Weighted-Average Key Assumptions and Fair Value Results for Shares Issued under ESPP

We use the Black-Scholes option pricing model to calculate the fair value of shares issued under the ESPP. The Black-Scholes model relies on a number of key assumptions to calculate estimated fair values. Expected volatility is based on the historical volatility of our common stock, and the expected term represents the period of time the ESPP purchase rights are expected to be outstanding and approximates the offering period. The risk-free interest rate is based on yields on U.S. Treasury Securities with a maturity similar to the estimated expected term of the ESPP purchase rights. We assume no expected dividends.

The following table sets forth the weighted-average key assumptions and fair value results for shares issued under the ESPP during the fiscal years ended September 30, 2024, 2023 and 2022:


	Year Ended September 30,
	2024			2023			2022
Expected dividend yield		0.00	%		0.00	%		0.00	%
Risk-free interest rate		5.33	%		3.95	%		0.41	%
Expected volatility		50.45	%		65.71	%		69.58	%
Expected life (in years)		0.50			0.50			0.50
Weighted-average fair value of shares issued (per share)	$	6.69		$	9.73		$	27.54

Schedule of Stock-based Compensation

The amounts included in the Consolidated Statements of Operations related to stock-based compensation are as follows (dollars in thousands):


	Year Ended September 30,
	2024		2023		2022
Cost of connected services	$	288	$	445	$	499
Cost of professional services		2,345		3,258		3,267
Research and development		10,449		17,167		10,196
Sales and marketing		2,199		3,454		3,569
General and administrative		8,392		16,442		6,545
Restructuring and other costs, net		—		—		4,000
Total	$	23,673	$	40,766	$	28,076