A recursive algorithm for selling at the ultimate maximum in regime-switching models
AA recursive algorithm for selling at theultimate maximum in regime-switchingmodels
Yue Liu
School of Finance and EconomicsJiangsu UniversityZhenjiang 212013P.R. China
Nicolas Privault
School of Physical and Mathematical SciencesNanyang Technological University21 Nanyang LinkSingapore 637371
September 20, 2018
Abstract
We propose a recursive algorithm for the numerical computation of the op-timal value function inf t ≤ τ ≤ T IE (cid:104) sup ≤ s ≤ T Y s /Y τ (cid:12)(cid:12) F t (cid:105) over the stopping times τ with respect to the filtration of a geometric Brownian motion Y t with Markovianregime switching. This method allows us to determine the boundary functions ofthe optimal stopping set when no associated Volterra integral equation is avail-able. It applies in particular when regime-switching drifts have mixed signs, inwhich case the boundary functions may not be monotone. Key words:
Optimal stopping; Markovian regime switching; non-monotone freeboundary; recursive approximation.Mathematics Subject Classification (2010):
The study of optimal stopping of Brownian motion as close as possible to its ultimatemaximum has been initiated in Graversen, Peskir and Shiryaev [3]. For geometricBrownian motion, the optimal prediction problem V t = inf t ≤ τ ≤ T IE (cid:20) sup ≤ s ≤ T Y s Y τ (cid:12)(cid:12)(cid:12) F t (cid:21) (1.1)1 a r X i v : . [ m a t h . P R ] F e b f selling at the ultimate maximum over all ( F ts ) s ∈ [ t,T ] -stopping times τ ∈ [ t, T ] hasbeen solved in [2] by Du Toit and Peskir when the asset price ( Y t ) t ∈ IR + is modeled bya geometric Brownian motion and ( F ts ) s ∈ [ t,T ] is filtration generated by ( B s − B t ) s ∈ [ t,T ] ,see [10] for background on optimal stopping and free boundary problems, and Chap-ter VIII therein for ultimate position and maximum problems.This framework has recently been extended in Liu and Privault [7] to the regime-switching model dY t = µ ( β t ) Y t dt + σ ( β t ) Y t dB t , ≤ t ≤ T, (1.2)driven by a finite-state, observable continuous-time Markov chain ( β t ) t ∈ IR + with statespace M := { , , . . . , m } independent of the standard Brownian motion ( B t ) t ∈ IR + ona filtered probability space (Ω , ( F t ) t ∈ IR + , P ), where ( F t ) t ∈ IR + is the filtration generatedby ( B t ) t ∈ IR + and ( β t ) t ∈ IR + and µ : M −→
IR, and σ : M −→ (0 , ∞ ) are deterministicfunctions.Regime-switching models were introduced by Hamilton [5] in the framework of timeseries, in order to model the influence of external market factors. European optionshave been priced in continuous time regime-switching models by Yao, Zhang andZhou [12] using a successive approximation algorithm. Optimal stopping for optionpricing with regime switching has been dealt with in e.g. Guo [4] and Le and Wang [6].It has been shown in particular in [2] that the boundary function b ( t ) is nonincreasingand continuous in t ∈ [0 , T ] and satisfies a Volterra integral equation of the form G ( t, b ( t )) = J ( t, b ( t )) − (cid:90) Tt K ( t, r, b ( t )) dr, (1.3)0 ≤ t ≤ T , with given terminal condition b ( T ), where the functions J ( t, x ) and K ( t, r, x ) are specified in [2].Under regime switching, the optimal boundary functions depend on the regime stateof the system, and they may not be monotone if the drift coefficients ( µ ( i )) i ∈M have2witching signs, cf. Figures 3 and 4 in Section 5. Essentially, a boundary function in-creases when there is sufficient time to switch from a state with negative drift to a statewith positive drift and to remain there until maturity, and is decreasing otherwise.We refer to [8] and [9] for other optimal settings that involve non monotone boundaryfunctions.In the regime switching setting however, no Volterra equation such as (1.3) is availablein general for boundary functions, cf. Section 5 and Remark 5.5 of [7]. In addition,the free boundary problem in the regime switching case consists in a system of inter-acting PDEs, making its direct solution more difficult, cf. Proposition 5.2 in [7]. InBuffington and Elliott [1] a free boundary problem has been solved under an order-ing assumption on the boundary functions in the two-state case, see Assumption 3.1therein, however this condition may not hold in general in our setting, cf. Figure 4below, and their method is specific to American options.In this paper we construct a recursive algorithm for the numerical solution of (1.1)in the regime-switching model (1.2), that includes the case where the drifts ( µ ( i )) i ∈M have nonconstant signs. Our algorithm has a linear complexity O ( n ) in the number n of time steps, hence in the absence of regime switching it also performs faster thanthe resolution of the Volterra equation, which has a quadratic complexity O ( n ) dueto the evaluation of the integral in (1.3), cf. Section 5.We start by recalling the main results of [7]. From Lemma 2.1 in [7], the optimalvalue function V t in (1.1) can be written as V t = V ( t, ˆ Y ,t /Y t , β t ) , where the function V : [0 , T ] × [1 , ∞ ) × M → IR + is given by V ( t, a, j ) := inf t ≤ τ ≤ T IE (cid:20) Y τ max( aY t , ˆ Y t,T ) (cid:12)(cid:12)(cid:12) β t = j (cid:21) (1.4)= inf t ≤ τ ≤ T IE (cid:20) G (cid:18) τ, β τ , Y τ max (cid:16) aY t , ˆ Y t,τ (cid:17)(cid:19) (cid:12)(cid:12)(cid:12) β t = j (cid:21) , (1.5)3or t ∈ [0 , T ], j ∈ M , a ≥
1, with ˆ Y s,t := max r ∈ [ s,t ] Y r , 0 ≤ s ≤ t ≤ T , and G ( t, a, j ) := IE (cid:104) max (cid:16) a, ˆ Y t,T /Y t (cid:17) (cid:12)(cid:12)(cid:12) β t = j (cid:105) , t ∈ [0 , T ] , j ∈ M . (1.6)Here, the infimum is taken over all ( F ts ) s ∈ [ t,T ] -stopping times τ , where F ts := σ ( B r − B t , β r : t ≤ r ≤ s ), s ∈ [ t, T ]. From Proposition 3.1 in [7], given β t = j ∈ M andˆ Y ,t /Y t = a ∈ [1 , ∞ ), t ∈ [0 , T ], the optimal stopping time for (1.1), or equivalentlyfor (1.5), is the first hitting time τ D ( t, a, j ) := inf (cid:40) r ≥ t : (cid:32) r, ˆ Y ,r Y r , β r (cid:33) ∈ D (cid:41) of the stopping set D := (cid:8) ( t, a, j ) ∈ [0 , T ] × [1 , ∞ ) × M : V ( t, a, j ) = G ( t, a, j ) (cid:9) (1.7)by the process ( r, ˆ Y ,r /Y r , β r ) r ∈ [ t,T ] .The stopping set D defined in (1.7) is closed, and its shape can be characterized as D = { ( t, y, j ) ∈ [0 , T ] × [1 , ∞ ) × M : y ≥ b D ( t, j ) } in terms of the boundary functions b D ( t, j ) defined by b D ( t, j ) := inf { x ∈ [1 , ∞ ) : ( t, x, j ) ∈ D } , t ∈ [0 , T ] , j ∈ M , cf. Proposition 3.2 of [7].If the condition µ ( j ) ≥ j ∈ M , then t (cid:55)→ b D ( t, j ) may not bedecreasing, cf. Figure 4 below. On the other hand, µ ( j ) ≤ j ∈ M leads to b D ( t, j ) = 1, t ∈ [0 , T ], j ∈ M , which corresponds to immediate exercise, cf. Proposi-tion 5.3 in [7].In this paper we construct a recursive algorithm for the numerical solution of theoptimal stopping problem (1.1), by determining the stopping set D from the valuesof V ( t, a, j ) and G ( t, a, j ), cf. Theorem 2.1 and Lemma 4.1 below. As this approach4oes not rely on the Volterra equation (1.3), it allows us in particular to determinethe boundary function b D ( t, j ) without requiring the condition µ ( j ) ≥ j ∈ M as in [7], cf. for example Figure 4 below. In addition we do not rely on closed formexpressions as in [2] as they are no longer available in the regime-switching setting.Our algorithm extends the method of [12] as it applies not only to the computation ofexpectations, but also to optimal stopping problems. However it differs from [12], evenwhen restricted to expectations IE[ φ ( Y T )] of payoff functions φ ( Y T ), where ( Y t ) t ∈ [0 ,T ] follows (1.2). In particular, the recursion of [12] is based on the jump times of theMarkov chain ( β t ) t ∈ IR + whereas we apply a discretization of the time interval [0 , T ],and our algorithm requires the Monte Carlo method only for the estimation of (2.3)below. In the sequel we let δ n := T /n , t nk := kδ n , k = 0 , , . . . , n , T n := ( t n , t n , . . . , t nn ), and (cid:100) s (cid:101) n := min (cid:8) t ∈ T n : t ≥ s (cid:9) , s ∈ [0 , T ] , n ≥ . In the following Theorem 2.1, which is proved in Section 3, the function V n ( t, a, j ) iscomputed by the backward induction (2.3) starting from the terminal time T . Theorem 2.1 ( i ) For all t ∈ [0 , T ] , j = 1 , , . . . , m and a ≥ , the solution V ( t, a, j ) of (1.4) satisfies V ( t, a, j ) = lim n →∞ V n ( (cid:100) t (cid:101) n , a, j ) , (2.1) where V n ( t nk , a, j ) is the discrete infimum V n ( t nk , a, j ) := inf t nk +1 ≤ τ n ≤ T IE (cid:34) ˆ Y ,T Y τ n (cid:12)(cid:12)(cid:12) ˆ Y ,t nk Y t nk = a, β t nk = j (cid:35) , k = 0 , , . . . , n − , (2.2) taken over all T n -valued stopping times τ n , and V n ( T, a, j ) := V ( T, a, j ) = a . ( ii ) The value of V n ( t nk , a, j ) in (2.2) can be computed by the backward induction V n (cid:0) t nk − , a, j (cid:1) = IE (cid:34) G (cid:32) t nk , ˆ Y ,t nk Y t nk , β t nk (cid:33) ∧ V n (cid:32) t nk , ˆ Y ,t nk Y t nk , β t nk (cid:33) (cid:12)(cid:12)(cid:12) ˆ Y ,t nk − Y t nk − = a, β t nk − = j (cid:35) , (2.3)5 or k = 1 , , . . . , n , under the terminal condition V n ( T, a, j ) = G ( T, a, j ) = a , a ≥ ,where G ( t, a, j ) in defined in (1.6) . In addition, by the following Theorem 2.2 proved in Section 4, we provide a way toapproximate the function G ( t, a, j ) used in (2.3). In the sequel we denote by ϕ r ( x, y ) := (cid:114) π (2 y − x ) r / e − (2 y − x ) / r , ≤ x ≤ y, r ∈ (0 , T ] , (2.4)the joint probability density function of (cid:18) B r , sup ≤ s ≤ r B s (cid:19) , and we let Q := [ q i,j ] ≤ i,j ≤ m denote the infinitesimal generator of ( β t ) t ∈ [0 ,T ] , and define u ( j ) := µ ( j ) /σ ( j ) − σ ( j ) / , j ∈ M . (2.5)Next, we show in Theorem 2.2 that G is approximated by a limiting sequence ( G n ) n ∈ IN given by the backward induction (2.6) below. Theorem 2.2
For any t ∈ [0 , T ] and j ∈ M we have G ( t, a, j ) = lim n →∞ G n ( (cid:100) t (cid:101) n , a, j ) , where the limit is uniform in a ≥ and G n ( t nk , a, j ) is defined by the backward induc-tion G n ( t nk − , a, j ) = (2.6) e q j,j δ n (cid:90) ∞ (cid:90) y −∞ e ( u ( j )+ σ ( j )) x − u ( j ) T/ (2 n ) G n ( t nk , a ∨ ( σ ( j ) y ) − σ ( j ) x, j ) ϕ δ n ( x, y ) dxdy + m (cid:88) i =1 i (cid:54) = j q j,i (cid:90) δ n e q j,j r (cid:90) ∞ (cid:90) y −∞ e ( u ( j )+ σ ( j )) x − u ( j ) r/ G n ( t nk , a ∨ ( σ ( j ) y ) − σ ( j ) x, i ) ϕ r ( x, y ) dxdydr,k = 1 , , . . . , n , with the terminal condition G n ( T, a, j ) = a , j ∈ M , a ≥ . In the particular case of constant drift µ and volatility σ cf, Theorems 2.1 and 2.2 alsoprovide an alternative numerical solution in the geometric Brownian motion modelof [2]. In this case, ( V n ( t nk − , a )) k =1 , ,...,n is computed from (2.3) by the backwardinduction V n (cid:0) t nk − , a (cid:1) = (cid:90) ∞ (cid:90) y −∞ G (cid:16) t nk , e σ ( log aσ ∨ y − x ) (cid:17) ∧ V n (cid:16) t nk , e σ ( log aσ ∨ y − x ) (cid:17) e λx − λ δ n / ϕ δ n ( x, y ) dxdy, G ( t, a ) = IE (cid:104) a ∨ e σS λT − t (cid:105) = (cid:90) ∞ (cid:90) y −∞ e (log a ) ∨ ( σy )+ λx − λ ( T − t ) / ϕ T − t ( x, y ) dxdy, (2.7)for all t ∈ [0 , T ] and a ≥
1, where S λt := max ≤ s ≤ t ( B s + λs ), λ := µ/σ − σ/
2, and ϕ r ( x, y )is given by (2.4). In the general regime switching setting, the function G ( t, a ) in (2.7)can be estimated by Monte Carlo, while in the absence of regime switching it can becomputed in closed form, cf. (2.7) in [2].In Sections 3 and 4 we prove Theorems 2.1 and 2.2. Numerical illustrations arepresented in Section 5 with and without regime switching. We observe in particularthat boundary functions may not be monotone when the drift coefficients µ ( j ), j ∈ M ,have different signs. ( i ) First, we note that for any ( F ts ) s ∈ [ t,T ] -stopping time τ ∈ [ t, T ] we haveIE (cid:34) ( aY t ) ∨ ˆ Y t,T Y τ (cid:12)(cid:12)(cid:12) β t = j (cid:35) = IE a ∨ (cid:16) ˆ Y t,T /Y t (cid:17) Y τ /Y t (cid:12)(cid:12)(cid:12) ˆ Y ,t Y t = a, β t = j = IE (cid:34) ˆ Y ,T Y τ (cid:12)(cid:12)(cid:12) ˆ Y ,t Y t = a, β t = j (cid:35) , (3.1) t ∈ [0 , T ], j ∈ M , a ∈ [1 , ∞ ), since ˆ Y ,t /Y t is conditionally independent of (cid:32) Y t Y τ , ˆ Y t,T Y τ (cid:33) = (cid:18) exp (cid:18) − (cid:90) τt σ ( β r ) d ˜ B r (cid:19) , exp (cid:18) sup t ≤ v ≤ T (cid:90) vt σ ( β r ) d ˜ B r − (cid:90) τt σ ( β t ) d ˜ B r (cid:19)(cid:19) given β t , where ( ˜ B v ) v ∈ [0 ,T ] is the drifted Brownian motion˜ B v := B v + (cid:90) v u ( β r ) dr, v ∈ [0 , T ] , (3.2)and u ( j ) := µ ( j ) /σ ( j ) − σ ( j ) / j ∈ M , is defined in (2.5). Hence by (1.4) we have V ( t nk , a, j ) = inf t nk ≤ τ ≤ T IE (cid:34) ˆ Y ,T Y τ (cid:12)(cid:12)(cid:12) ˆ Y ,t nk Y t nk = a, β t nk = j (cid:35) inf t nk ≤ τ n ≤ T IE (cid:34) ˆ Y ,T Y τ n (cid:12)(cid:12)(cid:12) ˆ Y ,t nk Y t nk = a, β t nk = j (cid:35) ≤ inf t nk +1 ≤ τ n ≤ T IE (cid:34) ˆ Y ,T Y τ n (cid:12)(cid:12)(cid:12) ˆ Y ,t nk Y t nk = a, β t nk = j (cid:35) = V n ( t nk , a, j ) ,k = 0 , , . . . , n − j ∈ M , a ≥
1, where we used (2.2) and the infimum is takenover all T n -valued discrete stopping times τ n . Therefore by the continuity of V ( t, a, j )with respect to t , cf. e.g. [10], Chap III, § § V ( t, a, j ) = lim n →∞ V ( (cid:100) t (cid:101) n , a, j ) ≤ lim inf n →∞ V n ( (cid:100) t (cid:101) n , a, j ) . (3.3)( ii ) On the other hand, by (3.1) we havelim sup n →∞ V n ( (cid:100) t (cid:101) n , a, j ) = lim sup n →∞ inf (cid:100) t (cid:101) n + δ n ≤ τ n ≤ T IE (cid:34) ˆ Y ,T Y τ n (cid:12)(cid:12)(cid:12) ˆ Y , (cid:100) t (cid:101) n Y (cid:100) t (cid:101) n = a, β (cid:100) t (cid:101) n = j (cid:35) = lim sup n →∞ inf (cid:100) t (cid:101) n + δ n ≤ τ n ≤ T IE (cid:34) ( aY (cid:100) t (cid:101) n ) ∨ ˆ Y (cid:100) t (cid:101) n ,T Y τ n (cid:12)(cid:12)(cid:12) β (cid:100) t (cid:101) n = j (cid:35) = lim sup n →∞ m (cid:88) l =1 (cid:2) e ( (cid:100) t (cid:101) n − t ) Q (cid:3) j,l inf (cid:100) t (cid:101) n + δ n ≤ τ n ≤ T IE (cid:34) ( aY (cid:100) t (cid:101) n ) ∨ ˆ Y (cid:100) t (cid:101) n ,T Y τ n (cid:12)(cid:12)(cid:12) β (cid:100) t (cid:101) n = l (cid:35) ≤ lim sup n →∞ inf (cid:100) t (cid:101) n + δ n ≤ τ n ≤ T m (cid:88) l =1 (cid:2) e ( (cid:100) t (cid:101) n − t ) Q (cid:3) j,l IE (cid:34) ( aY (cid:100) t (cid:101) n ) ∨ ˆ Y (cid:100) t (cid:101) n ,T Y τ n (cid:12)(cid:12)(cid:12) β (cid:100) t (cid:101) n = l (cid:35) = lim sup n →∞ inf (cid:100) t (cid:101) n + δ n ≤ τ n ≤ T IE (cid:34) ( aY (cid:100) t (cid:101) n ) ∨ ˆ Y (cid:100) t (cid:101) n ,T Y τ n (cid:12)(cid:12)(cid:12) β t = j (cid:35) , (3.4) t ∈ [0 , T − δ n ], j ∈ M , a ∈ [1 , ∞ ), where Q = [ q i,j ] ≤ i,j ≤ m is the infinitesimalgenerator of ( β t ) t ∈ [0 ,T ] . Next, we note that for every stopping time τ ∈ [ t, T ] we have |(cid:100) τ (cid:101) n − τ | < /n , hence ( (cid:100) τ (cid:101) n ) n ≥ converges to τ uniformly in L ∞ (Ω) and pointwise.Hence we havelim n →∞ IE (cid:34) ( aY (cid:100) t (cid:101) n ) ∨ ˆ Y (cid:100) t (cid:101) n ,T Y (cid:100) τ ∨ ( t + δ n ) (cid:101) n (cid:12)(cid:12)(cid:12) β t = j (cid:35) = IE (cid:34) lim n →∞ ( aY (cid:100) t (cid:101) n ) ∨ ˆ Y (cid:100) t (cid:101) n ,T Y (cid:100) τ ∨ ( t + δ n ) (cid:101) n (cid:12)(cid:12)(cid:12) β t = j (cid:35) , (3.5) t ∈ [0 , T − δ n ], for any stopping time τ ∈ [ t, T ], where we applied Lebesgue’s dominatedconvergence theorem based on the bounds (3.8) and (3.9) stated at the end of this8ection. Hence from (3.4) and (3.5) we find, for any stopping time τ ∈ [ t, T ],lim sup n →∞ V n ( (cid:100) t (cid:101) n , a, j ) ≤ lim sup n →∞ inf (cid:100) t (cid:101) n + δ n ≤ τ n ≤ T IE (cid:34) ( aY (cid:100) t (cid:101) n ) ∨ ˆ Y (cid:100) t (cid:101) n ,T Y τ n (cid:12)(cid:12)(cid:12) β t = j (cid:35) ≤ lim n →∞ IE (cid:34) ( aY (cid:100) t (cid:101) n ) ∨ ˆ Y (cid:100) t (cid:101) n ,T Y (cid:100) τ ∨ ( t + δ n ) (cid:101) n (cid:12)(cid:12)(cid:12) β t = j (cid:35) = IE (cid:34) lim n →∞ ( aY (cid:100) t (cid:101) n ) ∨ ˆ Y (cid:100) t (cid:101) n ,T Y (cid:100) τ ∨ ( t + δ n ) (cid:101) n (cid:12)(cid:12)(cid:12) β t = j (cid:35) = IE (cid:34) ( aY t ) ∨ ˆ Y t,T Y τ (cid:12)(cid:12)(cid:12) β t = j (cid:35) = IE (cid:34) ˆ Y ,T Y τ (cid:12)(cid:12)(cid:12) ˆ Y ,t Y t = a, β t = j (cid:35) , where we applied (3.1) and the pathwise continuity of ( Y t ) t ∈ [0 ,T ] . Hence by (2.2), weobtainlim sup n →∞ V n ( (cid:100) t (cid:101) n , a, j ) ≤ inf t ≤ τ ≤ T IE (cid:34) ˆ Y ,T Y τ (cid:12)(cid:12)(cid:12) ˆ Y ,t Y t = a, β t = j (cid:35) = V ( t, a, j ) ,t ∈ [0 , T − δ n ], which completes the proof of (2.1) by (3.3).( iii ) In order to prove (2.3) for 0 ≤ k ≤ l ≤ n , we consider an optimal stopping time τ ( t nl ) n such thatIE (cid:34) ˆ Y ,T Y τ ( tnl ) n (cid:12)(cid:12)(cid:12) ˆ Y ,t nk Y t nk = a, β t nk = j (cid:35) = inf t nl ≤ τ n ≤ T IE (cid:34) ˆ Y ,T Y τ n (cid:12)(cid:12)(cid:12) ˆ Y ,t nk Y t nk = a, β t nk = j (cid:35) , (3.6)where the infimum is taken over the discrete T n -valued stopping times τ n , and theexistence of τ ( h ) n is guaranteed by Corollary 2.9 of [10] as in Proposition 3.1 of [7]. Wenote the inductionIE (cid:34) ˆ Y ,T Y τ ( tnk ) n (cid:12)(cid:12)(cid:12) ˆ Y ,t nk Y t nk , β t nk (cid:35) = IE (cid:34) ˆ Y ,T Y t nk (cid:12)(cid:12)(cid:12) ˆ Y ,t nk Y t nk , β t nk (cid:35) ∧ IE (cid:34) ˆ Y ,T Y τ ( tnk +1) n (cid:12)(cid:12)(cid:12) ˆ Y ,t nk Y t nk , β t nk (cid:35) = G (cid:32) t nk , ˆ Y ,t nk Y t nk , β t nk (cid:33) ∧ V n (cid:32) t nk , ˆ Y ,t nk Y t nk , β t nk (cid:33) , (3.7)9 = 0 , , . . . , n − a ≥
1, where V n and G are defined in (2.2) and (1.6) respectively.By (3.6), this yields V n (cid:0) t nk − , a, j (cid:1) = IE (cid:34) ˆ Y ,T Y τ ( tnk ) n (cid:12)(cid:12)(cid:12) ˆ Y ,t nk − Y t nk − = a, β t nk − = j (cid:35) = IE (cid:34) IE (cid:34) ˆ Y ,T Y τ ( tnk ) n (cid:12)(cid:12)(cid:12) ˆ Y ,t nk Y t nk , ˆ Y ,t nk − Y t nk − = a, β t nk , β t nk − = j (cid:35) (cid:12)(cid:12)(cid:12) ˆ Y ,t nk − Y t nk − = a, β t nk − = j (cid:35) = IE (cid:34) IE (cid:34) ˆ Y ,T Y τ ( tnk ) n (cid:12)(cid:12)(cid:12) ˆ Y ,t nk Y t nk , β t nk (cid:35) (cid:12)(cid:12)(cid:12) ˆ Y ,t nk − Y t nk − = a, β t nk − = j (cid:35) = IE (cid:34) G (cid:32) t nk , ˆ Y ,t nk Y t nk , β t nk (cid:33) ∧ V n (cid:32) t nk , ˆ Y ,t nk Y t nk , β t nk (cid:33) (cid:12)(cid:12)(cid:12) ˆ Y ,t nk − Y t nk − = a, β t nk − = j (cid:35) ,k = 1 , , . . . , n , where we applied (3.7), the Markov property of ( ˆ Y ,t /Y t , β t ) t ∈ [0 ,T ] andthe relation V n ( T, a, j ) = G ( T, a, j ). (cid:3) We close this section with the proof of the two bounds used for (3.5) above.(a) Letting ˇ Y t,T := min t ≤ v ≤ T Y v , we check that, for any stopping time τ and a ≥
1, wehave the boundmax (cid:32) aY (cid:100) t (cid:101) n Y (cid:100) τ ∨ ( t + δ n ) (cid:101) n , ˆ Y (cid:100) t (cid:101) n ,T Y (cid:100) τ ∨ ( t + δ n ) (cid:101) n (cid:33) ≤ a ˆ Y t,T Y (cid:100) τ ∨ ( t + δ n ) (cid:101) n ≤ a ˆ Y t,T ˇ Y t,T , (3.8)in which the right hand side is integrable for all t ∈ [0 , T − δ n ].(b) On the other hand we have IE (cid:104) ˆ Y t,T / ˇ Y t,T (cid:12)(cid:12)(cid:12) β t = j (cid:105) < ∞ since, using the driftedBrownian motion ( ˜ B v ) v ∈ [0 ,T ] defined in (3.2) we have, using the Cauchy-Schwarzinequality,IE (cid:34) ˆ Y t,T ˇ Y t,T (cid:12)(cid:12)(cid:12) β t = j (cid:35) = IE (cid:20) e sup t ≤ r ≤ T (cid:82) rt σ ( β v ) d ˜ B v − inf t ≤ r ≤ T (cid:82) rt σ ( β v ) d ˜ B v (cid:12)(cid:12)(cid:12) β t = j (cid:21) (3.9) ≤ (cid:115) IE (cid:20) e t ≤ r ≤ T (cid:82) rt σ ( β v ) d ˜ B v (cid:12)(cid:12)(cid:12) β t = j (cid:21) IE (cid:20) e − t ≤ r ≤ T (cid:82) rt σ ( β v ) d ˜ B v (cid:12)(cid:12)(cid:12) β t = j (cid:21) ≤ (cid:115) IE (cid:20) e t ≤ r ≤ T (cid:82) rt σ ( β v ) d ˜ B v (cid:12)(cid:12)(cid:12) β t = j (cid:21) < ∞ , where we conclude to finiteness by conditioning and use of the density (2.4).10 Proof of Theorem 2.2
We start with two lemmas.
Lemma 4.1
For all k = 1 , , . . . , n , j ∈ M and a ≥ , we have G ( t nk − , a, j ) = (4.1) e q j,j δ n (cid:90) ∞ (cid:90) y −∞ e ( u ( j )+ σ ( j )) x − u ( j ) T/ (2 n ) G ( t nk , a ∨ ( σ ( j ) y ) − σ ( j ) x, j ) ϕ δ n ( x, y ) dxdy + m (cid:88) i =1 i (cid:54) = j q j,i (cid:90) δ n e q j,j r (cid:90) ∞ (cid:90) y −∞ e ( u ( j )+ σ ( j )) x − u ( j ) r/ G (cid:0) t nk − + r, a ∨ ( σ ( j ) y ) − σ ( j ) x, i (cid:1) ϕ r ( x, y ) dxdydr. Proof.
Let ˜ P denote the probability measure defined by d ˜ P d P := exp (cid:18) − (cid:90) T u ( β r ) dB r − (cid:90) T u ( β r ) dr (cid:19) , where u ( j ), j ∈ M , is defined in (2.5), and ( ˜ B r ) r ∈ [0 ,T ] is the standard Brownian motionunder ˜ P defined in (3.2). From the definition (1.6) of G ( t, a, j ) we have G ( t, a, j ) = IE (cid:34) a ∨ exp (cid:18) sup t ≤ s ≤ T (cid:90) st σ ( β r ) d ˜ B r (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β t = j (cid:35) (4.2)= ˜IE (cid:20) e (log a ) ∨ sup t ≤ s ≤ T (cid:82) st σ ( β r ) d ˜ B r + (cid:82) Tt u ( β r ) dB r + (cid:82) Tt u ( β r ) dr (cid:12)(cid:12)(cid:12) β t = j (cid:21) = ˜IE (cid:20) e (log a ) ∨ sup t ≤ s ≤ T (cid:82) st σ ( β r ) d ˜ B r + (cid:82) Tt u ( β r ) d ˜ B r − (cid:82) Tt u ( β r ) dr (cid:12)(cid:12)(cid:12) β t = j (cid:21) = IE (cid:20) e (log a ) ∨ sup t ≤ s ≤ T (cid:82) st σ ( β r ) dB r + (cid:82) Tt u ( β r ) dB r − (cid:82) Tt u ( β r ) dr (cid:12)(cid:12)(cid:12) β t = j (cid:21) , (4.3)which allows us to remove the drift component in the supremum sup t ≤ s ≤ T (cid:82) st σ ( β r ) dB r .Next, using (4.3) we write G ( t nk , a, j ) = Φ n ( t nk , a, j ) + Υ n ( t nk , a, j ) , j ∈ M , a ≥ , (4.4)whereΦ n ( t nk , a, j ) := IE (cid:34) e (log a ) ∨ sup tnk ≤ s ≤ T (cid:82) stnk σ ( β r ) dB r + (cid:82) Ttnk u ( β r ) dB r − (cid:82) Ttnk u ( β r ) dr { T ( t nk ) >t nk +1 } (cid:12)(cid:12)(cid:12) β t nk = j (cid:35) , (4.5)11ith T ( t ) := inf { s ≥ t : β s (cid:54) = β t } for any t ∈ IR + , andΥ n ( t nk , a, j ) := IE (cid:34) e (log a ) ∨ sup tnk ≤ s ≤ T (cid:82) stnk σ ( β r ) dB r + (cid:82) Ttnk u ( β r ) dB r − (cid:82) Ttnk u ( β r ) dr { T ( t nk ) ≤ t nk +1 } (cid:12)(cid:12)(cid:12) β t nk = j (cid:35) . (4.6)By (4.5) we have, for any k = 0 , , . . . , n − j ∈ M , and a ≥ n ( t nk , a, j ) = e q j,j δ n (cid:90) ∞ (cid:90) y −∞ IE e (log a ) ∨ σ ( j ) y ∨ σ ( j ) x + sup tnk +1 ≤ s ≤ T (cid:82) stnk +1 σ ( β r ) dB r + (cid:82) Ttnk +1 u ( β r ) dB r + u ( j ) x − (cid:82) Ttnk +1 u ( β r ) dr − u ( j ) δ n / × ϕ δ n ( x, y ) dxdy = e q j,j δ n (cid:90) ∞ (cid:90) y −∞ IE (cid:34) e (((log a ) ∨ ( σ ( j ) y ) − σ ( j ) x ) ∨ sup tnk +1 ≤ s ≤ T (cid:82) stnk +1 σ ( β r ) dB r + σ ( j ) x + (cid:82) Ttnk +1 u ( β r ) dB r + u ( j ) x − (cid:82) Ttnk +1 u ( β r ) dr − u ( j ) δ n / (cid:35) × ϕ δ n ( x, y ) dxdy = e q j,j δ n (cid:90) ∞ (cid:90) y −∞ e ( u ( j )+ σ ( j )) x − u ( j ) T/ (2 n ) G ( t nk +1 , e (log a ) ∨ ( σ ( j ) y ) − σ ( j ) x , j ) ϕ δ n ( x, y ) dxdy, (4.7)where in the first equality we used the fact that the time to the first jump of ( β s ) s ∈ [ t, ∞ ) after t is exponentially distributed with parameter − q j,j > β t = j , cf. e.g. § n ( t nk , a, j ), k = 0 , , . . . , n − j ∈ M , and a ≥
1, by (4.6)we see thatΥ n ( t nk , a, j ) = m (cid:88) i =1 i (cid:54) = j q j,i (cid:90) δ n e q j,j r (cid:90) ∞ (cid:90) y −∞ (4.8)IE (cid:34) e (log a ) ∨ ( σ ( j ) y ) ∨ ( σ ( j ) x + sup tnk + r ≤ s ≤ T (cid:82) stnk + r σ ( β z ) dB z + (cid:82) Ttnk + r u ( β z ) dB z + u ( j ) x − (cid:82) Ttnk + r u ( β z ) dz − u ( j ) r/ (cid:12)(cid:12)(cid:12) β t nk + r = i (cid:35) ϕ r ( x, y ) dxdydr = m (cid:88) i =1 i (cid:54) = j q j,i (cid:90) δ n e q j,j r (cid:90) ∞ (cid:90) y −∞ e ( u ( j )+ σ ( j )) x − u ( j ) r/ G ( t nk + r, e (log a ) ∨ ( σ ( j ) y ) − σ ( j ) x , i ) ϕ r ( x, y ) dxdydr, where we used the conditional probability distribution P ( T ∈ dt, β T = i | β = j ) = [0 , ∞ ) ( t ) q j,i e q j,j t dt, i (cid:54) = j ∈ M , − q i,i of the first jumptime T of the Markov chain ( β t ) t ∈ IR + started at i ∈ M and the transition matrix( − q i,j { i (cid:54) = j } /q i,i ) i,j ∈M of the embedded Markov chain, cf. e.g. § (cid:3) Lemma 4.2
For any j ∈ M , the function t (cid:55)→ G ( t, a, j ) is uniformly continuous in t ∈ [0 , T ] , uniformly in a ≥ , i.e. lim ε → sup | t − s |≤ ε sup a ≥ | G ( t, a, j ) − G ( s, a, j ) | = 0 , j ∈ M . (4.9) Proof.
By (4.2), for all a ≥ | G ( t, a, j ) − G ( s, a, j ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) IE (cid:34) a ∨ exp (cid:18) sup t ≤ v ≤ T (cid:90) vt σ ( β r ) d ˜ B r (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β t = j (cid:35) − IE (cid:34) a ∨ exp (cid:18) sup s ≤ v ≤ T (cid:90) vs σ ( β r ) d ˜ B r (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β s = j (cid:35) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) IE (cid:34) exp (cid:18) sup t ≤ v ≤ T (cid:90) vt σ ( β r ) d ˜ B r (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β t = j (cid:35) − IE (cid:34) exp (cid:18) sup s ≤ v ≤ T (cid:90) vs σ ( β r ) d ˜ B r (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β s = j (cid:35) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (4.10)hence it suffices to show the continuity in t ∈ [0 , T ] of the above bound. Similarly to(4.3), we haveIE (cid:34) exp (cid:18) sup t ≤ v ≤ T (cid:90) vt σ ( β r ) d ˜ B r (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β t = j (cid:35) = IE (cid:20) e sup t ≤ v ≤ T (cid:82) vt σ ( β r ) dB r + (cid:82) Tt u ( β r ) dB r − (cid:82) Tt u ( β r ) dr (cid:12)(cid:12)(cid:12) β t = j (cid:21) ,t ∈ [0 , T ], j ∈ M . Next, for any n ≥ (cid:100) t (cid:101) n ≤ s ≤ T (cid:90) s (cid:100) t (cid:101) n σ ( β r ) dB r = sup (cid:100) t (cid:101) n ≤ s ≤ T (cid:90) st σ ( β r ) dB r − (cid:90) (cid:100) t (cid:101) n t σ ( β r ) dB r ≤ sup t ≤ s ≤ T (cid:90) st σ ( β r ) dB r − inf t ≤ s ≤ T (cid:90) st σ ( β r ) dB r , and similarly by replacing σ ( β r ) with u ( β r ), thusexp (cid:32) sup (cid:100) t (cid:101) n ≤ s ≤ T (cid:90) s (cid:100) t (cid:101) n σ ( β r ) dB r + (cid:90) T (cid:100) t (cid:101) n u ( β r ) dB r − (cid:90) T (cid:100) t (cid:101) n u ( β r ) dr (cid:33) is upper bounded byexp (cid:18) sup t ≤ s ≤ T (cid:90) st (2 σ ( β r ) + u ( β r )) dB r − inf t ≤ s ≤ T (cid:90) st (2 σ ( β r ) + u ( β r )) dB r (cid:19) , P -integrable as in (3.9). Therefore, by dominated convergence we findlim s (cid:38) t IE (cid:20) e sup s ≤ v ≤ T (cid:82) vs σ ( β r ) dB r + (cid:82) Ts u ( β r ) dB r − (cid:82) Ts u ( β r ) dr (cid:12)(cid:12)(cid:12) β s = j (cid:21) = lim s (cid:38) t m (cid:88) l =1 (cid:2) e ( s − t ) Q (cid:3) j,l IE (cid:20) e sup s ≤ v ≤ T (cid:82) vs σ ( β r ) dB r + (cid:82) Ts u ( β r ) dB r − (cid:82) Ts u ( β r ) dr (cid:12)(cid:12)(cid:12) β s = l (cid:21) = lim s (cid:38) t IE (cid:20) e sup s ≤ v ≤ T (cid:82) vs σ ( β r ) dB r + (cid:82) Ts u ( β r ) dB r − (cid:82) Ts u ( β r ) dr (cid:12)(cid:12)(cid:12) β t = j (cid:21) = IE (cid:20) e sup t ≤ v ≤ T (cid:82) vt σ ( β r ) dB r + (cid:82) Tt u ( β r ) dB r − (cid:82) Tt u ( β r ) dr (cid:12)(cid:12)(cid:12) β t = j (cid:21) and similarly, lim s (cid:37) t IE (cid:20) e sup s ≤ v ≤ T (cid:82) vs σ ( β r ) dB r + (cid:82) Ts u ( β r ) dB r − (cid:82) Ts u ( β r ) dr (cid:12)(cid:12)(cid:12) β s = j (cid:21) (4.11)= IE (cid:20) e sup t ≤ v ≤ T (cid:82) vt σ ( β r ) dB r + (cid:82) Tt u ( β r ) dB r − (cid:82) Tt u ( β r ) dr (cid:12)(cid:12)(cid:12) β t = j (cid:21) . Combining (4.10) and (4.11) we conclude to Lemma 4.2 by a classical uniform conti-nuity argument. (cid:3)
Finally, we proceed to the proof of Theorem 2.2. Let∆ nk := max j ∈M sup a ≥ | G n ( t nk , a, j ) − G ( t nk , a, j ) | , k = 0 , , . . . , n, (4.12)with ∆ nn = 0. By (2.6), (4.1) and (4.12) we have∆ nk − ≤ e q j,j δ n ∆ nk max j ∈M (cid:90) ∞ (cid:90) y −∞ e ( u ( j )+ σ ( j )) x − u ( j ) δ n / ϕ δ n ( x, y ) dxdy (4.13)+ max j ∈M sup a ≥ m (cid:88) i =1 i (cid:54) = j q j,i (cid:90) δ n e q jj r (cid:90) ∞ (cid:90) y −∞ e ( u ( j )+ σ ( j )) x − u ( j ) r/ × | G n ( t nk , a ∨ ( σ ( j ) y ) − σ ( j ) x, i ) − G ( t nk − + r, a ∨ ( σ ( j ) y ) − σ ( j ) x, i ) | ϕ δ n ( x, y ) dxdydr,k = 1 , , . . . , n , where | G n ( t nk , a ∨ ( σ ( j ) y ) − σ ( j ) x, i ) − G ( t nk − + r, a ∨ ( σ ( j ) y ) − σ ( j ) x, i ) |≤ | G n ( t nk , a ∨ ( σ ( j ) y ) − σ ( j ) x, i ) − G ( t nk , a ∨ ( σ ( j ) y ) − σ ( j ) x, i ) | + | G ( t nk , a ∨ ( σ ( j ) y ) − σ ( j ) x, i ) − G ( t nk − + r, a ∨ ( σ ( j ) y ) − σ ( j ) x, i ) | ∆ nk + ε nk − , k = 1 , , . . . , n, a ≥ , with ε nk := max i ∈M sup a ≥ tnk ≤ s
1, hence∆ nk ≤ c (cid:32) δ n n − (cid:88) i = k ε ni (cid:33) , k = 0 , , . . . , n, and max k =0 , ,...,n ∆ nk = max k =0 , ,...,nj ∈M sup a ≥ | G n ( t nk , a, j ) − G ( t nk , a, j ) | ≤ c (cid:18) max k =0 ,...,n − ε nk (cid:19) which tends to 0 as n tends to infinity by (4.9) in Lemma 4.2. Consequently we havelim n →∞ sup a ≥ | G ( (cid:100) t (cid:101) n , a, j ) − G n ( (cid:100) t (cid:101) n , a, j ) | = 0for any 0 ≤ t ≤ T , j ∈ M , and by Lemma 4.2 it follows that G ( t, a, j ) = lim n →∞ G ( (cid:100) t (cid:101) n , a, j ) = lim n →∞ G n ( (cid:100) t (cid:101) n , a, j ) , uniformly in a ≥
1, for all j ∈ M and t ∈ [0 , T ]. (cid:3) Numerical results
In this section we present numerical estimates obtained from Theorems 2.1 and 2.2for the boundary functions b D ( t, j ) := inf { x ∈ [1 , ∞ ) : ( t, x, j ) ∈ D } , t ∈ [0 , T ] , j ∈ M , of the stopping set D defined in (1.7), in the case of two-state Markov chains with M = { , } .( i ) Constant drift.In the absence of regime switching, the recursive algorithm of Theorems 2.1 and 2.2 isapplied in Figure 1 to the computation of the value functions V ( t, a, j ) and G ( t, a, j )with T = 1, σ = 0 . µ = 0 . n = 50, and δ n = T /n = 0 . DV(t,a)G(t,a)b(t) 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8a 0 0.2 0.4 0.6 0.8 1t 1 1.5 2 2.5
Figure 1: Value functions computed from Theorems 2.1 and 2.2.Figure 1 allows us in particular to visualize the stopping set D defined in (1.7) andthe continuation set C = (cid:8) ( t, a ) ∈ [0 , T ] × [1 , ∞ ) : V ( t, a ) < G ( t, a ) (cid:9) .In Figure 2 the recursive method is compared to the solution of the Volterra integralequation (1.3) by dichotomy for the computation of the boundary function b ( t ).16 b ( t ) t recursion algorithmVolterra equation Figure 2: Boundary function computed from Theorems 2.1 and 2.2 vs (1.3).As shown in Figure 2, the recursive and Volterra equation methods yield similar levelsof precision. However, increasing the number n of time steps will make the Volterraequation method perform slower relative to the recursion method, due to the quadraticcomplexity of the former and to the linear complexity of the latter.( ii ) Drifts with switching signs.Figure 3 presents the graphs of the value functions obtained from the recursive algo-rithm of Theorems 2.1 and 2.2 with µ (1) = 0 . µ (2) = − . σ (1) = 0 . σ (2) = 0 . T = 0 . n = 100, δ n = T /n = 0 .
05, and Q = (cid:20) − . . − (cid:21) . DV(t,a,1)G(t,a,1)b D (t,1) 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4a 0 0.1 0.2 0.3 0.4 0.5 t 1 1.1 1.2 1.3 1.4 1.5 DV(t,a,2)G(t,a,2)b D (t,2) 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4a 0 0.1 0.2 0.3 0.4 0.5 t 1 1.1 1.2 1.3 1.4 1.5 Figure 3: Value functions under drifts of mixed signs.Figure 3 also allows us to visualize the stopping set D and the continuation set C = (cid:8) ( t, a, j ) ∈ [0 , T ] × [1 , ∞ ) × M : V ( t, a, j ) < G ( t, a, j ) (cid:9) . V ( t, a, j ) = G ( t, a, j ) when V ( t, a, j ) and G ( t, a, j ) are very close to each other. Weobserve that the corresponding boundary function t (cid:55)→ b D ( t,
1) starting from state 1 isnot monotone. Precisely, when time t is close to 0 it is better to exercise early becauseone may switch to state 2 after the average time 1 /q , = 0 .
4, in which case the drifttakes the negative value µ (2) = − .
2. On the other hand, when t increases up to0 . t (cid:55)→ b D ( t,
1) tends to increase as it makes more sense to wait sincewe may stay at state 1 with µ (1) = 0 . /q , = 0 . T − t . D (t,1)b D (t,2) Figure 4: Boundary functions under drifts of mixed signs.The boundary functions are plotted in Figure 4 with spline smoothing. Starting fromstate 2 we observe the usual decreasing boundary t (cid:55)→ b D ( t, T , since in this case we should exercise immediately as theaverage time 1 /q , = 0 . T − t untilmaturity.Until time 0 . t such that b ( t, < ˆ Y ,t /Y t = a < b D ( t, . b ( t, < ˆ Y ,t /Y t = a < b D ( t, References [1] J. Buffington and R.J. Elliott. American options with regime switching.
Int. J. Theor. Appl.Finance , 5(5):497–514, 2002.
2] J. du Toit and G. Peskir. Selling a stock at the ultimate maximum.
Ann. Appl. Probab. ,19(3):983–1014, 2009.[3] S.E. Graversen, G. Peskir, and A.N. Shiryaev. Stopping Brownian motion without anticipationas close as possible to its ultimate maximum.
Teor. Veroyatnost. i Primenen. , 45(1):125–136,2000.[4] X. Guo. An explicit solution to an optimal stopping problem with regime switching.
J. Appl.Probab. , 38(2):464–481, 2001.[5] J.D. Hamilton. A new approach to the economic analysis of non-stationary time series.
Econo-metrica , 57:357–384, 1989.[6] H. Le and C. Wang. A finite time horizon optimal stopping problem with regime switching.
SIAM J. Control Optim. , 48(8):5193–5213, 2010.[7] Y. Liu and N. Privault. Selling at the ultimate maximum in a regime switching model. PreprintarXiv:1508.06770v2, 2016.[8] G. Peskir and F. Samee. The British put option.
Appl. Math. Finance , 18(6):537–563, 2011.[9] G. Peskir and F. Samee. The British call option.
Quant. Finance , 13(1):95–109, 2013.[10] G. Peskir and A. Shiryaev.
Optimal stopping and free-boundary problems . Lectures in Mathe-matics ETH Z¨urich. Birkh¨auser Verlag, Basel, 2006.[11] N. Privault.
Understanding Markov Chains - Examples and applications . Springer, 2013.x+354 pp.[12] D.D. Yao, Q. Zhang, and X.Y. Zhou. A regime-switching model for European options.
Inter-national Series in Operation Research and Management Science , 94:281–300, 2006., 94:281–300, 2006.