Calibrating a Monthly Random Walk from Annual Price-Change Volatility

Summary

The document poses a calibration question for a stock modeled as a monthly random walk with independent, identically distributed changes. Each monthly change is assumed to equal either a positive or negative fixed amount with equal probability, giving the model zero expected annual change. The stated annual standard deviation of the total price change is 108 points, and the author asks how to choose the step size.

The setup illustrates how to relate a per-period shock to the dispersion of a sum of independent shocks. With twelve monthly steps, the annual variance is the sum of the monthly variances, so the step magnitude can be inferred from the stated annual standard deviation. However, the document contains only the question and no answer or validation. Its assumptions also simplify real stock returns: it models absolute price changes rather than percentage returns, and it does not address changing volatility, drift, or dependence between months.

Key ideas

• The model assigns equal probability to a fixed positive or negative monthly price change.
• Equal-probability symmetric steps imply zero expected change over the year.
• For independent monthly shocks, annual variance accumulates as the sum of monthly variances.
• The question provides an annual standard deviation but does not include a proposed solution or evaluate the model’s assumptions.

Tags

# Random Walk choosing constant $g$


# Random Walk choosing constant $g$












I am looking at a stock, say stock X and I am simulating it by a random walk. It is only simulated once every month, where $t$ represents the month. I am letting $S_0$ represent the value of the stock at the beginning of the year, $S_0$. I have the following:

$S_t - S_0 = X_1 +X_2+...+X_t$

$X_1,...$ is iid sequence of variables.

I am expecting that this stock will increase by $0$ this year and am modeling this as

$P(x_i = g) = \frac{1}{2}, P(x_i = -g) = \frac{1}{2}$

where $g > 0$ is a constant depending on $i$. I have found that assuming that in this year the stock has zero expectation gain so $E[S_{365} - S_0] = 0$. I have calculated the std of the change of the stock this year and it is 108 points (108 dollars). I am unsure of how large to take $g$.

Currently I am thinking it could be about 25 but I am not sure as this seems pretty high.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.