On predicting the ultimate maximum for exponential Lévy processes

We consider a problem of predicting of the ultimate maximum of the process over a finite interval of time. Mathematically, this problem relates to a particular optimal stopping problem. In this paper we discuss exponential Lévy processes. As the Lévy processes, we discuss α-stable Lévy processes, 0 < α ≤ 2, and generalized hyperbolic Lévy processes. The method of solution uses the representations of these processes as time-changed Brownian motions with drift. Our results generalize results of papers [10] and [24].


Introduction
Throughout this paper we consider optimal stopping problems where process S = (S t ) t≤T , T < ∞, is an exponential Lévy process S t = e Ht .
These two problems were discussed primarily in papers [10] and [24] in connection with a problem of optimal stock liquidation.In both papers it is supposed that the stock price S is evaluated as a geometric Brownian motion, dS t = rS t dt + σS t dB t , S 0 = 1, t ≤ T, where B = (B t ) t≤T is a Brownian motion.In [24], the authors consider problem (1.1) in cases r ≥ σ 2 /2 and r ≤ 0. In the first case, the solution of (1.1), i.e. a stopping time 0 ≤ τ * ≤ T such that is τ * = T , and the optimal liquidation strategy for the seller of the asset is "buy and hold".If r ≤ 0, then τ * = 0 and the optimal strategy is "stop immediately".The authors of [10] discuss the case 0 < r < σ 2 /2 in (1.1) and prove that the solution is τ * = 0 there.Moreover, they solve (1.1) independently when r ∈ [σ 2 /2, σ 2 ) and consider problem (1.2) for all ratios between r and σ deriving that its solution τ * = T if r ≥ σ 2 , τ * = 0 if r ≤ 0 and proving that there exists an increasing boundary function which determines the optimal stopping time if 0 < r < σ 2 .
Concerning other works at the same direction, let the authors mention paper [12], where the problem of minimizing of square-quadratic error between a Brownian motion and its ultimate maximum is discussed, papers [4] and [13] in which for the logarithmic utility function and a Brownian motion with randomly changing drift the ultimate maximum of the process is detected, [9], where geometric and arithmetic averages are discussed instead of the maximum in problems (1.1) and (1.2), and works [7] and [8], in which authors solve an infinite time horizon problem of stopping as close as possible to the zero hitting time considering a mean-reverting diffusion process.
As we mentioned above, instead of a geometric Brownian motion, in this paper we discuss exponential Lévy processes, which are very popular as a model of dynamics of assets in mathematical finance (among others, see, for example, recent papers [1], [5], [15], [26] on pricing and hedging theory).Our results correspond to the exponentials of the Lévy processes, both problems (1.1) and (1.2), logarithmic and linear utilities.On empirical tests which support a suggestion that log-returns of financial assets have α-stable or generalized hyperbolic distributions we refer to papers [11], [17], [20].
The paper is constructed as follows.Section 2 is dedicated to α-stable Lévy processes H with drift and problem (1.1) is solved in Section 2 in case of positive drift.All 0 < α ≤ 2 are discussed.Our result extends a result of [24], see Remark 2.1 and Theorem 2.3.In Section 3, we consider a time-changed Brownian motion.The results give full solution of problem (1.1) and extend results on (1.2) (when the optimal stopping time is 0 or T ) which are obtained by [10] for geometric Brownian motion.Proofs are set in Section 4. In comparison with a recent work [16], the proofs do not depend on the particular change of time.The proof of Theorem 2.3 is based on well-known timechanged representation of α-stable Lévy processes and properties of an extra function introduced in the proof of Theorem 2.1 from [24].The proof of Theorem 3.2 is simpler and straightly exploits the results of [10] and [24].The paper is completed by a list of literature.
If X (γ) is a positive random variable with Laplace transform where T (t) an α/2 -stable subordinator with Law( T (1)) = Law(X (α/2) ). (2.3) Remark 2.1.The proof of decomposition (2.2) is given e.g. in [23] and consists in straight calculation of the characteristic function of B T (t) which appears to be equivalent to (2.1).
Throughout this section, we model the price process of the asset S by the exponential Lévy process of the symmetric α-stable Lévy process with drift, i.e.
Such defined process H can be used as a model of evolution of log-returns of stock prices, see [20].
Remark 2.2.Keeping in mind studying of work [24], one could observe that a geometric Brownian motion with σ = 1 is S t = e Ht , where Setting µ = r − 1 2 , we are able to include the 2-stable symmetric Lévy process, a Brownian motion, in framework of (2.4) with Z t = B t .
Recalling proofs of results for a geometric Brownian motion ( [10] and [24]), one could observe that the proofs are based on exploiting the closed form expression of the density of the maximum of the Brownian motion.We are able to obtain the next result on the exponentials of the stable Lévy processes without knowledge of the distribution of their maxima.Theorem 2.3.Assume that H is an α-stable symmetric Lévy process with 0 < α ≤ 2 and drift µ.If µ ≥ 0, the solution of (1.1) is time T .Example 2.4.If α = 2, the 2-stable symmetric Lévy process is a Brownian motion B = (B t ) t≤T .As it is mentioned above, if the price of the asset is supposed to be a geometric Brownian motion, i.e.

Time-changed Brownian motion
Let H = (H t ) t≤T be a time-changed Brownian motion with drift, i.e.
Remark 3.1.Such defined time-changed Brownian motion determines a pure discontinuous Lévy process in many cases.This fact is obtained by straightforward calculation of its characteristic function for a particular change of time, see paper [2] and monograph [23] on generalized hyperbolic processes, and examples below.The proof of the next theorem is based on (3.1) and the results of [10] and [24].
The density function where K 1 is modified Bessel function of the second type.The NIG process (H t ) t≥0 is defined as a Lévy process such that H 1 has density (3.2).It is known, see for details [23], that for a Brownian motion B = ( Bs ) s≥0 , a change of time and an independent Brownian motion B = (B t ) t≥0 process H t can be represented in form Therefore, solutions of (1.1) and (1.2) for a NIG process do not depend on parameters α and δ, due to Theorem 3.2.
Despite the fact that parameters β ∈ R, σ > 0 and ν reflect only indirectly such parameters of the VG distribution as variance, skewness and kurtosis (it can be shown by straightforward calculation of moments of Y ), we immediately use such parametrization of the VG process as above since it is usually used in literature, see [14], [17], [19]).
Together with the NIG process, the VG process is often used as a model of distribution of market returns.The symmetric VG distribution was primarily studied in [17] and [18].In [19], the general case of VG process with application to option pricing was discussed.For further investigations on the VG process, see [14] and [25].
Solutions of (1.1) and (1.2) for a VG process are determined by conditions of Theorem 3.2 on β and σ.
Remark 3.5.In case of logarithmic utility, problems (1.1) and (1.2) can be rewritten as respectively, with q = 1.For q = 2 these problems were discussed in [12] for a Brownian motion.Their result was extended to all q > 0 by [21].For q = 1 and a Brownian motion with spontaneously changing drift, see [4] and [13].
Assume that H is a Lévy process which has decomposition where µ ∈ R, β ∈ R, σ > 0 and stochastic change of time (in the sense of definition (a)-(b), p.109, [22]) ϕ satisfies condition Since H is a Lévy process (3.3), it is submartingale if EH t ≥ 0 and it is supermartingale if EH t ≤ 0. Keeping in mind Hunt's stopping time theorem ((A.2), p.60, [22]) and Wald identity ((3.2.5), p.61, [22]), we conclude that solution of both problems (1.1) and (1.2) for logarithmic utility here is and In particular, for the VG process the solutions are time T if µ ≥ −β and time 0 if µ ≤ −β.

Proofs
Proof of Theorem 2.3.Set for t ≥ 0 It can be proved in the way of the proof of Proposition 5.1 from [24] that at the same and hence Keeping in mind arbitrariness of s and x, we conclude from (4.6) that and taking into account (4.2) and (4.5), we conclude that τ * = T is optimal.
Remark 4.1.One can easily observe that the method of the proof of Theorem 3.2 is not suitable for Theorem 2.3.Indeed, it tends to problem (1.1) for Brownian motion with time-dependent drift instead of Brownian motion with constant drift, and there are no results in the way of results of papers [10] and [24] for it.

S
u = e Ht .