Expected Time a Lognormal Asset Spends Inside a Price Corridor

Summary

The document derives the expected number of observation dates on which an asset price falls between two bounds. It treats the count as a sum of indicator variables and uses linearity of expectation, so the calculation needs the probability of being inside the corridor at each date, not the joint distribution across dates. Under geometric Brownian motion with constant volatility and deterministic interest rates, each probability is the difference between two lognormal tail probabilities, expressed through the standard normal cumulative distribution function.

Key ideas

• The expected count of in-corridor dates equals the sum of the individual date probabilities.
• Under constant-volatility lognormal dynamics, each date probability can be calculated from two normal cumulative distribution values.
• The expected count does not require modeling dependence among the indicators across dates.
• The stated closed-form calculation assumes constant volatility and deterministic rates.
• For general stochastic volatility, the document says there is no simple method and Monte Carlo may be needed.

Tags

# Expected number of days inside a corridor


# Expected number of days inside a corridor












Is there a simple (ish) approximation for the expected number of steps a random walk is within a set of bounds over a given time period? - in particular if i presume log normal and constant vol.

If i want to do it with something like local/stochastic/both vol, then is there a way to do it without resorting to MC/diffusion?

## Answer by Gordon (score 3)

https://quant.stackexchange.com/a/27850

This looks to me like a range accrual. Let $t_1, \ldots, t_n$, where $0 < t_1 < \cdots < t_n$ be business days that are being considered. We compute \begin{align*} E\left(\sum_{i=1}^n \pmb{1}_{b_1 < S_{t_i} < b_2} \right) &=\sum_{i=1}^n E\left(\pmb{1}_{b_1 < S_{t_i} < b_2} \right)\\ &=\sum_{i=1}^n \left[E\big(\pmb{1}_{S_{t_i} > b_1}\big) -E\big(\pmb{1}_{S_{t_i} \ge b_2} \big) \right]\\ &=\sum_{i=1}^n \left[\Phi\big(d_2^i(b_1)\big) - \Phi\big(d_2^i(b_2)\big)\right], \end{align*} where, for positive constant $K$, \begin{align*} d_2^i(K) = \frac{\ln\frac{S_0}{K} + \big(r-\frac{1}{2}\sigma^2 \big) t_i}{\sigma \sqrt{t_i}}. \end{align*} For general stochastic volatility, there is no simple method. Monte Carlo may be needed.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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