TRUE MARTINGALES FOR UPPER BOUNDS ON BERMUDAN OPTION PRICES UNDER JUMP-DIFFUSION PROCESSES. Helin Zhu Fan Ye Enlu Zhou

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1 Proceedings of the 213 Winter Simuation Conference R. Pasupathy, S.-H. Kim, A. Tok, R. Hi, and M. E. Kuh, eds. TRUE MARTINGALES FOR UPPER BOUNDS ON BERMUDAN OPTION PRICES UNDER JUMP-DIFFUSION PROCESSES Hein Zhu Fan Ye Enu Zhou Industria & Enterprise Systems Engineering University of Iinois at Urbana-Champaign Urbana, IL 6181, USA ABSTRACT Fast pricing of American-stye options has been a difficut probem since it was first introduced to financia markets in 197s, especiay when the underying stocks prices foow some jump-diffusion processes. In this paper, we propose a new agorithm to generate tight upper bounds on the Bermudan option price without nested simuation, under the jump-diffusion setting. By expoiting the martingae representation theorem for jump processes on the dua martingae, we are abe to construct a martingae approximation that preserves the martingae property. The resuting upper bound estimator avoids the nested Monte Caro simuation suffered by the origina prima-dua agorithm, therefore significanty improves the computationa efficiency. Theoretica anaysis is provided to guarantee the quaity of the martingae approximation. Numerica experiments are conducted to verify the efficiency of our proposed agorithm. 1 INTRODUCTION Pricing American-stye derivatives (which is essentiay an optima stopping probem has been an active and chaenging probem in the ast thirty years, especiay when the underying stocks prices foow some jump-diffusion processes, as they become more and more critica to investors. To present time, various jump-diffusion modes for financia modeing have been proposed to fit the rea data in financia markets, incuding: (i the norma jump-diffusion mode, see Merton (1976; (ii the jump modes based on Levy processes, see Cont and Tankov (23; (iv the exponentia jump diffusion modes, see Kou (28. A these modes are trying to capture some interesting features of the market behaviour that cannot be we expained by the pure-diffusion modes, such as the heavy-tai risk suffered by the market. In genera, cosed-form expressions for the American-stye derivatives can hardy be derived under these jump-diffusion modes due to the mutipe exercise opportunities and the randomness in the underying asset price caused by both jumps and diffusions. Hence, various numerica methods have been proposed to tacke the American-stye option pricing probems under the jump-diffusion modes, incuding: (i soving the free boundary probems via attice or differentia equation methods, e.g. Feng and Linetsky (28; (ii quadratic approximation and piece-wise exponentia approximation methods, e.g. Kou and Wang (24. A thorough study on jump-diffusion modes for asset pricing has been done by Kou (28. In a broader sense, an eegant overview of financia modes under jump processes is provided in Cont and Tankov (23. Another cass of widey-used methods is the Monte Caro simuation-based method, which has been successfuy impemented on option pricing probems under the pure-diffusion modes, see Longstaff and Schwartz (21, Tsitsikis and van Roy (21. They are abe to approximate the continuation vaues by regression on certain basis functions sets (caed function bases, which eads to good suboptima /13/$ IEEE 113

2 exercise strategies and ower bounds on the exact option price. Moreover, their methods bypass the curse of dimensionaity and scae we with the number of underying variabes, working efficienty for highdimensiona probems under the pure-diffusion modes. Though these methods can be naturay adapted to option pricing probems under the jump-diffusion setting, two key questions regarding the effectiveness of these methods remain to be addressed: (i how to choose the function bases for regression. (ii how to measure the quaity of the ower bounds. The second question is partiay addressed by the dua approach proposed independenty by Rogers (22, Haugh and Kogan (24, and Anderson and Broadie (24. They are abe to generate the upper bounds on the option price by soving the associated dua probem, which is obtained by subtracting the payoff function by a dua martingae adapted to a proper fitration. In theory, if the dua martingae is the Doob-Meyer martingae part of the option price process, namey the optima dua martingae, then the resuting upper bound equas the exact option price. In practice, the optima dua martingae is not avaiabe, but good approximations of it can generate tight upper bounds. With the access to the upper bounds, the quaity of suboptima exercise strategies or ower bounds coud be measured empiricay by ooking at the duaity gaps. Gasserman (24 provides an eegant and thorough overview of the duaity theory for option pricing probems. A ot work has been done foowing the duaity theory. To name a few, Ye and Zhou (213b appy the prima-dua approach with partice fitering techniques to optima stopping probems of partiay observabe Markov processes. Desai et a. (212 consider an additiona path-wise optimization procedure in constructing the dua martingaes for optima stopping probems. Rogers (27, Brown et a. (21 generaize the duaity theory to genera discrete-time dynamic programming probems and provide a broader interpretation of the dua martingae. From Brown et a. (21 s perspective, the dua martingae can be regarded as the penaty for the access to the future information (information reaxation and different degrees of reaxation resut in different eves of upper bounds. In particuar, the dua martingaes constructed by Haugh and Kogan (24, Anderson and Broadie (24 can be interpreted as perfect information reaxation, which means the option hoder has access to a the future prices of the underying assets. Ye and Zhou (212 consider an additiona path-wise optimization technique in constructing the penaties for genera dynamic programming probems. Ye and Zhou (213a aso deveop the duaity theory for genera dynamic programming probems under a continuous-time setting. The numerica effectiveness of the prima-dua approach has been demonstrated in pricing mutidimensiona American-stye options. A possibe deficiency of the agorithm is the heavy computation (quadratic in the number of exercisabe periods, caused by the nested simuation in constructing the dua martingae. To address the computationa issue, Beomestny et a. (29 propose an aternative agorithm to generate approximations of optima dua martingae via non-nested simuation under the Wiener process setting. By expoiting the martingae representation theorem on the optima dua martingae driven by Wiener processes, they are abe to approximate the optima dua martingae through regressing the integrand on some function bases at finite number of time points. The resuting approximation preserves the martingae property and generates a true upper bound. More importanty, their agorithm avoids nested Monte Caro simuation and is inear in the number of exercisabe periods. In this paper, we wi generaize Beomestny et a. (29 s idea of true martingae to Bermudan option pricing probems under jump-diffusion processes and provide a new perspective in understanding the structure of the optima dua martingae, which faciitates us to construct good approximations of it. According to our knowedge, we are among the first to ever consider estimating the upper bounds on American-stye option price under the jump-diffusion modes. In a greater detai, we have made the foowing contributions. Motivated by Beomestny et a. (29, we propose a new agorithm, which is referred as the true martingae agorithm (T-M agorithm, to compute the upper bounds on the Bermudan option price under the jump-diffusion modes. The resuting approximation (caed true martingae approximation preserves the martingae property, therefore generates true upper bounds on the 114

3 Bermudan option price. Moreover, compared with the prima-dua agorithm proposed by Anderson and Broadie (24 (A-B agorithm, our proposed T-M agorithm avoids the nested Monte Caro simuation and scaes ineary with the exercisabe periods, and hence achieves a higher computationa efficiency. We investigate the numerica effectiveness of Longstaff and Schwartz (21 s east-squares regression approach (L-S agorithm for Bermudan option price under the jump-diffusion modes. In particuar, we find that by incorporating the European option price under the corresponding pure-diffusion mode (referred as the non-jump European option price into the function basis of the L-S agorithm, the quaity of the induced suboptima exercise strategies and the ower bounds can be significanty improved. Motivated by the expicit structure of the optima dua martingae (Theorem 1, we propose a function basis that can be empoyed in our proposed agorithm to obtain upper bounds on the option price. This function basis is aso derived based on the non-jump European option price, which is critica to the true martingae approximation and hence the quaity of the true upper bounds. By impementing our agorithm together with the A-B agorithm on severa sets of numerica experiments, the numerica resuts demonstrate that both methods can generate tight and stabe upper bounds on option price of the same quaity; however, we observe that our agorithm is much more efficient than the A-B agorithm in practice due to the reief from nested simuation. To summarize, the rest of this paper wi be organized as foows. In section 2, we describe the Bermudan option pricing probem under genera jump-diffusion modes and review the dua approach. We deveop the true martingae approach and provide error anaysis and convergence anaysis of it in section 3. Section 4 focuses on the detaied T-M agorithm and its numerica advantages. Numerica experiments are conducted in section 5 to verify the computationa efficiency of the T-M agorithm. Concusion and future directions are given in section 6. 2 MODEL FORMULATION 2.1 Prima Probem In this paper, we consider a specia case of asset price modes jump-diffusion processes, i.e., the asset price X(t satisfies the foowing stochastic differentia equation (SDE: dx (t = b(t,x (tdt + σ (t,x (tdw (t + J (t,x (t,yp (dt,dy, (1 R d where t [,T, X(t = [X 1 (t,...,x n (t is a random process with a given initia deterministic vaue X( = X R n, W(t = [W 1 (t,...,w nw (t is a standard vector Wiener process, P(dt,dy is the Poisson random measure (see Definition 2.18 in Cont and Tankov (23 defined on [,T R d R d+1 with the intensity measure µ(dt dy, the coefficients b, σ and J are functions b : R R n R n, σ : R R n R n R n w and J : R R n R d R n satisfying mid continuity conditions (such as uniformy Lipschitz continuous or Hoder continuous. Throughout F = {F t : t T } is the augmented fitration generated by the Wiener process W(t and the Poisson random measure P. We consider a Bermudan option based on X(t that can be exercised at any date from the time set Ξ = {T,T 1,...,T J }, with T = and T J = T. Given a pricing measure Q and the fitration F, when exercising at time Ξ, the hoder of the option wi receive a discounted payoff H Tj := h(,x (, where h(, is a Lipschitz continuous function. Our probem is to evauate the price of the Bermudan option, that is, to find Prima : V = supe [h(τ,x (τ X ( = X, (2 τ Ξ where τ is an exercise strategy (in this case, a stopping time adapted to the fitration {F Tj : j =,...,J } taking vaues in Ξ, V represents the Bermudan option price at time T given the initia asset price X. 115

4 2.2 Review of Dua Approach Let M = {M Tj : j =,...,J } with M = be a martingae adapted to the fitration {F Tj : j =,...,J } and M represents the set of a such martingaes. Anderson and Broadie (24, Haugh and Kogan (24 show that the dua probem ( [ ( Dua : inf E HTj M Tj X ( = X (3 M M max j J yieds the exact option price V. Moreover, if we et M in (4 be the Doob-Meyer martingae part of the discounted Bermudan price process VT j, denoted by MT j, then the infimum in (4 is achieved. Precisey, we have: V = [ E max j J max j J (H Tj M Tj X ( = X. (4 In practice, the optima dua martingae is not accessibe to us. Nevertheess, we can sti obtain an upper bound with an arbitrary M M via [ V up ( (M = E HTj M Tj X ( = X. (5 It is reasonabe to expect that, if M Tj is the martingae induced by a good approximation, V Tj, of the option price process VT j, then M Tj is cose to the optima dua martingae MT j and the resuting upper bound V up (M shoud be cose to the exact option price V. Specificay, suppose V = {V : j =,...,J } is some approximation of V = {VT j : j =,...,J }. Consider the foowing Doob-Meyer decomposition: V Tj = V + M Tj +U Tj, j =,...,J, (6 where V is the approximation of the Bermudan option price at time T and U Tj is the residua predictabe process. Then we can obtain the martingae component M Tj in principe via the foowing recursion: M Tj+1 = M Tj +V Tj+1 E Tj [ VTj+1, with M =, (7 where E Tj [ means the conditiona expectation is taken with respect to the fitration F Tj, i.e., E Tj [ = E Tj [ FTj. Haugh and Kogan (24, Anderson and Broadie (24 both use the above theoretica resut as the starting point of their agorithms to the upper bounds. The difference between their approaches ies in the ways of generating dua martingaes. Nevertheess, due to the nested simuation in approximating the conditiona expectation in (7, both of their agorithms ose the martingae property. Thus the resuting upper bounds are not guaranteed to be true upper bounds. Furthermore, the nested simuation requires huge computationa effort. Under imited computationa resources, this approach might not be reaistic. In next section, we wi deveop an aternative approach to address these issues. 3 TRUE MARTINGALE APPROACH VIA NON-NESTED SIMULATION In this section, we wi deveop an approach that is fundamentay different from previous approaches by Haugh and Kogan (24, Anderson and Broadie (24. By expoiting the specia structure of martingaes jointy driven by the Wiener measure and the Poisson random measure, we are abe to construct an approximation of M without nested simuation, and thus preserves the martingae property. The foowing generaized martingae representation theorem provides the intuitive idea in understanding the unique structure of such martingaes. 116

5 Theorem 1 (Martingae Representation Theorem Fix T >. Let {W(t : t T } be a n w -dimensiona Wiener process and P be a Poisson random measure on [,T R d with intensity µ(dt dy, independent from W(t. If M = {M Tj : j =,...,J } is a ocay square-integrabe (rea-vaued martingae adapted to the fitration {F Tj : j =,...,J } with deterministic initia vaue M =, then there exist a predictabe process φ : Ω [,T R n w and a predictabe random function ψ : Ω [,T R d R such that Tj M Tj = φ s dw (s + Tj ψ (s,y P (ds,dy, R d j =,...,J, (8 where P is the compensated Poisson random measure induced by P. Remark 1 The proofs of a theorems, coroaries and propositions in this paper are provided in Zhu et a. (213. Inspired by Theorem 1 and foowing Beomestny et a. (29 s work, if one tries to approximate the martingae M Tj, a natura idea is to first estimate the integrands φ t and ψ (t,y in Tj M Tj = φ t dw (t + Tj ψ (t,y P (dt,dy, j =,...,J, (9 R d at a finite number of time and space points. Then an approximation of M Tj wi be represented via φ t and ψ (t,y using the Ito sum (simiar to the Riemann sum. We introduce a partition π = {t : =,1,...,L } on [,T such that t =, t L = T and π Ξ, and a partition A = {A k : k =,1,...,K } on R d such that {[t,t +1 A k } are µ-measurabe disjoint subsets and K A k = R d. Then P([t,t +1 A k = t +1 t A k P (ds,dy is a Poisson random variabe (regarded as k=1 Poisson increment, and P([t,t +1 A k = t +1 t A k P (ds, dy is the corresponding compensated Poisson random variabe (regarded as compensated Poisson increment. We denote the magnitude of partitions π and A as π and A respectivey, i.e., π = max (t t 1 and A = max A < L 1 k K k f (ydy. From (6, we have V Tj+1 V Tj = ( M Tj+1 M Tj + ( UTj+1 U Tj, j =,...,J. (1 Combining with the Ito sum of M Tj+1 in (9, we have V Tj+1 V Tj φ t (W (t +1 W (t + t <+1 K t <+1 k=1 ψ (t,y k P([t,t +1 A k +U Tj+1 U Tj, (11 where y k A k is a representative vaue, and we wi keep using this notation thereafter. Mutipying both sides of (11 by the Wiener process increment (W (t +1 W (t and taking conditiona expectations with respect to the fitration F t, we obtain φ t 1 t +1 t E t [ (W (t+1 W (t V Tj+1, Tj t < +1, (12 where we use the F -predictabiity of U, the independent increment property of W(t and the independence between W and P. Simiary, if we mutipy both sides of (11 by the compensated Poisson random variabe P([t,t +1 A k and take the conditiona expectations with respect to the fitration F t, we can obtain ψ (t,y k 1 [ µ ([t,t +1 A k E t P([t,t +1 A k V Tj+1, t < +1,1 k K. (13 117

6 Motivated by expressions (12 and (13, we denote the approximation of φ t and ψ (t,y k by φt π,a ψ π,a (t,y k respectivey, which are defined as foows: φt π,a := 1 [ π E t ( π W V Tj+1, Tj t < +1, (14 and ψ π,a 1 [ (t,y k := µ ([t,t +1 A k E t P([t,t +1 A k V Tj+1, t < +1,1 k K, (15 where π and π W represent the increments of time t and the Winer process W(t respectivey, i.e. π = (t +1 t and π W = (W +1 W. Therefore we can construct an approximation of M Tj, denoted by M π,a, as K t < k=1 M π,a := φt π,a ( π W + t < and ψ π,a (t,y k P([t,t +1 A k. (16 The construction procedure of M π,a can be summarized in the foowing Agorithm 1. Agorithm 1 Construction of the Martingae Approximation M π,a Step 1: Express M Tj as an integra of φ(t and ψ(t,y via (9. Step 2: Approximate φ t by φt π,a via (14 and ψ (t,y k by ψ π,a (t,y k via (15 respectivey. Step 3: Construct the approximation of M Tj, denoted by M π,a, via (16. Notice that M π,a = {M π,a : j =,...,J } remains to be a martingae adapted to the fitration {F Tj : j =,...,J }, based on its structure. We formay state this resut in the foowing theorem. Theorem 2 (True Martingae If an approximation of M, denoted by M π,a, is constructed using Agorithm 1, then M π,a is sti a martingae adapted to the fitration {F Tj : j =,...,J }. According to Theorem 2, if we pug M π,a in (5, it is easy to see that V up ( M π,a is a true upper ( M π,a is an unbiased expectation for a bound on the Bermudan option price V up in the sense that V vaid upper bound. Moreover, if we adopt the L-S agorithm to sove the prima probem (3, we wi obtain a suboptima exercise strategy τ. Exercising τ aong a certain [ number of sampe paths yieds an approximation V Tj of the Bermudan option price at time via V Tj = E Tj H τ j, where τ j means the stopping time τ takes vaue greater than or equa to j. Due to the tower property of conditiona expectations, we can rewrite (14 and (15 as and φt π,a := 1 [ π E t ( π W H τ j+1, Tj t < +1, (17 ψ π,a 1 [ (t,y k := µ ([t,t +1 A k E t P([t,t +1 A k H τ j+1, t < +1,1 k K. (18 Through this we avoid the computation of conditiona expectations in (14 and (15, which woud incur in (16 via non-nested simuation, and hence significanty improve the computationa efficiency. The foowing theorem provides the convergence anaysis of the above true martingae approximation. nested simuation in impementation. Therefore we can estimate M π,a Theorem 3 Let M Tj be the martingae component of V Tj = v (,X Tj and M π,a be its approximation obtained via Agorithm 1, where v(, are Lipschitz continuous functions. Then there exists a constant C > such that [ E max j J Mπ,A M Tj 2 C π. 118

7 According to the reationship between M and V up (M in (5, we can immediatey obtain the foowing coroary on the quaity of uppers bounds. Coroary 4 Under the assumptions of Theorem 3, we have V up (Mπ,A V up (M 2 C π. 4 TRUE MARTINGALE ALGORITHM We wi formay describe the T-M agorithm based on the construction of the martingae approximation M π,a in section 3. The outine of the T-M agorithm consists of four steps in order: generating a suboptima exercise strategy τ, approximating the integrands φ π,a and ψ π,a, constructing the martingae approximation M π,a, and generating true upper bounds V up ( ˆM π,a on the option price. First, et s start with generating the suboptima exercise strategy τ. It not ony provides the ower bound, but aso pays an important roe in approximating the integrands φ π,a and ψ π,a. We adopt the L-S agorithm to generate the suboptima exercise( strategy τ and the corresponding approximation of option price process V Tj at time, of the form V Tj = v,x π, A, where π π, A A are empoyed to simuate the discretized asset price process {X π, A }. Second, et us approximate the integrands φ π,a and ψ π,a. To avoid confusion, we denote { ( φ t π,a = 1 π E t [( π W v +1,X π, A T j+1, t < +1 [ ( ψ π,a 1 (t,y k = µ([t,t +1 A k E t P([t,t +1 A k v +1,X π, A +1, t < +1,1 k K as the counterparts of φt π,a and ψ π,a (t,y k respectivey, under the discretized asset price X π, A +1. Inspired by Longstaff and Schwartz (21 s east-squares regression approach to approximating the continuation vaues, we appy a simiar regression technique to approximate φ π and ψ ( π,a. Specificay, ( the function bases chosen to regress φ t π,a and ψ π,a (t,y k are row function vectors ρ W t,x π, A t = ρi W (t,x π, A t ( ( i=1,...,i 1 and ρ P t,y k,x π, A t = ρi P (t,y k,x π, A t respectivey, where I 1 and I 2 are the dimensions of i=1,...,i 2 the function bases. If we simuate N independent sampes of Wiener increments π W, denoted by { π nw : = 1,...L,n = 1,...,N}, and N independent sampes of Poisson increments P([t,t +1 A k, denoted by {P n ([t,t +1 A k : = 1,...,L,k = 1,...,K,n = 1,...,N}, and based on which we construct the sampe paths of the asset price {X π, A t,n } =,...,L,n=1,...,N, then we can obtain the regressed coefficients ˆα t and ˆβ t,k, for t < +1 and 1 k K, via { N ( ( ˆα t = arg min π nw α R I π H τ j+1 X π, A T 1 n=1 j+1,n ρ W t,x π, A t,n α 2} { N ( ( ˆβ t,k = arg min P n ([t,t +1 A k β R I µ([t 2,t +1 A k H τ j+1 X π, A +1,n ρ P t,y k,x π, A t,n β 2}, (19 n=1 where we empoy the tower property to avoid nested simuation, as described in (17 and (18. Therefore we can compute the approximations of the integrands φ t π,a and ψ π,a (t,y k, denoted by ˆφ π,a (t,x and ˆψ π,a (t,y k,x respectivey, via ˆφ π,a (t,x = ρ W (t,x ˆα t and ˆψ π,a (t,y k,x = ρ P (t,y k,x ˆβ t,k. (2 Next, with fixed ˆα and ˆβ, we construct an approximation of M π,a, denoted by ˆM π,a, by combining the approximation ˆφ π,a and ˆψ π,a of the integrands with the Euer scheme of system (1. Precisey, we have ( ˆM π,a := ˆφ π,a t < K t < k=1 t,x π, A t ( π W + ˆψ π,a ( t,y k,x π, A t P([t,t +1 A k. (21 119

8 Obviousy, ˆM π,a remains to be a martingae adapted to the fitration {F Tj : j =,...,J }. Consequenty, V up ( ˆM π,a is a true upper bound on the Bermudan option price V. Finay, et s estimate V up ( ˆM π,a via (5 by simuating a new set of N independent sampe paths {X π, A n : n = 1,..., N}. Precisey, an unbiased estimator for V up ( ˆM π,a is given as foows: ( ˆV up ˆM π,a = 1 N N n=1 where ˆM π,a,n represents the reaization of ˆM π,a these steps in the foowing Agorithm 2. max j J [ ( h,x π, A,n ˆM π,a,n, (22 aong the sampe path X π, A. We can formay summarize Agorithm 2 True Martingae Agorithm Step 1: Appy the L-S agorithm to generate a suboptima exercise strategy τ. Step 2: Simuate N independent sampes of Wiener increments π W and N independent sampes of Poisson increments P([t,t +1 A k, for =,...,L 1 and k = 1,...,K ; construct the sampe paths of the asset price {X π, A t,n } =,...,L,n=1,...,N. Step 3: Obtain the parameters ˆα = { ˆα t } =,...,L and ˆβ = { ˆβ(t,k} =,...,L,k=,...,K via east-squares regression (19 by exercising τ aong the sampe paths {X π, A t,n } =,...,L,n=1,...,N. Step 4: Simuate a new set of N independent sampe paths {X π, A t,n } =,...,L,n=1,..., N; compute ˆφ π,a and ˆψ π,a via (2; construct the martingae approximation ˆM π,a via (21; obtain an unbiased estimator ˆV up ( ˆM π,a for the true upper bound on the Bermudan option price V via (22.,n 5 NUMERICAL EXPERIMENTS In this section, we wi conduct numerica experiments to iustrate the computationa efficiency of our proposed T-M agorithm on a Bermudan option pricing probem under a jump-diffusion mode. The exact mode we consider here fas into the cass of jump-diffusion modes (see Merton (1976 and Kou (28 reviewed in section 1. Specificay, the asset prices evove as foows: ( dx (t P(t X (t = (r δdt + σdw (t + d (V i 1, (23 i=1 where r is the constant discount factor, δ is the constant dividend, σ is the constant voatiity, X(t = [X 1 (t,...,x n (t represents the asset price with a given initia price X, W(t = [W 1 (t,...,w n (t is a Wiener process, P(t is a Poisson process with intensity λ, and {V i } is a sequence of independent identicay distributed (i.i.d. nonnegative random variabes such that J = og(v is the jump ampitude with p.d.f. f (y. Here J can foow a norma distribution (see (Merton 1976 or an exponentia distribution (see Kou (28 or various other reasonabe distributions. For simpicity, we assume J foows a one-dimensiona (d = 1 norma distribution N(m,θ 2. We aso assume W(t, P(t and J are mutuay independent. To connect dynamics (23 with the jump-diffusion mode (1 we have mainy focused on, we shoud construct a Poisson random measure P such that dynamics (23 can be easiy transformed to an equivaent dynamics jointy driven by the Wiener measure and the Poisson random measure. The foowing proposition provides an intuitive criterion in seecting such a Poisson random measure P by expicity defining the intensity function µ (dt dy for the unique P induced by a compound Poisson process. Proposition 5 (Proposition 3.5 in Cont and Tankov (23 Let S(t t> be a compound Poisson process with intensity λ and jump size distribution f. Then the Poisson random measure P S induced by S(t t> on [, R d has intensity measure µ (dt dy = λ f (ydydt. 12

9 According to Proposition 5, for a compound Poisson process S(t, the compensated Poisson random measure P S induced by S(t can be simuated by P S = P S λ f (ydydt. Athough X(t satisfying (23 is not a compound Poisson process, S(t = og(x(t is usuay a compound Poisson process, and thus its Poisson random measure P S (t,y can be easiy simuated according to Proposition 5. Now if we incorporate P S into the asset-price dynamics (23, we can obtain an equivaent dynamics as dx (t X (t = (r δdt + σdw (t + yp S (t,y. (24 R d Unfortunatey, the soution to asset dynamics (23 or (24 is not uniquey determined in the risk-neutra sense, caused by the incompeteness of the market under the jump-diffusion setting. However, we can construct pricing measures Q s P such that the discounted price ˆX(t is a martingae under Q s (c.f. Chapter 1 in Cont and Tankov (23. Here we wi adopt the construction method proposed by Merton (1976. That is, changing the drift of the Wiener process but eaving other components of (23 unchanged to offset the jump resuts in a risk-neutra measure Q M, which is a generaization of the unique risk-neutra measure under the Back-Schoes mode. Therefore, the soution under Q M can be easiy derived and efficienty simuated. Precisey, the soution to the asset-price dynamics (23 is given by: [ P(t X (t = X exp µ M t + σw M (t + J i, t >, (25 where µ M = r δ 1 2 σ 2 E [ e J i 1 is the new drift, W M (t is a standard vector Wiener process and J i s are the i.i.d. random variabes according to J. Given the equivaence of (23 and (24, we can perform the Euer scheme on an equidistant partition π with π =.1 and a continuousy equi-probabiistic partition A on R d with A =.1 to simuate the Wiener increments {W t }, the Poisson random measure increments P([t,t +1 A k, and the resuting sampe paths of X(t = exp(s(t according to (25. We consider a Bermudan Min-Puts on n assets, whose evoution is given by (25. In particuar, at any time t Ξ = {T,T 1,...,T J }, the option hoder has the right to exercise his option to receive the payoff h(x (t = (SK min(x 1 (t,...,x n (t +. The maturity time of the option is T = 1 and can be exercised at 11 equay-spaced time points, i.e., = j T /1, j =,...,1. Our objective is to sove the Bermudan option pricing probem by providing a ower bound and an upper bound on the exact option price. 5.1 Suboptima Exercise Strategies and Lower Bounds First, et s adopt the L-S agorithm to generate a suboptima exercise strategy τ by regressing the continuation vaues on certain function bases, and compute the corresponding benchmark ower bound. Anderson and Broadie (24 propose a function basis consisting of a monomias of underying asset prices with degrees ess than or equa to three, the European min-put option with maturity T, its square and its cube, since the European option under the pure-diffusion mode has a cosed-form that can be fast numericay evauated (see Zhu et a. (213 for expicit formua. For the Bermudan option pricing probem under the jump-diffusion mode (23, the corresponding European option sti has a cosed-form expression (see Zhu et a. (213 for expicit formua. However, it is extremey difficut to be evauated because of the infinite sum and the integra in the cosed-form. Naturay, we try to approximate it directy by an European option under a cosey-reated pure-diffusion mode. Surprisingy, the most intuitive one, i.e., the European option under the pure-diffusion mode derived simpy by removing the jump part of (23 works extremey we in our numerica experiments. To avoid confusion, we refer to it as non-jump European option. Now the function basis we choose incudes a monomias of underying asset prices with degrees ess than or equa to three, the non-jump European option with maturity T, its square and its cube. With this basis, we impement the L-S agorithm, and obtain suboptima exercise strategies τ s and the corresponding ower bounds, as shown in Tabe 1. i=1 121

10 5.2 Upper Bound by True Martingae Approach Now et s impement our proposed T-M agorithm (Agorithm 2 described in section 4. Notice that we have addressed amost a the impementation detais except the choices of partitions π and A, and the bases ρ W and ρ P. First of a, the choice of partition π is essentia to baance the tradeoff between the quaity of the true martingae approximation and the computationa efficiency. It has to be sufficienty sma to reduce the overa mean square error between the true martingae approximation and the objective martingae, but not too sma so that the computationa effort for obtaining martingae approximation M π,a is much ess than the computationa effort for obtaining the inner sampe paths in A-B agorithm. In fact, a good way to achieve this tradeoff is to perform the regression on a rough partition in the beginning, and then interpoate them piece-wisey constant to a finer partition. To maximize the effect of this two-ayer regressioninterpoation technique, we choose to perform the regression procedure on the origina exercisabe dates Ξ = {T,T 1,...,T J 1} and interpoate the regression coefficients piece-wisey constant to the partition π of the Euer scheme. Secondy, the choice of the partition A is ess restrictive than the choice of π since π wi dominate the error between the martingae approximation and the objective martingae (see Theorem(3 regardess of the choice of A. For the sake of convenience, we et A = A. Therefore the compensated Poisson increments { P([t,t +1 A k } in (19 are obtained immediatey from the simuation of X π, A, and µ ([t,t +1 A k in (19 equas λ.1.1 (see Proposition 5. Specificay, we obtain { ˆα Tj, j =,...,J 1} and { ˆβ Tj,k, j =,...,J 1,k = 1,...,K } via the regression (19, and set ˆα t = ˆα Tj for t [,+1 and ˆβ t,k = ˆβ Tj,k for t [,+1,k = 1,...,K. Finay, the choice of the bases ρ W and ρ P affects the accuracy of the martingae approximation ˆM π,a. Notice that European option price is a good basis function for the L-S agorithm. Inspired by Theorem 1, if we appy Ito s emma on the European option price, the resuting integrands shoud be good candidates for basis functions. Precisey, we have C M ( Tj C M ( u,x u ;T Rd j t,x t ; = h(xtj X t X u σdwu M Tj [ C M ( u,x u e y ; C M ( u,x u ; P S (du,dy, (26 t wherec M (t,x t ; is the European option price with maturity under pricing measure Q M. After simpe numerica tests, we find out, for t [,+1 and 1 k K, ρ W (t,x t consisting of 1, CBS (t,x t ;+1 X X t and C BS (t,x t ;T X X t, ρ P (t,y k,x t consisting of 1,C BS (t,x t e y k;+1 C BS (t,x t ;+1 andc BS (t,x t e y k;t C BS (t,x t ;T yied the tightest upper bounds, where C BS (t,x t ; is the European option price under the corresponding pure-diffusion mode and y k A k is a representative vaue. We report the numerica resuts on the ower bounds by the L-S agorithm, the benchmark upper bounds by the A-B agorithm and the true upper bounds by the T-M agorithm in Tabe 1. The sma gaps between the ower bounds and the true upper bounds indicate that the T-M agorithm is quite effective in terms of generating tight true upper bounds. The sma ength of the confidence intervas of the true upper bounds indicates that T-M agorithm generates good approximations of the optima dua martingaes. The CPU time ratios indicate that T-M agorithm achieves a much higher numerica efficiency. It is instructive to theoreticay compare the computationa compexity of the T-M agorithm and the A-B agorithm, since the CPU time ratios in Tabe 1 are quite different for 1-dimensiona probems and 2-dimensiona probems. We know that the tota CPU time is mainy consumed by simuating sampe paths and evauating the basis functions. When n = 1, the time for simuating sampe paths wi significanty dominate the time for evauating the basis functions because the basis functions, which are European options and their derivatives, reduce to the c.d.f. s of a standard norma distribution, and hence can be evauated extremey fast. Therefore, the CPU time ratio wi be in the order of the ratio between the numbers of sampe paths simuated in both agorithms, which is consistent with the resut ( 1:4. However, when n 2, the basis functions reduce to infinite integras of the c.d.f. s of a standard norma distribution, which are reativey time-consuming to evauate. Therefore, the CPU time ratio shoud be the ratio between the 122

11 tota evauation times of the basis functions in both agorithms. For the A-B agorithm, the tota evauation times is in the order of (N 2 N 3 J J ; for the T-M agorithm, the tota evauation times is in the order of ( N L K. The ratio of the atter versus the former is around 1:2, which is in the approximatey same order of the resut ( 1:9. We can expect the CPU time ratios (T-M agorithm versus A-B agorithm to remain stabe if the dimension of the probem increases, and even further decrease when the number of exercisabe periods increases. See Zhu et a. (213 for more detaied discussion. Tabe 1: Bounds (with 95% confidence intervas for Bermudan Min-put options. The payoff of the minput option is: (SK min(x 1 (t,...,x n (t +. The parameters are: SK = 4,r = 4%,δ =,σ = 2%,m = 6%,θ = 2%,T = 1,J = 1. The jump intensity λ is 1 or 3 and the initia price is X = (X,...,X with X =36, as shown in the tabe. We use N = sampe paths to estimate the regression coefficients to determine the suboptima exercise strategy, and we use N = sampe paths to estimate the coefficients ˆα and ˆβ. We use N 1 = 1 5 sampe paths to determine the ower bounds. For the impementation of the A-B agorithm, we use N 2 = 1 3 outer sampe paths and N 3 = inner sampe paths to determine the benchmark upper bounds and the confidence intervas of appropriate ength. For the impementation of the T-M agorithm, we use N = sampe paths to determine the true upper bounds and the confidence intervas of appropriate ength. Lower Bound Upper Bound Benchmark U-B CPU Time Ratio n λ X (L-S agorithm (T-M agorithm (A-B agorithm (T-M vs A-B ± ± ±.38 1: ± ± ±.53 1: ± ± ±.4 1: ± ± ±.57 1:9 6 CONCLUSION AND FUTURE DIRECTIONS In this paper, we propose a true martingae agorithm (T-M agorithm to fast compute the upper bounds on the Bermudan option prices under the jump-diffusion modes, as an aternative approach for the cassic A-B agorithm proposed by Anderson and Broadie (24, especiay when the computationa budget is imited. The theoretica anaysis of our agorithm proves and the numerica resuts verify that our agorithm generates stabe and tight upper bounds with significant reduction of computationa effort. Moreover, we expore the structure of the optima dua martingae for the dua probem and provide an intuitive understanding towards the construction of good approximations of the optima dua martingae over the space of a adapted martingaes. Furthermore, from the information reaxation point of view (see Brown et a. (21, we can gain an intuitive understanding towards the structure of the optima penaty function. It inspires us to construct good penaty functions over the space of feasibe penaty functions for genera dynamic programming probems, which is sti an open area to expore (see Ye and Zhou (212 for some initia exporation. REFERENCES Anderson, L., and M. Broadie. 24. Prima-Dua Simuation Agorithm for Pricing Mutidimensiona American Options. Management Science 5 (9: Beomestny, D., C. Bender, and J. Schoenmakers. 29. True Upper Bounds for Bermudan Products via Non-Nested Monte Caro. Mathematica Finance 19: Brown, D., J. Smith, and P. Sun. 21. Information Reaxations and Duaity in Stochastic Dynamic Programs. Operations Research 58 (4: Cont, R., and P. Tankov. 23. Financia Modeing with Jump Processes, Voume 2. Chapman & Ha/CRC. 123

12 Desai, V., V. Farias, and C. Moaemi Pathwise Optimization for Optima Stopping Probems. Management Science. Feng, L., and V. Linetsky. 28. Pricing Options in Jump-diffusion Modes: An Extrapoation Approach. Operations Research 56 (2: Gasserman, P. 24. Monte Caro Methods in Financia Engineering. Springer. Haugh, M., and L. Kogan. 24. Pricing American Options: A Duaity Approach. Operations Research 52 (2: Kou, G. 28. Jump-diffusion Modes for Asset Pricing in Financia Engineering. Chapter 2 of Handbooks in Operations Research and Management Science. Kou, G., and H. Wang. 24. Option Pricing Under a Doube Exponentia Jump-diffusion Mode. Management Science: Longstaff, F., and E. Schwartz. 21. Vauing American Options by Simuation: A Simpe Least-Squares Approach. The Review of Financia Studies 14 (1: Merton, R Option Pricing When Underying Stock Returns are Discontinuous. Journa of Financia Economics 3 (1: Rogers, L. 22. Monte Caro Vauation of American Options. Mathematica Finance 12 (3: Rogers, L. 27. Pathwise Stochastic Optima Contro. SIAM J.Contro Optimization 46 (3: Tsitsikis, J., and B. van Roy. 21. Regression Methods for Pricing Compex American-Stye Options. IEEE Transactions on Neura Networks 12 (4: Ye, F., and E. Zhou Parameterized Penaties in the Dua Representation of Markov Decision Processes. In Decision and Contro (CDC, 212 IEEE 51st Annua Conference on, IEEE. Ye, F., and E. Zhou. 213a. Information Reaxation and Duaity in Controed Markov Diffusions. Working Paper. Ye, F., and E. Zhou. 213b. Optima Stopping of Partiay Observabe Markov Processes: A Fitering-based Duaity Approach. IEEE Transactions on Automatic Contro.. To Appear. Zhu, H., F. Ye, and E. Zhou Fast Estimation of True Bounds on Bermudan Option Prices Under Jump-diffusion Processes. Working Paper. AUTHOR BIOGRAPHIES HELIN ZHU is a second-year Ph.D student in the Department of Industria and Enterprise Systems Engineering, University of Iinois at Urbana-Champaign. His research interest is on stochastic contro. His emai address is hzhu11@iinois.edu. FAN YE is a third-year Ph.D student in the Department of Industria and Enterprise Systems Engineering, University of Iinois at Urbana-Champaign. His research interest is on stochastic contro. His emai address is fanye2@iinois.edu. ENLU ZHOU is an Assistant Professor in the Department of Industria and Enterprise Systems Engineering at the University of Iinois at Urbana-Champaign. She received the B.S. degree with highest honors in eectrica engineering from Zhejiang University, China, in 24, and received the Ph.D. degree in eectrica engineering from the University of Maryand, Coege Park, in 29. Her research interests incude stochastic contro and simuation optimization, with appications towards financia engineering. Her emai address is enuzhou@iinois.edu and her web page is 124