at p osition N as
N j
j
y N y
j
d
S e
N
j
The sto ck price at no de j is calculated as
y j y
d
S e
j
When we have determined the sto ck price for the no des we will determine the discretisa
tion in time b etween the current time and the maturity date Howmany time steps N are
i
reasonable to cho ose The mo del works with any number of steps greater than but of
N
j
course one step won t give an accurate value Ihavechosen N which makes the mo del
i
converge fast Then the time step t is given by
T
t
N
i
We havechosen a spread of the sto ck price for the grid b ecause we want the value at the
current time no de t y to dep end of all values at all no des on this interval In this mo del
k
Ihavechosen Cranc Nicolson as the discretization mo del this is an implicit metho d b ecause
it requires knowing the values not only one time step forward but also the values ab oveand
below the present value This gives us an equation system that includes all the equations
given at each no de one step back from maturity That is the value at the current time no de
t y will always more or less dep end on the upp ermost value at maturity
k
If wetake equation and generate an equation for each j where j N we will get
j
an equation system with the app earance as b elow
C
i N
j
C
i N
j
p p p C
u m d i N
j
p p p C
u m d i N
j
p p p C
u m d i
p p p C
u m d i
p p C
u m i
C
i
u
p C p C p C
u i N m i N d i N
j j j
p C p C p C
u i N m i N d i N
j j j
p C p C p C
u i N m i N d i N
j j j
p C p C p C
u i m i d i
p C p C p C
u i m i d i
p C p C p C
u i m i d i
p C p C
u i m i
Boundary conditions
The upp er and the lower equation in the system ab ove are results of two b oundary conditions
First the upp er one is a result of the assumption that the rate of change of the option value
with resp ect to the sto ck price is far from the current asset price i e
C
if S S
j y
k
S
gives
C C
i N i N u
j j
where
S S
u N N
j j
The barrier gives the lower b oundary condition On the barrier the option values are equal
to zero Applied on equation at the center p osition j gives
C
i j
C
i j
and the following equation
p C p C p C p C
u i m i u i m i
The option values are known at maturity date this gives the nal b oundary condition nec
essary for starting the calculation
We solve the equation system for each time step starting at the maturity date to get the
option values one time step back in time The equation system to solve is tri diagonal which
makes the calculation more e cient with resp ect to calculation time
Next we are going to lo ok at the third nite di erence mo del this one is implicit but not
fully centered
Implicit not fully centered
We will nally take a lo ok at another implicit nite di erence mo del The implementation is
made almost the same way as for the one ab ove using Cranc Nicolson approximations The
di erence is the way of approximating the derivatives I am going to show you that this
mo del is not as fast as the Cranc Nicolson though it is interesting to see the di erence in
convergence for the mo dels Even though a fully centered mo del as the one using Crank
Nicolson is faster with resp ect to the number of discretizations it is not necessarily faster
with resp ect to calculation time The equation system for the Cranc Nicolson implementa
tion is more complex whichmaymake a di erence Moreover the result retrieved from this
mo del was presented in a working pap er by I Io e and M Io e but the Implementation
I will presentbelowdoesnot converge as fast as their similar implementation The reason
for this is hard to say since they have not describ ed their implementation
To get a clear idea how the derivatives is computed we lo ok at gure that illustrates
this As you see standing at time step i the option price one time step forward is calculated
Figure An Implicit Finite Di erence mo del
from the option prices at p osition i as well as from the option prices at p osition i This is
the implicit b ehaviour in this mo del Wewant to calculate the option prices from values we
do not have explicitly therefore wehave to solve an equation system that is derived
below
j y
The continuous option value at time i t and asset price S e i e C i t S is
i j i j
approximated to the value in the grid at p osition i j
C C i t S C
i j i j
Before presenting that equation we approximate the following derivatives as
C C C
i j i j
t t
C C C
i j i j
y y
and
C C C C
i j i j i j
y y
Which gives us the following equation when approximating the continuous derivatives in the
transformed Black Scholes PDE eq
p C p C p C C
u i j m i j d i j i j
where
p t
u
y y
p t r y
m
y
and
t p
d
y y
Nowwe can write the equation system
C
i N u
j
p p p C C
u m d i N i N
j j
p p p C C
u m d i N i N
j j
p p p C C
u m d i i
p p p C C
u m d i i
p p C C
u m i i
As describ ed b efore in section this system is solved using an algorithm solving tri diagonal
equation systems e ciently This is imp ortant if using a function based on Gaussian elimi
nation the calculation will b e muchslower For further description of the implementation I
direct you to the previous section The results retrieved from the di erent mo dels can b e
viewed in next section where we make a comparison b etween all the Finite Di erence mo dels
Comparison between the Finite Di erence Mo dels
As I mentioned in the b eginning of this pap er the main idea when comparing di erent mo dels
is not the numb er of time steps required to reach a certain level of accuracy but to compare
the total calculation time Though there are some exceptions when comparing one of the
mo dels presented in this pap er with a similar mo del prop osed in some other pap er the lack
of calculation time data makes this legit In the working pap er by M Io e and I Io e
the implementation is not presented so I only have the results and have b een unable to
repro duce the exact results The results from the two similar mo dels are presented in table
We can see that the results retrieved from the mo del prop osed by Io e and Io e converge
faster than those coming from my non fully centered implicit mo del My implementation
pro duce results less accurate than their implementation so we come to conclusion that it is
p ossible to make a more e cient implementation Therefore I should compare the mo del
prop osed by M Io e and I Io e and not my non fully centered implicit mo del with other
di erentmodelstoevaluate which one that is the fastest with resp ect to the calculation time
Anyway the di erence is not that big and we can use the results retrieved here to continue
the evaluation pro cess
N
j
When generating the results presented in table I used the ratio N which I be
i
lieveIo eandIo ehavechosen in their mo del to o But if wecho ose a greater N we get a
i
b etter result i e if wecho ose N N the result will converge faster with resp ect to calcu
i j
lation time though N will not b e the same in the two cases This b ehaviour is probably
j
caused by the approximation of the derivatives the accuracy dep ends more on how many time
prop erties there are than howmany discretizations of the asset price that is done see table
Number of Down and Out Down and Out
time partitions Call price Call price
Io e Io e implicit not fully centered
Accurate value
Table Comparison b etween the implicit mo del prop osed by Io e and Io e and a non
fully centered implicit mo del presented in section
As I mentioned ab ove the explicit mo del collapse into the trinomial mo del if we try to
put b oth the current asset price and the barrier on a layer of no des Therefore this mo del
will not be taken into consideration here b ecause it will be treated in section where the
b est one of all the di erenttyp es of mo dels is compared Here wewant to nd which one of
the two implicit mo dels that is pro ducing the fastest convergence with resp ect to time
In table we can see the di erence when comparing option prices with resp ect to calcula
tion time retrieved from di erent mo dels Here wehave implemented an non fully centered
N
j
implicit mo del with a ratio N as well as one with a ratio N N The third mo del
i i j
shown in the table is the Implicit mo del using Crank Nicolson approximations
Six decimals is way to exact when pricing options but when evaluating which one that
pro duces the b est results wehavetolookatthismany decimals b ecause as youseeinthe
right column the price has already converged to four decimals after ms Even if weim
prove the non fully centered mo del as in the middle column we still have a great di erence
this mo del do esn t converge to four decimals b efore calculation time greater than ms
We arealsogoingto lo ok at a graphical representation of the results retrieved from these
mo dels in gure Finally we can app oint the mo del using Cranc Nicolson approximations
to be the fastest of the mo dels presented in this section To see comparisons b etween the
di erent typ es of mo dels please go to section Next we are going to lo ok at the Monte
Carlo simulation mo del
Calculation Down and Out Down and Out Down and Out
time ms Call price Call price Call price
Implicit section Implicit section Implicit Cranc Nicolson
N
j
N N N
i i j
Table Comparison b etween two di erent implicit mo dels the non fully centered and one
with Cranc Nicolson which is fully centered Two di erent implementations of the non fully
centered mo del are present The comparison is made with resp ect to calculation time
Figure Graphs representing two implicit mo dels one using central di erence and the
other Cranc Nicolson approximations to the derivatives
Monte Carlo Simulation
Monte Carlo simulation is a di erentway of estimating an option s price It di ers from the
tree approaches earlier describ ed For this mo del wemakemanysimulations and the option
value is the average of the outcomes It is well known that the Monte Carlo simulation
converge slow compared to the tree approaches for pricing path dep endent derivatives Hull
for example says The main problem in using Monte Carlo simulation to value path
dependent derivatives is that the computation time necessary to achieve the required level of
accuracy can beunacceptably high Furthermore American style path dep endent derivatives
cannot b e handled
In the following section I am going to show as an educationally example the theory be
hind the mo del and how the mo del can b e implemented
We will makeM simulations when pricing an option for each simulation we ll simulate the
asset price s path with N time steps from the start at current time t until maturity t
i T
The value of the option C for the j th simulation at time t will then b e
j
C maxS K
j N
i
if the barrier is not crossed and
C
j
otherwise i e if S B That go es for a Down and out call option
i j
When we have simulated the price M times we get the option price as the average of all
the outcomes
M
X
C C
j
M
j
How is the asset price path simulated then Because wehave divided the path into N parts
i
wewanttosimulate a jump at each time step as
S S e
i j i j
Here is a sample from a standard normal distribution and is calculated as
r q t
When all simulations are made we get the option price from equation
Figure shows simulated paths with data for the standard barrier option contract
On the vertical axis we see the asset price and on the horizontal axis is the time presented
in years We see that if the barrier is crossed in a simulation that simulation ends and a
horizontal line is drawn in the graph until maturity
Figure simulated paths On the vertical axis we see the asset price and on the
horizontal axis is the time presented in years We see that if the barrier is crossed in a
simulation that simulation ends and a horizontal line is drawn in the graph until maturity
In gure we can see the slow convergence of this mo del On the horizontal axis is the
calculation time presented in milliseconds while the option value is presented on the vertical
axis As you see even a calculation that takes over seconds do esn t give a go o d result
Figure Monte Carlo Simulation of value for the standard barrier option contract Option
prices on the vertical axis for di erent calculation times on the horizontal axis
In the comparison in the coming section this mo del will b e neglected b ecause of its very slow
convergence
Comparison
Wehave earlier in each section that treats di erent mo dels done comparisons to nd the b est
mo del of eachtyp e Wehave come to conclusion that wehave one binomial mo del where we
tryto ndthebestnumb er of time steps N to eliminate the oscillating b ehaviour section
i
Furthermore the trinomial mo del with a stretch parameter as presented in section will
also b e considered in this evaluation The MonteCarlo Simulation will we neglect b ecause
of the very slow convergence section Finally the Implicit nite di erence mo del using
Cranc Nicolson will be shown to be the b est mo del for pricing single barrier options see
section
We will see b oth graphs and tables with data that we compare to and from whichwe make
our decisions
Evaluation of the best suited numerical mo del for pricing the
standard contract
Wewill start the evaluation by comparing the binomial tree mo del with the trinomial tree
mo del We are not only going to lo ok at prices retrieved from a x calculation time but also
in whichway the mo dels converge We start with lo oking at a graphical representation of
the results in gure here we see that the mo dels converge almost equally fast but that
the trinomial mo del converge in a less oscillating way than the binomial tree An other dis
advantage for the binomial tree mo del is that it is not p ossible to cho ose whatever number
of time discretizations That is if wewanttomake a calculation at a certain numb er of time
steps the mo del will cho ose a time step so that the nearest higher b est p ointishit
Figure Comparison between the trinomial tree mo del and the binomial tree mo del
Option prices is displayed on the vertical axis for di erent power rate on the horizontal
axis
From gure we come to conclusion that the trinomial tree mo del is the b est of those
Now we are going to compare the trinomial tree mo del with the nite di erence mo del
Here wecho ose to put the real calculation time on the x axis The pattern is not as easy to
see as b efore that is b ecause the calculation time is varying from time to time i e making
two calculations with the same settings can result in di erent calculation times measured by
the computer Here wemake calculations with a small di erence in the p ower rate whichis
why the pattern is a bit di use
Figure shows the option price with resp ect to calculation time for the trinomial tree mo del
and the nite di erence mo del
Figure Comparison between the trinomial tree mo del and the nite di erence mo del
using Cranc Nicolson Option prices is displayed on the vertical axis for di erent calculation
time on the horizontal axis
Lo oking at the two graphs we see that the nite di erence mo del converges fast compared to
the trinomial tree mo del Furthermore the nite di erence mo del do es not oscillate much
the visible oscillating b ehaviour is gone after ms calculation time We are nally going
to see data of the di erent mo dels in table From this we can draw no further conclusions
then ab ove but we can again app oint that the nite di erence mo del have converged to the
th decimal already after ms For pricing a standard contract describ ed in section
with current asset price S equal we see that the nite di erence mo del is the fastest and
most suitable mo del
We haveseen that the nite di erence mo del using Cranc Nicolson approximations to the
derivatives is without doubt the fastest mo del for pricing the standard barrier options con
tract section But what will the result b e if we are far from or very close to the barrier
Is the nite di erence mo del equally fast in these cases Those are questions that we are
going to answer next
Evaluation of the best suited numerical mo del for pricing con
tracts with asset price very close to the barrier
Here we are going to see what happ ens when the current asset price gets very close to the
barrier Wehave already seen that the nite di erence mo del is very fast when we are close
to the barrier compared to other mo dels this mo del is extraordinary and gives correct results
within a fraction of time used by those But what happ ens if wegoeven closer to the barrier
In table we can see the results retrieved from the di erent mo dels when the barrier lies at
all other parameters are the same as for the standard contract
The closed form is due to Boyle and Tian here I have presented a more ex
act result but I can not con rm the accuracy b ecause I have not tested the calculation with
double precision
As you see calculations at a very short amount of time are imp ossible to do This is b ecause
the mo dels are not validforavery small amount of disctetizations E g for the binomial
mo del is the smallest amount of time steps For the trinomial mo del by Ritchken to
Calculation Down and Out Down and Out Down and Out
time ms Call price Call price Call price
Binomial tree Trinomial tree Finite Di erence
Table Comparison between the binomial tree trinomial tree and the nite Di erence
mo dels with resp ect to calculation time for the standard option contract
Calculation Down and Out Down and Out Down and Out
time ms Call price Call price Call price
Binomial tree Trinomial tree Cranc Nicolson
Table Comparison between the binomial tree trinomial tree and the nite di erence
mo dels with resp ect to calculation time for the standard barrier contract with current asset
price at
obtain non negative probabilities is the minimum numb er of time steps needed for this
contract The nite di erence mo del has not exactly the same problem for this mo del we
have a criteria that the numb er of discretizations b etween the barrier and the current asset
price must be at least one This makes the number of discretizations between the barrier
and the upp ermost asset price level very big For this contract the number of discretiza
tions in price will be at least and b ecause wehavethe condition that the number of
time steps shall b e the half of that it is set to If weallowtwo discretizations b etween
the barrier and the current asset price there will b e twice as many steps in the asset price etc
In gure you can see the results presented one graph for each mo del We can clearly
Figure Comparison between the Binomial Tree mo del Trinomial tree mo del and the
Finite Di erence mo del using Cranc Nicolson
see that the nite di erence mo del is the fastest though it isn t as fast as for the standard
contract tested ab ove in section The closer the barrier the asset price comes the slower
will the mo del b ecome
Evaluation of the b est suited numerical mo del for pricing con
tracts with asset price far away from the barrier
The nite di erence mo del has b een shown to b e the b est suited mo del for pricing the stan
dard contract and contracts where the asset price lie very close to the barrier Here wewill
vary the asset price to see if the mo del is equally accurate for a barrier option contract with
the asset price far from the barrier The asset price that will b e evaluated is
In table as well as in gure we can see that the trinomial mo del has a very slow
convergence when the asset price is far from the barrier Both the binomial mo del and the
nite di erence mo del showmuch b etter convergence Lo oking at the table it seems like the
binomial mo del and the nite di erence mo del converges equally fast After ms b o oth
themodelshave converged to the th decimal but the b ehaviour of the binomial tree mo del
is still very oscillating which do es not show in the table Figure shows us a magni ed
picture of the two mo dels b ehaviour wenow see the great di erence in the wayofconvergence
The nite di erent mo del using Cranc Nicolson has then been shown to be the fastest
and most suitable mo del for pricing all kinds of single barrier options
Calculation Down and Out Down and Out Down and Out
time ms Call price Call price Call price
Binomial tree Trinomial tree Cranc Nicolson
Table Comparison between the binomial tree trinomial tree and the nite di erence
mo dels with resp ect to calculation time for the standard barrier conteract with current asset
price at
Figure Comparison b etween the binomial tree mo del trinomial tree mo del and the nite
di erence mo del using Cranc Nicolson
Figure Comparison between the binomial tree mo del and the nite di erence mo del
shows that the nite di erence mo del converge more stict han the binomial tree mo del
Computing Hedge Sensitivities
Hedge sensitivities is often used by traders to understand the b ehaviour of the contract they
p ossess How will the contingent claim price change when the price of the underlying asset
changes Howmuch will the contingent claim price rise if the underlying asset rise one unit
The gives us the answer to these questions There are several similar ratios that are inter
esting for the trader to know Ihave gathered some of them together here b elow
Some of the ratios can be taken from one single calculation to do that some data need
to b e saved during the calculation In each section treating di erent mo dels the wayofcal
culating these ratios from out the tree or the grid is describ ed For those ratios that can not
b e computed directly from the tree or grid one more calculation has to b e made How this
is done and how the di erent hedge sensitivities is describ ed can b e read b elow
Delta
Delta is the rate of change of an option s value with resp ect to the underlying sto ck price
Roughly ifwe know the delta then we can predict howmuch the option s value will change
when the underlying s price changes by a certain amount This parameter is computed di
rectly from the tree for the binomial and trinomial tree mo dels and from the grid for the Finite
Di erence mo del for the MonteCarlo simulation mo del an extra calculation ought to b e done
In continuous time this parameter is in fact the partial derivativeofthe option value with
resp ect to the asset price
Gamma
Gamma is the rate of change of the Delta with resp ect to the underlying s sto ck price
This parameter is taken directly from the tree for the binomial and trinomial tree mo dels
likewise it can b e calculated from the grid of the Finite Di erence mo del but for the Monte
Carlo simulation mo del an extra calculation oughttobedone
In continuous time this parameter is in fact the second partial derivative of the option value
with resp ect to the asset price
Omega
Omega is the rate of change of the Gamma with resp ect to the underlying s sto ck price This
parameter is taken directly from the tree for the binomial and trinomial tree mo dels for the
MonteCarlo simulation mo del an extra calculation ought to b e done
In continuous time this parameter is in fact the third partial derivative of the option value
with resp ect to the asset price
Theta
Theta is the rate of change of the value of an option with resp ect to time For example it
tells us how the option value is going to change if the time to maturitychange Theta can
be calculated from out the tree for the binomial and trinomial mo del or from out the grid
for the Finite Di erence mo del for the MonteCarlo simulation mo del an extra calculation
oughttobedone
In continuous time this parameter is in fact the partial derivativeofthe option value with
resp ect to the time t
Vega
Vega isthe rate of change of the value of an option with resp ect to volatility That is how
will the option value change if the market b ecomes more volatile A high v indicates that
an option s price is very sensitive to small changes in the volatility of the underlying asset
The v has to be calculated with an extra calculation To do this we rst save the old
option value C and the old volatilityvalue After this wechange the volatility with
old old
one p ercent then we calculate a new option value C with this new
new old new
parameter Finally we compute the v as the rate of change as
C C
old new
v
old new
In continuous time this parameter is in fact the partial derivativeofthe option value with
resp ect to the volatility
Rho
Rho is the rate of change of the value of an option with resp ect to the risk free interest rate
This hedge parameter tells us how sensitive the option value is to changes in the interest
rate high indicates high sensitivity
As for the Vega the can not be computed from the tree or the grid Here we have to
make an extra calculation in the same way as for the Vega First wesave the option C
old
value calculated with the prop er Rho r Then we change the interest rate one p ercent
old
to r r and calculate the option value C Having all these values we can
new old new
calculate the as
C C
old new
v
r r
old new
In continuous time this parameter is in fact the partial derivativeofthe option value with
resp ect to the interest rate
Including Discrete Dividends
In real life applications it is necessary to takelumpy dividends into consideration If there is
a discrete dividend payment on an asset during the option s life the value will dramatically
di er from the value for the same option with no dividend payment even if the dividend
payment is small I will presentaway of dealing with lumpydividendpayments so that no
changes have to b e done in the tree for binomial mo dels or in the grid for Finite Di erence
and Trinomial mo dels
What we are going to do is adjusting the asset prices in the tree grid with the dividend
payments The asset prices will be lowered with the sum of all dividend payments during
the option s life discounted to current time with the interest rate Todothiswe rst found a
vector D with elements from to N each element representing a time steps At each time
i
step welock if there are any dividend payments if there are we put them in the vector at its
prop er p osition E g if there is a payment at time step i we ll put the value of the payment
at p osition i in the vector D The dividend payment time t doesseldomhitatime
div idend
step i t exactly so wehavetodosomekindofapproximation to put the dividend payment
date in the vector D That is a dividend payment V will b elong to the closest time
div idend
step i and we ll put the discounted dividend value in the vector at that p osition
r it t
div idend
D e V ti t ti
i div idend div idend
Nowwe compute the discounted cumulativevalue of all the future dividend payments in a
new vector D also with elements from to N Each element i in the vector is calculated
sum i
as b elow
N
i
X
i
rkt
D e D
sum k
k i
We can nally calculate the asset price spread at maturity date for the binomial metho d and
for the whole grid for the Trinomial and the Finite Di erence mo del We ll do this in the
same way as b efore describ ed for each mo del by computing the lowest asset price in the
spread S and subtract the value D from it Wenowhavea newlowest asset price
low sum
S S D
low sum
low
from whichwe compute the spread in a normal way If the option has the Europ ean style we
don t havetochange anything else The option s price will fall out as b efore on the current
time But if we re dealing with American style options wehave to adjust the options price
at the exercise check We ll then add the discounted cumulative dividend value at that time
step in the following way
i
C maxC S D K
i j i j i j sum
This way of dealing with lumpy dividends is very easy to understand as you see in fact it
is also very easy to implement in the calculation pro cess There are no limitations in how
many dividends that can b e added to this mo del but the option price converges much slower
if many dividends is present That is imp ortant to rememb er and it s b etter informing the
traders ab out it It is also p ossible to make an adaptive system that cho oses the p ower rate
necessary
Conclutions and Results
Wehave seen that the implicit nite di erence mo del using Cranc Nicolson approximation of
BlackandScholes partial di erential equation is the fastest mo del for pricing single barrier
options section This mo del is fast robust easy to understand and easy to implement
Robust b ecause the stabilityis indep endent of the number of discretizations Futhermore
even if the current asset price lies close to the barrier we get a correct result in a short amount
of calculation time which other mo dels presented in this pap er do es not manage to do
Di culties in the evaluation pro cess
There have b een some di culties in the evaluation pro cess When getting a mo del from a
working pap er published in some journal it is often di cult to repro duce the results Often
there is no description how to implement the mo del and there is often a lack of evident
theoretically asp ects to o Therefore the results retrieved from my implementations is not
always exactly the same as those presented in the pap er In this particular case we try to
compare the mo dels by calculation time Firstly almost no other makes their comparisons
with resp ect to calculation time therefore I ve b een forced to implement all di erent mo dels
to make the comparison Otherwise I could have taken the results directly from the pub
lished pap ers though it would probably b e almost imp ossible to use the values b ecause the
computer con guration and p erformance would have to b e exactly the same for me and for
the one used to get the results in the pap er
Notes ab out the Finite Di erence mo del
The Finite Di erence Mo del is the b est suited mo del for pricing single barrier options though
when the current asset price is close to the barrier even this mo del b ecomes slow It is p ossi
ble to makeaninterp olation in that case to sp eed up the calculation but in real life options
with asset prices closer to the barrier than when the current asset price is ab out
is really not interesting to calculate b ecause that the asset is traded with at maximum one
decimal
I m going to present the results for the standard contract section in two di erentways
so that the results presented in this pap er can be used in future comparisons First one
table with di erentvalues with resp ect to the numb er of discretizations second one table
with values with resp ect to calculation time
To get an idea of the b ehaviour of the option price close to the barrier we are going to lo ok
at the delta see section for di erentvalues of the current asset price In gure b elow
the horizontal axis shows the current asset price and the verical axis shows the value We
can see from the gure that the option value gets more and more sensitiveforchanges in the
current asset price the closer it is to the barrier this is a typical b ehaviour for the barrier
option
Calculation Down and Out
time ms Call price
Finite Di erence
Table The option value for the standard barrier option contract see section is presented
N
j
for di erent calculation time the asset price is here N
i
Discretizations Down and Out
in asset price Call price
Finite Di erence
Table The option value for the standard barrier option contract see section is presented
N
j
for di erentnumb er of discretizations of the asset price here N
i
Figure Graph representing the delta value with resp ect to current asset price all other
data as for the standard barrier contract
Further Development
The standard barrier option contract used in this pap er see section is just one of many
di erent typ es of single barrier option contracts Adjusting the nite di erence mo del to
these contracts is a development pro cess that has to b e done Furthermore make the option
pricing functions more e cient is evident For example the function for solving tri diagonal
equation systems maybe improved One can also use a p erformance monitoring to ol that
computes the time sp entindi erent partitions of the calculation functions to see where the
greatest e ort should b e used to maximize the calculation sp eed
All mo dels treated in this pap er are used for pricing single barrier options where the barrier
is constant This is the most common typ e of barrier option contract but there exist many
other typ es such as those with non constant barriers or those with two barriers or combina
tions of them
First I would like to recommend further developing of the mo dels for double barrier op
tion pricing That is b ecause I think these will be traded more in the future We have
already lo oked at a mo del which is originally constructed for pricing double barrier options
I think the Finite Di erence Mo del using Cranc Nicolson for approximating the deriva
tives with the implementation and interp olation suggested in that pap er will give the b est
results Other pap er I ve found where information and pricing mo dels for double barrier
options can b e found are Intermark s working pap ers and
The barrier option contract with non constant barrier is treated in several publications
Peter Ritchken for example is evaluating barrier options with exp onential b oundaries
I don t think trading with non constant barrier option contracts will b e common in a near
future b ecause of its complexity it is not easy to intuitively see how the contract is going
to develop when the price of the underlying asset is changing Anyway to b e prepared for
the future is not a waste of time So I recommend nding a fast mo del for the non constant
barrier contract
App endix Ab out the evaluation to ol built in Java
The evaluation to ol is built for the purp ose to price di erent typ es of derivatives The
program is mo delled in away suchthat it is easy to extend with new derivatives without
having much programming exp erience The graphical interface of the program is copied from
the calculation mo dule in Extra Extra is a trader program develop ed by ABN Amro Soft
ware AB At the moment only standard options can b e valued with Extra Therefore I have
extended the interface so that other typ es of derivatives can b e calculated in the same mo dule
Mainly the program consist in three di erent parts the inputpanel where we write the
contract sp eci cation the output eld where we get all the output data such as option values
nally wehave the graphwriter that makes it p ossible to have a graphical representation of
the mo dels b ehaviour
On the inputpanel we sp ecify the data of the option contract that is going to be valued
typical data here is the underlying asset s current price time to maturityandvolatility etc
There are some elds here that is common for all di erenttyp es of derivativecontracts each
typ e of derivative has also its own input elds for the barrier option this is for example which
typ e of barrier option that is b eing valued i e down and out or down and in etc Other
inputs as interest rate and dividends are added using dialog b oxes
In the output eld we get the results from the calculations they are presented in a list
with one row for each calculation In addition to the option s value we also get values of
hedge sensitivities Furthermore the program measures the calculation time and displays
this in the output eld it is measured and displayed in milliseconds This is a limitation of
the Java language we can t get a more exact measurement as for example in nanoseconds
A more accurate calculation time can b e interesting when dealing with very fast mo dels as
the Binomial mo del for pricing standard options For the barrier option on the other hand
this is less imp ortant In the output eld is also some of the input data displayed so that we
know whichcontract that got a sp eci c value
The graphwriter is a very useful to ol where we can test the options b ehaviour That is
we display all option values from the output eld in a diagram with the option value on
the vertical axis and the power rate or calculation time on the horizontal axis The most
interesting to see is how the value of the derivative converges with higher calculation time
and then esp ecially compared to other mo dels We can also place the power rate on the
horizontal axis When doing this we get a more exact picture of the mo del b ecause the
power rate do esn t give an alternating error as with the calculation time i e a measurement
of the calculation time is not exact b ecause of the load on the computer Furthermore to b e
able to compare twodi erent mo dels using the p ower rate we need to weightthenumber of
discretizations so that the mo dels have the same calculation time for the same p ower rate
We can also display the b ehaviour of the hedge sensitivities for di erent levels of the current
asset price for a certain contract
How to add new features to the program
As I mentioned in the intro duction to this pap er the program is built in Java This makes
the program very easy to extend b ecause it s easy to change old classes or add new classes
to the program without need to recompile the whole program
The structure of the program makes it easy to nd and change the classes In gure
we see where di erent parts of the program is placed In Source is all source co de les
placed to makechanges to the program copy this folder and rebuild the program using for
Figure The program structure of the Evaluation To ol built in Java The folder Calculator
includes all parts of the program i e the program sp eci c class les placed in Lib and the
Java Runtime environment placed in Jre A launcher program Calculator exe is placed
in Bin All source co de for the program is placed in Source The two marked folders in
the middle is platform dep endent and has to b e changed if the application shall run under
an other op erating system
example JDK bySunMicrosystems Inc When doing this make sure all les in the folder
C al cul ator nLibn excluding exev is linked to the pro ject
Java is an OS indep endent programming language but to make the program easy to use
I have made it semi dep endent That is to run the program on any other platform than
the Windows NT some changes has to b e done to the program The launcher here
called Calculator exe has to be rewritten for the new platform What this program do es
is to launch the Jre exe and feed this with the lo cation of all the class les and libraries
Furthermore the Java Runtime environment Jre has to b e changed for this application if
it shall run under an other op erating system
Then howdowe add new features as for example a new derivative I ll take an example
we wantto be able to evaluate the b est suited numerical mo del for pricing Asian Options
using this to ol We start with copying all java lesinthefolderSource to a new pro ject in a
developmentenvironment e g JDK What we then have to do is making a new Class called
AsianOption this class inherit from the base class Option that on the other hand inherit its
functionality from the class Derivative Weoverride the calculation functions that wewant
to use as for example calculateBT to make a calculation with the binomial tree mo del We
can nowmake the inputpanel for this sp eci c instrument as I mentioned earlier wehavea
base class with all common inputs from whichwe inherit to the new panel called for example
AsianPanel To this panel we add input elds sp eci c to this instrument Finally to add
this panel to the program wehavetomakeachange in one le This le is called Control
Panel java
When all the program is written compiled and debugged we can add the new class les
to the existing program in its prop er p osition Don t forget to add the new source les in
the Source folder so that the program can b e rebuilt next time someone want to extend the
functionality of the program
The develop ers environment
I have built a develop er environment with automatic generation of co de for adding new
derivatives to the program Here you just op en a dialog b ox from the to olbar and typ e the
name of the new derivative You can on the same time cho ose the number of additional
input elds you want then you cho ose names for those variables The application will then
automatically generate les with the co de for the new derivative It constructs a new panel
asso ciated with it and adds it to the evaluation to ol Finally it compiles the newly created
and changed les and op ens the le with the calculation functions for editing
To add functionality to the mo del you typ e co de inside the pre generated functions When
done just press save followed by compile
A more detailed description of how to add new derivatives to the application can b e found in
The DerivativeEvaluation To ol Develop ers Guide which b elongs to ABN Amro Software
AB and can not b e distributed with this pap er
Bibliography
References
Peter Ritchken
On Pricing Barrier Options
The Journal of Derivatives Volume Number Winter
Phelim P Boyle and Yisong Sam Tian
An explicit nite di erence approach to the pricing of barrier options
Applied Mathematical Finance
G Io e and M Io e
Application of Finite Di erence Metho d for Pricing Barrier Options
Intermark s Working Pap ers Last mo di ed June http www intermarkit com
Boyle P and S H Lau
Bumping up against the Barrier with the Binomial Metho d
Journal of Derivatives pp
Black F and M Scholes
The Pricing of Options and Corp orate Liabilities
Journal of Political Economy Vol No pp
Schwartz E S
The Valuation of Warrants Implementing a New Approach
Journal of Financial Economics pp
Emanuel Derman and Ira j Mani and Neil Chriss
Implied Trinomial Trees of the Volatility Smile
Yuh Bauh Lyun
Very Fast Algorithms for Barrier Option Pricing and the Ballot Problem
The Journal Of Derivatives Volume Numb er Spring
N K Chidambaran and stephen Figlewski
Streamlining Monte Carlo Simulation with the Quasi Analytic Metho d Analysis of a
path dep endent Option Strategy
Journal of Derivatives winter
Mark Broadie and Paul Glasserman and Gautam Jain
Enhanced monte Carlo Estimates for American o option Prices
The Journal of Derivatives Volume Number Fall
Silvio Galanti and Alan JUNG
Low Discrepanay Sequences Monte Carlo Simulation of Option Prices
Journal of Derivatives Volume Number Fall
A More Accurate Finite Di erence Approximation for the valuation of options
Journal of Financeial and Quantitative Analysis Decemb er pp
Black F and M Scholes
Finite Di erence Metho ds and Jump Pro cesses Arising in the Pricing of Contingent
Claims ASynthesis
Journal of Financial and Quantitative Analysis Septemb er pp
Hull J and White A
Valuing Derivative Securities Using the Explicit Finite Di erence Metho d
Journal of Financial and Quantitative Analysis March pp
Les Clewlow and Chris Strickland
Implementing Derivatives Mo dels
ISBN
John Wiley and Sons
John C Hull
Options Futures and other Derivatives
ISBN
Prentice Hall Inc
Neil A Chriss
Black Scholes and Beyond Option Pricing Mo dels
ISBN
McGraw Hill
G Io e
Di erences in theoretical and actual prices of double Kno ck Out and binary range FX
options
Intermark s Working Pap ers Last mo di ed June http www intermarkit com
Io e M Io e G Krell D
Pricing formulas for Double Kno ck Out and Binary Range Options
Intermark s Working Pap ers Last mo di ed June http www intermarkit com