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Simulated Annealing of the Simple Moving Average Strategy on Apple Stock Chart
Introduction
In financial markets, success is rarely about perfect prediction. Instead, traders thrive by adapting to changing market conditions and exploiting statistical advantages wherever they appear.
One fundamental approach in trading is to use technical indicators, such as moving averages, to systematically generate buy and sell signals.
However, even a simple strategy like a Simple Moving Average (SMA) crossover relies heavily on the choice of its parameters, particularly the window length used to calculate the average. Selecting the wrong window can significantly reduce profits, while the right window can amplify returns.
Instead of manually testing different parameters or relying on heuristics, we can turn to a mathematical optimization method inspired by physics: Simulated Annealing (SA).
This approach allows us to explore a wide range of possible SMA windows, balancing exploration and exploitation, and ultimately finding a near-optimal configuration that maximizes cumulative returns.
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The Simple Moving Average Strategy
Before diving into simulated annealing, it is essential to understand the strategy we are optimizing.
The Simple Moving Average (SMA) strategy is one of the most widely used technical indicators in trading:
The SMA is calculated as the average of the last
Nclosing prices.A buy signal is triggered when the stock price crosses above the SMA.
A sell signal is triggered when the stock price crosses below the SMA.
The parameter
N— the length of the moving average window — is critical: too short, and the strategy becomes noisy; too long, and it lags behind trends.
Mathematically, the SMA at time t with window N is:
The Simple Moving Average Function
where P_{t-i} represents the closing price i days before t.
Once signals are generated, the cumulative return of the strategy can be calculated as:
The Cumulative Returns Function
where:
r_t = \frac{P_t — P_{t-1}}{P_{t-1}}is the daily return,s_{t-1} \in {0,1}indicates whether we are in a long position (1 = holding, 0 = out of market) based on the previous day’s signal.
This cumulative return serves as the objective function that we aim to maximize.
Simulated Annealing: A Trading Optimization Tool
Simulated annealing is a probabilistic optimization technique inspired by the physical process of annealing in metallurgy, where materials are heated and then cooled slowly to reduce defects and reach a stable state. The algorithm adapts this idea for optimization problems:
Start with an initial solution — in our case, an SMA window.
Iteratively propose a small change (a neighboring window).
Accept the new solution if it improves the objective function.
Occasionally accept worse solutions based on a probability that decreases over time — this prevents the algorithm from getting stuck in local maxima.
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The Mathematics Behind Simulated Annealing
The key equations in simulated annealing are:
Acceptance probability for worse solutions:
The Acceptance Probability
where:
\Delta E = R_{new} — R_{current}is the difference in cumulative return between the proposed and current solution.Tis the “temperature,” a parameter that starts high and decreases gradually according to a cooling schedule:
The Cooling Schedule
The cooling schedule controls the balance between exploration (high temperature) and exploitation (low temperature).
Early in the process, larger changes are more likely to be accepted, allowing the algorithm to explore widely. Later, the algorithm converges, refining the best-found solution.
Applying Simulated Annealing to SMA Optimization
AAPL Adjusted Close Price Chart
In our experiment, we applied simulated annealing to optimize the SMA window for a stock over a six-month historical period. Here’s the workflow:
Initialization: Randomly select an initial SMA window within a plausible range (e.g., 2–60 days).
Iteration: For each step:
Propose a new window by adding a small random change to the current window.
Compute the cumulative return for the new window.
Accept the new window if it improves the return or probabilistically if it worsens it.
Update the temperature according to the cooling schedule.
3. Convergence: Continue until a predetermined number of iterations or until temperature approaches zero.
4. Multi-start refinement: Run multiple SA processes with different random seeds and select the best overall SMA window.
Insights from the Optimization
Several key observations emerged from this process:
The best SMA window is not necessarily intuitive; it often lies in a non-obvious part of the search space.
Simulated annealing allows the algorithm to escape local maxima, finding solutions that traditional gradient-like methods may miss.
The optimized SMA consistently generated buy/sell signals that aligned well with price trends, resulting in superior cumulative returns compared to arbitrarily chosen windows.
Best SMA overlay with buy/sell signals on stock prices
A quick scan of the objective function across all SMA windows before optimization helps to visualize the rugged landscape that SA must navigate.
Objective function landscape before SA
Why Simulated Annealing Works for Trading
Stock price movements are noisy and non-linear, making many traditional optimization methods unreliable. Simulated annealing excels in this environment because:
It does not rely on derivatives or smooth objective functions.
It balances exploration and exploitation, enabling robust search across complex landscapes.
It is simple to implement and highly adaptable to different trading strategies, including SMAs, exponential moving averages, or multi-parameter indicators.
Conclusion: From Physics to Profitable Decisions
Simulated annealing offers a bridge between the stochastic world of markets and structured optimization. By carefully tuning the SMA window through this method, traders can enhance the reliability and profitability of their strategies, even when underlying price movements are unpredictable.
While it does not guarantee success on future data, it provides a systematic, mathematically grounded way to explore trading parameter spaces and uncover hidden opportunities.
Final objective curve with optimized SMA highlighted
Simulated annealing is not just a computational curiosity, it is a practical tool that translates the principles of physics into actionable trading strategies.
Check out this GitHub Repository for the full code implementation of the Simulated Annealing Algorithm.