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Stock Warrants | 3 Months Ended | 12 Months Ended
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Mar. 31, 2023 | Dec. 31, 2022
Stock Warrants |  |
Stock Warrants |
4. | Stock Warrants
On December 20, 2018, the Company issued a non-dilutive warrant that expires on December 20, 2025. The warrant has an exercise price of $1,250,000 and is exercisable into (i) shares of common stock equal to five percent (5%), calculated on a post-exercise basis, of the fully diluted capitalization of the Company, as of the date or dates of exercise, plus (ii) shares of preferred stock of each class or series of preferred stock of the Company equal to five percent (5%), calculated on a post-exercise basis, of the total issued and outstanding number of preferred shares of the Company, as of the date or dates of exercise. At March 31, 2023 and December 31, 2022, this warrant was exercisable into a total of 1,700,297 and 1,697,525 shares, respectively, of the Company’s capital stock. The fair value of this warrant was $7,384,084 and $7,392,041 at March 31, 2023 and December 31, 2022, respectively, and was estimated using a Black-Scholes valuation model with the following assumptions: ​
​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
​ |  | March 31, |  | December 31, |
​ |  | 2023 |  | 2022 |
Estimated per-share fair value of common stock | ​ | $ | 5.01 | ​ | $ | 5.01 | ​
Expected term | ​ |  | 2.7 | years |  | 3.0 | years
Volatility | ​ |  | 26.01 | % |  | 25.07 | %
Dividend rate | ​ |  | 0.0 | % |  | 0.0 | %
Discount rate | ​ |  | 3.87 | % |  | 4.24 | %
​ At both March 31, 2023 and December 31, 2022, the Company estimated the fair value of its common stock by reference to the price per share at which it was currently selling shares of preferred stock. Since the Company’s preferred stock is convertible into two shares of common stock, the estimated fair value of common stock was determined to be one-half the price per share of its recent sales of preferred stock. In October 2020, the Company issued a warrant to a vendor in exchange for platform and technology services provided to the Company in connection with its offering of Series B Preferred Stock. This warrant is exercisable into shares of Series B Preferred Stock equal to 2% of the total number of shares of Series B Preferred Stock issued to investors in connection with the Company’s offering of Series B Preferred Stock. The exercise price of this warrant is $5.01, and the warrant expires in October 2025. At both March 31, 2023 and December 31, 2022, the warrant was exercisable into 116,457 shares of Series B Preferred Stock, and the estimated value of the warrant liability was $121,200 and $127,059 at March 31, 2023 and December 31, 2022, respectively. In February 2019, the Company entered into a warrant agreement that provided the holder with the right to acquire $1,000,000 worth of shares of the Company’s capital stock upon the occurrence of the Company raising $5,000,000 in an equity financing. As a result of the Series A Preferred Stock issuances in 2020, this threshold was achieved, and the warrant is now exercisable into 273,972 shares of Series A Preferred Stock at a price of $3.65 per share. This warrant expires in February 2024. As the Company issues additional shares of common or preferred stock, the estimated fair value of the warrant liability is expected to increase. | 8.Stock Warrants On December 20, 2018, the Company issued a non-dilutive warrant that expires on December 20, 2025. The warrant has an exercise price of $1,250,000 and is exercisable into (i) shares of common stock equal to five percent (5%), calculated on a post-exercise basis, of the fully diluted capitalization of the Company, as of the date or dates of exercise, plus (ii) shares of preferred stock of each class or series of preferred stock of the Company equal to five percent (5%), calculated on a post-exercise basis, of the total issued and outstanding number of preferred shares of the Company, as of the date or dates of exercise. At December 31, 2022 and December 31, 2021, this warrant was exercisable into a total of 1,697,525 and 1,385,724 shares, respectively, of the Company's capital stock. The fair value of this warrant was $7,392,041 and $4,021,810 at December 31, 2022 and 2021, respectively, and was estimated using a Black-Scholes valuation model with the following assumptions: ​
​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
​ |  | December 31, |  | December 31, |
​ |  | 2022 |  | 2021 |
Estimated per-share fair value of common stock | ​ | $ | 5.01 | ​ | $ | 3.76 | ​
Expected term | ​ |  | 3.0 | years |  | 4.0 | years
Volatility | ​ |  | 25.07 | % |  | 30.3 | %
Dividend rate | ​ |  | 0.0 | % |  | 0.0 | %
Discount rate | ​ |  | 4.24 | % |  | 1.2 | %
​ At both December 31, 2022 and 2021, the Company estimated the fair value of its common stock by reference to the price per share at which it was currently selling shares of preferred stock. Since the Company's preferred stock is convertible into two shares of common stock, the estimated fair value of common stock was determined to be one-half the price per share of its recent sales of preferred stock. In October 2020, the Company issued a warrant to a vendor in exchange for platform and technology services provided to the Company in connection with its offering of Series B Preferred Stock. This warrant is exercisable into shares of Series B Preferred Stock equal to 2% of the total number of shares of Series B Preferred Stock issued to investors in connection with the Company's offering of Series B Preferred Stock. The exercise price of this warrant is $5.01, and the warrant expires in October 2025. At December 31, 2022 and 2021, the warrant was exercisable into 116,457 and 34,870 shares of Series B Preferred Stock, respectively, and the estimated value of the warrant liability was $127,059 and $65,426, respectively. In February 2019, the Company entered into a warrant agreement that provided the holder with the right to acquire $1,000,000 worth of shares of the Company’s capital stock upon the occurrence of the Company raising $5,000,000 in an equity financing. As a result of the Series A Preferred Stock issuances in 2020, this threshold was achieved, and the warrant is now exercisable into 273,972 shares of Series A Preferred Stock at a price of $3.65 per share. This warrant expires in February 2024. As the Company issues additional shares of common or preferred stock, the estimated fair value of the warrant liability is expected to increase.

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