Ok, thats what I thought might have happened. I assume your Father has transferred this property to you for $150,000. What has apparently occurred is what is called a "bargain sale".
Essentially, you have two transactions where one part is considered a gift and the other is considered to be a sale.
Your father effectively completed a gift to you of $300,000 (i.e. $450k-150K or 2/3 of the property). He must file a gift tax return (i.e. Form 709) for the year of transfer and report the gift. No gift tax will be required to be paid, however, he will use up $300k of his $1,000,000 lifetime gift exemption. Also, he will be treated as completing a sale of 1/3 (i.e. $150k/$450k) of the property. If he meets the requirements under Sec. 121 he may exclude any gain from be recognized and taxed. To be able to use the exclusion he must have owned and used the property as his principal residence for at least two out of five years as of the date of transfer. If he did not use the property as a residence then this part of the transaction is taxable to him to the extent of any gain recognized. To calculate his gain he needs to determine his cost basis in the property and split it between the two transactions. 1/3 of the cost basis applies to the $150k sale and 2/3 of the cost basis applies to the gift.
For your purposes, any sale of the property by you is potentially taxable to you. Your cost basis in the property is equal to the sum of the purchase price (i.e. $150k) plus 2/3 of your Father's original cost basis (i.e. the portion of your Father's cost basis allocable to the gift portion of the transfer). When you receive a gift you retain the giftor's cost basis and holding period.
Were you to sell the property within one year of transfer (I assume you will not qualify to use the Sec. 121 exclusion I discussed above), you will have a reportable transaction. Specifically, 2/3 of the gain on the sale of the property will be treated as long-term capital gain (i.e. the gifted portion you received) and 1/3 will be treated as short-term capital gain. The long-term capital gain is subject to a maximum Federal tax rate of 15%. The short-term gain may be offset by other short-term losses. Any short-term gain left after offset by any short-term losses will be treated as ordinary income (i.e. the same as dividends and interest) subject to your marginal tax rates. If you hold the property for more then one year from the date of transfer, then all the gain will be treated as long-term capital gain. Don't forget NY taxes.
Whether your Father reports the transaction as I described above is irrelevant to you. You will be required by law to report any sale of the property as I described above. So get the basis information (and his date of purchase) from your Father so you can properly report your subsequent transaction.