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Numerical Models for Pricing Barrier Options

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Numerical Models for Pricing Barrier Options Numerical Mo dels for Pricing Barrier Options Oskar Milton th March Contents Abstract Variable sp ecication Intro duction Briey ab out options What is a barrier option Upandout call Upandin call Downandout call Downandin call Upandout put Upandin put Downandout put Downandin put Double barrier option Delimitations Standard option contract BlackScholes analytic formula and theory Binomial Tree Mo del CoxRossRubinstein Binomial Mo del Trigeorgis Mo del Calculating a barrier options 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 Dierence Mo del Explicit approach Implementation Stability and convergence Implicit CrancNicolson Construction of the grid Boundary conditions Implicit not fully centered Comparison b etween the Finite Dierence 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 Diculties in the evaluation pro cess Notes ab out the Finite Dierence 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 dierenttyp 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 Dierence Mo del with CrancNicolson the results is accurate due to earlier works but here is the results presented with a higher precision this is not scientically proved and shall just b e used as a help to understand the very fast convergence of this mo del Variable sp ecication 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 Stij ij in a tree or a grid Standard deviation of return volatilityofas sigS 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 cKT Europ ean call price with strike price K and time to maturityT pKT 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 Cii ij 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 straightforward 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 ie 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 Dsumi 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 riskfree 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 Ive b een commissioned by ABNAmro Software AB to evaluate dierentnumerical 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 Dierence Mo del Monte Carlo Simulation When comparing dierent mo dels for pricing derivatives it is normally the tree or grid size ie 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 timecritical 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 calculationtime for the dierent mo dels When I say that one mo del is faster than another it is normally with resp ect to the calculationtime that I have made the comparison To get a clear idea of the behaviour of the dierent mo dels the diagrams are displayed with option value on the vertical axis and computerp ower or real calculation time on the horizontal axis The computerp ower is a relative factor that indicates how much time a calculation of an options 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 options 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 dierent mo dels in the same diagram and makes it easy to compare the convergence of the dierent 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 its accuracy near the barrier The to ol is constructed so that implementation and testing of dierentderivatives are made easy and fast This was already from the start a pronounced goal b ecause ABNAmro 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 ABNAmro Software AB and can not b e distributed with this pap er Briey 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
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