In the constantly changing financial markets, investors and traders need to trade using optimal trading strategies. In this article, we optimize the expected cost and the execution risk to develop an optimal trading strategy for risk-averse investors. In this context, we study a second order convex function that incorporates both transient and permanent market impacts in the price path rule of motion. By the use of dynamic programming, we get a closed form solution of the unconstrained problem, in some particular cases for the risk-averse investor.
into account the transaction size and an information factor, which was calculated as a percentage of
the price without any impact. [2], proposed linear price impact functions and integrated an arithmetic
random walk to model the dynamics of execution prices, using the no-efect price as a benchmark.
[1], introduced the initial mathematical optimization model for inscribe the optimal trading problem.
Further, other researcher, including, [7],[2],[3],[4],[5], explored the inquires related the optimal trading
for single asset. [6], introduced the jump difusion model to capture the unpredictable efects of large
transactions on prices during execution. In this model, the purpose is to provide a better understanding
of the fluctuations in asset values during significant transactions. However [ 8]; and [9], proposed
the use of stochastic dominance for all types of investors in portfolio optimization.Using the same
approach, [5], analyzed optimal trading strategies. In addition, [10], extended the linear price impact
with information price model of [1], referred to as the AR(
In this paper, we extend the AR(
In existing literature, no-impact price dynamics are often described as arithmetic random
walks.According to this assumption, the price of the asset in any given period will be determined
by the price of the asset in the previous period, reflecting a first-order autoregressive AR(
Consider an investor with a goal to buy a significant amount of shares, let’s call it ¯, within a specific time period, say [0, ]. This time frame is split into smaller intervals, each with a length of Δ = , and the ℎ time frame is equal to Δ.
Let’s use to represent the shares acquired during the interval ℎ at a price of , and will be the remaining shares to buy at time ℎ.
Now, the investor wants to minimize the expected cost of getting ¯ shares while keeping a handle on a predetermined risk level. This perspective on risk is inspired by the work of Almgren and Chriss (2000) and Huberman and Stanzl (2005). So, in a nutshell, the investor’s challenge is to navigate this buying process in a way that minimizes costs and manages the associated risks efectively. The problem (P1)
min {}=1 ︃( ∑︁ =1
)︃ 1 = , = − 1 − − 1 ∀ = 1, 2, ..., , +1 = 0.
In outlining the dynamics of the stock price for this dynamic programming problem, let’s denote ˜ as the observed stock price at ℎ time. To simplify, we’ll assume that market prices follow geometric Brownian motion. This allows us to express the market price at ℎ time as follows: A trader’s decision-making involves not only the current stock price but also factors in the broader market conditions. We represent this by introducing the market information variable, denoted as . This variable captures the potential influence of evolving market conditions, encompassing aspects like variations in buy-sell volume or private information about the security.
To model this variable, we adopt an autoregressive process of order 2 that is (AR(
= 1− 1 + 2− 2 +
Here, is a white noise process with zero mean and variance 2. At ℎ time, the trader observes the price ˜ and market information and decides to trade shares. We assume that the price impact of trading is a linear function of and . Therefore, the average execution price of trading shares at ℎ time is given by:
= ˜ + + +
The average execution price of trading shares at ℎ is determined by the observed market price ˜ , the quantity of shares traded , and the market information . Additionally, there is an uncertainty component denoting the noise in the acquisition price, characterized by a white noise process with zero mean and variance 2.
The motion of price expressed in equ. (
In solving the given problem using dynamic programming (DP) method, denoted as (P1), we leverage the insight presented in (Bertsimas and Lo (1998)), According to this approach, the portion of the optimal trading strategy (, +1, ..., ) during any time period , corresponding to time periods to , must be optimal for these time intervals.
In the context of an auto regressive model of order 2 (AR(
Consider the problem (P1), to be solved by DP method.
DP Problem subject to
min (∑︁ ) {}=1 =1 1 = , = − 1 − − 1 ∀ = 1, 2, ..., +1 = 0.
(
=1 be the optimal cost of execution corresponding to trades into periods .
During the final trading period , the optimal strategy is to execute all the remaining shares to ifnish the trade. i.e. = . Thus is given as follows: (˜,− 1,− 2,,− 1,) = min[]
= (˜ + + )
The Bellman Equation relating − − +1, is given as follows:
− (˜− ,− − 1,− − 2,− ,− − 1,− )
= min − [− − + − +1(−˜+1,− ,− − 1,− +1,− ,− +1)] (
= 2 + 2− 1(1 − )221 2 + 2− 1(1 − )1 2 − 2− 1(1 − )1
+ 2− 1 + 2− 1 − 2− 1
1, = − 1 − 2− 1 + ( + (1 − )1)− 1 − 2(1 − )1 ((1 − )1 − )− 1
− ( + 2(1 − )1)− 1 − 1
2, = − 1 − − 1 − 2− 1 − − 1 1 + − 1 + ℎ− 1 1 − − 1(1 − )1
5, = 2− 1 − − 1 − 3(1 − )1
(
˜ − = − 1− + − 1− + − 1− − 1 + − 1− − 1 + − 1− − (˜− ,− − 1,− − 2,− ,− − 1,− ) = ˜2− + − − ˜ + − − − 1 + ˜− − − 1 + ˜− − + 2− + 2− − 1 (11) ˜ 2 + ℎ− − + − − 1− + − − 1− + − + − − 1− + 2− − 1 + − − 1− − 1 + − − 1−
Examining the influence of a change in on the strategy is another noteworthy aspect. In the price path, determines the degree to which our trade has a lasting impact on the market price. To illustrate this, we conducted a numerical example involving the purchase of 1000 shares, using simulation parameters from [10].
= 20, 1 = 0.5, = 1000, Price process and market information have been considered as ∼ (0, 2). Figure 1 represent the optimal trading strategy with . We have also investigated how strategies change based on whether the market is expected to be bullish or bearish. We consider the three market categories.
In addition, we analyzed how strategies difer when market expectations are bullish or bearish.we examined three types of market situations by taking diferent value of . There are diferent market scenario mention as follows in figure 3 and figure 4 as follows: Additionally, we have examined how trading strategies adapt to diferent market expectations, whether bullish or bearish. We are considering three types of market scenario.
To develop complex methods for trading in financial markets optimally, this research proposes a
novel nonlinear programming approach. In a geometric Brownian motion under AR (
The research presents simulations and empirical validations that show the efectiveness of our suggested solution. In a variety of market conditions, the model consistently outperforms other trading strategy optimization techniques.However, we also acknowledge that more empirical the market information’s are necessary to validate and improve our model in a variety of asset classes and market situations given the complex of financial markets.