Valuation of Asian Option

Transcription

1 Mälardales Uversty västerås Mathematcs ad physcs departmet Project aalytcal face I Valuato of Asa Opto Q A T077 Jgjg Guo89003-T07

2 Cotet. Asa opto Prcg Mote Carlo smulato Cocluso Refereces Appedx

3 . Asa opto Defto of Asa opto always emphaszes the gst that the payoff depeds o the average prce of the uderlyg asset over a certa perod of tme as opposed to at maturty. That s why Asa opto could also be recogzed as average value opto. Ths type of opto cotract emerges ad gets developed because t teds to cost less tha regular Amerca ad Europea optos. It s also kow that a Asa opto ca protect a vestor from the volatlty rsk that comes wth the market. There are two basc forms of Asa opto, amely Average Prce Opto ad Average Strke Opto. The former dcates to pay the dfferece betwee the strke ad the average of uderlyg prce, ad the latter meas takg the average prce over specfed perod as the strke of the opto ad pays the dfferece betwee ths strke ad uderlyg market prce. These could be sorted as follow. types payoff Europea opto Call max(s(t)-k,0) Put max(k-s(t),0) Average prce opto Call max(a(t)-k,0) Put max(k-a(t),0) Average strke opto Call max(s(t)-a(t),0) Put max(a(t)-s(t),0) There are three terms that must be specfed. ) The averagg perod. E.g. the last three moth, the etre term of opto etc. 2) The samplg frequecy. E.g. daly, weekly, aually etc. 3) The averagg method. smple arthmetc average N A S ( t ) j N j 2geometrc average A( T ) S 3weghted average.arthmetc weghted average AT ( ) ws w.geometrc weghted average 3

4 ww 2w w w2 w A T S S2 S ( ) where w deotes the weght. Basc math kowledge fers that geometrc gves more accurate soluto. Yet the arthmetc average s more wdely used. 2. Prcg Three ma approaches exst to prcg Asa optos. ) Europea-style optos based o geometrc averages ca be prced by adaptg the aalytcal models. The reaso s f uderlyg prce s assumed to follow log ormal dstrbuto, the ts geometrc average s also log ormal dstrbuted. 2) The soluto above does ot apply whe t s based o arthmetc averages, because the arthmetc average wll ot be log ormal dstrbuted eve f the uderlyg prce s. It s oly possble to make approxmato. 3) Mote Carlo smulato could be used o matter what ts style or averagg method s. As ths smulato requres tesve calculato t s mostly performed by computer program. 3. Mote Carlo Smulato Mote Carlo smulato produces results by costructg a stochastc model, where the am soluto s the expectato of the model, ad the use a large umber of sample results to approxmate the am soluto. The law of large umbers esures that sample mea coverges to the populato mea almost surely as. That s the reaso why the sample mea s used as a estmate of populato mea for large samples. Prcg wth Mote Carlo method s assumed to be rsk-eutral world. Frstly to geerate a path of uderlyg prce stochastcally, the compute the expectato of returs, at last dscout t wth rsk-free terest rate. I ths report, Mote Carlo smulato s troduced to value a Asa relatve opto, whch performs o two or more uderlyg assets, ad the payoff s determed by the retur of oe or more assets relatve the retur of some other assets. To make t smpler the commo case of two uderlyg assets s take. The prce ad payoff are gve by d rt Q rt Q A (0) e E ( T) e E max,0 t S ( t0) where deotes the weght for each retur. I ths smple case s ad 2 s -. Ad S( t j) s the prce of uderlyg asset at tme t j, d s the umber of uderlyg assets. The Asa sum A s defed by N A S ( t ), based o smple arthmetc average. j N j 4

5 The t gves us A A rt Q S( t0) S2( t0) 2 (0) e E max,0 I our case there are two varables S ad S 2,whch are both uder Black-Scholes model wth o dvded, ad satsfy ds rs dt S dz t t t t where r s rsk-free terest rate, σ s the stadard devato of retur o the asset prce, dz t s the stadard Browa moto uder Q. So the arthmetc average prce of Asa opto s rt Q C e E0 max A( T) K,0 Where A(T) s arthmetc mea of prce,k s the strke Prce, T s tme to maturty, ad E s the expectato at t=0 uder Q. Because the effectve ubased estmato of E s sample mea, we ca rewrte the equato as followg, ˆ rt C( T, ) e max ˆ A ( T ) K,0 Where A (T) s the -th smulated arthmetc mea prce, s the umber of smulato tmes. Now we ca wrte some scrpts o MATLAB to calculate the prce of Asa Optos by Mote-Carlo smulato. Frst s to smulate paths of asset prce, ad the to calculate the optos prce. (As performed appedx) Now we have our program for the calculato, the we assume some datum to calculate the prce ad aalyze the propertes. opto Curret Prce Strke Prce Iterest Rate Tme to Volatlty Maturty $00 $0 0. year $00 $0 0.2 year $00 $ year $00 $0 0. year 0.2 The we substtute the datum to the fucto, we ca get the results quckly, opto Number of Steps Number of Smulatos Opto prce Cofdece terval Cofdece Legth of cofdece terval (-.735,4.5728) 95% ( ,7.9559) 95% (5.025,0.869) 95%

6 (6.69,8.4840) 95% (8.2502,0.248) 95% (0.6903,3.3772) 95% (2.3402,3.0745) 95% It s easy to see that, wth the crease of the umber of smulatos, the legth of cofdece tervals become smaller ad smaller, whch meas the results of smulatos are more ad more accurate. So we could obta a opto prce very close to ts real prce. I opto 2, 3 ad 4, we chaged some values of the parameters so that we have the chace to aalyze the effects caused by those parameters roughly. Sce we get dfferet prces by usg the exactly same values opto, we could t get some geeral rule about how the other parameters affect the prce by a sgle result. We eed to do more calculatos. The reaso s every tme we use ths fucto the MATLAB, the system wll ot create the exactly same radom samples. But we ca prove we ca get more accurate prce wth Mote-Carlo smulato as the umber of radom samples close to fty. 4. Cocluso Asa Optos are commoly traded o curreces ad commodty products whch have low tradg volumes. They are optos that esure to ts buyer a average retur at a lesser rsk. Mote-Carlo smulato s a very useful tool whe prcg the Asa opto... Also whe we try to get some solutos, we eed to cotrol all the varables ad summarze the geeral results based o a vast umber of trals. 5. Refereces Lecture otes Aalytcal Face, Ja R. M. Röma 2 Kjma, M Stochastc processes wth applcato to face, Chapma & Hall/CRC, Boca Rato. ISBN: Joh.C.Hull Optos, Futures ad Other Dervatves Pearso Educato; 6th Iteratoal edto, ISBN: Appedx Matlab scrpt. fucto [Prce,CoIterval] = aapaprcemc (CurretPrce,StrkePrce,... ItRate,TmeToMaturty,StdDevato,CallOrPut,NumOfSteps,NumOfPaths) % AAPAPRICEMC prcg arthmetc average prce of Asa opto wth % Mote-Carlo Smulato % PARAMETERS: % Prce s the opto prce calculated wth Mote-Carlo Smulato % CoIterval s the 95% cofdece terval for the estmato of the prce % CurretPrce s the curret prce of the uderlyg asset % StrkePrce s the strke prce of the opto % ItRate s the rsk-free terest rate 6

7 % TmeToMaturty s the tme to maturty % StdDevato s the stadard devato of the retur o the prce of the uderlyg asset % CallOrPut : f the opto s a call, CallOrPut s ; % f the opto s a put, CallOrPut s 0; % default, CallOrPut s ; f (CallOrPut~=0)&&(CallOrPut~=) error(sprtf('callorput should be or 0.')) ed Paths = prcepaths(curretprce,itrate,tmetomaturty,stddevato,... NumOfSteps,NumOfPaths); f CallOrPut PayOff = max(mea(paths(:,2:(numofsteps+)),2)-strkeprce,0); else PayOff = max(strkeprce-mea(paths(:,2:(numofsteps+)),2),0); ed [Prce,VarOfPrce,CoIterval]=ormft(exp(-ItRate*TmeToMaturty)*PayOff) ed fucto Paths = prcepaths ( CurretPrce, ExpRate, TmeToMaturty,StdDevato, NumOfSteps, NumOfPaths) % PrcePaths smulatg paths of asset prce whch follows the geometrc % Browa moto % % PARAMETERS: % Paths s NumOfPaths prce paths smulated wth NumOfSteps steps; % CurretPrce s the curret prce of the asset; % ExpRate s the expected rate of retur o the asset prce; % TmeToMaturty s the tme to maturty; % StdDevato s the stadard devato of retur o the asset prce; DeltaT = TmeToMaturty/NumOfSteps; Drft = (ExpRate-StdDevato^2/2)*DeltaT; Volatlty = StdDevato*sqrt(DeltaT)*rad(NumOfSteps,NumOfPaths); Icremets = Drft+Volatlty; LogPaths = cumsum([log(curretprce)*oes(numofpaths,),icremets],2); Paths = exp(logpaths); ed 7