Proceedigs of the 2015 Iteratioal Coferece o Operatios Excellece ad Service Egieerig Orlado, Florida, USA, September 10-11, 2015 Threshold Fuctio for the Optimal Stoppig of Arithmetic Orstei-Uhlebeck Process Alo Dourba Faculty of Idustrial Egieerig ad Maagemet Techio - Israel Istitute of Techology Haifa 32000, Israel alodo@tx.techio.ac.il Liro Yedidsio Faculty of Idustrial Egieerig ad Maagemet Techio - Israel Istitute of Techology Haifa 32000, Israel liroy@ie.techio.ac.il Abstract Mea reversio processes ca be foud i the core dyamics of umerous vital applicatios. I a mea revertig process, the value of the process teds to revert back to a log-ru average value. I this paper, we study the optimal stoppig problem of the widely used stochastic mea revertig process the arithmetic Orstei Uhlebeck process. We cosider a sigle item that eeds to be purchased withi a give deadlie, where its price process follows the arithmetic Orstei Uhlebeck process. Methods to deal with this problem so far have focused o a ifiite time horizo, which led to a costat threshold policy. We itroduce a optimal policy for the fiite time horizo case. The optimal policy is based o a recursive method to calculate a time-variat threshold fuctio that represets the optimal stoppig decisio as a fuctio of time. As a sub-routie of our method we develop explicit terms for the crossig time probability ad the overshoot expectatio of the mea revertig process. The, we use these terms i the Bellma's equatio of our model to costruct the threshold fuctio for the purchasig decisio. Fially, we aalyze the threshold fuctio behavior with respect to its differet parameters ad derive meaigful properties. 1. Itroductio The optimal stoppig time problem of the Orstei-Uhlebeck (O-U process relates to a plethora of fields icludig applicatios i fiace (see Bessembider et al. 1995, Pidyck ad Rubifeld 1991, Schwartz 1997, Ekstrom et al. 2011, biology (Ricciardi ad Sacerdote 1991, ad physics (Tateo et al. 1995. I fiacial applicatios the process is typically used to model a price process whose dyamics exhibit some degree of mea reversio to a costat level. Examples of such models iclude the price process of commodities that is assumed to follow the geometric O-U process (see Bessembider et al. 1995, Pidyck ad Rubifeld 1991, Schwartz 1997, ad the spread process of two related assets uder a pairs tradig strategy that is assumed to follow the arithmetic O-U process (Ekstrom et al. 2011. I these kids of models the problem of the optimal stoppig time that determies the time to trigger a actio, e.g., buy or sell the mea revertig asset, is crucial. Despite its mai role, may aspects of the problem still lack a closed form solutio. The geeral solutio of this problem is determied i the form of a threshold policy that sets the price values across time that bouds the actio regio (Va Moerbeke 1976. For the case of ifiite horizo, a costat threshold value ca be calculated explicitly by solvig a free boudary problem (Peskir ad Shiryaev 2006. However, for the case of fiite horizo o explicit aalytic solutio for calculatig the optimal policy is kow, but oly umeric methods that are based o biomial ad triomial trees (Nelso ad Ramaswamy 1990, Hull ad White 1994, Mote Carlo simulatio (Logstaff ad Schwartz 2001, ad o solvig partial differetial equatios usig fiite differece method (Schwartz 1977. Other related sub-problems of the optimal stoppig are the crossig time probability ad the overshoot expectatio of the O-U process with respect to some threshold level. These sub-problems preset major challeges as well, ad do ot have explicit aalytic solutios except for a few special cases. For the crossig time probability there are three mai approximatio methods: the eigevalue expasio based method, the itegral represetatio, ad the 3-dimesioal Bessel Bridge (Alili et al. 2005. All of these methods are limited to apply for a costat threshold oly. I the discrete form of the O-U process (also kow as the first order autoregressive, AR(1, process, the overshoot expectatio over a threshold does ot have a kow closed form solutio as well, 672
ad may be derived usig Mote Carlo simulatio or by approximatig the process to a autoregressive process with expoetial stochastic terms (Jacobse ad Jese 2007. This work itroduces a ew approach to determie the optimal policy for purchasig a mea revertig asset whose price process follows the arithmetic O-U process whe there is a fiite plaig horizo. The optimal policy is determied by a simple threshold fuctio that divides the price regio ito a cotiuatio regio ad a stoppig regio i which the asset is purchased. The threshold calculatio method is based o solvig the value fuctio of the model accordig to the Bellma equatio whe decomposig the stochastic terms of the price process ito a simplified form of a stadard multivariate ormal variable. Uder this settig the crossig time probability ad overshoot expectatio of the process are derived ad the threshold ca be calculated recursively. This procedure outputs a simple threshold fuctio, which is icreasig towards the ed of the plaig horizo. To the best of our kowledge, a threshold fuctio for the arithmetic O-U model has yet to be established. The rest of the paper is orgaized as follows: We describe the model i sectio 2. We itroduce the method for calculatig the threshold fuctio i sectio 3. Lastly, we aalyze the threshold fuctio properties i sectio 4. 2. Model We cosider a fiite time horizo with legth T. A sigle item has to be purchased by the ed of the time horizo. Our goal is to miimize the expected purchase cost of the item. The item s price, deoted by x t, fluctuates accordig to a arithmetic Orstei-Uhlebeck (O-U process: dx t = K(θ x t dt + σ dw t, (1 where θ is the log ru price average, K is the revertig rate, dt is the time iterval legth, ad σ is the degree of volatility. The stochastic term of the process, deoted by W t, is modeled i the form of a Wieer process. Note that i this geeral model we allow the price process to reach a egative level, as it may represet the spread of two related assets i a pairs tradig model (Ekstrom et al. 2011. Let π deote a possible purchasig policy, ad let V π (x t, t deote the expected purchasig cost uder policy π. We will defie π as the optimal policy that miimizes the expected purchasig cost. That is, π = arg mi V π (x t, t, ad V (x t, t = V π (x t, t. Accordig to Va Moerbeke (1976, it is well kow that the optimal policy is i the form of a icreasig threshold fuctio, b(t. Hece, the optimal stoppig problem ca be reduced to fidig the threshold fuctio. However, i the case of a fiite time horizo, o closed solutio for the threshold fuctio has bee foud. 3. The Threshold Policy The followig sectio characterizes the structure of the optimal policy. The optimal policy is based o a threshold fuctio over time that states the followig purchasig policy: buy if the item s price drops below the threshold, otherwise wait. A atural way to calculate the threshold fuctio is by discretizig the price process ad applyig Bellma s equatio: We get that at the ed of the time horizo V (x T, T = x T, as at time T the decisio maker has o choice other tha purchasig the item at price x T. I order to simplify otatio we defie i the discrete form: N = T/dt, = t/dt ad t i = i dt for 0 i N, ad let x 0, x t1, x t2,..., x tn deote the discretized item price series. Whe modelig the process i the discrete form, it is importat to ote that the costrait 1 dtk < 1 must be added i order to keep the statioary property of the process. Accordig to the Bellma equatio, we get that for 0 t < t N : V (x t, t = mi {E[V (x t+1, t +1 x t ], x t }, (2 ad the threshold value at time t is set to the value b(t that satisfies: V (b(t, t = b(t. (3 At the ed of the time horizo, if the item has ot yet bee purchased it is purchased at ay price. Hece, we ca set the threshold value at the ed of the horizo to ifiity: b(t N =. I additio, it ca be easily see that i the last period before the ed of the time horizo, t N 1, eq. (3 is satisfied whe the process is at its log ru average level. Therefore, we get that at time t N 1 the threshold fuctio equals: b(t N 1 = θ. 3.1 Crossig Time Probability ad Overshootig Expectatio Let τ deote the first crossig time of the process x t with its respective threshold fuctio, b(t: τ = if {t 0: x t b(t}. We defie P t (x 0 as the probability that the first crossig time occurs at time t, whe the curret price is x 0 : P t (x 0 = Pr (τ = t x 0, 673
ad the overshoot expectatio, E t (x 0, as the expectatio of the item s price, coditioed that the first crossig time occurs at time t: E t (x 0 = E(x t x 0, τ = t. Note that i the cotiuatio regio, where x 0 > b(0, we get that: V (x 0, 0 = P t1 (x 0 E t1 (x 0 + (1 P t1 (x 0 E[V (x t1, t 1 x 0, x t1 > b(t 1 ] or for a geeral t: N = P ti (x 0 E ti (x 0, i=1 N V (x t, t = P ti (x t E ti (x t. i=+1 (4 3.2. Decompositio to Multivariate Normal Variables Accordig to eq. (1 x t = x t 1 + dx t 1. By usig (1 recursively, this equatio ca be exteded to represet the item s price at time t, based o a kow startig price at time 0: x t = θ + (x 0 θ(1 dt K. (5 +σ (1 dt K i 1 dw t i. i=1 Therefore x t distributes as a ormal radom variable: x t N (μ, σ xt x 2 t, where its expectatio, μ, ad its variace, σ xt x 2 t, equal: μ = θ + (x xt 0 θ(1 dt K, (6 2 = Var [σ (1 dt K i 1 dw t i ] (7 σ xt i=1 1 (1 dt K2 2 = dt σ 1 (1 dt K 2. Let z t = x t μ x t be the correspodig stadardized ormal radom variable of x σ t. That is, z t N(0,1. As xt x ti, 1 t i t are correlated, we defie the correspodig symmetric covariace matrix of z ti, 1 t i < t by Σ t. That is, Σ t = cov (z ti, z tj for 1 t i, t, where for t i : cov(z ti, z tj = dt σ2 (1 dt K j i σ xti σ xtj 1 (1 dt K2i 1 (1 dt K 2. (8 Propositio 1. where P t (x 0 = F( B t (x 0 ; Σ t F( B t (x 0 ; Σ t, (9 B t (x 0 = [β t1 (x 0, β t2 (x 0,..., β t (x 0 ], B t (x 0 = [β t1 (x 0, β t2 (x 0,..., β t (x 0 ], β ti (x 0 = b(t i [θ + (x 0 θ (1 dtk ], (10 σ xt β ti (x 0 = { β t i (x 0, 1 i 1, i =, ad F(a; Σ is the cumulative distributio fuctio of a stadard multivariate ormal variable with a covariace matrix Σ, over the trucatio vector a R. Proof. P t (x 0 = Pr (x t b(t, x t 1 > b(t 1,..., x t1 > b(t 1 x 0 674
= Pr (z t β t (x 0, z t 1 > β t 1 (x 0,..., z t1 > β t1 (x 0 x 0. By the law of total probability we get: P t (x 0 = Pr (z t 1 β t 1 (x 0,..., z t1 β t1 (x 0 x 0 Pr (z t β t (x 0, z t 1 β t 1 (x 0,..., z t1 β t1 (x 0 x 0. = F( B t (x 0 ; Σ t F( B t (x 0 ; Σ t. Let where A t (x 0 = [α t1 (x A t 0 = [α t1 (x 0,..., α tj 1 (x 0,..., α tj 1 (x0, α (x0 tj+1,.., α t (x 0 ], (x0, (x0,.., (x α tj+1 α t 0 ], α ti (x 0 = (β ti (x 0 Cov (z ti, z tj β tj (x 0 / 1 Cov (z ti, z tj 2, 1 i, j, i j, (x α ti 0 = { α t (x i 0, 1 i 1, 1 j, i j,, i = ad let M t be the 1 1 first-order partial correlatio matrix of z ti, for 1 i, j, i j, whe removig the cotrollig variable z tj. That is: where M t = ρ ti,t k., 1 i, j, k, i, j k, ρ ti,t k. = cov(z t i, z tk cov (z ti, z tj cov (z tk, z tj 1 cov (z ti, z tj 2 1 cov (z tk, z tj 2. By usig Tallis equatio to calculate the coditioal expectatio of a elemet from a stadard multivariate ormal variable (Tallis 1961, we derive the terms for calculatig E t (x 0. First we use Tallis equatio to calculate the terms E(x t x 0, τ > t ad E(x t x 0, τ > t 1 ad get: E(z t x 0, τ > t = j=1 E(z t x 0, τ > t 1 = j=1 cov cov (z t, z tj φ (β tj (x 0 F (A t (x 0 ; M t, (11 F(B t (x 0 ; Σ t (z t, z tj φ (β tj (x 0 F (x (A t 0 ; M t, (12 F(B t (x 0 ; Σ t where φ(x is the probability desity fuctio (PDF of a stadard ormal variable. Therefore, E(x t x 0, τ > t σ cov xt = μ + j=1 (z t, z tj φ (β tj (x 0 F (A t (x 0 ; M t xt, F(B t (x 0 ; Σ t E(x t x 0, τ > t 1 σ cov (z xt = μ + j=1 t, z tj φ (β tj (x 0 F (x (A t 0 ; M t xt. F(B t (x 0 ; Σ t The, by the law of total expectatio, usig eqs. (9, 13, 14 we get that: E t (x 0 = [E(x t x 0, τ > t 1 (1 P t (x 0 E(x t x 0, τ > t ]. (15 P t (x 0 3.3 Threshold Calculatio The threshold values ca be calculated by satisfyig eq. (3, where by eqs. (9, 15 the value fuctio ca be derived. However, at time t eqs. (9, 15 require the values of b(t i for < i N as a iput. Therefore, we obtai b(t by followig a recursive procedure that starts with the calculatio of b(t N =, b(t N 1 = θ, ad the proceeds to lower idexes of t sequetially. I each step the value of b(t i that satisfies eq. (3 is derived by a simple search that ca be desiged for ay precisio desired. Figure 1 exemplifies the threshold fuctio, (13 (14 675
b(t, for differet parameters. 4. Threshold Properties I this sectio we itroduce some meaigful properties of the threshold fuctio. Lemma 1. θ is the itercept of the threshold fuctio. Proof. By the defiitio of the price process i eq. (1, the oly term of θ that affects the process is the differece betwee θ ad x t. That is, oly this differece affects the process ad ot the price level itself. Lemma 2. b(t is liear i σ. Proof. Accordig to Lemma 1, θ acts oly as the itercept of the threshold. Hece, without loss of geerality, we -0.2 0 5 10 15 20-0.7 b(t -1.2 K=0.8 K=0.6 K=0.4 K=0.2-1.7-2.2 t Figure 1: The threshold fuctio for θ = 0, σ = 1, dt = 1, across K = 0.2, 0.4, 0.6, 0.8 for a time horizo of T = 20. may assume that θ = 0 for this proof. Let us assume that Assumptio 1: for 1 i N, b(t i is liear i σ. Next, we prove that uder Assumptio 1, b(t 0 is liear i σ as well. Let x t deote the price process where σ = 1, which follows the process dx t = K(θ x t dt + dw t. Respectively, let τ, z ti, b (t i, β ti (x 0, B ti (x 0, B ti (x 0, Σ ti, P t i (x 0, E t (x 0, V (x 0, t 0 deote the correspodig x t process-related terms of τ, z ti, b(t i, β ti (x 0, B ti (x 0, B ti (x 0, Σ ti, P ti (x 0, E t (x 0, V (x 0, t 0. Accordig to eqs. (7, 8, cov (z ti, z tj, for 1 i, j is idepedet i σ. Lookig carefully at x t, the geeral price process with a volatility degree, σ, i compariso to x t, we get that accordig to eq. (10 uder Assumptio 1, β ti (σ x 0 = β t i (x 0 for 1 i N. Therefore, F( B t (σ x 0 ; Σ t = F( B t (x 0 ; Σ t, ad F( B t (σ x 0 ; Σ t = F( B t (x 0 ; Σ t. Hece, accordig to eq. (9 we get that P t (σ x 0 = P t (x 0 uder Assumptio 1. Similarly, accordig to eqs. (11, 12 we get that uder Assumptio 1 E(z t σ x 0, τ > t = E(z t x 0, τ > t, ad 676
E(z t σ x 0, τ > t 1 = E(z t x 0, τ > t 1. Hece, accordig to eqs. (6, 7, 13, 14, we get that uder Assumptio 1, E(x t σ x 0, τ > t = E(x t σ x 0, τ > t, ad E(x t σ x 0, τ > t 1 = E(x t σ x 0, τ > t 1, ad therefore accordig to eq. (15, E t (σ x 0 /σ = E t (x 0. Fially, we get that accordig to eq. (4, uder Assumptio 1, V (σ x 0, t 0 /σ = V (x 0, t 0. By the defiitio of the threshold value, V (b (t 0, t 0 = b (t 0, ad therefore, V (σ b (t 0, t 0 = σ b (t 0. That is, the threshold values are liear i σ uder Assumptio 1. Moreover, at the ed of the time horizo, b(t N 1 = 0 for ay σ, ad trivially satisfies: V (σ b (t N 1, t N 1 = σ b (t N 1. Therefore b(t N 1 is i fact liear i σ, which makes the iductio assumptio valid. Figure 2 shows the liear behavior of the threshold fuctio for differet deadlies. Lemma 3. For a costat ratio of dt 1, b(t is liear i dt ad i K Proof. Accordig to eqs. (6, 7, 8, the parameters: dt, K, σ appear oly i two forms: dt K ad dt σ 2, or ca be easily be maipulated to appear i the forms of: dt K ad 1 K σ2. Therefore, it is straight forward to apply the same iductio procedure as i Lemma 2 to show that b(t is liear i dt, ad 1, uder a costat value of dt K. K. K 10 b(t 9.8 9.6 9.4 9.2 9 8.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 σ t=8 t=7 t=6 t=5 t=4 t=3 t=2 t=1 Figure 2: Threshold values as a fuctio of σ, with θ = 10, dt = 1, K = 0.2, T = 8 at differet times: t = 1, 2, 3, 4, 5, 6, 7, 8. 5. Coclusio I this paper, we itroduced a simple aalytical method to solve the optimal stoppig problem of a arithmetic O-U process. The optimal policy is i the form of a simple threshold fuctio, which idicates whether the process should be stopped. The method is based o fidig explicit terms for the related crossig time probability ad 677
overshoot expectatio by decomposig the terms to the form of multivariate ormal variables. We show that the threshold fuctio is liear i σ, ad for a costat dt K value it is liear i dt, ad 1. The techique developed i this research could provide practical tools i fiacial istrumets ad other O-U related areas such as biology ad physics. Refereces Alili, L., Patie, P., ad Pederse, L., Represetatios of the first hittig time desity of a Orstei-Uhlebeck process, Stochastic Models, vol. 21, pp. 967-980, 2005. Bessembider, H., Cougheour, J., Sequi, P., ad Smoller, M., Mea reversio i equilibrium asset prices: evidece from futures Term structure, The Joural of Fiace, vol. 50, o. 1, pp. 361-375, 1995. Ekstrom, E., Lidberg, C., ad Tysk, J., Optimal liquidatio of a pairs trade, Advaced Mathematical Methods for Fiace, pp. 247-255, 2011. Hull, J., White, A., Numerical procedures for implemetig term structure models I: sigle-factor models, The Joural of Derivatives, vol. 2, o. 1, pp. 7-16, 1994. Jacobse, M., Jese, T., Exit times for a class of piecewise expoetial Markov processes with two-sided jumps, Stochastic processes ad their applicatios, vol. 117, pp. 1330 1356, 2007. Logstaff, F., Schwartz, E., Valuig America Optios by Simulatio: A Simple Least-Squares Approach, The Review of Fiacial Studies, vol. 14, o. 1, pp. 113 147, 2001. Nelso, D., Ramaswamy, K., Simple Biomial Processes as Diffusio Approximatios i Fiacial Models, The Review of Fiacial Studies, vol. 3, o.3, pp. 393-430, 1990. Peskir, G., Shiryaev, N., Optimal Stoppig ad Free-Boudary Problems, Lectures Math. ETH Zurich, Birkhauser, Basel, 2006. Pidyck, R., Rubifeld, D., Ecoometric Models ad Ecoomic Forecasts, McGraw-Hill, 3, 1991. Ricciardi, L., Sacerdote, L., The Orstei-Uhlebeck process as a model for euroal activity, Biological Cyberetics, vol. 35, o. 1, pp. 1-9, 1991. Schwartz, E., The valuatio of warrats: Implemetig a ew approach, Joural of Fiacial Ecoomics, vol. 4, pp. 79-94, 1977. Schwartz, E., The stochastic behavior of commodity prices: implicatios for valuatio ad hedgig, The Joural of Fiace, vol. 5, o. 3, pp. 923-973, 1997. Tallis, G., The Momet Geeratig Fuctio of the Trucated Multi-Normal Distributio, Joural of the Royal Statistical Society. Series B (Methodological, vol. 23, o. 1, pp. 23-29, 1961. Tateo, T., Doi, S., Sato, S., Ricciardi, L., Stochastic phase lockigs i a relaxatio oscillator forced by a periodic iput with additive oise: A first-passage-time approach, Joural of Statistical Physics, vol. 78, o. 3-4, pp. 917-935, 1995. Va Moerbeke, P., O optimal stoppig ad free boudary problems, Archive for Ratioal Mechaics ad Aalysis, vol. 60, o. 2, pp. 101-148, 1976. Biography Alo Dourba is a Idustrial Egieerig Ph.D. studet i the Idustrial Egieerig faculty at the Techio Israel Istitute of Techology. His research is coducted uder the supervisio of Prof. Liro Yedidsio. He eared his B.Sc. ad M.Sc. i Idustrial Egieerig from the Techio Israel Istitute of Techology. His mai research iterests are stochastic problems i Logistics ad Supply Chai Maagemet ad Reveue Maagemet. Alo teaches courses i Stochastic Models ad Project Maagemet. Liro Yedidsio is a assistat professor at the Faculty of Idustrial Egieerig ad Maagemet at the Techio Israel Istitute of Techology. His research iterests lie i discrete optimizatio, NP-hard problems, ad Approximatio algorithms. He did his Ph.D. at Be-Gurio Uiversity of the Negev, Israel, ad his Post-Doc at MIT Massachusetts Istitute of Techology. Liro is both a Kreitma Fellow ad a Fulbright Fellow. Liro was ackowledged umerous times as a outstadig lecturer ad wo the very prestigious Yaai Prize for Excellece i Teachig. Liro teaches courses i Logistics ad Supply Chai Maagemet, Productio ad Service Operatio, ad Itroductio to Schedulig. Liro is curretly ivolved i collaborative research projects with Elbit Systems Ltd. ad with Marvell Techology Group Ltd. He also acted as a cosultat for Teva Pharmaceutical Idustries Ltd. K 678