International Journal of Financial Studies
Article Optimal Timing to Trade along a Randomized Brownian Bridge
Tim Leung 1,* ID , Jiao Li 2 and Xin Li 3
1 Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA 2 APAM Department, Columbia University, New York, NY 10027, USA; [email protected] 3 Bank of America Merrill Lynch, One Bryant Park, New York, NY 10036, USA; [email protected] * Correspondence: [email protected]
Received: 31 December 2017; Accepted: 3 August 2018; Published: 30 August 2018
Abstract: This paper studies an optimal trading problem that incorporates the trader’s market view on the terminal asset price distribution and uninformative noise embedded in the asset price dynamics. We model the underlying asset price evolution by an exponential randomized Brownian bridge (rBb) and consider various prior distributions for the random endpoint. We solve for the optimal strategies to sell a stock, call, or put, and analyze the associated delayed liquidation premia. We solve for the optimal trading strategies numerically and compare them across different prior beliefs. Among our results, we find that disconnected continuation/exercise regions arise when the trader prescribe a two-point discrete distribution and double exponential distribution.
Keywords: speculative trading; Brownian bridge; optimal stopping; variational inequality
MSC: 60G40; 62L15; 91G20; 91G80
JEL Classification: C41; G11; G13
1. Introduction One fundamental problem faced by all traders is to determine when to sell an asset or financial derivative over a given trading horizon. The optimal trading decision depends crucially on the trader’s subjective belief of the distribution of the asset price in the future and the observed price fluctuations. By monitoring the asset price evolution over time, the trader decides whether to sell the asset or derivative at the prevailing market price, or continue to wait until a later time. In this paper, we tackle this problem by constructing models that reflect the two major features: (i) the trader’s market view on the terminal asset price distribution and the uninformative noise embedded in the asset price dynamics, and (ii) the timing option that gives rise to an optimal stopping problem corresponding to the trader’s asset or derivative liquidation. In order to describe the asset price evolution, we present a randomized Brownian bridge (rBb) model, whereby the log-price of the asset follows a Brownian bridge with a randomized endpoint representing the random terminal log-price. In turn, the trader’s prior belief and learning mechanism can be encoded in the drift of the log-price process. Our model is adapted from the novel work by Cartea et al.(2016). They model the asset’s mid-price as a randomized Brownian bridge, and solve a trading problem that maximizes the expected trading revenue by placing either limit or market orders while penalizing the running inventory. In comparison, the underlying asset price in our model is an exponential rBb with different randomized endpoints, with examples including discrete, normal, and double exponential distribution. In addition, we introduce an optimal stopping approach to the trader’s
Int. J. Financial Stud. 2018, 6, 75; doi:10.3390/ijfs6030075 www.mdpi.com/journal/ijfs Int. J. Financial Stud. 2018, 6, 75 2 of 23 liquidation problem, which is applicable not only to selling the underlying asset, but also options written on it. To solve the trader’s optimal stopping problem, we devise and apply a number of analytical tools. We first define the optimal liquidation premium that represents the additional value from optimally waiting to sell, as opposed to immediate liquidation. Indeed, as soon as this premium vanishes, it is optimal for the trader to sell. On the other hand, if this premium is always strictly positive, then the trader finds it optimal to wait through maturity. Therefore, the optimal liquidation premium provides not only new financial interpretations, but also another avenue of analytical investigation of the optimal stopping problem. Among our results, we identify the conditions under which it is optimal to immediately liquidate, or hold the asset/option position through expiration. Furthermore, we prove that the optimal strategy to liquidate a long-call–short-put position is identical to the optimal strategy to sell the underlying stock under the same trading horizon. This timing parity holds for any distribution of the randomized endpoint. Moreover, we derive the variational inequality associated with the optimal stopping problem, and present a finite-difference method to solve for the optimal trading boundaries. In the literature, Brownian bridges have been used to represent market uncertainty or uninformative noise (see e.g., Brody et al.(2009), Hughston and Macrina(2012), and Macrina(2014)). The randomized Brownian bridge model in this paper belongs to the information-based approach to pricing and trading. Among related studies, Brody et al.(2008) and Filipovi´cet al.(2012) study information-based models where the asset prices are computed via conditional expectation with respect to an information process. Randomized Brownian bridges, or their variations, have great potential applicability in a number of finanical applications. For example, while futures prices are supposed to be equal to the spot price in theory, it is observed that some commodity futures prices do not exactly converge to the corresponding spot prices upon maturity (see Guo and Leung(2017)). Brennan and Schwartz(1990) and Dai et al.(2011) investigate arbitrage strategies on stock index futures where the index arbitrage basis is assumed to follow a Brownian Bridge. One can also look for more potential applications in index tracking and exchange-traded funds (see Leung and Santoli(2016)). The randomized endpoint provides added flexibility in modeling random shocks to the asset price on a future date, which is applicable to events such as Federal Reserve announcements, and earnings surprises (see e.g., Johannes and Dubinsky(2006) and Leung and Santoli(2014)). As for optimal stopping problems involving Brownian bridges, Ekström and Wanntorp(2009) consider optimal single stopping of a Brownian bridge or odd powers of a Brownian bridge, without discounting. Ekström and Vaicenavicius(2017) use an optimal stopping approach to maximize the expected value of a Brownian bridge with an unknown pinning point. The optimal double problem is further studied in Baurdoux et al.(2015), where they maximize the expected spread between the payoffs at the entry and exit times where the underlying follows a Brownian bridge. In related works on optimal stopping problems for securities trading under a finite horizon setting, Leung and Ludkovski(2011, 2012) introduce the concept of delayed purchase premium and analyze the problem of purchasing equity European and American options in an incomplete market, where the investor’s belief on asset price dynamics may differ from prevalent market views. Leung and Liu(2012) analyze the delayed purchase premium associated with credit derivatives under a multi-factor intensity-based default risk framework, and derive the optimal trading strategies. In Leung and Shirai(2015), the authors study the optimal timing to sell an asset or option subject to a path-dependent risk penalty. The rest of the paper is structured as follows. In Section2, we present the randomized Brownian bridge model for the underlying asset price, show how different prior beliefs are encapsulated in the stochastic differential equation corresponding to the price dynamics, and illustrate the price behaviors through Monte Carlo simulation. In Section3, we formulate and analyze the optimal liquidation problem and present numerical results to illustrate the optimal trading strategies. In Section4, Int. J. Financial Stud. 2018, 6, 75 3 of 23 we summarize the numerical algorithm used for solving the optimal stopping problems under different settings in this paper. A number of proofs are collected in Section6.
2. Prior Belief and Price Dynamics
The model consists of a single asset whose positive price process is denoted by (St)0≤t≤T. The trader in our model specifies a prior belief on the future distribution of the stock price at a fixed future time T. We denote Xt = log St to be the log-price of the asset, with X0 being the stock’s initial log-price. The trader’s belief is described by a real-valued random variable D to be realized at time T so that the terminal log-price is given by
XT = X0 + D. (1)
To avoid arbitrage, we require that D have finite second moment and P(D > rT) ∈ (0, 1), where r is positive risk-free interest rate, under the historical probability measure P. As time progresses, new information arrives in the form of changes in asset price. The price fluctuation can also be viewed as noise prior to the realization of the terminal log-price D. We model the log-price process as a randomized Brownian bridge. To this end, we first let (βt)0≤t≤T be a standard Brownian bridge
t β = B − B , t ∈ [0, T], (2) t t T T where (Bt)0≤t≤T is a standard Brownian motion. The process β starts at 0 and ends at 0. It can be viewed as uninformative noise, which vanishes both at times 0 and T. We assume that the processes B and thus β are independent of the random variable D. Then, the log-price process is given by
t X = X + σβ + D, t ∈ [0, T], (3) t 0 t T where σ > 0 is a constant parameter. The trader observes the price evolution of the asset, and the corresponding filtration F ≡ (Ft)0≤t≤T is generated by the log-price process X. In other words, the trader does not directly observe the standard Brownian bridge β over time. Mathematically, this means that βt is not adapted to F. The financial interpretation is that β reflects the uninformative noise in the markets such as market sentiments and rumors. Without direct observation of βt, the trader is unable to decompose Xt into βt and D at any time t < T. At time T, the trader observes the realization of D and thus XT. The parameter σ allows us to control effect of β on the log-price t fluctuation. By inspecting Equation (3), the ratio T can be viewed as the rate at which the value of D is revealed, going linearly from fully hidden at time 0 to fully revealed at time T. By Proposition 1 of Cartea et al.(2016), the log-price process satisfies the stochastic differential equation (SDE)
dXt = A(t, Xt)dt + σdWt, (4) for t ∈ [0, T], where (Wt)0≤t≤T is an F-adapted standard Brownian motion under the probability measure P. The drift term is a(t, X ) − (X − X ) A(t, X ) = t t 0 , (5) t T − t Int. J. Financial Stud. 2018, 6, 75 4 of 23 where R ∞ z z x−X0 − z2 t dF(z) −∞ exp σ2(T−t) 2Tσ2(T−t) a(t, x) := E[D|Xt = x] = , (6) R ∞ z x−X0 − z2 t dF(z) −∞ exp σ2(T−t) 2Tσ2(T−t) and F(·) is the cumulative distribution function of the random variable D. We refer to Appendix A of Cartea et al.(2016) for the derivation of Equation (4). The F-Brownian motion Wt appearing in (4) can be considered as the market information accessible by the trader. Unlike βt, the value of Wt contains real information relevant to the risky asset price. Equations (4)–(6) represent how the price innovations reflect the probability distribution of D and market information flow. The incomplete information in (3) is reformulated as a complete information by projecting the log-price innovations onto the observable filtration. In that sense, the information-based approach is more flexible to add additional interpretation and intuition based on observable price process, as we present in the following sessions of the paper. To understand the conditional expectation in (6), we start with the definition
Z ∞ E[D|Xt = x] = zπt(z)dz, (7) −∞ where πt(z) is the conditional probability density or mass function for the random variable D defined by
d πt(z) = P(D ≤ z|Xt = x). dz Using Bayes formula, the conditional probability density is given by
p(z)ρ (x|D = z) π (z) = Xt , (8) t R ∞ ( ) ( | = ) −∞ p z ρXt x D z dz where p(z) denotes the probability density or mass function for D, and ρXt (x|D = z) denotes the conditional density function for the random variable Xt given D = z. According to (3), for any fixed t ∈ [0, T), Xt given D = z is Gaussian, i.e., t 2 t Xt ∼ N X + z , σ (T − t) . D=z 0 T T
Therefore, we can express the conditional probability density for Xt as ! 1 T(x − X − t z)2 ( | = ) = − 0 T ρXt x D z q exp . t(T−t) 2σ2t(T − t) σ 2π T
Substituting this expression into (8), we have
p(z) z x−X0 − z2 t exp σ2(T−t) 2Tσ2(T−t) πt(z) = . (9) R ∞ p(z) z x−X0 − z2 t dz −∞ exp σ2(T−t) 2Tσ2(T−t)
Finally, Equation (6) follows after we substitute (9) into (7). The asset price is an exponential rBb defined by St = exp(Xt). By Ito’s lemma, we obtain the SDE
σ2 dS = A(t, X ) + S dt + σS dW , 0 ≤ t ≤ T. (10) t t 2 t t t Int. J. Financial Stud. 2018, 6, 75 5 of 23
In this paper, we will consider three different distributions for D: two-point discrete distribution, normal distribution, and double exponential distribution. The first one is discrete while the other two are continuous distributions. Compared to the normal distribution, the double exponential distribution is capable of generating two heavier tails that can be symmetric or asymmetric. Next, let us illustrate the effect of the distribution of D on the asset price dynamics.
Example 1. Suppose that the trader’s prior belief on the future log-price follows a two-point discrete distribution defined by
( δu with probability pu, Dδ = (11) δd with probability pd = 1 − pu, with mean and variance, respectively,
E[Dδ] = δu pu + δd pd, (12) 2 2 Var[Dδ] = δu pu(1 − pu) + δd pd(1 − pd) − 2δuδd pu pd. (13)
Then, the drift term A(t, x) of Xt in (4) is given by
1 δ u(t, x) + δ d(t, x) A(t, x) = u d − (x − X ) , (14) T − t u(t, x) + d(t, x) 0 where x − X0 2 t u(t, x) = pu exp δu − δ , (15) σ2(T − t) u 2Tσ2(T − t)
x − X t d(t, x) = p exp δ 0 − δ2 . (16) d d σ2(T − t) d 2Tσ2(T − t)
A special case of Example1 is when D is a constant δ. Then, the log-price process (4) is simply a Brownian bridge that starts at X0 and ends at X0 + δ. Its SDE is given by
X + δ − X dX = 0 t dt + σdW , 0 ≤ t ≤ T. (17) t T − t t
Example 2. Suppose that the trader’s prior belief on the future log-price follows a normal distribution with 2 mean µ and variance σD, i.e.,