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A Class of Itô Diffusions with Known Terminal Value and Specified Optimal Barrier

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A Class of Itô Diffusions with Known Terminal Value and Specified Optimal Barrier mathematics Article A Class of Itô Diffusions with Known Terminal Value and Specified Optimal Barrier Bernardo D’Auria 1,2,∗ and Alessandro Ferriero 3,4 1 Department of Statistics, Madrid University Carlos III (UC3M), Avenida de la Universidad 30, 28911 Leganés (Madrid), Spain 2 UC3M-BS Institute of Financial Big Data (IFiBiD), Calle Madrid 135, 28903 Getafe (Madrid), Spain 3 Department of Mathematics, The Autonomous University of Madrid (UAM), Campus de Cantoblanco, 28049 Madrid, Spain; [email protected] 4 Institute of Mathematical Sciences (ICMAT), Campus de Cantoblanco, 28049 Madrid, Spain * Correspondence: [email protected] Received: 13 November 2019; Accepted: 8 January 2020; Published: 14 January 2020 Abstract: In this paper, we study the optimal stopping-time problems related to a class of Itô diffusions, modeling for example an investment gain, for which the terminal value is a priori known. This could be the case of an insider trading or of the pinning at expiration of stock options. We give the explicit solution to these optimization problems and in particular we provide a class of processes whose optimal barrier has the same form as the one of the Brownian bridge. These processes may be a possible alternative to the Brownian bridge in practice as they could better model real applications. Moreover, we discuss the existence of a process with a prescribed curve as optimal barrier, for any given (decreasing) curve. This gives a modeling approach for the optimal liquidation time, i.e., the optimal time at which the investor should liquidate a position to maximize the gain. Keywords: Hamilton–Jacobi–Bellman equation; optimal stopping time; Brownian bridge; liquidation strategy MSC: 60G40; 60H30; 91B26 1. Introduction In this paper, we study the optimal stopping-time problems related to an Itô diffusion, fXsg, for time s in [t, 1], modeling for example an investment gain, for which the terminal value, say at time s = 1, is a priori known. This could be the case of an insider trading [1–3] or of the pinning at expiration of stock options [4–7]. Roughly speaking, the class of stochastic processes subject of our study is defined by bringing the infinite horizon mean-reverting Ornstein–Uhlenbeck process with constant parameters to a finite horizon. We solve and provide explicit solutions to the optimal stopping-time problems associated with this class, which contains as particular case but it is not limited to, the optimal stopping time associated with the Brownian bridge X dX = − s ds + dB , s 2 [t, 1] , with X = x . s 1 − s s t In this case, it is known (see [8,9]) that the optimal stopping time of supfE[XtjXt = x] : t stopping time in [t, 1]g =: V(x, t) Mathematics 2020, 8, 123; doi:10.3390/math8010123 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 123 2 of 14 is the one given by p t∗ := inffs 2 [t, 1] : Xs ≥ b∗ 1 − sg , ≈ for an appropriatep constant b∗ 0.84. In other words, if the stock price is equal to the optimal barrier b∗ 1 − t at time t then Xt = V(Xt, t) and thus the stock price is equal to the maximum gain in expectation. We also present different processes whose optimal barrier has the same form as the optimal barrier of the Brownian bridge. They represent a catalogue of possible alternatives to the Brownian bridge which in practice could offer a better fitting to real applications. Moreover, we discuss the existence of a process with a prescribed curve as optimal barrier, for any given (decreasing) curve. This gives a modeling approach for the optimal liquidation time, i.e., the optimal time at which the investor should liquidate a position to maximize the gain. More precisely, an investor takes short/long positions in the financial market based on her view about the future economy; for example, the real estate price is believed to increase in the long term, the inflation is believed to increase in the short term, the value of a certain company is believed to reduce in the medium term, etc. If Xs is the market price at a time s of a product of the market which is the object of the investor’s view, the investor’s view can be modeled as a deterministic function g(s) which represents the “right” expected value at time s, assigned by the investor to the product at the time t < s when the position needs to be taken. One may think of g as the short/medium/long period view. Of course, if the evolution of the economy induces the investor to distrust her initial view, the position is liquidated. Otherwise, if the investor maintains her view over time, we are interested in answering the following question: what is the optimal time at which the investor should liquidate her position to maximize the gain? Similar problems have been studied in the recent literature. As mentioned at the beginning, the optimal stopping time associated with the case in which fXsg evolves like a Brownian bridge 2 [ ] (finite horizon, i.e., s 0, 1 ) was originallyp investigated in [8] and later with a different approach in [9], and its optimal barrier is equal to b∗ 1 − s. In [10], the authors study the double optimal stopping-time problem, i.e., a pair of stopping times t1 < t2 such that the expected spread between the payoffs at t1 and t2 is maximized, still associated with the case in which fXsg evolves like a Brownian bridge. The two optimal barriersp which define the two optimalp stopping times are found to have the same shape as in [9], i.e., b1 1 − s, b1 ≈ −0.56, and b2 1 − s, b2 = b∗. In [11], the authors study the optimal stopping problem when fXsg evolves like a mean-reverting Ornstein–Uhlenbeck process (infinite horizon, i.e., s 2 [0, ¥)), but in the presence of a stop-loss barrier, i.e., a level B < 0 such that the position is liquidated as Xs ≤ B. Furthermore, they study the dependency of the optimal barrier (which is a level b > 0 in this case) to the parameters of the Ornstein–Uhlenbeck process and to the stop-loss level B. The contribution of this paper consists of giving solutions for optimal stopping problems related to a class of non-homogeneous diffusion processes. Generally, such solutions are not easy to obtain in explicit form, and having them is useful in modeling procedures. The paper is organized as follows. In Section2, we introduce the class of processes studied in our work. In Sections3 and4, we find the optimal barrier by using the Hamilton–Jacobi–Bellman equations and prove the optimality. In Section5, we give a simple application and, in Section6, we discuss the results contained in the paper. 2. The Formulation of the Problem Our problem can be formulated as follows. Let us consider a mean-reverting Ornstein–Uhlenbeck process with constant coefficients q ≥ 0, s > 0: dZt = −qZtdt + sdBt , t > 0 , Mathematics 2020, 8, 123 3 of 14 where fBtg is the standard Brownian motion. The mean of Zt converges to 0 and its variance converges to s2/(2q), as t goes to ¥, which is infinite if q = 0. We want to adapt this model, which has infinite time horizon, to a finite time horizon. Without loss of generality, we assume the final time to be t = 1. We define the function b(s) := g(s) − g(1) , s 2 [0, 1] , (1) that will be used below to make the time-change, where g is the function that carries the view of the future in the model. The function b is assumed to be a non-negative decreasing continuous function, therefore differentiable almost everywhere. We map s in [0, 1] to t in [0, ¥] by t = − ln[b(s)/b(0)], we define Xs := b(s)Z− ln[b(s)/b(0)] + g(1) and we rewrite the Ornstein–Uhlenbeck process as X − g(1) X − g(1) d s = −q s d[− ln b(s)] + sd[B ] , s 2 (0, 1]. b(s) b(s) − ln b(s) Indeed, one can verify that X − g(1) h i d s = d Z = −qZ d[− ln[b(s)/b(0)]] + sd[B ] b(s) − ln[b(s)/b(0)] − ln[b(s)/b(0)] − ln[b(s)/b(0)] X − g(1) = −q s d[− ln b(s) + ln b(0)] + sd[B − B ] b(s) − ln b(s) − ln b(0) X − g(1) = −q s d[− ln b(s)] + sd[B ]. b(s) − ln b(s) As, by Lemma1, X1 = g(1), the quantity Z− ln[b(s)/b(0)] = [Xs − g(1)]/b(s) represents the deviation of the market price to the final value in proportion to the deviation of the “right” value to the final value, at time s. As X − g(1) dX b0(s) d s = s − (X − g(1)) b(s) b(s) s b2(s) p 0 2 p 2 and d[B− ln b(s)] = [− ln b(s)] dBs, if we relabel the parameters by q := a and s := 2/b , we have that s 0 0 s 0 dXs b (s) 2 Xs − g(1) b (s) 2 −b (s) − (Xs − g(1)) = a ds + dBs. b(s) b2(s) b(s) b(s) b2 b(s) By multiplying by b(s) the two terms, we then obtain the equation for fXsg: 0 s 0 2 b (s) −2b (s)b(s) dXs = (1 + a )(Xs − g(1)) ds + dBs, s 2 (t, 1), Xt = x, (2) b(s) b2 starting at any time t in (0, 1).
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