Stock-Based Compensation Expense and Shareholders' Equity (Details 3) |
9 Months Ended | |
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Apr. 30, 2025 |
Apr. 30, 2024 |
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| Stockholders' Equity and Stock-based Compensation [Abstract] | ||
| 3-year historical volatility | 36.00% | 35.00% |
| 3-year risk free rate | 3.40% | 4.50% |
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- Definition Stockholders' equity and stock-based compensation. No definition available.
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- Definition The estimated measure of the percentage by which a share price is expected to fluctuate during a period. Volatility also may be defined as a probability-weighted measure of the dispersion of returns about the mean. The volatility of a share price is the standard deviation of the continuously compounded rates of return on the share over a specified period. That is the same as the standard deviation of the differences in the natural logarithms of the stock prices plus dividends, if any, over the period. Reference 1: http://www.xbrl.org/2003/role/disclosureRef
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- Definition The risk-free interest rate assumption that is used in valuing an option on its own shares. Reference 1: http://www.xbrl.org/2003/role/disclosureRef
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