Block Trading: Building up a Stock Position Under a Regime Switching Model

Abstract

This work is intended to systematically characterize the problem of block trading. The author characterizes it as a constrained optimal purchasing (or control) problem, on behalf of the vendee. The purpose of this work is to investigate the optimal purchasing strategies and give quantitative reference information on the pricing of block trade. Noting that large block of stock trading may result in the price fluctuation and different market environment may lead to distinct investing strategies, the controlled diffusion model with regime switching is adopted, which can be used to represent distinct environment of financial market, and the model is modified to allow the price impact, by allowing the drift depend on the purchase rate. Where the switching regimes can be completely observable or unobservable (hidden). Which is a significant difference in contrast to most market models in the existing literature. The objective is to maximize the total discounted shares of the underlying stock, subject to the expected fund for building up corresponding position with an upper bound. This work mainly focuses on the completely observable case, as to the case with hidden regimes, this work just briefly talk about how to reduce the unobservable model to a completely observable one. In the completely observable case, to solve this constrained control problem, the Lagrange multiplier methods and the dynamic programming approaches are used. Though optimality is obtained, what is still lacking is an explicit solution to the optimality equation. To overcome this difficulty, the two loop approximation scheme is given. For fixed Lagrange multiplier, the inner loop is to obtain optimal strategies, which is based mainly on the finite difference approximation. While, the outer loop is a stochastic recursive approximation algorithm to get the optimal Lagrange multiplier. Convergence results are obtained for both the inner and the outer approximations. The quantitative pricing information is related to the threshold levels of the optimal purchasing strategy. Numerical examples are also provided to illustrate our results. Finally, as to the unobservable case, by using the well known Wonham filter approach, one can reduce the unobservable case to a completely observable one.

Appendix

Definition A.1

For each $i\in \mathcal {M}$ and fixed λ, V (x,i,λ) ∈ C([0,∞)) (the space of all continuous functions on [0,∞)) is a viscosity solution of (3.4), if,

• (a) V (x,i,λ) is a viscosity supersolution of (3.4) on [0,∞); i.e.,

  $$\rho \varphi(x_{0})\geq\sup_{u\in\mathbb{U}}\left\{\mathcal{L}^{u}\varphi(x_{0})+a(x_{0},u,\lambda)\right\}, $$

  holds for all φ(⋅) ∈ C²([0,∞)) with C²([0,∞)) as the family of all functions on [0,∞) which are twice conotinuously differentiable with respect to x, and all x₀ ∈ S such that V (x,i,λ) − φ(x) has a local minimum at x₀.

• (b) and V (x,i,λ) is also a viscosity subsolution of (3.4) on [0,∞); i.e.,

  $$\rho \varphi(x_{0})\leq\sup_{u\in\mathbb{U}}\left\{\mathcal{L}^{u}\varphi(x_{0})+a(x_{0},u,\lambda)\right\}, $$

  for all φ(⋅) ∈ C²([0,∞)), and x₀ ∈ S such that V (x,i,λ) − φ(x) has a local maximum at x₀.

The viscosity solution method will see its application in characterizing the convergence property of the approximation structure later in Section 4.

Definition A.2

A sequence {π_m}⊆π is said to be weakly convergent to π ∈π, if for any square integrable function f(x,i) defined on S, and bounded continuous function g(x,i,u) defined on $S\times \mathbb {U}$,

$$\lim_{m\rightarrow\infty}{\int}_{0}^{\infty}f(x,i){\int}_{\mathbb{U}}g(x,i,u)\pi_{m}(du|x,i)dx ={\int}_{0}^{\infty}f(x,i){\int}_{\mathbb{U}}g(x,i,u)\pi(du|x,i)dx, $$

for each $i\in \mathcal {M}$.

Proof of Theorem 3.4

The idea of the proof is similar to the corresponding part in Pemy et al. (2008). For any stopping time τ, the dynamic programming principle implies that for each $i\in \mathcal {M}$,

$$ V(x,i,\lambda)=\max_{u\cdot\in\mathcal{A}}E\left\{{\int}_{0}^{\tau}e^{-\rho t}a(X(t),u(t),\lambda)dt+e^{-\rho \tau}V(X(\tau,i,\lambda))\right\}. $$

(A.1)

Let $\alpha (0)=i_{0}\in \mathcal {M}$, X(0) = x₀ ∈ [0,∞). Let φ(x) ∈ C²([0,∞)) such that V (x,i₀,λ) − φ(x) has a local minimum at x₀ in N(x₀), a neighborhood of x₀. Without loss of generality we can assume that V (x₀,i₀,λ) − φ(x₀) = 0. Let τ be the first jump time of α ⋅. Moreover, let τ₀ ∈ (0,τ] be a stopping time such that for 0 ≤ t ≤ τ₀, X(t) ∈ N(x₀). Consider the control $u(t)=u\in \mathbb {U}$, for t ∈ [0,τ₀], and u a constant such that $u\cdot \in \mathcal {A}$. Moreover, for t ∈ [0,τ₀], V (X(t),i,λ) − φ(X(t)) ≥ 0.

Given 0 < 𝜖 ≤ τ₀, using (A.1), we have

$$ V(x_0,i_0,\lambda)\geq E{\int}_{0}^{\epsilon}e^{-\rho s}a(X(s),u(s),\lambda)ds+E[e^{-\rho\epsilon}V(X(\epsilon),i_0,\lambda)]. $$

(A.2)

Define

$$ \psi(x,i,\lambda)= \left\{\begin{array}{ll} \varphi(x)+V(x_0,i_0,\lambda)-\varphi(x_0)&\!\!\displaystyle \text{ if \(i=i_{0}\)},\\ V(x,i,\lambda)&\!\!\displaystyle \text{ if \(i\neq i_{0}\)}. \end{array}\right. $$

(A.3)

Using Dynkin’s formula, we have

$$\begin{array}{rl} &\!\!\displaystyle Ee^{-\rho \epsilon}\psi(X(\epsilon),i_{0},\lambda)-\psi(x_{0},i_{0},\lambda)\\ =&\!\!\displaystyle {\int}_{0}^{\epsilon}e^{-\rho s}\left\{\left(\mu(i_{0})X(s)+X(s)\phi(i_{0},u(s))\right)\frac{\partial \varphi(X(s))}{\partial x}+\frac{1}{2}\sigma^{2}(i_{0})X^{2}(s)\frac{\partial^{2}\varphi(X(s))}{\partial x^{2}}\right.\\ &\!\!\left.\displaystyle+Q\psi(X(s),\cdot,\lambda)(i_{0})-\rho\psi(X(s),i_{0},\lambda)\right\}ds. \end{array} $$

Recall that x₀ is a local minimum of V (x,i₀,λ) − φ(x). Then for 0 ≤ t ≤ 𝜖, we have

$$ V(X(t),i_0,\lambda)\geq \varphi(X(t))+V(x_0,i_0,\lambda)-\varphi(x_0)=\psi(X(t),i_0,\lambda). $$

(A.4)

It follows that

$$ \begin{array}{rl} &\!\!\displaystyle Ee^{-\rho \epsilon}V(X(\epsilon),i_{0},\lambda)-V(x_{0},i_{0},\lambda)\\ \geq&\!\!\displaystyle E{\int}_{0}^{\epsilon}e^{-\rho s}\left\{\left(\mu(i_{0})X(s)+X(s)\phi(i_{0},u(s))\right)\frac{\partial \varphi(X(s))}{\partial x}+\frac{1}{2}\sigma^{2}(i_{0})X^{2}(s)\frac{\partial^{2} \varphi(X(s))}{\partial x^{2}}\right.\\ &\!\!\left.\displaystyle+Q\psi(X(s),\cdot,\lambda)(i_{0})-\rho V(X(s),i_{0},\lambda)\right\}ds. \end{array} $$

(A.5)

Moreover, (A.4) implies that

$$Q\psi(X(s),\cdot,\lambda)(i_{0})\geq QV(X(s),\cdot,\lambda)(i_{0}). $$

Then from (A.5) we have

$$ \begin{array}{rl} &\!\!\displaystyle Ee^{-\rho \epsilon}V(X(\epsilon),i_{0},\lambda)-V(x_{0},i_{0},\lambda)\\ \geq&\!\!\displaystyle E{\int}_{0}^{\epsilon}e^{-\rho s}\left\{\left(\mu(i_{0})X(s)+X(s)\phi(i_{0},u(s))\right)\frac{\partial \varphi(X(s))}{\partial x}+\frac{1}{2}\sigma^{2}(i_{0})X^{2}(s)\frac{\partial^{2} \varphi(X(s))}{\partial x^{2}}\right.\\ &\!\!\left.\displaystyle+QV(X(s),\cdot,\lambda)(i_{0})-\rho V(X(s),i_{0},\lambda)\right\}ds. \end{array} $$

(A.6)

Combining (A.2) and (A.6), we obtain

$$\begin{array}{rl} E&\!\!\displaystyle{\int}_{0}^{\epsilon} e^{-\rho s}\left\{\left(\mu(i_{0})X(s)+X(s)\phi(i_{0},u(s))\right)\frac{\partial \varphi(X(s))}{\partial x}+\frac{1}{2}\sigma^{2}(i_{0})X^{2}(s)\frac{\partial \varphi(X(s))}{\partial x^{2}}\right.\\ &\!\!\left.\displaystyle+a(X(s),u(s),\lambda)+ QV(X(s),\cdot,\lambda)(i_{0})-\rho V(X(s),i_{0},\lambda)\right\}ds\leq0 \end{array} $$

By letting 𝜖 → 0 enables us to conclude that

$$ \begin{array}{rl} \max_{u\in \mathbb{U}}\left[\right.&\!\!\displaystyle(\mu(i_0)x_0+\phi(i_0,u))\frac{\partial \varphi(x_0)}{\partial x}+\frac{1}{2}\sigma^{2}x_{0}^2\frac{\partial^2 \varphi(x_0)}{\partial x^2}+a(x_0,u,\lambda)\\ &\!\!\left.\displaystyle +QV(x_0,\cdot,\lambda)(i_0)\right]-\rho V(x_0,i_0,\lambda)\leq0. \end{array} $$

(A.7)

Thus, V (x,i,λ) is a viscosity supersolution.

Next, we show that V (x,i,λ) is viscosity subsolution on S for fixed λ and $i\in \mathcal {M}$. Assume otherwise, for $i_{0}\in \mathcal {M}$ there exists x₀ ∈ [0,∞) and δ > 0 such that for $u\in \mathbb {U}$,

$$ \begin{array}{rl} &\!\!\displaystyle(\mu(i_0)x_0+\phi(i_0,u))\frac{\partial \varphi(x_0)}{\partial x}+\frac{1}{2}\sigma^{2}x_{0}^2\frac{\partial^2 \varphi(x_0)}{\partial x^2}+a(x_0,u,\lambda)\\ &\!\!\displaystyle +QV(x_0,\cdot,\lambda)(i_0)-\rho V(x_0,i_0,\lambda)\leq-\delta \end{array} $$

(A.8)

in the neighborhood of N(x₀), where φ(x) ∈ C²([0,∞)) such that V (x,i₀,λ) − φ(x) attains its maximum at x₀ in N(x₀). Without loss of generality, we can assume that V (x₀,i₀,λ) − φ(x₀) = 0. let u(⋅) be an admissible control. Let τ be the first jump time of the process α(⋅). Let τ₀ ≤ τ be a stopping time such that for 0 ≤ s ≤ τ₀ ≤ τ, X(s) ∈ N(x₀) and V (X(s),i₀,λ) − φ(X(s)) ≤ 0. Then for 0 ≤ 𝜖 ≤ τ₀, we have

$$\begin{array}{rl} L(x_{0},i_{0},u(\cdot),\lambda)\leq&\!\!\displaystyle E{\int}_{0}^{\epsilon}e^{-\rho s}a(X(s),\alpha(s),u(s),\lambda)ds+Ee^{-\rho \epsilon}V(X(\epsilon),i_{0},\lambda)\\ \leq&\!\!\displaystyle E{\int}_{0}^{\epsilon}e^{-\rho s}a(X(s),\alpha(s),u(s),\lambda)ds+Ee^{-\rho \epsilon}\varphi(X(\epsilon)). \end{array} $$

Recall that x₀ is the local maximum of V (x,i₀,λ) − φ(x). Then using (A.3), for 0 ≤ t ≤ 𝜖, we have

$$V(X(t),i_{0},\lambda)\leq\varphi(X(t))+V(x_{0},i_{0},\lambda)-\varphi(x_{0})=\psi(X(t),i_{0},\lambda). $$

Similarly, we have

$$Q\psi(X(s),\cdot,\lambda)(i_{0})\leq QV(X(s),\cdot,\lambda)(i_{0}). $$

Thus we have

$$\begin{array}{rl} L(x_{0},i_{0},u(\cdot),\lambda)\leq& \displaystyle E{\int}_{0}^{\epsilon}e^{-\rho s}\left(-\delta+\rho V(X(s),i_{0},\lambda) - (\mu(i_{0})X(s)+ \phi(i_{0},u(s))X(s))\frac{\partial \varphi(X(s))}{\partial x}\right.\\ &\!\!\left.\displaystyle-\frac{1}{2}\sigma^{2}(i_{0})X^{2}(s)\frac{\partial^{2} \varphi(X(s))}{\partial x^{2}}-QV(X(s),\cdot,\lambda)(i_{0})\right)ds+Ee^{-\rho \epsilon}\varphi(X(\epsilon))\\ \leq&\!\!\displaystyle E{\int}_{0}^{\epsilon}e^{-\rho s}\left(-\delta+\rho \psi(X(s),i_{0},\lambda)-(\mu(i_{0})X(s)+\phi(i_{0},u(s))X(s))\frac{\partial \varphi(X(s))}{\partial x}\right.\\ &\!\!\left.\displaystyle-\frac{1}{2}\sigma^{2}(i_{0})X^{2}(s)\frac{\partial^{2} \varphi(X(s))}{\partial x^{2}}-Q\psi(X(s),\cdot,\lambda)(i_{0})\right)ds+Ee^{-\rho \epsilon}\psi(X(\epsilon),i_{0},\lambda). \end{array} $$

Taking the supremum over all admissible control u(⋅) we have

$$V(x_{0},i_{0},\lambda)\leq -E{\int}_{0}^{\tau}e^{-\rho s}\delta ds+V(x_{0},i_{0},\lambda). $$

This contradicts the fact that δ > 0. Therefore V (x,i₀,λ) is a viscosity subsolution.

Finally the uniqueness of viscosity solution can be obtained as in Jakobsen and Karlsen (2006). □

Let ν_n be a sequence of positive numbers so that

$$\nu_{n}\rightarrow 0 \quad \text{and}\quad \frac{1}{\nu_{n}}\sum\limits_{j=m(t_{n}+t+l\nu_{n})}^{m(t_{n}+t+(l + 1)\nu_{n})-1}\epsilon_{j}/\delta_{j}\rightarrow1, $$

as n →∞. Here l is a positive integer.

Lemma A.3

For eachl ≥ 1 and fixedλ,we have

$$\begin{array}{rl} &\!\!\displaystyle\frac{1}{\nu_{n}}\sum\limits_{j=m(t_{n}+t+l\nu_{n})}^{m(t_{n}+t+(l + 1)\nu_{n})-1} E_{m(t_{n}+t+l\nu_{n})}\zeta_{n}\\ =&\!\!\displaystyle\frac{1}{\nu_{n}}\sum\limits_{j=m(t_{n}+t+l\nu_{n})}^{m(t_{n}+t+(l + 1)\nu_{n})-1} E_{m(t_{n}+t+l\nu_{n})}{\int}_{0}^{\infty}{\int}_{\mathbb{U}}e^{-\rho t}ud\pi^{n}(\lambda,u|X_{n}(t),\alpha(t))dt\\ \rightarrow&\!\!\displaystyle\bar{\zeta}=E{\int}_{0}^{\infty}{\int}_{\mathbb{U}}e^{-\rho t}ud\pi(\lambda,u|X_{n}(t),\alpha(t))dt, \end{array} $$

in probability,

$$\begin{array}{rl} &\!\!\displaystyle\frac{1}{\nu_{n}}\sum\limits_{j=m(t_{n}+t+l\nu_{n})} ^{m(t_{n}+t+(l + 1)\nu_{n})-1}E_{m(t_{n}+t+l\nu_{n})}\xi_{n}\\ =&\!\!\displaystyle\frac{1}{\nu_{n}}\sum\limits_{j=m(t_{n}+t+l\nu_{n})}^{m(t_{n}+t+(l + 1)\nu_{n})-1} E_{m(t_{n}+t+l\nu_{n})}{\int}_{0}^{\infty}{\int}_{\mathbb{U}}e^{-\rho t}X_{n}ud\pi^{n}(\lambda,u|X_{n}(t),\alpha(t))dt\\ \rightarrow&\!\!\displaystyle\bar{\xi}=E{\int}_{0}^{\infty}{\int}_{\mathbb{U}}e^{-\rho t}X(t)ud\pi(\lambda,u|X_{n}(t),\alpha(t))dt, \end{array} $$

in probability. Where (X(⋅),π(λ,⋅)) is the weak limit of (X_n(⋅),πⁿ(λ,⋅). And E_n is the conditional expectation with respect to $\mathcal {G}_{n}$, the σ-algebra generated by {ξ_j : j ≤ n}.

Proof

First we derive the tightness of the pair (X_n(⋅),πⁿ(λ,⋅)). since {πⁿ(λ,⋅)} is in π, the sequence is tight. Now we show that the associated controlled diffusion X_n(⋅) is tight in $D([0,\infty ):\mathbb {R})$, the space of functions that are right continuous with left limits endowed with the Skorohod topology. As in (Yin and Zhu 2010, p31, Proposition 2.3), by using Itô isometry and the Gronwall inequality, detailed computations lead to

$$\sup_{0\leq s\leq t\leq T}{E_{s}^{n}}|X_{n}(t)|^{2}\leq C_{n}(s,T), $$

for any T > 0 and s ≤ T, with sup0≤s≤TEC_n(s,T) < ∞. Within this proof ${E_{s}^{n}}$ denotes the expectation with respect to $\mathcal {F}_{s}^{n}=\sigma \{W_{n}(t): t\leq s\}$. Thus, for any η > 0,t ≥ 0 and 0 ≤ s ≤ η, we have

$$ \begin{array}{rl} &\!\!\displaystyle {E_{t}^{n}}|X_{n}(t+s)-X_{n}(t)|^{2}\\ \leq&\!\!\displaystyle 2{E_{t}^{n}}|{\int}_{t}^{t+s}X_{n}(r)(\mu(\alpha(r))dr+\sigma(\alpha(r))dW_{n}(r)|^{2}\\ &+{E_{t}^{n}}|{\int}_{t}^{t+s}{\int}_{\mathbb{U}}\phi(\alpha(r),X_{n}(r),u)d\pi^{n}(\lambda,u)dr|^{2}\\ \leq&\!\!\displaystyle 2 \tilde{C}_{n}(\eta,s,T), \end{array} $$

(A.9)

with $\tilde {C}_{n}(\eta ,s,T)$ satisfies that

$$\lim_{\eta\rightarrow0}\limsup_{n\rightarrow\infty}E\tilde{C}_{n}(\eta,s,T)= 0. $$

Then by the tightness criterion (Ethier and Kurtz 2005, Chapter 3, Corollary 8.6), see also (Kushner and Yin 2003, p230, Theorem 3.3), we conclude that X_n(⋅) is tight in $D([0,\infty ):\mathbb {R})$. Thus (X_n,πⁿ(λ,⋅)) is tight in $D([0,\infty ):\mathbb {R})\times {\Pi }$.

Since (X_n,πⁿ(λ,⋅)) is tight, it is sequentially compact. So we can extract weakly convergent subsequences. Select such a sequence still denoted by (X_n,πⁿ(λ,⋅)). Denote its limit by (X(⋅),π(λ,⋅)). By Skorohod representation, we assume that (X_n,πⁿ(λ,⋅)) converges to (X(⋅),π(λ,⋅)) with probability 1. Consider the sequence {ξ_n}, we can deduce the following estimation

$$E|\xi_{n}|\leq{\int}_{0}^{\infty}{\int}_{\mathbb{U}}e^{-\rho t}E^{1/2}|X_{n}(t)|^{2}d\pi^{n}(\lambda,u|X_{n}(t),\alpha(t))dt<\infty. $$

Next, {ξ_n} is an independent sequence because the sequence is generated by integrating w.r.t. independent Brownian motions. By the weak convergence, the Skorohod representation, and the dominated convergence theorem,

$$\begin{array}{rl} \displaystyle &\!\!\displaystyle\frac{1}{\nu_{n}}\sum\limits_{j=m(t_{n}+t+l\nu_{n})}^{m(t_{n}+t+(l + 1)\nu_{n})-1} E_{m(t_{n}+t+l\nu_{n})}{\int}_{0}^{\infty}{\int}_{\mathbb{U}}e^{-\rho t}X_{n}ud\pi^{n}(\lambda,u|X_{n}(t),\alpha(t))dt\\ \rightarrow&\!\!\displaystyle\bar{\xi}=E{\int}_{0}^{\infty}{\int}_{\mathbb{U}}e^{-\rho t}X(t)ud\pi(\lambda,u|X_{n}(t),\alpha(t))dt, \ \text{ in probability as } n \to \infty . \end{array} $$

Similarly, we can derive the other assertion of this lemma. The desired result follows. □