A direct method for solving optimal stopping problems for Lévy processes

Transcription

1 A direct method for solving optimal stopping problems for Lév processes arxiv: v1 [math.pr] 14 Mar 2013 E.J. Baurdoux This version: March 15, 2013 Abstract We propose an alternative approach for solving a number of well-studied optimal stopping problems for Lév processes. Instead of the usual method of guess-and-verif based on martingale properties of the value function, we suggest a more direct method b showing that the general theor of optimal stopping for strong Markov processes together with some elementar observations impl that the stopping set must be of a certain form for the optimal stopping problems we consider. The independence of increments and the strong Markov propert of Lév processes then allow us to use straightforward optimisation over a realvalued parameter to determine this stopping set. We illustrate this approach b appling it to the McKean optimal stopping problem (American put), the Novikov Shiraev optimal stopping problem and the Shepp Shiraev optimal stopping problem (Russian option). Ke words: optimal stopping, Lév processes. MSC 2000 subject classification: 60J75, 60G40, 91B70. 1 Introduction Lév processes have stationar and independent increments and the satisf the strong Markov propert. In this paper we shall not make use of an further properties of Lév processes and instead refer the interested reader to the monographs [8] and [14]. Lév processes form a surprisingl rich class with applications in a wide variet of fields including biolog, insurance and mathematical finance. In the latter, Lév processes have also been popular for studing optimal stopping problems. We consider optimal stopping problems of the form V(x) = supe x [e qt G(X t )]. Here G is a given function called the pa-off function, q > 0 is to be thought of as the discount rate and X is a Lév process defined on a filtered probabilit space (Ω,F,(F t ) t 0,P x ) satisfing Department of Statistics, London School of Economics. Houghton street, London, WC2A 2AE, United Kingdom. e.j.baurdoux@lse.ac.uk 1

2 the natural conditions (see [9]), with P x (X 0 = x) = 1 with E x denoting the related expectation operator. Finall, T denotes the set of all [0, ]-valued stopping times with respect to (F t ) t 0. To exclude trivialities we assume that the paths of X are not monotone. For Brownian motion, and, more generall, diffusions, optimal stopping problems are often solved b finding a solution to the related free-boundar problem (see for example p in [20]). For a general Lév process, this method might not alwas be feasible as the infinitesimal generator is now an integro-differential operator, and, unless further assumptions are made on the jump distribution (see for example [12]) in most cases one resorts to a so-called verification lemma of the following form (with an integrabilit assumption in case G is unbounded). Suppose that τ T and denote V (x) = E x [e qτ G(X τ )]. Then the pair (τ,v ) is a solution to the optimal stopping problem if V (x) G(x) for all x R and if the process {e qt V (X t )} t 0 is a right-continuous supermartingale. The proof of such a verification lemma is straightforward, see for example Lemma 11.1 in [14]. However, in general, it is not obvious how τ should be chosen. For continuous processes, a method based on a change of measure was proposed in [5], see also [7]. This was extended to certain optimal stopping problems for Lév processes with one-sided jumps in [4]. However, this method hinges on the fact that the underling process does not overshoot the boundar of the stopping region. We consider general Lév processes and the approach we propose boils down to the following elementar steps. 1. Based on general theor of optimal stopping together with some elementar observations, show that the stopping set can be parameterised b a real number, sa, and denote the corresponding stopping time τ(). 2. Use the stationarit and independence of the underling process to find an expression for V(x,), the expected paoff corresponding to τ() under P x. 3. Maximise V(x,) over where we are free to choose a convenient value for x. We shall illustrate this method b solving three examples. Firstl, we consider the McKean optimal stopping problem (American put) with G(x) = (K e x ) + with K > 0. Secondl, we stud the so-called Novikov Shiraev optimal stopping problem with G(x) = (x + ) n. Finall, we consider the Shepp Shiraev optimal stopping problem (Russian option). 2 The McKean optimal stopping problem The value function of the McKean optimal stopping problem (or American put when q is the risk less rate) is given b V(x) = supe x [e qτ max(k e Xτ,0)], (2.1) where K > 0 strike price, q > 0 discount rate and X the underling Lév process. This optimal stopping problem was first solved in [16] in the case when the underling is a Brownian motion. Denote X t = inf 0 s t X t and let e q be an independent, exponentiall distributed random variable with parameter q. Also, for R, denote b τ the first passage time of X below, i.e. τ = inf{t > 0 : X t < }. In [17] the random walk proof from [11] was extended to the case of a general Lév process. In [1] an alternative proof was given based on a verification lemma. The solution is as follows. 2

3 Theorem 1. An optimal stopping time for (2.1) is given b τ = inf{t > 0 : X t < }, with exp( ) = KE[e X eq ]. To prove this we shall make use of the following result, which is (6.33) on p.176 of [14]. As the proof is straightforward we include it here for completeness. Lemma 1. For q,β 0 E x [e qτ +βx τ 1{τ < } ] = e βxe[eβx eq 1{Xeq >x}] E[e βx eq ]. Proof. Since 1 {Xeq <} = 1 {τ <e we get b conditioning on F q} τ [ ] [ ] E e βx eq 1{Xeq <} = E e βx eq 1{τ <e q} = E [1 {τ <eq} eβx τ E [e β(x eq X ) ]] τ Fτ. On the event {τ < e q} and given F τ it holds that X eq X τ = inf (X s X τ 0 s e ) q ( = min X τ, inf τ s e q X s = inf τ s e q (X s X τ ) d = inf 0 s e qτ X s, ) X τ and the fact that X τ = X τ. Here X denotes an independent cop of X. Furthermore, due to the lack of memor propert of the exponential distribution (P(e q s > t e q > s) = P(e q > t)) it follows that on {τ < e q } we have inf 0 s eqτ X d s = X eqτ = X eq. Hence [ ] [ ] [ ] E e βx eq 1{Xeq <} = E 1 {τ <e q} EβX τ E e βx eq = E [e qτ +βx τ ] E [ e βx eq ] Proof of Theorem 1. The proof will follow the three steps as set out in the introduction. Classical theor of optimal stopping (see for example Corollar 2.9 in [20]) implies that an optimal stopping time τ (x) exists and is of the form τ (x) = inf{t 0 : X t D} under P x with D = { R : V() = (K e ) + }. Since (K x) + is convex it follows that V(logx) is convex well. As V(logK) > 0 (consider τ = inf{t : X t logk/2}) and q > 0 this implies 3

4 that D = (, ] for some < logk. For < logk denote b V(x,) the expected pa-off corresponding to τ when X 0 = x, i.e. V(x,) := E x [e qτ max(k e X τ,0)]. From Lemma 1 it follows that V(x,) = ) ] E [(KE[e X eq ]e x+x eq 1 {Xeq >x} E[e X eq ] As we are looking for an optimal choice of (which we know is independent of x) we are looking to maximise ) ] E [(E[e X eq ]e X eq 1 X {Ke eq. <e } To maximise this expected value the indicator should be 1 precisel when the random variable in round brackets is positive. This implies that to exp( ) = KE[e X eq ]. Remark 1. For specific classes of Lév processes the expression for x and the value function become more explicit. For example, see [2] and [6] when the Lév process is assumed to have no positive jumps. 3 The Novikov Shiraev optimal stopping problem Next, we consider the pa-off function G(x) = (x + ) ν with ν > 0, i.e. V(x) = supe x [e qτ (X τ + )ν ]. (3.2) with q > 0. We shall assume throughout this section that x ν Π(dx) < (3.3) (1, ) whereπdenotesthelévmeasureofx. Thisconditionissufficient toguaranteethate[x ν e q ] <. The Novikov Shiraev optimal stopping problem was first solved in both a random walk and Lév process case in [19] in which the authors extended their results from the setting ν N in [18]. The proof is based is based on a verification lemma. Note that for a general Lév processes (3.2) was solved in [15] for ν N again using a verification lemma. See also [21]. The solution to this optimal stopping problems is given in terms of the so-called Appell functions. Here we mention some of their properties without proof and refer to [19] for further details. Appell functions can be defined inductivel. For > 0 and s < 0 define Q s () = 1 Γ(s) u s1 0 e u E[e uxeq ] du and let Q 0 () = 1 for an > 1. Then for s (0,ν) we define Q s () via d dx Q s(x) = sq n1 (x) 4

5 and E[Q s (X eq )] = 0. This expectation can be shown to be finite because of assumption (3.3). It follows then that E x [Q s (X eq )] = x s. (3.4) We are now read to state the solution to (3.2) as in [19]. Theorem 2. Let ν > 0 and suppose X is a Lév process satisfing (3.3). Then an optimal stopping time for (3.2) is given b τ + a(ν) = inf{t > 0 : X t > a(ν)} where a(ν) denotes the positive solution to the equation Q ν (x) = 0. Proof. Again, instead of appling a verification lemma we use a direct approach. We invoke the general theor of optimal stopping to conclude that due to assumption (3.3) there exists an optimal stopping time which is given b the first hitting time of the set D = {x R : V(x) = (x + ) ν }. Note that D R + and that D, again due to (3.3). We follow arguments similar to those in [10] to deduce that for > x > 0 (and making explicit the dependence of optimal stopping times on the starting point and using that τ () ma not be optimal under P x ) it holds that V() ( + ) V(x) [ ( E e qτ () (( +Xτ ()) + ) ν ν (x + ) ν ν (x+x τ ()) + ) ν x ν E [ e qτ () ( ((1+X τ ()/) + ) ν ((1+X τ ()/x) + ) ν)] 0. Hence, for an x D we have that V(x) = x ν and thus also that D when > x. Therefore we conclude that D = [a, ) for some a > 0. The strong Markov propert, stationarit and independence of increments and (3.4) now lead to E x [Q ν (X eq)1 {Xeq a} ] = E x[e qτ+ a X ν τ + a 1 {τ + a < } ] (3.5) for an a > 0, where X t = sup 0 s t X s. It suffices now to maximise this over a to find the optimal stopping set D. For this, we refer to Lemma 1 in [19] which states that for each ν > 0 there exists a(ν) such that Q ν (x) 0 for 0 < x < a(ν),q ν (a(ν)) = 0 and Q ν (x) is increasing for x > a(ν). Just as in the case of the McKean optimal stopping problem, maximising (3.5) over a is now straightforward as we should choose a such that the indicator function is equal to 1 onl when X eq is such that Q(X eq ) 0, i.e. we should choose a = a(ν). Remark 2. In [19] the authors also consider the paoff function G(x) = 1 e x+ (see also Exercise 11.2 in [14]). For this paoff function we also readil deduce that for > x > 0 (stopping is not optimal when X t < 0) V()V(x) E[e qτ () (G( +X τ ())G(x+X τ ()))] = E[e qτ () (e (x+x τ ()) + e (+X τ ()) + )] e x e, 5 )]

6 from which it follows that the stopping region is again of the form [x, ). The equivalent of Lemma 1 now is [( ) ] ( ) ] E x [e qτ+ a 1e X τ a + 1 {τ + a < } = E x 1 exeq 1 E[e Xeq {Xeq a} ] from which we deduce immediatel that x = log(e[e Xeq ]). 4 The Shepp Shiraev optimal stopping problem The Shepp Shiraev optimal stopping problem (or Russian option) is given b V(x) = supe[e qτ+(xτ x) ]. (4.6) Here we assume that ψ(1) := log(e[e X 1 ]) < and q > (ψ(1) 0) so that V will be finite. It was proposed and solved first for a Brownian motion in [22]. Later, in [23] an alternative method was described which was based on a change of measure dp 1 dp = e Xtψ(1)t Ft under which (4.6) is transformed into an optimal stopping problem for the one-dimensional strong Markov process Y x t = (x X t )X t as V(x) = supe 1 [e (qψ(1))τ+y x τ ], (4.7) where E 1 denotes the expectation under P 1. Similar to the previous cases we can show that the stopping set (for Y) is of the form [z, ) for some z 0. Indeed, for > x > 0 we get [ V()V(x) E e qτ () ( e X τ () e x X τ () )] e e x. To get an expression for the expected paoff corresponding to stopping times τ x + the method we used earlier does not work now since the reflected process does not have independent increments. Instead we shall consider spectrall negative Lév processes (i.e. those with no positive jumps and the paths of which are not monotone). When X is of bounded variation we shall denote the drift b d. When q d, (4.6) is trivial since in this case qt+(x t x) is a decreasing process and hence stopping immediatel is optimal. Therefore we shall assume q < d when X is of bounded variation. (4.8) Scale functions are ubiquitous when it comes to fluctuation theor for spectrall negative Lév processes. For r 0 define the scale function W (r) (x) on [0, ) as the unique continuous function such that e λx W (r) 1 (x)dx = ψ(λ)r 0 for λ 0 such that ψ(λ) > r. We set W (r) (x) = 0 for x < 0. Furthermore, define Z (r) (x) = 1+r x 0 W(r) ()d. The following result was proved in [3], again using a verification lemma. 6

7 Theorem 3. Let X be a spectrall negative Lév process satisfing (4.8) and let q > ψ(1) 0. The stopping set for (4.6) is given b [x, ) where x is the unique solution to the equation Z (q) (x) = qw (q) (x) and V(x) = e x Z (q) (x x). Proof. Having alread established that the stopping set is of the form [x, ), it suffices now to maximise the the expected paoff corresponding to first passage times Tz x = inf{t > 0 : Y t x > z}. In [3] it was shown that ( ) V(x,z) := E (1) [e qtx z +Y x Tz x ] = e x Z (q) (z x)w (q) (z x) qw(q) (z)z (q) (z). W (q) (z)w (q) (z) Here we have implicitl assumed that the Lév measure of X has no atoms when X is of bounded variation, since otherwise W (q) will be differentiable onl almost everwhere and we would have to resort to left derivatives instead. To maximise over z we are free to choose x = 0 We find f(z) := V(0,z) = Z(q) (x)w (q) (z)q(w (q) (z)) 2. W (q) (z)w (q) (z) For notational convenience we assume that W (q) is twice differentiable on (0, ) (note that from (8.22) and(8.23) in [14] it follows that W (q) (x) is log-concave and hence twice differentiable almost everwhere. In fact, W (q) is a C 2 function on (0, ) when X has a Gaussian component, see [13]). We then find that f (z) = (Z(q) (z)qw (q) (z))(w (q) (x)w (q) (z)w (q) (z)). (W (q) (z)w (q) (z)) 2 Under the assumption (4.8) and q > ψ(1) it holds that g(z) := Z (q) (z) qw (q) (z) satisfies g(0) > 0,g (z) < 0 and g( ) = so there is a unique x such that g(x ) = 0. The logconcavit of W (q) allows us to deduce that f (z) 0 for x < x and f (z) 0 for x > x. It is therefore optimal to choose z = x leading to V(x) = e x Z (q) (x x). References [1] L. Alili and A.E. Kprianou: Some remarks on the first passage of Lév processes, the American put and smooth pasting. Ann. Appl. Probab. 15, (2005). [2] F. Avram, T. Chan and M. Usabel On the valuation of costant barrier options under spectrall one-sided exponential Lév models and Carr s approximation for American puts. Stochast. Process. Appl (2002). [3] F. Avram, A.E. Kprianou and M.R. Pistorius Exit problems for spectrall negative Lév processes and applications to (Canadized) Russian options. Ann. Appl. Probab (2004). [4] E.J. Baurdoux Examples of optimal stopping via measure transformation for processes with one-sided jumps. Stochastics (2007) 7

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