Computers and Mathematics with Applications. The evaluation of barrier option prices under stochastic volatility

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1 Coputers and Matheatics with Applications 64 () Contents lists aailable at SciVerse ScienceDirect Coputers and Matheatics with Applications journal hoepage: The ealuation of barrier option prices under stochastic olatility Carl Chiarella a, Boda Kang b,, Gunter H. Meyer c a Finance Discipline, UTS Business School, Uniersity of Technology, Sydney, Australia b Finance Discipline, UTS Business School, Uniersity of Technology, Sydney, PO Box 3, Broadway, NSW 7, Australia c School of Matheatics, Georgia Institute of Technology, Atlanta, United States a r t i c l e i n f o a b s t r a c t Article history: Receied 9 January Receied in reised for Noeber Accepted 8 March Keywords: Barrier option Stochastic olatility Continuously onitored Discretely onitored Free boundary proble Method of lines This paper considers the proble of nuerically ealuating barrier option prices when the dynaics of the underlying are drien by stochastic olatility following the square root process of Heston (993) [7]. We deelop a ethod of lines approach to ealuate the price as well as the delta and gaa of the option. The ethod is able to efficiently handle both continuously onitored and discretely onitored barrier options and can also handle barrier options with early exercise features. In the latter case, we can calculate the early exercise boundary of an Aerican barrier option in both the continuously and discretely onitored cases. Elseier Ltd. All rights resered.. Introduction Barrier options are path-dependent options and are ery popular in foreign exchange arkets. They hae a payoff that is dependent on the realized asset path ia its leel; certain aspects of the contract are triggered if the asset price becoes too high or too low during the option s life. For exaple, an up-and-out call option pays off the usual ax(s K, ) at expiry unless at any tie during the life of the option the underlying asset has traded at a alue H or higher. In this exaple, if the asset reaches this leel (fro below, obiously) then it is said to knock out and becoe worthless. Apart fro out options like this, there are also in options which only receie a payoff if a certain leel is reached, otherwise they expire worthless. Barrier options are popular for a nuber of reasons. The purchaser can use the to hedge ery specific cash flows with siilar properties. Usually, the purchaser has ery precise iews about the direction of the arket. If he or she wants the payoff fro a call option but does not want to pay for all the upside potential, belieing that the upward oeent of the underlying will be liited prior to expiry, then he ay choose to buy an up-and-out call. It will be cheaper than a siilar anilla call, since the upside is seerely liited. If he is right and the barrier is not triggered he gets the payoff he wanted. The closer that the barrier is to the current asset price then the greater the likelihood of the option being knocked out, and thus the cheaper the contract. Barrier options are coon path-dependent options traded in the financial arkets. The deriation of the pricing forula for barrier options was pioneered by Merton [] in his seinal paper on option pricing. A list of pricing forulas for one-asset barrier options and ulti-asset barrier options both under the geoetric Brownian otion (GBM) fraework can be found in the articles by Rich [] and Wong and Kwok [3], respectiely. Gao et al. [4] analyzed option contracts with both knock-out barrier and Aerican early exercise features. Zan et al. [5] hae discussed the oscillatory behaior of the Corresponding author. E-ail addresses: carl.chiarella@uts.edu.au (C. Chiarella), boda.kang@uts.edu.au (B. Kang), eyer@ath.gatech.edu (G.H. Meyer). 898-/$ see front atter Elseier Ltd. All rights resered. doi:.6/j.cawa..3.3

2 C. Chiarella et al. / Coputers and Matheatics with Applications 64 () Crank Nicolson ethod for pricing barrier options, and they applied the backward Euler ethod in order to aoid unwanted oscillations. Deriatie securities are coonly written on underlying assets with return dynaics that are not sufficiently well described by the GBM process proposed by Black and Scholes [6]. There hae been nuerous efforts to deelop alternatie asset return odels that are capable of capturing the leptokurtic features found in financial arket data, and subsequently to use these odels to deelop option prices that better reflect the olatility siles and skews found in arket traded options. One of the classical ways to deelop option pricing odels that are capable of generating such behaior is to allow the olatility to eole stochastically, for instance according to the square-root process introduced by Heston [7]. The ealuation of barrier option prices under the Heston stochastic olatility odel has been extensiely discussed by Griebsch [8] in her thesis. Howeer, there are certain drawbacks in the ealuation of the Barrier option prices under SV using either tree or finite difference ethods, these include the fact that the conergence is rather slow and it takes ore effort to obtain accurate hedge ratios. Yousuf [9,] hae deeloped a higher order soothing schee for pricing barrier options under stochastic olatility. The ethod is stable and conerges rapidly which oercoe soe drawbacks of the finite difference ethods. But those papers do not discuss how to handle the possible early exercise features of the barrier option pricing proble. It turns out that another well known ethod, the ethod of lines is able to oercoe those disadantages. In this paper, we introduce a unifying approach to price both continuously and discretely onitored barrier options without or with early exercise features. Specifically, except for Aerican style knock-in options, we are able to price all other kinds of European or Aerican barrier options using the fraework deeloped here. The reainder of the paper is structured as follows. Section outlines the proble of both continuously and discretely onitored barrier options where the underlying asset follows stochastic olatility dynaics. In Section 3 we outline the basic idea of the ethod of lines approach and ipleent it to find the price profile of the barrier option. A nuber of nuerical exaples that deonstrate the coputational adantages of the ethod of lines approach are proided in Section 4. Finally we discuss the ipact of stochastic olatility on the prices of the barrier option in Section 5 before we draw soe conclusions in Section 6.. Proble stateent-barrier option with stochastic olatility Let C(S,, τ) denote the price of an up-and-out (UO) call option with tie to aturity τ written on a stock of price S and ariance that pays a continuously copounded diidend yield q. The option has strike price K and a barrier H. Analogously to the setting in [7], the dynaics for the share price S under the risk neutral easure are goerned by the stochastic differential equation (SDE) syste 3 ds = (r q)sdt + SdZ, d = κ (θ )dt + σ dz, where Z, Z are standard Wiener processes and E(dZ dz ) = ρdt with E the expectation operator under a particular risk neutral easure. In (), r is the risk free rate of interest. In () the paraeter σ is the so called ol-of-ol (in fact, σ is the ariance of the ariance process ). The paraeters κ and θ are respectiely the rate of ean reersion and long run ariance of the process for the ariance. These are under the risk-neutral easure and are related to the corresponding quantities under the physical easure by a paraeter that appears in the arket price of olatility risk. 4 We are also able to write down the aboe syste () () using independent Wiener processes. Let W = Z and = ρw + ρ W where W and W are independent Wiener processes under the risk neutral easure. Then, Z () () Strictly speaking, Aerican style knock-in options could be priced nuerically as well. But the approach will be ore coplicated than that indicated in this paper. In fact, let us take an Aerican up-and-in option C ui (S,, τ, H) as an exaple. If H is the upper barrier, then we would hae τ C ui (S,, τ, H) = C(H,, τ ξ)p(h,, ξ S, )dξ d ; where C(H,, τ ξ) is a standard Aerican option with stock price H, ariance and tie to aturity τ ξ and p(h,, ξ S, ) is the transition density (Greens function) of the two diensional processes (S, ). Hence, we could price C(H,, τ ξ) using the ethod of lines for certain quadrature points on and ξ. But then we would need to work out the alue of the Greens function p(h,, ξ S, ) on the corresponding quadrature points as well and then ealuate the two diensional nuerical integral aybe using the sparse grid approach. Thus, it is hard to ipleent the detailed approach in this paper to price Aerican-style knock in options directly. Note that τ = T t, where T is the aturity date of the option and t is the running tie. 3 Of course, since we are using a nuerical technique we could in fact use ore general processes for S and. The choice of the Heston processes is drien partly by the fact that this has becoe a ery traditional stochastic olatility odel and partly because a copanion paper on the ealuation of European copound options under stochastic olatility uses techniques based on a knowledge of the characteristic function for the stochastic olatility process, which is known for the Heston process (see []), and can be used for coparison purposes. 4 In fact, if it is assued that the arket price of risk associated with the uncertainty driing the ariance process has the for λ, where λ is a constant (this was the assuption in [7]) and κ P, θ P are the corresponding paraeters under the physical easure, then κ = κ P + λσ, θ = κp θ P κ P +λσ. In this forulation, the choice of a risk neutral easure coes down to deciding the paraeters. This could for instance be done by a calibration procedure.

3 36 C. Chiarella et al. / Coputers and Matheatics with Applications 64 () the dynaics of S and can be rewritten in ters of independent Wiener processes as ds = (r q)sdt + S(ρdW + ρ dw ), (3) d = κ (θ )dt + σ dw. (4) The price of a barrier option under stochastic olatility at tie to aturity τ, C(S,, τ), can be forulated as the solution to a partial differential equation (PDE) proble. We need to sole the PDE for the alue of the barrier option C(S,, τ) gien by C τ = KC rc, on the interal τ T, where the Kologoro operator K is gien by (5) K = S + ρσ S S S + σ + (r q) S S + (κ (θ ) λ), (6) where λ is the constant appearing in the equation for the arket price of olatility risk, which as stated in Footnote 4 is of the for λ. Both the terinal and boundary conditions need to be specified depending on the detailed specifications of the barrier options. Note that the Fichera function for the tie discretized pricing equation (5) is h(s,, τ) = [(r q)s (S + σ ρs/)]n + [κ(θ ) λ (σ ρ/ + σ /)]n. On S = we hae (n, n ) = (, ) and h(,, τ) =. This eans the pricing equation (5) has to hold for all paraeters. We note that C = is the solution of this equation. Howeer on = we hae (n, n ) = (, ) so that h(s,, τ) = κθ σ /. The Fichera theory says that if h(s,, τ) < then one CAN ipose a Dirichlet condition on =. Howeer, one can also ipose lots of other boundary conditions. In particular, one can require that the pricing equation (5) holds at = een if h <, because the pricing equation defines a Venttsel boundary condition for which the Eq. (5) has a unique solution. Hence we can always sole the pricing equation (5) at = with C(,, τ) as boundary condition regardless of the Feller condition. This akes good sense financially because the proble does not change in any way when the paraeters are perturbed slightly but the Fichera function changes sign. Both at S = and = the pricing equation is the natural boundary condition for which the solution can be expected to be continuous with respect to the paraeters in the equation. In the following discussions, we assue the Feller condition holds in each one of the following cases. To obtain a consistent approxiation of Eq. (5) near =, we fit a quadratic polynoial through the option prices in the neighborhood of, say,, and 3, and then use it to extrapolate a alue for. This quadratic extrapolation also insures that the differential equation (5) holds with =... Continuously onitored barrier option A continuously onitored barrier option is an option which is onitored all the tie between the current tie and the aturity of the option at tie T. The option with or without early exercise features, has the terinal condition C(S,, ) = (S K) +. (7) The doain for the up and out call option is < S < H, < <, < τ < T. (8) The boundary conditions for the barrier option without the early exercise features are: C(,, τ) = ; C(H,, τ) = ; li C (S,, τ) =. (9) () () The option with early exercise features has the free (early exercise) boundary condition C(b(, τ),, τ) = b(, τ) K, when b(, τ) < H () where S = b(, τ) is the early exercise boundary for the barrier option at tie to aturity τ and ariance, and there also hold the sooth-pasting conditions C li S b(,τ) S =, li C S b(,τ) =. (3)

4 C. Chiarella et al. / Coputers and Matheatics with Applications 64 () In the aboe case, C(S,, τ) = S K, b(, τ) < S < H. Howeer, if we cannot find a b(, τ) < H then C(H,, τ) = H K, because technically, for the knock-out eent and the exercise date to be well defined, the option contract is defined in a way such that when the asset price first touches the barrier, the option holder has the option to either exercise or let the option be knocked out. Since we assue the rebate is equal to zero, the option should be exercised once the asset price touches the barrier... Discretely onitored barrier option A discretely onitored barrier option is an option which is onitored only at discrete dates t t < t <,, < t N T, while the option is not onitored at other ties. The option with or without early exercise features, has the terinal condition C(S,, ) = (S K) +. The doain for the up and out call option is: (, H), τ {T tn, T t S N,..., T t }, (, S ax ), otherwise, and < <, < τ < T. The boundary conditions for the barrier option without early exercise features are: C(,, τ) = ; (5) C(H,, τ) =, τ {T t N,..., T t }; (6) li C(S,, τ) =, τ {T t N,..., T t }; (7) S li C (S,, τ) =. A discretely onitored barrier option with the early exercise feature, at the onitoring ties τ {T t N,..., T t }, has the free (early exercise) boundary condition C(b(, τ),, τ) = b(, τ) K, when b(, τ) < H (9) where b(, τ) is the early exercise boundary for the barrier option at tie to aturity τ and ariance, and the soothpasting conditions C li S b(,τ) S =, In the aboe case, we hae C(S,, τ) = S K, li C S b(,τ) =. b(, τ) < S < H so that C(S,, τ) is known oer < S < H. If there is no such b(, τ) then for the sae reason as the case for the continuously onitored option, C(S,, τ) ust satisfy C(H,, τ) = H K. At all other ties τ {T t N,..., T t }, standard Aerican option free boundary conditions apply. Before going into details of the aluation, the following relations between the payoffs of barrier options and anilla options are pointed out. The in out parity for European barrier options, naely knock-in + knock-out = anilla; allows us to consider only the faily of knock-out options for the aluation using the ethod of lines (MOL) since we are able to price anilla options under Heston odel using the analytic solution fro Heston [7]. In the next section, we are going to discuss the details of coputing the up-and-out barrier option prices by ipleenting the MOL. 3. Method of lines (MOL) approach In this section, we will proide the details of the ipleentation of the Method of Lines for Eq. (5) on the coputational doain < S < S < H, < < ax, < τ < T (4) (8) ()

5 38 C. Chiarella et al. / Coputers and Matheatics with Applications 64 () for continuously onitored barrier options and < S < S < H, τ {T t N,..., T t }; < S < S < S ax, τ {T t N,..., T t }; < < ax, < τ < T, for discretely onitored barrier option. The key idea behind the ethod of lines is to approxiate a PDE with a syste of ordinary differential equations (ODEs), the solution of which can be obtained with ODE techniques. When olatility is constant, the syste of ODEs is obtained by discretizing tie. For the PDE (5), we ust in addition discretize the ariance,. S is retained as independent ariable. We begin by setting = where =,,,..., M. Typically we will set the axiu ariance to be M = %. Furtherore, we discretize the tie to expiry according to τ n = n τ, where n =,,,..., N and τ N = T. We denote the option price along the ariance line and tie line τ n by C(S,, τ n ) = C n (S), and set V(S,, τ n ) C(S,, τ n ) S = V n (S), which is of course the option delta at the particular grid point. We now select finite difference approxiations for the deriatie ters with respect to. For the second order ter, at the grid point (S,, τ n ) we use the standard central difference schee () C = C n + C n + C n. ( ) Siilarly for the cross-deriatie ter at the grid point (S,, τ n ), we use the central difference approxiation () C S = V n + V n. Since the coefficients of the second order deriatie ters go to zero as, we use an upwinding finite difference schee for the first order deriatie ter (see []), such that, at the grid point (S,, τ n ) we hae C n + C n if α C = β, C n C n if > α (4) β, where α = κ θ and β = κ + λ. If the coefficients of the -deriaties, especially close to =, do not hae diagonal doinance then the axiu principle does not apply to the discrete equations and oscillatory solutions ight arise. Hence upwinding helps to stabilize the finite difference schee with respect to. Next we ust select a discretization for the tie deriatie. Initially we use a standard backward difference schee for 3 tie steps, gien at the grid point (S,, τ n ) by C τ = C n C n. τ This approxiation is only first order accurate with respect to tie. For the case of the standard Aerican put option, it is known fro Meyer [3] that the accuracy of the ethod of lines increases considerably by using a second order approxiation for the tie deriatie, specifically C τ = 3 C n C n τ C n C n τ. Thus we initiate the ethod of lines solution by using (5) for the first seeral tie steps, and then switch to (6) for all subsequent tie steps. For a discretely onitored barrier option that we will discuss below, we switch back to the backward difference schee (5) for three tie steps right after each onitoring tie and then switch to (6) before the next onitoring tie. Applying () (6) to the PDE (5), we now need to sole a syste of second order ODEs at each tie step and ariance grid point. For the first few tie steps, the ODE syste at the grid point = and τ = τ n is S d C n ds + ρσ S V n + V n + σ C n + C n + C n + α β C n + C n ( ) + α β C n + C n + C n + (r q)s dc n ds (3) (5) (6) rc n C n C n =, (7) τ

6 C. Chiarella et al. / Coputers and Matheatics with Applications 64 () and for all subsequent tie steps the ODE syste has the for S d C n + ρσ ds S V n + V n + σ C n + C n + C n + α β C n + C n z ( ) + α β C n + C n + C n + (r q)s dc n ds rc n 3 C n C n + C n C n =. (8) τ τ We require two boundary conditions in the direction, one at and the other at M. For large alues of, the rate of change of the option price with respect to conerges to zero. So for sufficiently large alues of, one can treat this rate of change as zero without any ipact on the accuracy of the solution at other alues of. Thus we set dc/d = ( M+ M )/(d) = and sole (7) also for = M. This akes the boundary condition approxiation second order. In this case we hae a syste of M equations along the ariance boundary = M. To obtain a consistent approxiation of Eq. (5) near = we fit a quadratic polynoial through the option prices at, and 3, and then use it to extrapolate a alue for which then is used in (7) and (8) for =. It turns out that this proides a satisfactory estiate of the price along for the purpose of generating a stable solution for sall alues of. 5 After taking the boundary conditions into consideration, we ust sole a syste of M second order ODEs in S along the line segent (S,, t n ), S [S, H] or S [S, S ax ] depending on the properties of the barrier option for =,..., M and fixed t n. We sole this syste of ODEs iteratiely for increasing alues of, using the latest aailable estiates for C n, + C n, V n + and V n. The initial estiates for C n and V n n are siply C and V n, then we use the latest estiates for C n and V n found during the current iteration through the ariance lines. At a grid alue of S we continue to cycle through the lines until the change in the price between successie iterations falls below a tolerance of 8. We accept the last iterate as the solution C n (S) and proceed to the next tie leel. The syste of ODEs (7) and (8), after rearrangeent, aybe cast into the generic first order syste for dc n ds = V n, (9) dv n ds = A (S)C n + B (S)V n + P n (S), where, for exaple, for Eq. (7) A (S) = S σ + α β + r + τ and where P n (S) is a function of C n +, C n, V n +, V n, C n (r q)s, B (S) =, S (3) and C n that ay be inferred fro the RHS of (7) or (8). The restriction to the line segent < S < S < H or < S < S < S ax assures that the coefficients of (3) reain bounded. We sole the syste (9) (3) using the Riccati transfor, full details of which are proided by Meyer [3]. 6 Note that we are only able to apply the Riccati transfor to the syste (9) (3) proided that both equations are treated as ODEs. We use an iteratie technique in which the price (C n ) and the deriatie (V n ) ters are updated until the price profile conerges. The Riccati transforation is gien by C n (S) = R (S)V n (S) + W n (S), where R and W are solutions to the initial alue probles dr ds = B (S)R (S) A (S)(R (S)), R (S ) =, (3) dw n = A (S)R (S)W n ds R (S)P n (S), W n (S ) =. (33) Note that the coefficients in (3) depend on whether (7) or (8) applies, but for each case and for a constant tie step, Eq. (3) is independent of tie and needs to be coputed only once for each. To obtain V n (S) required for (3) we need to sole the ODE (34),7 dv n ds = A (S)(R (S)V n + W n (S)) + B (S)V n + P n (S), (34) (3) 5 See [4] for ore discussion and justification for this procedure for handling the boundary conditions at = for stochastic olatility odels. 6 Chapter of Meyer [3] has the ost detailed description of the ethod. The integration of the differential equations with the trapezoidal rule is sensible because the ethod is second order and therefore consistent with the difference quotients used in the and t direction, except for the upwinding ter. Its adantage is easy counication with the solution being generated fro the preceding tie leel. An adaptie integrator in S would require interpolation of functions stored only at the esh points. 7 It is certainly true that the grid should be selectiely refined. Our code does that in the S direction when we sole Eqs. (3) (34), although not adaptiely. We hae ore points near the strike and near the barrier. Howeer, we hae not systeatically studied the conergence of the iteration when we hae un-eenly spaced lines. Our sense is, howeer, that the nuber of iterations required for conergence depends on the sallest distance between lines, not on the total nuber of lines. The nuber of points along lines has a direct influence on run-ties but does not influence the nuber of iterations required.

7 4 C. Chiarella et al. / Coputers and Matheatics with Applications 64 () Fig.. One sweep of the solution schee on the τ grid. The stencil for the typical point is displayed in Fig.. subject to an initial condition which depends on the properties and the specifications of the barrier options: For continuously onitored barrier options without early exercise opportunities, we sole (33) for increasing alues of S, ranging fro S < S < H. Using the fact that C n (H) = we obtain fro (3) the terinal condition V n (H) = W n (H) R (H) and then integrate (34) fro S = H to S = S. For continuously onitored barrier options with early exercise opportunity we integrate (3), (33) fro S to S ax and onitor the function φ(s) = R (S) + W n (S) (S K). If φ(s ) = for soe S (S, H) then S is the early exercise boundary b(, t n ) = b n at the grid point (, t n ). In general φ(s) will change sign at ost once on [S, H]. b n will change during the iteration but will conerge as the prices conerge. Once b n is found we integrate (34) backward fro bn toward S subject to the initial condition V(b n ) =. If φ(s) has no zero in [S, H) then there is no early exercise below the barrier and we sole (34) subject to V n (H) = H K W n(h). R (H) In fact, at any tie to aturity τ, if the asset hits the barrier H, then the option will be exercised, naely, C(H,, τ) = H K, according to the Riccati transfor (3) we hae C n (H) = R (H)V n (H) + W n (H) = H K. For discretely onitored barrier options without early exercise features, the procedures to sole the PDE are siilar to those for the continuously onitored counterpart, but we should change back to standard Euler backward tie difference for 3 steps after each onitoring tie and then switch to the second order schee until the next onitoring tie. 8 The tie difference in the Riccati equation should be adjusted in a siilar anner as well. For discretely onitored options with early exercise features, we sole R fro the Riccati equation (3) and sole W n fro the forward sweep (33) as usual. We find the free boundary point S in the standard way as for the continuously onitored option but let b n = in{s, H} at each of the onitoring dates and update the corresponding option alue as well. At the non-onitoring dates, we set b n = S as the early exercise boundary alue which is used as the terinal alue fro which to work backward to sole Eq. (34) fro S = b n to S = S. In this case, we still need to change back to the standard Euler backward tie difference for 3 tie steps after each onitoring tie and then switch to the second order schee before the next onitoring tie. The tie difference in the Riccati equation should be adjusted in a siilar anner. In Fig. we illustrate one sweep through the grid points on the τ plane. In Fig. we show the stencil for the typical grid point in Fig., this essentially shows the grid point alues of C that enter the right-hand side of (3). Fig. 3 then illustrates the solution of (33) along a line in the S direction fro a typical grid point in the τ plane. (35) 8 Zan et al. [5] applied the backward Euler ethod in order to aoid unwanted oscillations in the Crank Nicolson schee. Here if the barrier condition holds then the delta is discontinuous at the barrier, so we need to restart the tie ealuation oer the next period fro a discontinuous initial delta. For the sae reason as Zan et al. [5] a backward Euler ethod is applied here for the first few tie steps.

8 C. Chiarella et al. / Coputers and Matheatics with Applications 64 () Fig.. Stencil for the typical grid point of Fig.. The stencil for C n depends on (C n, C n, C n, + C n, C n ). Fig. 3. Soling for the option prices along a (, τ n ) line. There is no proen rate of conergence for the aboe algorith. Its perforance ust be read off the tables of nubers in our exaples in Section 4. There is an analysis of the MOL line iteration for an elliptic free boundary odel proble in [5] but this proble does not contain a cross deriatie ter. For the tie discrete elliptic proble the iteration at least for the uncorrelated case of ρ = can be inferred to conerge. For ρ > conergence is only obsered. Stability of the iplicit two-leel tie discretization schee is known for the heat equation and obsered in our case. For a ore coprehensie analysis and deonstration of the conergence of MOL in option pricing context, for instance efficiency plots, we refer the reader to Chiarella et al. [4]. 4. Nuerical exaples To deonstrate the perforance of the ethod of lines outlined in Section 3 we ipleent the ethod for a gien set of paraeter alues shown in Table, 9 chosen to be consistent with the stochastic olatility paraeters being used by 9 Here we assue a continuous diidend yield, howeer in the case of discrete diidends the coputation would hae to be restarted with a shifted initial alue. That is straightforward since the stochastic olatility ters do not change at the ex-diidend tie.

9 4 C. Chiarella et al. / Coputers and Matheatics with Applications 64 () Table Paraeter alues used for the barrier option. The stochastic olatility (SV) paraeters are those used in Heston s original paper. Paraeter Value SV paraeter Value r.3 θ. q.5 κ. T.5 σ. K λ. ρ ±.5 H 3 Table Prices of the continuously onitored barrier option without early exercise features coputed using ethod of lines (MOL), finite difference (FD) and Monte Carlo siulation (MC). Paraeter alues are gien in Table, with ρ =.5 and =.. ρ =.5, =. S Runtie (s) Method (N, M, S pts ) 8 9 MOL (5,, 4) MOL (,, 64) FD (,, ) MC (4,,,, ) MC upper bound MC lower bound Table 3 Prices of the continuously onitored barrier option with early exercise features coputed using ethod of lines (MOL) and Monte Carlo siulation (MC). Paraeter alues are gien in Table, with ρ =.5 and =.. ρ =.5, =. S Runtie (s) Method (N, M, S pts ) 8 9 MOL (5, 5, 4) MOL (,, 44) MOL (,, 64) MOL (, 4, 9) MC (,, 5,,, ) MC upper bound MC lower bound Heston [7] and which hae been standard in any papers undertaking nuerical studies of stochastic olatility odels. We use weekly onitoring frequency for the discretely onitored options. The nuber of iterations is an iportant concept for the MOL, based on the coputational experience, we found that at each tie step the prices profile will conerge to 8 after a axiu of 4 iterations independent of the type of the options. Also in order to check and benchark the results and to deonstrate the perforance of the MOL, we use seeral aailable ethods, such as the Finite Difference (FD) ethod (see [6]), Fourier Cosine Expansion (COS) ethod (see [8,9]) together with the Monte Carlo Siulation ethod (see []) to work out the prices of different kinds of barrier options to copare the prices fro the MOL. Here we hae chosen those ethods as they are the best ones in calculating the prices and hedge ratios with respect to different barrier options with different features as the benchark or alternatie approach. Fro Tables 5 we can see that the MOL is ery efficient in producing the barrier option prices and it is also iportant to note that the MOL produces hedge ratios, such as deltas, gaas to the sae leel of accuracy as the prices theseles. In fact, delta and gaa are aailable fro the differential equations. Delta (V n ) is the solution of ODE (34) but Gaa is calculated fro the right hand side of Eq. (34) which is a direct calculation and does not require nuerical differentiation. The iteratie schee will not stop running until the price profile conerges to a certain accuracy, howeer based on the Riccati transfor equation (3) the conergence of delta (V n ) should be faster than that of the price. The adantage of The source code for all ethods was ipleented using NAG Fortran with the IMSL library running on the UTS, Faculty of Business F&E HPC Linux Cluster which consists of 8 nodes running Red Hat Enterprise Linux 4. (64 bit) with 3.33 GHz, 6 MB cache Quad Core Xeon X547 Processors with 333MHz FSB 8 GB DDR-667 RAM. We eployed a ariant of the COS ethod ainly to cater for the Heston odel. This ersion of COS ethod has been presented in [7]. The lower efficiency of the COS ethod applied to pricing barrier options under the Heston odel is ainly because we hae to consider not only the transition of the spot price but also the transition of the stochastic ariance. Hence it becoes truly a two-diensional pricing proble that is dealt with in [7] by a cobination of a Fourier Cosine series expansion, as in [8,9], and high-order quadrature rules in the other diension. The nuerical results in [7] deonstrate that the run tie to price one single barrier option would be about 5 6 s. Our results are roughly coparable to theirs since the run tie in Table 4 is for pricing 5 different options.

10 C. Chiarella et al. / Coputers and Matheatics with Applications 64 () Table 4 Prices of the discretely onitored barrier option without early exercise features coputed using ethod of lines (MOL), Fourier Cosine expansion (COS) and Monte Carlo siulation (MC). Paraeter alues are gien in Table, with ρ =.5 and =.. ρ =.5, =. S Runtie (s) Method (N, M, S pts ) 8 9 MOL (5,, 4) MOL (,, 64) COS (,, ) MC (4,,,, ) MC upper bound MC lower bound Table 5 Prices of the discretely onitored barrier option with early exercise features coputed using ethod of lines (MOL) and Monte Carlo siulation (MC). Paraeter alues are gien in Table, with ρ =.5 and =.. ρ =.5, =. S Runtie (s) Method (N, M, S pts ) 8 9 MOL (5,, 4) MOL (, 5, 4) MOL (5, 5, 64) MC (,, 5,,, ) MC upper bound MC lower bound Price profile continuous Barrier without early exercise C(S,,T) S 6 8 Fig. 4. Price profile of a continuously onitored up-and-out call option without early exercise opportunities. the MOL is that the discontinuity of the gaa at the exercise boundary does not enter into the calculation along the line since we do not use difference quotients in S straddling the boundary. The C ter does straddle the early exercise boundary, but our nuerical experients indicate that it is better to base that difference approxiation on the intrinsic alue beyond the exercise boundary rather than soe sooth (aybe quadratic) extrapolation of C beyond the free boundary. It can also be seen fro the efficiency plots in [4] that delta and gaa can achiee the sae accuracy as the prices. Figs. 4 deonstrate that the MOL is able to produce both sooth option prices, early exercise boundaries and option deltas which are a part of the solution we hae after running the MOL. In fact, Tables 5 show that the prices of continuously onitored European up-and-call option produced fro the MOL are close to those prices generated fro the finite difference ethod but the MOL proides better hedge ratios; the prices of discretely onitored European up-and-call option produced fro the MOL are close to those prices generated fro the Fourier Cosine Expansion ethod but the MOL is ore efficient since the runtie of COS ethod shown in Table 4 are the tie to produce only 5 prices while the runtie of the MOL is the tie to hae prices of all grid points;

11 44 C. Chiarella et al. / Coputers and Matheatics with Applications 64 () Price profile discrete Barrier without early exercise C(S,,T) S 6 8 Fig. 5. Price profile of a discretely onitored up-and-out call option without early exercise opportunities. Early exercise boundary of continuous barrier option 3 5 b(,τ) τ.45.5 Fig. 6. Early exercise boundary of a continuously onitored up-and-out call option. Early exercise boundary of Discrete Barrier option 35 3 b(, τ) τ.45.5 Fig. 7. Early exercise boundary of a discretely onitored up-and-out call option (including 3 onitoring dates).

12 C. Chiarella et al. / Coputers and Matheatics with Applications 64 () Delta profile continuous Barrier without early exercise.. C S (S,) S 5 Fig. 8. Delta profile of a continuous onitoring up-and-out call option without early exercise opportunities. Delta profile Continuous Barrier with early exercise.8 C S (S,) S 5 Fig. 9. Delta profile of an up-and-out call option with early exercise opportunities. the prices of both continuously and discretely onitored Aerican up-and-call option produced fro the MOL are close to those prices generated fro the Monte Carlo siulation which ran considerably longer than the MOL. 5. Ipact of stochastic olatility on the prices of the barrier option In this section, we explore the ipact of stochastic olatility on the price profiles of Barrier options with arious features. We consider two odels for the underlying asset price: (i) the geoetric Brownian otion (GBM) odel of Black and Scholes [6] and Merton []; (ii) the stochastic olatility (SV) odel of Heston [7]. Here we ai to obsere the ipact that stochastic olatility has on the shape of the price profile, where the ariance of S is consistent for both odels. Setting the spot ariance to =. (corresponding to a olatility standard deiation of 33%) in the SV odel, we deterine the tie-aeraged ariance s for ln S oer the life of the option by using the characteristic function for the arginal density of x = ln S gien in []. By requiring that s be equal for both the odels, we then deterine the necessary paraeter olatility σ for the GBM to ensure that they both hae consistent ariance oer the tie period of interest. To atch the tie-aeraged ariance for the GBM and SV odels for a 6-onth option, the global olatilities, s, are 3.48% for ρ =.5, and 3.8% for ρ =.5. The alue of in the SV odel is %. Hence, the constant olatility σ in GBM is chosen to be 3.48% for ρ =.5, and 3.8% for ρ =.5 in all the following coparisons. The specification of each Monte Carlo siulation in the tables are the nubers in the parenthesis after MC which ean (No. of tie steps, No. of olatility leels, No. of siulations) for the options without early exercise opportunities and (No. of tie steps, No. of olatility leels, No. of early exercise opportunities, No. of siulations) for the options with early exercise opportunities, respectiely.

13 46 C. Chiarella et al. / Coputers and Matheatics with Applications 64 () Delta profile Discrete Barrier without early exercise.3.. C S (S,) S 6 8 Fig.. Delta profile of a discrete onitoring up-and-out call option without early exercise opportunities. Delta profile Discrete Barrier with early exercise.8 C S (S,) S 5 Fig.. Delta profile of a discrete onitoring up-and-out call option with early exercise opportunities. Fig.. The effect of stochastic olatility on continuously onitored European up-and-out call (UOC) option. The correlation is ρ =.5 and all other paraeter alues are as listed in Table. The at the oney UOC price under GBM is.497. Figs. 5 deonstrates the prices differences of difference types of barrier options under Heston stochastic olatility odel and those option prices under the standard Geoetric Brownian Motion.

14 C. Chiarella et al. / Coputers and Matheatics with Applications 64 () Fig. 3. The effect of stochastic olatility on discretely onitored European up-and-out call option. The correlation is ρ =.5 and all other paraeter alues are as listed in Table. The at the oney UOC price under GBM is Fig. 4. The effect of stochastic olatility on the continuously onitored Aerican up-and-out call option. The correlation is ρ =.5 and all other paraeter alues are as listed in Table. The at the oney UOC price under GBM is Fig. 5. The effect of stochastic olatility on discretely onitored Aerican up-and-out call option. The correlation is ρ =.5 and all other paraeter alues are as listed in Table. The at the oney UOC price under GBM is 8.35.

15 48 C. Chiarella et al. / Coputers and Matheatics with Applications 64 () Conclusion We hae studied the pricing of Barrier options under stochastic olatility using the Method of Lines. We also proide the Barrier option pricing results fro Finite Difference ethod, Fourier Cosine Expansion ethod and Monte Carlo Siulation approach as bencharks to the MOL. It turns out that the MOL is able to handle both continuously and discretely onitored options with or without early exercise opportunities. Hence we beliee this proides a unified fraework to efficiently price arious kinds of Barrier options with different kinds of properties. One ain adantage of the MOL is that it produces the hedge ratios of the option, naely the deltas and gaas, to the sae accuracy as the prices theseles within the sae tie frae. In future research, the knock-in option under stochastic olatility with early exercise features should be further inestigated. References [] R.C. Merton, Theory of rational option pricing, Bell Journal of Econoics and Manageent Science 4 (973) [] D.R. Rich, The atheatical foundation of barrier option-pricing theory, Adances in Futures and Options Research 7 (994) [3] H.Y. Wong, Y.K. Kwok, Multi-asset barrier options and occupation tie deriaties, Applied Matheatical Finance (3) (3) [4] B. Gao, J.Z. Huang, M. Subrahanya, The aluation of Aerican barrier options using the decoposition technique, Journal of Econoic Dynaics and Control 4 () [5] R. Zan, K. Vetzal, P. Forsyth, PDE ethods for pricing barrier options, Journal of Econoic Dynaics and Control 4 () [6] F. Black, M. Scholes, The pricing of corporate liabilites, Journal of Political Econoy 8 (973) [7] S. Heston, A closed-for solution for options with stochastic olatility with applications to bond and currency options, Reiew of Financial Studies 6 () (993) [8] S. Griebsch, Exotic option pricing in Heston s stochastic olatility odel, Ph.D. Thesis, Frankfurt School of Finance & Manageent, 8. [9] M. Yousuf, Efficient soothing of Crank Nicolson ethod for pricing barrier options under stochastic olatility, Proceedings in Applied Matheatics and Mechanics 7 (8) 8 8. [] M. Yousuf, A fourth-order soothing schee for pricing barrier options under stochastic olatility, International Journal of Coputer Matheatics 86 (6) (9) [] C. Chiarella, S. Griebsch, B. Kang, The ealuation of European copound option prices under stochastic olatility, Working Paper, Quantitatie Finance Research Centre, Uniersity of Technology Sydney, 9. [] D.J. Duffy, Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach, Har/Cdr ed., Wiley, 6. [3] G. Meyer, Pricing options and bonds with the ethod of lines, Georgia Institute of Technology, 9. [4] C. Chiarella, B. Kang, G.H. Meyer, A. Ziogas, The ealuation of Aerican option prices under stochastic olatility and jup-diffusion dynaics using the ethod of lines, International Journal of Theoretical and Applied Finance (3) (9) [5] G. Meyer, Free boundary probles with nonlinear source ters, Nuerische Matheatik 43 (984) [6] T. Kluge, Pricing deriaties in stochastic olatility odels using the finite difference ethod, Diploa Thesis, Chenitz Uniersity of Technology,. [7] F. Fang, C. Oosterlee, A Fourier-based aluation ethod for Berudan and barrier options under Heston s odel, Technical Report, Delft Uniersity of Technology, Delft Institute of Applied Matheatics,. [8] F. Fang, C.W. Oosterlee, A noel pricing ethod for European options based on Fourier-Cosine series expansions, SIAM Journal on Scientific Coputing 3 () (9) [9] F. Fang, C.W. Oosterlee, Pricing early-exercise and discrete barrier options by Fourier-Cosine series expansions, Nuerische Matheatik 4 () (9) 7 6. [] A. Ibanez, F. Zapatero, Monte Carlo aluation of Aerican options through coputation of the optial exercise frontier, Journal of Financial and Quantitatie Analysis 39 () (4) [] G. Cheang, C. Chiarella, A. Ziogas, The representation of Aerican options prices under stochastic olatility and jup diffusion dynaics, Quantitatie Finance, ifirst () 3 (to be printed).