Analytical Approximate Solutions for Stochastic Volatility. American Options under Barrier Options Models

Transcription

1 Aalytical Approximate Solutios for Stochastic Volatility America Optios uder Barrier Optios Models Chug-Gee Li Chiao-Hsi Su Soochow Uiversity Abstract This paper exteds the work of Hesto (99) ad itegrates the Richardso extrapolatio techique ad barrier optios models for developig stochastic volatility America optios aalytical approximate solutios. By usig large sample least-square Mote Carlo Simulatios as the bechmarks, we prove that our model is accurate ad efficiet from the results of umerical experimets. Fially, we show that our stochastic volatility America optio model is superior i pricig tha the traditioal costat volatility model from the empirical tests of Taiwa s put warrat market. Keywords: America Optio, Barrier Optio, Richardso extrapolatio, Stochastic Volatility.. Itroctio America optios are popular istrumets for tradig ad hedgig i fiacial markets. Several fiacial procts embedded with America optios, say, the covertible bod allowed the ivestor covertig a bod ito stocks before the maturity. As the ivestor of covertible bod ca exercise the coversio right before maturity, the coversio optio is therefore America-style. America optios are more expesive tha their Europea couterparties, sice the holders are allowed to exercise optios at ay time before the expiratio date. ricig * Chug-Gee Li is professor of fiace, Chiao-Hsi Su is master, both are at the Departmet of Fiacial Egieerig ad Actuarial Mathematics i Soochow Uiversity, Taiwa. Correspodig Author: Chug-Gee Li. Address: 56, Kuei-Yag Street, Sectio, Taipei, Taiwa. Tel: ext. 69. Fax: cgli@scu.e.tw ACKNOWLEDGEMENTS: This research was partially supported by a grat from the Natioal Sciece Coucil of Taiwa (NSC98-4-H--5).

2 America optios is extremely complex, sice the optimal-exercise policy is ukow. Therefore, o explicit closed-form solutio has bee derived for America optios. Although a fairly large literature exists o pricig ad hedgig America optios, the volatility is assumed as fixed or costat. However, empirical studies have show that the volatility i the real world is stochastic. Hece, a more reasoable way of evaluatig America optios is to relax the restrictio of costat volatility. Numerical methods are commoly used for pricig America optios. However, they are time cosumig ad computatioally more demadig. I cotrast, aalytical solutios are more accurate ad faster. They are formula forms which ca ituitively describe pheomea such as mootoe ad covergece that caot be proved by umerical methods. With the differetiatio of aalytical solutios, we ca also obtai delta, gamma ad other hedge ratios (Greeks). Hece, it is advatageous to have aalytical solutios for the value of a fiacial istrumet withi a give model. The purpose of this paper is to derive a aalytical approximate solutio for America optios with stochastic volatility by utilizig barrier optios. Barrier optios are cotiget claims whose value depeds upo their behavior at various boudaries. A dow-ad-out optio is oe of the prototypical barrier optios that will expire if the stock price ever falls below the kock-out barrier. Sice America optios are similar to barrier optios with the exercise boudary treated as the barrier ad the payoff as the rebate, we developed a discrete dow-ad-out optio with stochastic volatility model i the derivatio of the aalytical solutio uder the Hesto (99) framework. The model is further modified ito a discrete America optio with stochastic volatility. To appraise the pricig performace of our model, we coct umerical aalyses to verify the accuracy i compariso with the least-squares Mote Carlo simulatio (LSM) method. I additio, we compare the differece betwee stochastic volatility model ad costat volatility model. Empirical aalysis is also performed o fiacial derivatives with our derived aalytical solutio model. The results are very ecouragig. The paper is orgaized as follows. I chapter, we discuss more about stochastic volatility ad the methods used i evaluatig America optios. Chapter is the derivatio of the aalytical solutio for America optios with stochastic volatility usig barrier optios. Numerical aalyses o America optios are preseted i chapter 4. Empirical tests are give i chapter 5. We coclude i chapter 6.. Literature Review To derive the aalytical solutio for America optios, it is helpful to kow the developmet of stochastic volatility ad the pricig of America optios by umerical methods ad aalytical solutios. With these two backgroud kowledge, a good evaluatio model ca the be derived. From the past literature, researchers have attempted usig stochastic volatility to overcome the assumptio of costat volatility sice the real world demostrates a stochastic oe. Hull ad White (987) were the first to suggest usig the idea of stochastic volatility to relax the assumptio of costat volatility i Black-Scholes (97) model. I Hull ad White (987) model, the stock price ad the asset volatility followed their respective diffusio process. The greatest restrictio is the zero correlatio betwee both. The volatility of Hull ad White (987) model adopted the logormal process ad obtaied the power series approximatio of the Europea optio. Sice the, stochastic volatility has

3 bee widely paid attetio to. O the other had, Stei ad Stei (99) adopted a differet viewpoit. By obeyig the mathematical Orstei-Uhlebeck process ad usig a separate umerical itegratio, a closed-form solutio was derived. However, this model still fails to relax the ureasoable assumptio of o correlatio betwee stock price ad volatility. Of all, Hesto (99) model makes the most cotributio. Hesto (99) described the volatility of uderlyig asset as a dyamic oe ad derived a closed-form solutio for Europea optios. Hesto (99) model implemeted Fourier trasform techique ad characteristic fuctio to calculate the probability of i-the-moey whe the optio reaches its expiratio date. The diffusio process of stock price is the same as Black-Scholes (97) model while the closed-form is similar. The oly differet variable is the volatility that chages with time. The closed-form solutio for Europea call proposed by Hesto (99) allows correlatio betwee stock price ad volatility. This is crucial because the stock price ad the volatility are supposed to have mutual ifluece theoretically. Therefore, we adopted the framework of Hesto (99) model to calculate the stochastic volatility sice it is more coformed to the real world. Optios ca be priced usig aalytical formulas or umerical methods. Brea ad Schwartz (978) proposed the fiite-differece methods, while Cox, Ross ad Rubistei (979) used a backward solvig biomial model which ca circumvet the problem of premature exercise. However, as the optio market progresses, the uderlyig asset of may ew optios is o loger a sigle oe. The problem of path depedet may exist, so Tilley (99) applied Mote Carlo simulatio, recorded the paths ad used sortig to estimate the optimum exercise poit. Later, Barraquad ad Martieau (995) separated each path ito differet regios to calculate the probability. The, they carried out the pricig with a backward method similar to biomial model. Grat, Vora ad Weeks (996) used critical prices to decide for a early exercise or a holdig. Raymar-Zwecher (997) revised Barraquad-Martieau s (995) model by separatig the asset prices ito two dimesios. Logstaff ad Schwartz () applied the least-squares Mote Carlo simulatio method (LSM) to estimate the price of America optios. They used the expectatio method ad calculated if early exercise is doe o every poit of each path. Although the above metioed methods are flexible ad easy to apply, they are doe uder costat volatility with time-cosumig calculatios. Besides umerical methods, researchers have also developed aalytical solutios to price America optios. They are classified ito three groups. The first approach is usig the itegral method to calculate uder the risk eutral probability measure. Geske ad Johso (984) priced the America put by usig a modified America optio (Bermuda optio) with Geske (979) compoud optio formula. They used four exercisable dates for extrapolatio to obtai a approximate America optio that ca be cotiuously exercised at ay poit of time. Addig oe more exercisable date to icrease accuracy would mea calculatig oe more dimesioal ormal distributio itegral. I other words, it ca be computatioally time cosumig to calculate them with multidimesioal umerical itegratio methods. The secod approach is seekig solutio directly from Black ad Scholes (97) partial differetial equatio (DE). Baroe-Adesi ad Whaley (987) (BAW) developed a approximate aalytical formula for America optios. They used the quadratic approximatio approach to price America call optios. This cocept is adapted from the critical price of early exercise. BAW (987) thought that early exercise premium is a particular fuctio. Therefore, the price of America optios is equivalet to the price of the

4 Europea whe this early exercise premium is added. Although BAW (987) approximatio is fast, the serious drawback of it is the lack of accuracy especially uder log maturity optios. With the same cocept, Kim (99) assumed the stochastic process of the stock price to be Cox-Igersoll-Ross (985) process (CIR) model, ad solved the problem of early exercise with the critical price cocept. He proposed the recursive itegratio method to price America optios. To solve America put i a faster way, Huag, Subrahmayam ad Yu (996) proposed accelerated recursive itegratio method, usig four-poit Richardso extrapolatio to approximate. Similarly, Ju (998) also used such method i additio with the udgmet of critical price. The third approach is the use of dow-ad-out put optio by Igersoll (998) to price America optios. Sbuelz (4) also adopted the same cocept. Barrier optios ot oly udge the fial stock price o the expiratio date, but also cosider whether it hits the barrier before the expiratio date. A dow-ad-out put optio is valuable whe the stock price is below the strike price ad does ot hit the barrier. The characteristic of America optios is similar to barrier optios. Whe both optios hit a certai level, the value will begi to chage. For Igersoll (988) ad Sbuelz (4), whe usig a dow-ad-out put optio to price America optios, a rebate value will be added to the model. This is to satisfy the immediate exercise value whe the stock price hits the barrier. However, these models are still derived uder costat volatility.. The Model We ituitively thik that by developig a dow-ad-out put optio model uder the Hesto (99) framework, a dow-ad-out put optio with stochastic volatility model ca be derived to price America optios. However, difficulty arises i the calculatio of dow-ad-out probability uder the Hesto model. The accuracy of the probability caot be obtaied correctly; thus obstructig the developmet of the model. Therefore, we used aother alterative to circumvet the problem. With referece to Griebsch ad Wystup (8) paper i developig the model for fader optio evaluatio, it has directed us with a solutio. Uder the same Hesto (99) framework, Griebsch ad Wystup (8) used the characteristic fuctio proposed by Shepard (99) to calculate discrete probability distributio. This is equivalet to calculatig a discrete dow-ad-i probability. With this probability obtaied, we ca hece develop a discrete dow-ad-i put ad evetually a dow-ad-out put optio with stochastic volatility model. After addig a immediate exercise value, the model becomes a discrete America put optio with stochastic volatility (i.e.,,, ). The, we ca apply extrapolatio to approach ad obtai the aalytical solutio.. Hesto Model The diffusio process of the stock price ad the variace follows the settig of Hesto (99) stochastic volatility model ds S dt S dz () d dt dz () i is the America put optio (Bermuda optio) with i-th umber of exercisable time poits. 4

5 where is the rate of mea reversio, is the log term variace ad represets the volatility of variace. They are all costat parameters. z ad z are two correlated Browia motios. is the correlatio coefficiet of these two Browia motios. I equatio (), the stock price S is a geometric Browia motio ad its volatility chages with time. I equatio (), curret variace follows the square root process (Cox-Igersoll-Ross (985) process). Similar to Black-Scholes (97) formula, Hesto (99) guessed the solutio form of this Europea call with stochastic volatility rt,, C S K T SF Ke F () F iu l K u exp Re iu (4) where =,. K is the strike price, r represets the risk-free iterest rate, T deotes the time to maturity, Re stads for real part, fuctio u is the characteristic fuctio. F is the coditioal probability that the optio expires i-the-moey. The form of Europea put with stochastic volatility ca be obtaied through put-call parity.,,,, rt SKT C SKT Ke S. Discrete dow-ad-out put optio with stochastic volatility (5) I Griebsch ad Wystup (8) paper, the calculatio of the probability i the fader optio is the probability differece of the stock price calculated betwee high barrier H ad low barrier L (i.e., the probability of stock price uder high barrier H mius the probability of stock price uder low barrier L). By ust cosiderig the probability uder low barrier L, we ca obtai a discrete dow-ad-i probability. With it, a discrete dow-ad-i put optio with stochastic volatility ca be developed. Through i-put parity, a discrete dow-ad-out put optio with stochastic volatility ca be formed. Below is the illustratio of the discrete dow-ad-i probability calculatio. With the dimesioal characteristic fuctio ad the examples ad priciple provided by Shephard (99), we are able to calculate this probability distributio. Corollary We derived the followig probability distributio 4 o our ow. -dimesioal: 4 The detail of the derivatio is o Appedix B. 5 ue ic u Fcrob xt c (6) iu -dimesioal: For characteristic fuctio u, please refer to Appedix A.

6 F c, c rob x c, x c t t ic u,, u e u e 4 icu Re Re iu iu,, icu icu icu icu u u e u u e Re R (7) -dimesioal: F c, c, c rob x c, x c, x c t t t ic u,,,, u e u e icu Re Re iu iu icu,, u e Re 4 iu R Re icu icu u, u, e u, u, e,,,,,,,, icu icu icu icu icu icu u u e u u e Re 4 R icuicu icuicu u u e u u e Re d 4 u R,,,, u u u e u u u e Im u icu icuicu icu icuicu 4 (8) R icu icuicu icu icuicu u, u, ue u, u, ue u for, ( differet measures. Re : real part, Im : imagiary part). rob is the probability uder Through the probability distributio, the discrete dow-ad-i put optio with stochastic volatility 5 ca be calculated with a extesio of three discrete exercisable time poits.. A discrete America put with stochastic volatility Our motive is to evaluate America put optios with stochastic volatility. Therefore, the discrete dow-ad-out put with stochastic volatility model 6 eeds to be modified ito a 5 lease refer to equatios () ad (9) for the form of the model. 6 lease refer to equatios () ad () for the form of the model. 6

7 discrete America put with stochastic volatility (or Bermuda optio) ad apply extrapolatio to approach. For America put optios, the optimal early exercise is whe the stock price falls below or hits the critical price. The cocept of the critical price is equivalet to that of the barrier. I barrier optios, whe the stock price hits the barrier, the value of the optio will be zero. However, for America optios, whe the stock price hits the critical price, a immediate exercise value ca be obtaied. To avoid a ull value, we add this immediate exercise value to the discrete dow-ad-out put optio with stochastic volatility model so that a discrete America put with stochastic volatility (Bermuda optio) ca be formed. For America put optios, the value of immediate exercise is the differece betwee the strike price ad the critical price. Therefore, whe the stock price hits the critical price i the discrete dow-ad-out put optio with stochastic volatility model, we should add a put optio value as metioed i.. This put optio value is equivalet to the immediate exercise value 7. It ca be calculated by takig the critical price as the strike price. l S/ K r / T I Black-Scholes pricig formula d, Nd represets the T probability of i-the-moey coditio, ad the stock price is more tha the strike price o the expiratio date. We ituitively thik that by replacig the strike price K with the critical price i l S/ K, the obtaied probability of Nd ad Nd will become the probability of i-the-moey calculated with the critical price istead of the strike price. Critical price S is a importat elemet whe pricig America put optios with stochastic volatility because it represets the optimal exercise time. For America optios, the optimal-exercise policy ca be preseted as the exercise boudary i price-time space. The boudary partitioed the space ito a hold regio ad a exercise regio. I a put optio, early exercise occurs whe the stock price falls below or hits the critical price, ad there should be a holdig whe it is above the critical price. Therefore, critical price ca be determied whe the immediate exercise value is equivalet to the holdig value. This ca K S S, K, T for some S S ad ay be illustrated i a simple put optio below: T. A iitial value of the critical price will be give first. This value also stads for the iitial stock price. The, substitute the give value ito the model ad solve it iteratively with the bisectio method util the immediate exercise value is equivalet to the holdig value. The result obtaied will hece be the critical price..4 Aalytical Solutio for America ut with Stochastic Volatility After obtaiig a discrete America put with stochastic volatility model (Bermuda optio) ad a calculated critical price, three-poit Richardso extrapolatio is applied to evaluate America put optios with stochastic volatility. Let be a pure Europea put with stochastic volatility that ca oly be exercised at expiratio time T (Equivalet to equatio (6)). S, K, T (9) 7 lease refer to equatios (), () ad () for the form of the model. 7

8 Let di be a dow-ad-i put with stochastic volatility. A barrier H at time T /. Oly if the stock price hits or falls below the barrier, the optio is valuable. Whe it comes to expiratio time T, the optio must be valuable.,, Ke rob x H, x K Srob x H, x K rt di T / T T / T rt Ke F H K S F H K do is a dow-ad-out put with stochastic volatility ad ca be obtaied:,,,,, do di 8 () S K T H S K T () Let exercise be a Europea put with stochastic volatility. Expiratio time be T /. The critical price is take as the strike price at T / so that the value of the put will be the immediate exercise value. exercise S, ST /, T / () Let be a America put with stochastic volatility. Early exercise ca oly be udged at time T /. ( is equivalet to Bermuda optio),,,,,, S K T S S K T S () T / do T / exercise Sice the maturity iterval from T / to T is oly a pure Europea put (see Figure ), critical price S T / at T / ca be obtaied by usig bisectio method to solve iteratively. KS,, / T/ ST/ K T (4) Let di be a dow-ad-i put with stochastic volatility. There is a barrier H at T / ad T / respectively. Oly if the stock price hits or falls below the barrier, the optio is valuable. It has to be valuable whe it comes to expiratio time T. However, this probability must be cosidered i detail i three cases:. The stock price hits or falls below the barrier at T /. However, the stock price does ot hit at T / ad is above the barrier. Whe it comes to expiratio time T, it has to be valuable.. The stock price does ot hit at T /ad is above the barrier. However, it hits or falls below the barrier at T /. Whe it comes to expiratio time T, it has to be valuable.. The stock price hits or falls below the barrier at both T / ad T /. Whe it comes to expiratio time T, it has to be valuable. The above three coditios are all dow-ad-i put probabilities, so we ca use the cocept of set: x / H x / H x K xt/ H xt K xt/ H x T/ H xt K,,, rob T, T, T rob, rob,, F H K F H H K (5)

9 x / H x / H x K x T/ H xt K xt/ H x T/ H xt K,,, rob,, T T T rob, rob,, F HK F HHK (6) x H x H x K F H H K rob,,,, (7) T / T / T Thus, the probability of a dow-ad-i ( rob di ) is the sum of the three coditios: x / H x K x / H x K x H x H x K rob dirob T, T rob T, T rob,, T/ T/ T (8) for, di ca be obtaied: rob rob Ke S (9) rt di di di do is a dow-ad-out put with stochastic volatility ad ca be obtaied:,,,,,, S K T H H S K T () do di Let exercise be a Europea put with stochastic volatility. Expiratio time be T /. The critical price is take as the strike price so that the value of the put will be the immediate exercise value. exercise S, ST /, T / () Let exercise be a model of di. Expiratio time be T /. The critical price at T / is take as the barrier, ad the critical price at T / be the strike price. The value of the put is the same as the immediate exercise value. exercise T/ T/ T/ T/ T/ T/ T/ T/ T, T T, T rt Ke rob x S, x S S rob x S, x S Ke F S S S F S S rt / / / / () Let be a America put with stochastic volatility. Early exercise ca oly be udged at time T / ad T /. ( is equivalet to Bermuda optio),,,,,,,, SKTS S SKTS S () T / T / do T / T / exercise exercise Sice the maturity iterval from T / to T is oly a pure Europea put (See Figure ), critical price S T / at T / ca be obtaied by usig the bisectio method to solve iteratively. K S,, / T/ S T/ K T (4) 9

10 The critical price S T / at T / requires model to solve because the maturity iterval from T / to T is a represetatio of a model. It ca be exercised at either T / or T. Hece, critical price S T / ca still be obtaied by usig the bisectio method to solve iteratively. (See Figure ) KS S, K, T /, S (5) T/ T/ T/ Evetually, we derived, ad. By applyig the three-poit Richardso extrapolatio to approach 8, we ca obtai the aalytical solutio for the America put optios with stochastic volatility: 9 4 (6) 4. Numerical Aalyses Numerical aalyses are carried out o our model for America put optios with stochastic volatility ( ). The programs are writte i C++ ad MATLAB. Sice the least-squares Mote Carlo simulatio (LSM) (Logstaff ad Schwartz ()) model is ituitive, accurate, efficiet ad coveiet to apply, we set the value calculated by it as the bechmark. The, we compared the differece betwee stochastic volatility ad costat volatility. 4. Bechmark ad Settig of the arameters I the LSM method, we uified i adoptig, paths, repeatig times ad usig the tradig day as steps 9. The iitial hypothesis of the parameters is set as: strike price K =5, risk-free iterest rate r=5%, the rate of mea reversio =, the log term variace =%, volatility of variace =.5% ad variace =%. The settig of the parameters follows the Hesto (99) paper. The correlatio coefficiet is divided ito three coditios: =.5, =, =-.5. The stock price S is set as: i-the-moey (ITM) S = 45, at-the-moey (ATM) S = 5 ad out-the-moey (OTM) S =55. We assume that if the relative error i price calculated by both ad LSM method is less tha %, the is of high accuracy. The relative error is calculated by first usig the value of mius the value of LSM method, ad the differece obtaied would be divided by the value of LSM method. Relative error = LSM LSM (7) I terms of calculatig efficiecy, we ca use the time take by the LSM method over 8 The three-poit Richardso extrapolatio is referred from Geske ad Johso (984) paper. 9 We have tried settig both quadratic polyomial ad cubic polyomial for basis fuctio f. There is oly a slight differece betwee the LSM value of both polyomials. However, the time cosumed for cubic polyomial is much more tha expected. Therefore, we cosidered util quadratic polyomial. The basis fuctio f used i LSM model is f a as as a a4 a5s, a, a, a, a, a 4, a 5 are the parameters estimated by regressio.

11 the time take by. CU time Efficiecy = CU time LSM (8) 4. The Accuracy of the Aalytical Solutio Model We illustrate the accuracy of i ITM, ATM ad OTM respectively i Figure 4. The settig of the parameters is the same as the iitial settig except for the correlatio coefficiet which is set at.5. The time to maturity is from the sixth moth to the ith moth. It is see that all the prices of lie withi the 95% cofidece iterval (C.I). This shows that our aalytical solutio is accurate. Table to Table is the compariso betwee ad LSM method i ITM, ATM ad OTM respectively. The parameters are the same as the iitial settig. The optio with the shortest time to maturity is oe moth, ad the logest is three years. Accordig to the tables, the relative errors show are all less tha %. This proves that our model is very accurate i differet terms of optios. Regardless of the value of correlatio coefficiet, the value of the optios icreases with icreasig time to maturity. Isert Tables to about here I the speed of calculatio, the average time take for is 6.9s. While the average time take for the LSM method varies uder differet coditios of moeyess. I the aspect of ITM where early exercise is more likely, the average time take for a six-moth optio ad a oe-year optio is about s ad s respectively. It is about s ad 77.9s for ATM. I OTM, it is about s ad 96.s e to the low possibility of exercisig. Compared with the time take i calculatig a six-moth optio ad a oe-year optio by the LSM method, the efficiecy of is.e+ ad.e+ for ITM, 9.8E+ ad.9e+ for ATM, ad 7.8E+ ad.5e+ for OTM. Therefore, it proves that the speed of the aalytical solutio is much faster tha the LSM method. Table 4 shows the differece betwee the stochastic volatility model ad the costat volatility model GJ (Geske ad Johso (984)). The rage of variace varies from % to 6%. The time to maturity is half a year. The rest of the parameters are the same as the iitial settig. The results i Table 4 reveal that as variace icreases, the differece becomes greater. Iitially, we assume that stochastic volatility model is more realistic tha costat volatility model. The differeces show i Table 4 have verified our assumptio. The applicatio of costat volatility model i the past is ideed ureasoable. Therefore, it is more practical to adopt stochastic volatility model. If we set the time to maturity from a moth to a year, the figure would be too small for observatio. Moreover, the terms of the optios usually last from six moths to ie moths i the market; hece, we oly draw the tred from the sixth moth to the ith moth. The average time take of a oe-moth optio by usig the LSM method is about 4.74s for i-the-moey, 94.95s for at-the-moey ad 58.4s for out-the-moey. As the time to maturity of the optio icreases mothly, the average time take for calculatio also almost icreases i multiples of that iitial time.

12 Isert Table 4 about here 5. Empirical erformace of Aalytical Solutio Model To test our aalytical solutio i the real fiacial market applicatio, we perform empirical studies. First, we eed to decide o the type of derivatives to be evaluated. Next, we will do a estimatio of the parameters. With that, we will verify if stochastic volatility model is more realistic tha costat volatility model. 5. Resource of Iformatio Sice our aalytical solutio model evaluates o plai vailla America put optios with stochastic volatility, the empirical data used must comply with such coditio. Warrats ad optios are similar i that the two cotractual fiacial istrumets are discretioary ad have expiratio dates. Therefore, we decided to use the put warrats i Taiwa fiacial market as our empirical study. Thirty put warrats are carefully chose as samples. The time period selected for the put warrat cotracts is from Jauary 6 to May 9. Relevat iformatio such as the past market prices ad the past stock prices of the uderlyig assets correspodig to the warrats are collected from the Market Observatio ost System (M.O..S) ad Taiwa Ecoomic Joural (TEJ). We used moey market iterest rate C-9 as the risk-free iterest rate. Table 5 provides the iformatio of the thirty put warrats i Taiwa fiacial market, icludig their warrat code, uderlyig stock umber, ad the time to maturity. 5. arameters Estimatio Isert Table 5 about here To determie if the stochastic volatility model is more realistic tha the costat volatility model, we compared with the costat volatility model GJ proposed by Geske ad Johso (984). Both models applied three-poit Richardso extrapolatio to approach. For GJ, the most importat parameter is the estimatio of costat volatility. However, for, parameters such as the rate of mea reversio, the log term variace, volatility of variace, curret variace ad correlatio coefficiet are all cosidered importat.. Therefore, we follow the cocept ad methods foud i Bakshi, Cao ad Che (997) paper to estimate these importat parameters. Let N be the umber of the total put warrats. For each,..., N. Let T be the time to maturity of the -th put warrat, ad K is the strike price. Let ˆ T ; Kbe the market price (i.e., the observe price). T ; K is the aalytical solutio price (i.e., the model price). The differece betwee ˆ ad is a fuctio of the values take by,,,. For each, defie ad by The, fid ad parameter vector, to solve, ˆ T ; K T ; K (9) N SSE mi,, ()

13 By usig the Exhaustive Attack method iteratively util a set of parameters is obtaied whe SSE is at its smallest. The result will hece be the estimated parameters that are eeded to be substituted ito the models for the calculatio of the put warrat prices. 5. Empirical erformace We set the put warrat market prices obtaied rig large volume as the bechmark because the prices are more stable at that time. The, ad GJ are applied to calculate the prices of the thirty put warrats respectively. Fially, root mea squared error (RMSE) of the prices obtaied by each model will be calculated for compariso. The smaller the RMSE, the differece betwee the model price ad the put warrat market price is smaller. RMSE N ˆ ; ; T K T K N () Table 6 shows the market prices of the put warrats, the prices of the put warrats calculated by ad GJ ad the respective RMSE obtaied. It is see that the RMSE of the put warrat prices uder is. while the put warrat prices uder GJ is.9. The RMSE obtaied by is smaller tha that of GJ, provig that stochastic volatility model is more realistic ad efficiet for the actual market. Isert Table 6 about here 6. Coclusios Much of the literature o America optios is evaluated uder costat volatility. I this paper, we cosider the harder problem of derivig a aalytical solutio by usig barrier optios to evaluate America put optios with stochastic volatility. Our model proves to be accurate ad efficiet with relative error less tha % i umerical aalyses. From the empirical result, it is show that stochastic volatility model ideed performs better tha costat volatility model i evaluatig the value of fiacial derivatives. I other words, stochastic volatility model ca better illustrate the real market. Our aalytical solutio is practical because it ca be applied broadly o ay optios that satisfy the coditios of plai vailla America put. Besides pricig optios, hedge ratio (Greeks) is aother area that ca be further ivestigated. Theoretically, if differetiatio is performed o our model, other aalytical solutios for hedge ratio could also be derived. But still, we strive to develop the closed-form for America optios with stochastic volatility i future studies. I additio, Black ad Scholes (97) model has treated shareholders equity as a stadard call optio with corporate value as the uderlyig asset. Black ad Scholes (97) assumed the call optio to be path-idepedet. Istead of beig iflueced by the tred of the asset value withi the ratio, the profit ad loss (&L) of the optio will oly be ehaced o the expiratio date. Moreover, the equity value will become othig whe the corporate declares bakrupt because the corporate value is less tha the liability value rig the moitorig period. To modify such usuitable assumptio, the equity value ca be treated as a dow-ad-out barrier call optio with corporate value as the uderlyig asset, while the bod value as the strike price. The characteristic of path-depedet ca the be captured. Sice we have already derived a discrete barrier optio with stochastic volatility

14 model i this paper, future research ca be doe by applyig it o the evaluatio of the credit risks of a corporate. Refereces Bakshi, G., C. Cao, ad Z. Che, 997, Empirical erformace of Alterative Optio ricig Models, Joural of Fiace, 5, -49. Baroe-Adesi, G., ad R. E. Whalley, 987, Efficiet Aalytic Approximatio of America Optio Values, Joural of Fiace, 4, -. Barraquad, J., ad D. Martieau, 995, Numerical Valuatio of High Dimesioal Multivariate America Securities, Joural of Fiacial ad Quatitative Aalysis,,, Black, F., ad M. Scholes, 97, The ricig of Optios ad Corporate Liabilities, Joural of olitical Ecoomy, 8, Boyle,.., 977, Optios: A Mote Carlo Approach, Joural of Fiacial Ecoomics, 4, -8. Brea M. J., ad E. S. Schwartz, 978, Fiite Differece Methods ad Jump rocesses Arisig i the ricig of Cotiget Claims: A Sythesis, Joural of Fiacial ad Quatitative Aalysis,, September, Cox, J. C., J. E. Igersoll, ad S. A. Ross, 985, A Theory of the Term Structure of Iterest Rates, Ecoometrica, 5, Cox, J. C., S. A. Ross, ad M. Rubistei, 979, Optio ricig: A Simplified Approach, Joural of Fiacial Ecoomics, 7, 9-6. Geske, R., 979, The Valuatio of Compoud Optios, Joural of Fiacial Ecoomics, 7, 6-8. Geske, R., ad H. E. Johso, 984, The America ut Optio Valued Aalytically, Joural of Fiace, 9, Grat, D., G. Vora, ad D.E. Weeks, 996, Simulatio ad the Early-Exercise Optio roblem, Joural of Fiacial Egieerig, vol.5(), 7. Griebsch, S., ad U. Wystup, 8, O the Valuatio of Fader ad Discrete Barrier Optios i Hesto's Stochastic Volatility Model, Workig aper. Hesto, S., 99, A Closed-Form Solutio for Optios with Stochastic Volatility with Applicatios to Bod ad Currecy Optios, Review of Fiacial Studies, 6, 7 4. Huag, J., M. Subrahmayam, ad G. Yu, 996, ricig ad Hedgig America Optios: A Recursive Itegratio Method, Review of Fiacial Studies, 9, 77. Hull, J., ad A. White, 987, The ricig of Optios o Assets with Stochastic Volatilities, Joural of Fiace, 4, 8. Igersoll r, J. E., 998, Approximatig America Optios ad other Fiacial Cotracts Usig Barrier Derivatives, Joural of Computatioal Fiace,, 85. Ju, N., 998, ricig a America Optio by Approximatig Its Early Exercise Boudary 4

15 as a Multipiece Expoetial Fuctio, Review of Fiacial Studies,, Kim, I. J., 99, The Aalytic Valuatio of America Optios, Review of Fiacial Studies,, Logstaff, F. A., ad E. S. Schwartz,, Valuig America Optios by Simulatio: Simple Least-Squares Approach, Review of Fiacial Studies, 4, 47. Raymar, S., ad M. Zwecher, 997, Mote Carlo Estimatio of America Call Optios o the Maximum of Several Stocks, Joural of Derivatives, 5,, 7-. Sbuelz, A., 4, Aalytical America Optio ricig: The Flat-Barrier Lower Boud, Ecoomic Notes,, Shephard, N. G., 99, From Characteristic Fuctio to Distributio Fuctio: A Simple Framework for the Theory, Eco. Theory. Cambridge Uiversity ress, vol. 7(4), Stei, E. M., ad J. C. Stei, 99, Stock rice Distributios with Stochastic Volatility: A Aalytic Approach, Review of Fiacial Studies, 4, Tilley, J. A., 99, Valuig America Optios i a ath Simulatio Model, Trasactios of the Society of Actuaries, vol. 45, Appedix A Characteristic Fuctio Settig (Griebsch ad Wystup, 8) Griebsch ad Wystup (8) derive -variate characteristic fuctios of the log-spot prices l S t,, l St at times < t < < t = T i the Hesto (99) model uder two differet probability measures: X x,..., x be a radom vector ad u u,..., u be a vector of real umbers. The oit characteristic fuctio of radom variables x,..., t x t is defied by Let t t iux exp... t t u Ee iu x iu x d X (A) R is the probability measure fuctio of X. The fuctio X where u is a complex-valued cotiuous fuctio of the real variables u,..., u. Griebsch ad Wystup (8) derive the characteristic fuctio uder the risk-eutral measure rob ad the spot measure rob with the spot price as umeraire. I the Hesto (99) model as defied i () the oit characteristic fuctio of the logarithm of spot values X x,..., t x t at times t t... t T uder the risk-eutral measure rob is give by u,..., uexp iukhtk quk tk Bk Av k k k (A) 5

16 where we set Bk B tk tk, quk Ak, w u k (A) Ak A tk tk, quk Ak, w u k (A4) startig with A ad d f ht x r r t (A5) The fuctios A ad B above are defied as t v t (A6) d d da e e b a A A, a, b (A7) B B, a, b d l d (A8) with d b (A9) ad the fuctios w ad q as d d d e a e (A) wu iu iu (A) qu iu (A) The -variate characteristic fuctio uder the spot measure rob is give by u,..., uexp iukhtk quk tk t Bk Av k k k (A) with a differet defiitio of the fuctios A ad B tha for, amely Bk B tk tk, quk Ak, w u i (A4) k 6

17 startig with A Ak A tk tk, quk Ak, w u i (A5) k Appedix B. roof of corollary -dimesioal: F c icu u e u iu ic u ue iu ic u ue iu icu u e iu icu icu icu u e iu icu u e iu u e iu u e iu u e iu u e icu icu icu u e where u u u u iu iu 7

18 -dimesioal: u, u e icu icu u Re 4 F c, c F c F c u u, u e Re ic u,, icu u e u e 4 Fc, c Re Re iu iu icu icu ic u,, icu u e u e Fc, c Re Re 4 iu iu,, u u e u u e Re icu icu icu icu ic u,, u e u e 4 icu Re Re iu iu,, icu icu icu icu u u e u u e Re 8

19 -dimesioal: 4 icu icuicu u, u, ue Im u u u 8 F c, c, c 4 F c, c F c, c F c, c F c F c F c F c, c, c icu u,, e Re Re 8 4 iu icu,, u e Re iu ic u,,,, icu icu icu icu icu, u, e icu u e u e Re Re 4 iu iu u, u, e Re icu,,,,,,,, u, u, e iu icu u e u e Re Re 4 iu iu icu icu icu icu u u e u u e Re u icu,,,,,,,, icu u e u e Re Re 4 iu iu icu icu icu icu u u e u u e Re 4,,,, u u u e u u u e Im u u ic u ic u ic u ic u ic u ic u 9

20 ic u,,,, u e u e Fc c c icu,, Re Re iu 4 iu F c, c, c,, u icu Re 4 iu e u,, e, u, e 8 icu icu Re Re iu iu,, u icu Re iu e,,,, icu icu icu icu u u e u u e Re 4,,,, icu icu icu icu u u e u u e Re 4,,,, icuicu icuicu u u e u u e Re 4,,,, u u u e u u u e Im u icu icuicu icu icuicu 4 icu icuicu icu icuicu u, u, u e u, u, u e u ic u,,,, u e u e icu Re Re iu iu,, u icu Re 4 iu e,,,, icu icu icu icu u u e u u e Re 4,,,, icu icu icu icu u u e u u e Re 4,,,, icu icu icu icu u u e u u e Re 4 4 u, u, u e u, u, u e Im u icu icuicu icu icuicu,,,, u u u e u u u e u icu icuicu icu icuicu

21 S T (yr) Table Compariso betwee ad LSM for i-the-moey =.5 = =-.5 LSM relative LSM relative LSM relative std error std error std error % % 5.98 (.8) (.8) (.6).46% % 5.97.% 5.56 (.) (.) (.9).77% % % 5.64 (.) (.) (.).4% % % 5.85 (.) (.) (.).6% % 6.9.5% 6.68 (.7) (.7) (.4).5% % % 6.68 (.) (.7) (.5).% % 6.5.% 6.45 (.7) (.) (.4).65% % % 6.66 (.9) (.5) (.7).4% % % (.7) (.) (.).% % 7. -.% 6.94 (.9) (.) (.).6% % % 7.86 (.6) (.9) (.).6% % 7..5% 7.4 (.8) (.7) (.9).6% % % (.) (.4) (.5).47% % % 9.68 (.) (.6) (.5).76% Note: The parameters are set as follows: K =5, r =5%, =, =%, =.5%, =%. is the aalytical solutio derived i this paper, LSM is the least-squares Mote Carlo simulatio ad std is the stadard deviatio. LSM method uses, paths with the tradig day adopted for the steps ad times of repeatig. Calculatio for relative error: price mius LSM price ad the outcome is the divided by LSM price. The result is show i percetage.

22 S T (yr) Table Compariso betwee ad LSM for at-the-moey =.5 = =-.5 LSM relative LSM relative LSM relative std error std error std error %.64.7%.64 (.6) (.6) (.6).65% %.54.5%.5 (.9) (.7) (.6).8% %.78.%.76 (.9) (.9) (.9).78% %.76.5%.74 (.) (.) (.).94% %.9.97%.88 (.) (.4) (.5).59% %.666.5%.66 (.5) (.5) (.5).55% %.9.7%.9 (.) (.5) (.6).% % % 4.5 (.) (.) (.4) -.4% % 4.4.% 4.4 (.4) (.9) (.8).67% % % 4.5 (.7) (.9) (.8) -.56% % 4.7.5% 4.7 (.) (.5) (.6) -.5% % % (.5) (.7) (.) -.44% % % 6.7 (.4) (.) (.) -.86% % % 7.59 (.) (.4) (.).8% Note: The parameters are set as follows: K =5, r =5%, =, =%, =.5%, =%. is the aalytical solutio derived i this paper, LSM is the least-squares Mote Carlo simulatio ad std is the stadard deviatio. LSM method uses, paths with the tradig day adopted for the steps ad times of repeatig. Calculatio for relative error: price mius LSM price ad the outcome is the divided by LSM price. The result is show i percetage.

23 Table Compariso betwee ad LSM for out-the-moey =.5 = =-.5 S T (yr) LSM relative LSM relative LSM relative std error std error std error %.85.%.7 (.) (.) (.) -.48% %.7.5%.75 (.5) (.6) (.6) -.66% %.9.48%.4 (.7) (.7) (.8) -.9% %.4.6%.48 (.9) (.9) (.8) -.69% %.77.9%.78 (.8) (.) (.) -.% %.98.9%.5 (.) (.) (.) -.66% %.5.57%.97 (.) (.) (.4) -.% % %.5 (.) (.) (.) -.69% % %.7 (.) (.5) (.6) -.6% %.848.6%.95 (.) (.5) (.5) -.% %.9.4%.7 (.) (.) (.) -.% %.99.%.78 (.) (.5) (.5) -.6% % % 4.85 (.9) (.) (.4) -.6% % % 5.88 (.9) (.7) (.) -.45% Note: The parameters are set as follows: K =5, r =5%, =, =%, =.5%, =%. is the aalytical solutio derived i this paper, LSM is the least-squares Mote Carlo simulatio ad std is the stadard deviatio. LSM method uses, paths with the tradig day adopted for the steps ad times of repeatig. Calculatio for relative error: price mius LSM price ad the outcome is the divided by LSM price. The result is show i percetage.

24 Table 4 Compariso of differece betwee ad GJ uder differet variaces T =.5(yr) =.5 = =-.5 S GJ diff. GJ diff. GJ diff Note: The parameters are set as follows: K =5, r =5%, =, =%, =.5%. aalytical solutio derived i this paper. mius GJ. is the GJ is Geske ad Johso (984) model. Differece calculatio: 4

25 Warrat No. Table 5 Iformatio of ut Warrat i Taiwa Market Warrat code Uderlyig stock No. Time to maturity 588 Capital NN 894~9 58 Capital NJ 89~9 575 Capital MZ ~99 57 Capital MV ~ JS J ~9 458 YT JN 68 85~8 4 YTJB 84 84~8 84 SC DB 5 848~87 8 YT F 5 88~ YTCQ 49 75~ YTAB 7~ JSB ~ YCJD ~87 48 YCAW ~7 469 GCSCB ~ YCAS ~ YCAT 9 698~77 44 YCAR ~76 49 GCSCB ~75 4 YCA 7 68~7 4 YCAQ ~7 49 YCAL 48 68~7 48 GCSCB 9 689~7 74 YCF 8 5~ YCF5 9 5~ YCC7 58~ YCC5 57~ YCC6 57~ JS66 9 5~64 74 Masterlik ~657 5

26 Table 6 Empirical erformace of the Stochastic Volatility Model Warrat No. Warrat code Market price Costat volatility optio price Stochastic volatility optio price 588 Capital NN Capital NJ Capital MZ Capital MV JS J YT JN YTJB SC DB YT F YTCQ YTAB JSB YCJD YCAW GCSCB YCAS YCAT YCAR GCSCB YCA YCAQ YCAL GCSCB YCF YCF YCC YCC YCC JS Masterlik Root Mea Squared Error (RMSE).9. Note: The costat volatility optio price is calculated by Geske ad Johso (984) model The stochastic volatility optio price is calculated by the aalytical solutio. GJ. 6