Hopscotch and Explicit difference method for solving Black-Scholes PDE

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1 Mälardale iversity Fiacial Egieerig Program Aalytical Fiace Semiar Report Hopscotch ad Explicit differece method for solvig Blac-Scholes PDE Istructor: Ja Röma Team members: A Gog HaiLog Zhao Hog Cui 0

2 Table of Cotets Itroductio.... Objectives of the Semiar.... Brief Historical Review....3 Methodology... Theoretical Framewor Hopscotch Method Explicit Fiite Differece Method Blac-Scholes Formula Numerical Examples Coclusios Refereces... 9

3 Itroductio. Objectives of the Semiar The mai objectives of the semiar are to study the mechaism of the Hopscotch ad explicit differece methods for solvig Blac-Schoels PDE, ad to lear how to build applicatios of these models i Excel Visual basic applicatio (VBA). Our goal is to study how these differet methods fuctio i Excel/VBA for calculatig Europea optio prices, ad to compare the results from these methods ad Blac-Schoels model by usig graphs.. Brief Historical Review I 997, the Royal Swedish Academy of Scieces has decided to award the Ba of Swede Prize i Ecoomic Scieces i Memory of Alfred Nobel to Professor Robert C. Merto ad Myro S. Scholes. Robert C. Merto, Myro S. Scholes ad Fischer Blac (973) have developed a pioeerig formula for the valuatio of stoc optios. Their methodology has paved the way for ecoomic valuatios i may areas. It has also geerated ew types of fiacial istrumets ad facilitated more efficiet ris maagemet i society. Gordo (965) ad Gourlay (970) have itroduced a class of so called Hopscotch algorithms to solve parabolic ad elliptic partial differetial equatios i two or more state variables. The purpose of this paper is the to preset Hopscotch methods ad to demostrate how they ca be used to solve fiacial models with two state variables..3 Methodology I order to uderstad how the Hopscotch ad explicit fiite differece method wors, we have formulated a umerical example of Europea optios ad solved it by help of Excel/VBA. We have also used the Blac-Scholes model to calculate the price of the same optios i VBA. Hopscotch methods for two-state fiacial models (Article by Adam Kurpiel, Thierry R ocalli: 999)

4 Theoretical Framewor. Hopscotch Method Hopscotch method ca solve parabolic ad elliptic partial differetial equatios i two or more state variables but their utility i fiacial applicatios has ot yet bee realized. I will itroduce how the hopscotch method ca be used to solve fiacial models with two-state variables. The basic idea is to divide the mesh poits i the two-dimesioal x y mesh (ih, jh) as follows: i j odd i j eve The Hopscotch cosists of two sweeps. I the first sweep (ad subsequet odd-umbered sweeps) the mesh poits that are mared by a diamod, that is for which i j is odd, are calculated based o curret values (time level ) at the eighborig poits. It ca be defied as follows: = Δ x Δ y for () odd For the secod sweep at the same time level the same calculatio is used at odes mared with a circle. This secod sweep is fully implicit. The scheme is: = Δ x Δ y for () eve From this equatio we ca fid the value at time level as follows: 3

5 4 =.... y x y j i j i x j i j i h h h h I the secod ad subsequet eve-umbered time steps, the roles of the diamods ad circles are iterchaged. Now let K ad Ʈ be the exercise price ad the time to maturity of a Europea optio o the uderlyig asset price S (t). I the Blac-Scholes framewor, the call optio price C (Ʈ ; S) satisfies the followig equatio rc C bc C S σ τ s ss = = ) ( ) 0, ( K S S C The parameter b is the cost-of-carry rate. To solve this problem umerically usig Hopscotch methods, we have to add two boudary coditios for the extreme values S- ad S tae by the S variable. For S equal to S-, we chose the followig coditio 0 ),, ( = S y t u because the optio price teds to be zero whe the uderlyig asset price decreases.

6 I may cases, we do ot ow four boudary coditios. Sometimes, a simple guess is used as a prior for a boudary coditio. We may however use icorrect boudary coditios ad still cosider umerical solutios i the cetral regio. We must be careful ad we have to verify the behavior of the umerical solutio whe we chage the boudary fuctio.. Explicit Fiite Differece Method The mai goal of fiite differece techiques is solve umerically the Blac-Scholes equatio or oe if its variatios. The aim of build a umerical scheme for that equatio is ot to fid the solutio itself (we ow that Blac-Scholes for Europea optios has a aalytical solutio) but to exploit such scheme to solve more geeral equatios ad iequalities. A easy way to start is to impose a coordiate trasformatio that permits to simplify the BS equatio to oe of its variaces with costat coefficiets. Fiite Differeces As with ay class of optio, the price of the derivative is govered by solvig the uderlyig partial differetial equatio. The use of fiite differece methods allows us to solve these PDEs by meas of a iterative procedure. We ca start by looig at the Blac-Scholes partial differetial equatio: Where dv is the chage i the value of a optio, dt is a small chage i time. is the volatility of the uderlyig, S is the uderlyig price ad is the carry (r-d). By specifyig iitial ad boudary coditios, oe ca attai umerical solutios to all the derivatives of the Blac-Scholes PDE usig a fiite differece grid. The grid is typically set up so that partitios i two dimesios - space ad time (i our case, we would be looig at the asset price ad the chage i time): Oce the grid is set up, there are three methods to evaluate the PDE at each time step. The differece betwee each of the three methods is cotiget o the choice of differece used for time (i.e. forward, bacward or cetral differeces - more details i our fiacial mathematics glossary here). Cetral differeces are used for the space grid (S). 5

of all, the PDE as a remider: if we substitute x = l(s), the equatio becomes: Applyig the fiite differeces method, the above

7 Explicit Fiite Differeces Explicit FD uses forward differeces at each time ode t. By splittig the differetial equatio ito the time elemet ad space elemets, we ca apply forward differeces to time as follows: First of all, the PDE as a remider: if we substitute x = l(s), the equatio becomes: Applyig the fiite differeces method, the above equatio ca be broe dow ad approximated: becomes For the space grid, we ca apply cetral differeces for all order of derivatives: ad becomes ad becomes becomes Combiig the terms gives: 6

It ca also be show that the followig approximatio holds:.

8 Which is the same as: where the probabilities of each of the odes is: This case is actually equivalet to the triomial tree where probabilities ca be assiged to the lielihood of a up move, a dow move as well as o move. It ca also be show that the followig approximatio holds:.3 Blac Scholes Formula The Blac-Scholes formula calculates the price of a call optio to be: The the price of a put optio is: Where l σ T t σ T t St= price of uderlyig stoc K=Optio exercise price r= Ris free iterests rate 7

9 T=Expirig time t= Curret time N ( ) = Area uder the ormal distributio curve There are some assumptios of the model which are as followigs 3 : The Blac-Scholes model assumes that the optio ca oly be exercised o the expiratio date; It requires to use costat ad ow iterest rates, the ris-free rate such as the discout rate o.s Govermet Treasury Bills are usually used; It also assumes that the uderlyig stoc does ot pay divideds; It assumes the returs o the uderlyig stoc are ormally distributed. It assumes that the maret is efficiet. 3 Numerical Examples The uderlyig stoc price is 50 SEK today, ad the strie price of the optio is set as 50 SEK. The volatility of the maret has bee give as 0% ad the ris-free iterest rate is 5%. This is a Europea optio with maturity date of year from today. Please calculate the price of the call optio as well as the price of the put optios. By usig the Blac-Scholes model we ca calculate the both call ad put optio prices as followigs: The results show o the figure above have bee gaied by usig the Excel/VBA ad the details of the applicatio that build iside the VBA for this calculatio ca be foud i the separate excel file called as Semiar_BS_VBA

10 By usig Hopscotch ad explicit differece methods to build applicatios i Excel, we will get similar prices of the optios. These calculatios ad applicatios i VBA will be illustrated i a separated Excel file ad this will be show i the semiar presetatio. (This part is ot ready yet). 4 Coclusios Blac-Scholes theories are some of the most sigificat cotributios i the developmet of fiace theory. Blac-Scholes model is well-ow as sufficiet istrumet to price securities o the fiacial maret together with Hopscotch ad explicit fiite differece method. Today, it has become much more powerful with help of the Excel/VBA which maes the complicatig meth calculatios to be doe i a much simple way. 5 Refereces Hopscotch methods for two-state fiacial models (Article by Adam Kurpiel, Thierry R ocalli: 999) 9

Subject CT1 Financial Mathematics Core Technical Syllabus

Subject CT1 Financial Mathematics Core Technical Syllabus Subject CT1 Fiacial Mathematics Core Techical Syllabus for the 2018 exams 1 Jue 2017 Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig