Estimating Volatility from Brownian Passage Times
Summary
The document presents passage time as an alternative to estimating volatility from fixed-interval Brownian increments. Under a zero-drift Brownian motion with constant volatility, an expected passage time for traveling a specified distance provides a moment condition that can be rearranged to estimate volatility. The author then considers local volatility on a time grid: at each grid point, estimate volatility using the next passage time and the previous passage time, and average the two estimates. The backward estimate relies on time reversibility.
For changing volatility, passage times need not be identically distributed, so the constant-volatility estimator cannot simply be applied across the full sample. The discussion motivates local estimates but leaves the choice and size of grid intervals unresolved. Its displayed formulas are missing in the supplied text, and it gives no empirical results or practical validation. The method therefore depends on assumptions and implementation details that would need to be checked before use with market data.
Key ideas
- Fixed-distance passage times can provide a moment condition for estimating constant Brownian volatility.
- The passage-time approach measures elapsed time to move a set distance instead of return size over a fixed interval.
- Local volatility estimates can use both forward and backward passage times around each grid point.
- Time reversibility supports using backward passage times in the proposed estimator.
- When volatility varies, passage times may not be identically distributed, and grid selection remains unresolved.
Tags
Full text
# Estimating constant and local volatility based on passage times # Estimating constant and local volatility based on passage times Consider a Brownian motion `B_t` with constant instantaneous volatility `σ` and zero drift where `t` is larger than zero and the brownian motion is equal to zero in the beginning. We know that the increments are equal to and satisfy This leads to the moment condition: For which we can estimate `σ` based on a sample of returns. This is basically the same as close-to-close / traditional volatility. However, rather than measuring the size of Brownian increments over a fixed time interval, we can also measure the time it takes the Brownian motion to travel a given distance up or down. This can be written as: According to Borodin and Salminen (2002) we have the following moment condition Again we can calculate `σ`, when we re-arrange the formula: However, when volatility is not constant we can not assume that the observed passage times are identically distributed. According to this paper we can apply the following method to calculate the local volatility: Consider a fixed time grid that consists of `N`, not necessarily equispaced, intervals. Each interval is equal to At a given point, `t_i`, we need to look for the next passage time . Since the Brownian motion is time-reversible, we can also look backwards and find the previous passage time, . Visually this looks as follows: Next, we have two independent estimators of local volatility at `t_i`, namely and We take the average of the two and have an estimation of the local volatility at the point `t_i`. What I don't understand is that the `N` intervals are not necessarily equispaced intervals. Why is this the case? How to determine then the size of each individual interval?
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