# Optimal Timing to Trade along a Randomized Brownian Bridge

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## Abstract

**:**

## 1. Introduction

## 2. Prior Belief and Price Dynamics

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 3. Optimal Liquidation Problems

**Proposition**

**1.**

- 1.
- $G(u,x)>0$, $\forall (u,x)\in [t,\overline{T}]\times {\mathbb{R}}_{+}\Rightarrow {\tau}^{*}=\overline{T}$,
- 2.
- $G(u,x)\le 0$, $\forall (u,x)\in [t,\overline{T}]\times {\mathbb{R}}_{+}\Rightarrow {\tau}^{*}=t$.

**Proof.**

#### 3.1. Stocks

**Proposition**

**2.**

- (i) downward-sloping and concave in x if$${\sigma}_{D}>\sqrt{T}\sigma \phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}x\le q\left(t\right)-2,$$$${\sigma}_{D}<\sqrt{T}\sigma \phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}x\ge q\left(t\right)-1,$$
- (ii) downward-sloping and convex in x if$${\sigma}_{D}>\sqrt{T}\sigma \phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}q\left(t\right)-2\le x\le q\left(t\right)-1,$$
- (iii) upward-sloping and concave in x if$${\sigma}_{D}<\sqrt{T}\sigma \phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}q\left(t\right)-2\le x\le q\left(t\right)-1,$$
- (iv) upward-sloping and convex in x if$${\sigma}_{D}>\sqrt{T}\sigma \phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}x\ge q\left(t\right)-1,$$$${\sigma}_{D}<\sqrt{T}\sigma \phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}x\le q\left(t\right)-2,$$$$q\left(t\right):={X}_{0}-\frac{\mu {\sigma}^{2}T+(\frac{1}{2}{\sigma}^{2}-r)(t{\sigma}_{D}^{2}+T{\sigma}^{2}(T-t))}{{\sigma}_{D}^{2}-T{\sigma}^{2}}.$$

**Proposition**

**3.**

- (i)
- downward-sloping and concave in t if$${\sigma}_{D}<\sqrt{T}\sigma \phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}x\ge {X}_{0}-\frac{\mu {\sigma}^{2}T}{{\sigma}_{D}^{2}-T{\sigma}^{2}},$$
- (ii)
- downward-sloping and convex in t if$${\sigma}_{D}>\sqrt{T}\sigma \phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}x\ge {X}_{0}-\frac{\mu {\sigma}^{2}T}{{\sigma}_{D}^{2}-T{\sigma}^{2}},$$
- (iii)
- upward-sloping and concave in t if$${\sigma}_{D}>\sqrt{T}\sigma \phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}x\le {X}_{0}-\frac{\mu {\sigma}^{2}T}{{\sigma}_{D}^{2}-T{\sigma}^{2}},$$
- (iv)
- upward-sloping and convex in t if$${\sigma}_{D}<\sqrt{T}\sigma \phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}x\le {X}_{0}-\frac{\mu {\sigma}^{2}T}{{\sigma}_{D}^{2}-T{\sigma}^{2}}.$$

#### 3.2. Call and Put Options

**Lemma**

**1.**

**Proposition**

**4.**

**Proof.**

## 4. Numerical Implementation

## 5. Conclusions

## 6. Proofs

#### 6.1. Normal Distribution

#### 6.2. Double Exponential Distribution

#### 6.3. Proof of Propositions 2 and 3

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Path simulation of (4) with the prior belief on the future log-price following (

**a**) two-point discrete distribution and (

**b**) normal distribution. Parameters: (

**a**) ${\delta}_{u}=0.3,\phantom{\rule{4pt}{0ex}}{\delta}_{d}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}-0.3,\phantom{\rule{4pt}{0ex}}{p}_{u}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.5,\phantom{\rule{4pt}{0ex}}{p}_{d}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.5$; (

**b**) $\mu =0,\phantom{\rule{4pt}{0ex}}{\sigma}_{D}=0.3$. Common parameters: ${X}_{0}=1,\phantom{\rule{4pt}{0ex}}T=1,\phantom{\rule{4pt}{0ex}}\sigma =0.4$.

**Figure 2.**$A(t,x)$ under three different distribution with common Mean = 0 and Var = $0.36$ at $t=0.1$ (

**a**) and $t=0.8$ (

**b**). Parameters: (two-point) ${\delta}_{u}=-{\delta}_{d}=0.6,\phantom{\rule{4pt}{0ex}}{p}_{u}={p}_{d}=0.5$; (normal) $\mu =0,\phantom{\rule{4pt}{0ex}}{\sigma}_{D}=0.6$; (double exponential) $\theta =0,\phantom{\rule{4pt}{0ex}}{p}_{1}={p}_{2}=0.5,\phantom{\rule{4pt}{0ex}}{\lambda}_{1}={\lambda}_{2}=2.357$. Common parameters: ${S}_{0}=2.72({X}_{0}=1),\phantom{\rule{4pt}{0ex}}r=0.1,\phantom{\rule{4pt}{0ex}}\overline{T}=1,\phantom{\rule{4pt}{0ex}}T=1.1,\phantom{\rule{4pt}{0ex}}\sigma =0.4.$

**Figure 3.**Optimal boundaries for selling stock under two-point discrete distribution (

**a**,

**b**), normal distribution (

**c**) and double exponential distribution (

**d**). Parameters for each plot are as follows: (

**a**) ${\delta}_{u}=-{\delta}_{d}=0.1,\phantom{\rule{4pt}{0ex}}{p}_{u}={p}_{d}=0.5$; (

**b**) ${\delta}_{u}=-{\delta}_{d}=2,\phantom{\rule{4pt}{0ex}}{p}_{u}={p}_{d}=0.5$; (

**c**) $\mu =0,\phantom{\rule{4pt}{0ex}}{\sigma}_{D}=0.2$; (

**d**) $\theta =0$, ${p}_{1}={p}_{2}=0.5,\phantom{\rule{4pt}{0ex}}{\lambda}_{1}={\lambda}_{2}=10$. Common parameters: ${S}_{0}=2.72({X}_{0}=1),\phantom{\rule{4pt}{0ex}}r=0.1,\phantom{\rule{4pt}{0ex}}\overline{T}=1,\phantom{\rule{4pt}{0ex}}T=1.1$, $\sigma =0.4.$

**Figure 4.**Optimal trading boundaries for stock liquidation under the two-point discrete distribution (

**a**,

**b**), normal distribution (

**c**,

**d**) and double exponential distribution (

**e**,

**f**). Parameters: (

**a**) ${\sigma}_{D}=0.2$; (

**b**) $\mu =0$; (

**c**) ${\delta}_{u}=-{\delta}_{d}=0.1$; (

**d**) ${p}_{u}={p}_{d}=0.5$; (

**e**) ${\lambda}_{1}={\lambda}_{2}=10$; (

**f**) ${p}_{1}={p}_{2}=0.5$. Other common parameters are the same as Figure 3.

**Figure 5.**Optimal trading regions for stock liquidation under the two-point discrete distribution. The continuation regions are in yellow (light) color and the exercise regions are in blue (dark), and they are disconnected. Parameters: ${\delta}_{u}=-{\delta}_{d}=0.8,\phantom{\rule{4pt}{0ex}}{p}_{u}={p}_{d}=0.5$. Common parameters: ${X}_{0}=1,\phantom{\rule{4pt}{0ex}}r=0.1$, $\overline{T}=1,\phantom{\rule{4pt}{0ex}}T=1.1,\phantom{\rule{4pt}{0ex}}\sigma =0.4$.

**Figure 6.**Optimal trading regions for stock liquidation under the double exponential distribution. The continuation regions are in yellow (light) color and the exercise regions are in blue (dark), and they are disconnected. Parameters for (

**a**): $\theta =0,\phantom{\rule{4pt}{0ex}}{\lambda}_{1}={\lambda}_{2}=4.714,\phantom{\rule{4pt}{0ex}}{p}_{1}={p}_{2}=0.5$; parameters for (

**b**): $\theta =0,\phantom{\rule{4pt}{0ex}}{\lambda}_{1}={\lambda}_{2}=3.536,\phantom{\rule{4pt}{0ex}}{p}_{1}={p}_{2}=0.5$. Common parameters: ${X}_{0}=1,\phantom{\rule{4pt}{0ex}}r=0.1$, $\overline{T}=1,\phantom{\rule{4pt}{0ex}}T=1.1,\phantom{\rule{4pt}{0ex}}\sigma =0.4.$.

**Figure 7.**Optimal boundaries for selling European call with ${S}_{0}=105$ (

**a**) and put with ${S}_{0}=95$ (

**b**) under normal distribution. The strike price $K=100$ and maturity $T=1$, and the other parameters: $\mu =0,\phantom{\rule{4pt}{0ex}}{\sigma}_{D}=0.1,\phantom{\rule{4pt}{0ex}}r=0.1,\phantom{\rule{4pt}{0ex}}\overline{T}=1,\phantom{\rule{4pt}{0ex}}\sigma =0.4$. The green line is defined by $G=0$, and continuation region contains $G\ge 0$.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Leung, T.; Li, J.; Li, X.
Optimal Timing to Trade along a Randomized Brownian Bridge. *Int. J. Financial Stud.* **2018**, *6*, 75.
https://doi.org/10.3390/ijfs6030075

**AMA Style**

Leung T, Li J, Li X.
Optimal Timing to Trade along a Randomized Brownian Bridge. *International Journal of Financial Studies*. 2018; 6(3):75.
https://doi.org/10.3390/ijfs6030075

**Chicago/Turabian Style**

Leung, Tim, Jiao Li, and Xin Li.
2018. "Optimal Timing to Trade along a Randomized Brownian Bridge" *International Journal of Financial Studies* 6, no. 3: 75.
https://doi.org/10.3390/ijfs6030075