Trading with Confidence: A Fourier Transform and GARCH Model Strategy

Fourier Transform Analysis

Theoretical Background

The Fourier Transform is a mathematical technique that decomposes a function of time into its constituent frequencies. Named after the French mathematician Jean-Baptiste Joseph Fourier, this transformation provides a way to analyze signals in the frequency domain rather than the time domain. In the context of financial markets, the Fourier Transform allows us to identify cyclical patterns and periodic components that may not be immediately apparent when looking at price charts.

For computational efficiency, we use the Fast Fourier Transform (FFT), an algorithm that computes the DFT in \(O(N \log N)\) time instead of \(O(N^2)\).

Application to Financial Time Series

When applied to price data, the Fourier Transform reveals the dominant frequencies (cycles) present in the market. These cycles can range from short-term intraday patterns to longer-term seasonal effects. By identifying these cycles, we can potentially forecast future price movements based on the continuation of these patterns.

Our implementation follows these steps:

1. Detrending: Remove the linear trend from the price series to focus on oscillations around the trend.
2. FFT Application: Apply the Fast Fourier Transform to the detrended price series to obtain the frequency spectrum.
3. Dominant Frequency Identification: Identify the dominant frequencies by selecting the components with the highest magnitudes.
4. Signal Reconstruction: Reconstruct the signal using only the dominant frequencies, filtering out noise.
5. Forecasting: Project the reconstructed signal forward to predict future price movements.

GARCH Volatility Modeling

Theoretical Background

The Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model, introduced by Bollerslev in 1986, is an extension of the ARCH model developed by Engle. GARCH models are particularly well-suited for financial time series because they capture two key characteristics of market volatility:

1. Volatility Clustering: Periods of high volatility tend to be followed by high volatility, and periods of low volatility tend to be followed by low volatility.
2. Mean Reversion: Volatility tends to revert to a long-run average level over time.

In our strategy, we use a GARCH(1,1) model, which is often sufficient to capture the volatility dynamics of financial returns:

Application to Financial Time Series

The GARCH model provides several valuable insights for our trading strategy:

1. Volatility Forecasting: By estimating the parameters of the GARCH model, we can generate forecasts of future volatility, which is crucial for risk assessment and position sizing.
2. Risk Assessment: Higher forecasted volatility indicates greater uncertainty and risk, which may warrant smaller position sizes or wider stop-loss levels.
3. Regime Identification: Sudden changes in volatility patterns can signal regime shifts in the market, potentially requiring adjustments to the trading strategy.

Confidence Scoring System

Methodology

The confidence scoring system is the cornerstone of our strategy, integrating signals from both the Fourier Transform analysis and the GARCH volatility model to generate a comprehensive measure of trade quality. The system assigns a score between 0 and 100 to each potential trading opportunity, with higher scores indicating greater confidence in the trade's success.

Trade Execution Threshold

The strategy only executes trades when the confidence score exceeds a predefined threshold (typically 70). This filtering mechanism ensures that the strategy only takes positions when multiple factors align favorably, potentially improving the risk-adjusted return profile.

This approach allocates more capital to higher-conviction trades while maintaining a minimum position size for all trades that meet the threshold criteria.

QuantConnect Framework Integration

QuantConnect is a cloud-based algorithmic trading platform that provides researchers and traders with the tools to design, test, and deploy trading strategies. The platform offers several key advantages for implementing our Fourier Transform and GARCH model strategy:

• Data Access: QuantConnect provides high-quality minute-level data for a wide range of asset classes, including equities, forex, futures, and cryptocurrencies.
• Python Integration: The platform supports Python, allowing us to leverage powerful scientific computing libraries such as NumPy, SciPy, and the ARCH package for GARCH modeling.
• Backtesting Engine: QuantConnect's backtesting engine simulates trading with realistic assumptions about slippage, commission, and market impact.
• Live Trading Capabilities: Strategies can be seamlessly transitioned from backtesting to live trading with minimal code changes.

Our implementation follows QuantConnect's algorithm lifecycle, which consists of the following key methods:

• Initialize(): Sets up the algorithm, including data subscriptions, parameters, and initial state.
• OnData(): Processes incoming data and makes trading decisions.
• OnOrderEvent(): Handles order execution events.
• OnEndOfAlgorithm(): Performs final calculations and cleanup when the algorithm completes.

Custom Indicators

To implement our strategy, we created several custom indicators that extend QuantConnect's indicator framework:

FFT Indicator

class FourierTransformIndicator:
    def __init__(self, window_size=1024, num_components=10, forecast_periods=5):
        self.window_size = window_size
        self.num_components = num_components
        self.forecast_periods = forecast_periods
        self.price_history = []
        self.last_forecast = None
        self.forecast_direction = 0
        self.forecast_magnitude = 0
        self.signal_strength = 0
        
    def update(self, price):
        # Add the new price to our history
        self.price_history.append(price)
        
        # Keep only the most recent window_size prices
        if len(self.price_history) > self.window_size:
            self.price_history = self.price_history[-self.window_size:]
        
        # Only perform FFT analysis if we have enough data
        if len(self.price_history) < self.window_size:
            return {
                'forecast_direction': 0,
                'forecast_magnitude': 0,
                'signal_strength': 0,
                'forecast_values': None
            }
        
        # Perform FFT analysis
        return self._perform_fft_analysis()

GARCH Indicator

class GARCHModelIndicator:
    def __init__(self, window_size=2000, p=1, q=1, forecast_horizon=5):
        self.window_size = window_size
        self.p = p
        self.q = q
        self.forecast_horizon = forecast_horizon
        self.returns_history = []
        self.volatility_forecast = None
        self.model = None
        self.model_fit = None
        self.last_update_size = 0
        self.volatility_trend = 0
        self.volatility_level = 0
        
    def update(self, price):
        # Calculate return if we have at least one previous price
        if len(self.returns_history) > 0:
            log_return = np.log(price / self.returns_history[-1])
            self.returns_history.append(price)
        else:
            # First price point, no return yet
            self.returns_history.append(price)
            log_return = 0
        
        # Keep only the most recent window_size returns
        if len(self.returns_history) > self.window_size:
            self.returns_history = self.returns_history[-self.window_size:]
        
        # Only perform GARCH analysis if we have enough data
        if len(self.returns_history) < 100:
            return {
                'volatility_forecast': None,
                'volatility_trend': 0,
                'volatility_level': 0,
                'risk_score': 0,
                'volatility_confidence': 50
            }
        
        # Only refit the model every 20 new data points to save computation
        if len(self.returns_history) - self.last_update_size >= 20 or self.model_fit is None:
            self.last_update_size = len(self.returns_history)
            return self._perform_garch_analysis()
        else:
            # Return the last forecast if we're not refitting
            return {
                'volatility_forecast': self.volatility_forecast,
                'volatility_trend': self.volatility_trend,
                'volatility_level': self.volatility_level,
                'risk_score': self._calculate_risk_score(),
                'volatility_confidence': self.get_volatility_confidence()
            }

Confidence Scorer

class ConfidenceScorer:
    def __init__(self, fft_weight=0.6, garch_weight=0.4, threshold=70):
        self.fft_weight = fft_weight
        self.garch_weight = garch_weight
        self.threshold = threshold
        self.confidence_score = 0
        self.trade_signal = 0
        
    def calculate_confidence(self, fft_indicator, garch_indicator):
        # Get confidence components from each indicator
        fft_confidence = fft_indicator.get_signal_confidence()
        volatility_confidence = garch_indicator.get_volatility_confidence()
        
        # Calculate weighted confidence score
        self.confidence_score = (
            self.fft_weight * fft_confidence + 
            self.garch_weight * volatility_confidence
        )
        
        # Determine trade signal based on confidence and FFT direction
        if self.confidence_score >= self.threshold:
            self.trade_signal = fft_indicator.forecast_direction
        else:
            self.trade_signal = 0
        
        return {
            'confidence_score': self.confidence_score,
            'trade_signal': self.trade_signal,
            'fft_confidence': fft_confidence,
            'volatility_confidence': volatility_confidence,
            'threshold_met': self.confidence_score >= self.threshold
        }

Risk Management

Position Sizing

Position sizing is a critical aspect of our strategy, directly linked to the confidence scoring system. The base position size is set as a fraction of the portfolio (typically 1%), which is then scaled based on the confidence score:

def enter_position(self, direction, price, timestamp):
    # Calculate position size based on confidence
    if self.last_confidence_score < self.confidence_threshold:
        self.position_size = 0.0
    else:
        # Scale position size linearly from threshold to 100
        confidence_above_threshold = self.last_confidence_score - self.confidence_threshold
        max_confidence_range = 100 - self.confidence_threshold
        
        # Scale from 50% to 100% of base size based on confidence
        position_scale = 0.5 + 0.5 * (confidence_above_threshold / max_confidence_range)
        self.position_size = self.position_sizing_factor * position_scale
    
    # Set position state and execute the trade
    self.current_position = direction
    self.entry_price = price
    self.SetHoldings(self.symbol, self.position_size * direction)

Stop Loss and Take Profit

The strategy implements dynamic stop loss and take profit levels based on volatility forecasts from the GARCH model:

def update_stop_and_target(self, price, volatility):
    volatility_price_impact = price * volatility
    
    if self.current_position > 0:  # Long position
        self.stop_loss_price = max(
            self.stop_loss_price if self.stop_loss_price > 0 else 0,
            price - (self.stop_loss_multiplier * volatility_price_impact)
        )
        self.take_profit_price = price + (self.take_profit_multiplier * volatility_price_impact)
    else:  # Short position
        self.stop_loss_price = min(
            self.stop_loss_price if self.stop_loss_price > 0 else float('inf'),
            price + (self.stop_loss_multiplier * volatility_price_impact)
        )
        self.take_profit_price = price - (self.take_profit_multiplier * volatility_price_impact)