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stresstest maker model/optimal risk reward ratio/main.tex

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+\documentclass[conference]{IEEEtran}
+\usepackage{amsmath, amssymb, amsthm, bm}
+\usepackage{graphicx}
+\usepackage{cite}
+\usepackage{url}
+
+\newtheorem{definition}{Definition}
+\newtheorem{proposition}{Proposition}
+\newtheorem{corollary}{Corollary}
+
+\begin{document}
+
+\title{Optimal Take-Profit Ratio Under Spread, Slippage, and Commission}
+
+\author{\IEEEauthorblockN{algorembrant}
+\IEEEauthorblockA{July 18, 2026}
+\thanks{}}
+
+\maketitle
+
+\begin{abstract}
+This paper derives closed-form expressions for the optimal take-profit ratio in a single-asset long position, explicitly incorporating bid-ask spread, execution slippage, and proportional commissions. We formulate the realized profit and loss as functions of market parameters, derive the break-even take-profit level, and obtain the optimal risk-reward ratio that maximizes expected return for a given risk appetite. The analysis extends to recovery from consecutive losses, yielding a dynamic adjustment rule for the take-profit target.
+\end{abstract}
+
+\section{Introduction}
+Let $\mathcal{A}$ be a financial asset with tick size $\delta>0$ and bid-ask spread $s=\kappa\delta$, $\kappa\in\mathbb{Z}_{>0}$. Prices are quoted with precision $d$ (digits). We consider a buy order executed at the ask price $P_a$, with a stop-loss (SL) and take-profit (TP) order placed at distances $\lambda>0$ and $\mu>0$ from the effective entry price, respectively.
+
+Market frictions: (i) spread $s$ (bid-ask), (ii) slippage $\varepsilon$ at entry and exit (random, bounded), and (iii) proportional commission fees $c_e, c_x \in (0,1)$ for entry and exit.
+
+We derive the optimal TP distance $\mu^*$ and the corresponding risk-reward ratio $\psi^* = \mu^*/\lambda$ that maximises net profit for a given loss tolerance or recovery requirement.
+
+\section{Model Formulation}
+Let $P_b$ be the current bid price. Then $P_a = P_b + s$. Let $\varepsilon_e, \varepsilon_s, \varepsilon_t$ be the slippage at entry, stop-loss exit, and take-profit exit, respectively, with $|\varepsilon| \le \eta$ for some $\eta>0$. The effective entry price:
+\begin{equation}
+P_e = P_a + \varepsilon_e.
+\end{equation}
+Nominal SL and TP prices:
+\begin{align}
+P_{sl} &= P_e - \lambda, \quad P_{tp} = P_e + \mu.
+\end{align}
+Actual execution prices (including slippage and spread requirement):
+\begin{align}
+P_{sl}^{real} &= P_e - \lambda + \varepsilon_s + s, \\
+P_{tp}^{real} &= P_e + \mu + \varepsilon_t + s.
+\end{align}
+Let $B_0$ be the initial balance, $\gamma\in(0,1)$ the fractional risk per trade. The stake $S = \gamma B_0$. Entry fee $F_e = c_e S$, position size $Q = S(1-c_e)$. Exit fee $F_x = c_x Q R$, where $R$ is the gross return multiple.
+
+\section{Profit and Loss}
+The gross profit/loss:
+\begin{align}
+\Pi_g &= Q \frac{P_{tp}^{real} - P_e}{P_e} 
+       = Q \frac{\mu + \varepsilon_t + s}{P_e}, \\
+\Pi_l &= -Q \frac{P_e - P_{sl}^{real}}{P_e}
+       = -Q \frac{\lambda - \varepsilon_s - s}{P_e}.
+\end{align}
+Define effective loss multiple:
+\begin{equation}
+\rho = \frac{\lambda - \varepsilon_s - s}{P_e}.
+\end{equation}
+Then $\Pi_l = -Q \rho$. Net profit and loss:
+\begin{align}
+\Pi_g^{net} &= \Pi_g - F_e - F_x \\
+&= S(1-c_e)(1-c_x) \frac{\mu + \varepsilon_t + s}{P_e} - c_e S, \\
+\Pi_l^{net} &= -S(1-c_e)(1+c_x)\rho - c_e S.
+\end{align}
+
+\section{Breakeven Condition}
+Set $\Pi_g^{net} + \Pi_l^{net} = 0$. Solving for $\mu$:
+\begin{align}
+\mu_{BE} &= \frac{(1-c_e)(1+c_x)\rho + c_e}{(1-c_e)(1-c_x)} P_e - \varepsilon_t - s.
+\end{align}
+
+\section{Optimal Take-Profit}
+Let $k>0$ be the desired profit multiple relative to the net loss, i.e., $\Pi_g^{net} = k |\Pi_l^{net}|$. Then:
+\begin{align}
+\mu^* &= \frac{k (1-c_e)(1+c_x)\rho + (1+k)c_e}{(1-c_e)(1-c_x)} P_e - \varepsilon_t - s.
+\end{align}
+The optimal risk-reward ratio:
+\begin{equation}
+\psi^* = \frac{\mu^*}{\lambda}.
+\end{equation}
+
+\section{Recovery from Loss Streaks}
+Assume $L$ consecutive losses. The remaining balance:
+\begin{equation}
+B_L = B_0 \left[1 - \gamma (1-c_e)(1+c_x)\rho - \gamma c_e \right]^L.
+\end{equation}
+To recover $B_0$ with the next win, require $\Pi_g^{net} = B_0 - B_L$. Hence:
+\begin{equation}
+\mu_{rec} = \frac{(B_0 - B_L)/S + c_e}{(1-c_e)(1-c_x)} P_e - \varepsilon_t - s.
+\end{equation}
+The required risk-reward ratio:
+\begin{equation}
+\psi_{rec} = \frac{\mu_{rec}}{\lambda}.
+\end{equation}
+
+\section{Discussion}
+The derived formulas yield deterministic TP targets given stochastic slippage terms. In practice, $\varepsilon_e, \varepsilon_s, \varepsilon_t$ are random; we can take expectations (assuming zero mean) to obtain the expected optimal TP. The breakeven and optimal ratios increase with spread $s$, commission rates, and slippage magnitude, as expected. The recovery ratio scales linearly with the number of losses $L$.
+
+\section{Conclusion}
+We have presented a rigorous mathematical framework for computing the optimal take-profit ratio in the presence of spread, slippage, and commission. The results provide a foundation for systematic trade management and can be extended to short positions and multiple assets.
+
+\bibliographystyle{IEEEtran}
+\bibliography{references}
+
+\end{document}