An Excel spreadsheet containing over 900 days of daily data...

An Excel spreadsheet containing over 900 days of daily data on a number of different exchange rates and stock indices can be downloaded from the author's website: www-2.rotman.utoronto.ca/ hull/data. Choose one exchange rate and one stock index. Estimate the value of $\lambda$ in the EWMA model that minimizes the value of $\sum_i\left(v_i-\beta_i\right)^2$, where $v_i$ is the variance forecast made at the end of day $i-1$ and $\beta_i$ is the variance calculated from data between day $i$ and day $i+25$. Use the Solver tool in Excel. Set the variance forecast at the end of the first day equal to the square of the return on that day to start the EWMA calculations.

An Excel spreadsheet containing over 900 days of daily data on a number of different exchange rates and stock indices can be downloaded from the author's website:
www-2.rotman.utoronto.ca/ hull/data.
Choose one exchange rate and one stock index. Estimate the value of $\lambda$ in the EWMA model that minimizes the value of $\sum_i\left(v_i-\beta_i\right)^2$, where $v_i$ is the variance forecast made at the end of day $i-1$ and $\beta_i$ is the variance calculated from data between day $i$ and day $i+25$. Use the Solver tool in Excel. Set the variance forecast at the end of the first day equal to the square of the return on that day to start the EWMA calculations.

Key Concepts

Excel Solver Tool

Excel's Solver is an optimization add-in that can be used to find the best solution for linear, nonlinear, and integer programming problems. In the context of the EWMA variance forecasting model, the Solver tool is employed to adjust the lambda parameter such that the squared errors between the forecasted variance and the observed variance are minimized. This allows for an automated and efficient approach to fine-tuning the model based on the available data.

Exponential Weighted Moving Average (EWMA) Model

This model is used to estimate variances and volatilities in time series data by applying exponentially decaying weights to past observations. The EWMA methodology gives more importance to recent data points while gradually discounting older data, making it particularly useful in finance where recent market movements are more indicative of current volatility. By updating the variance forecast recursively using a decay factor, the EWMA model helps in capturing the dynamics of volatility over time.

Variance Forecasting

Variance forecasting involves predicting the future variability or risk in financial returns based on current and past information. In the context of the EWMA model, this is done by estimating a time-varying variance using an exponentially weighted average of past squared returns. Accurate variance forecasting is crucial for effective risk management and pricing in financial markets, as it helps to anticipate the amount of uncertainty and potential fluctuations in asset returns.

Parameter Optimization

Parameter optimization in this context refers to the process of identifying the optimal decay factor (lambda) that minimizes the discrepancy between forecasted variance and observed variance measures. This is typically achieved by minimizing a sum of squared errors, where the goal is to reduce the difference between the predicted values and the actual data. The optimization of parameters like lambda ensures that the model provides the most accurate representation of volatility dynamics.

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