There is no amputation of the hand except in the case of one who steals something worth one quarter of a dinar or more

It is proven in as-Saheehayn from the hadith of ‘Aa’ishah (may Allah be pleased with her) that the Prophet (blessings and peace of Allah be upon him) said: “The hand of the thief is not to be cut off except for one quarter of a dinar or more.” Narrated by al-Bukhaari (6789) and Muslim (4494). 

This hadith indicates that one quarter of a dinar is the minimum value for which the hand of the thief must be cut off. So whoever steals something worth one quarter of a dinar or more is to have his hand cut off, but if a person steals three dirhams, his hand is not to be cut off unless their value on the day of the theft is equal to one quarter of a dinar, because the basic standard is the dinar, not dirhams. The value of the dirham varies, so its value is to be checked against that of the dinar, and whether it was worth more or less, at the time of the theft. If it was equal in value to one quarter of a dinar, then the hand of the thief is to be cut off. 

It is on this basis that the hadith of Ibn ‘Umar (may Allah be pleased with him) is to be interpreted, in which he said that the Prophet (blessings and peace of Allah be upon him) cut off the hand of the thief for the theft of a shield, the value of which was three dirhams. Agreed upon. That is, because at that time three dirhams were equal in value to a quarter of a dinar. 

Imam ash-Shaafa‘i (may Allah have mercy on him) said concerning the minimum value of stolen goods (for which the hand of the thief is to be cut off), and that it is equal to the value of one quarter of a dinar: 

The hadith of Ibn ‘Umar is in harmony with the hadith of ‘Aa’ishah, because three dirhams at the time of the Prophet (blessings and peace of Allah be upon him) and afterwards were equal in value to one quarter of a dinar… 

So if a thief was caught, and the value of what he stole on the day that he stole it was worked out, and was equivalent to one quarter of a dinar, his hand would be cut off, but if it fell short of the value of one quarter of a dinar, his hand would not be cut off. 

So the basic measure is one quarter of a dinar. If dirhams rise in value to the point that two dirhams become equal in value to a dinar, then the hand of the thief is to be cut off for one quarter of a dinar, even if it is worth half a dirham. If dirhams fall in value to the point that one dinar becomes equivalent to a hundred dirhams, the hand of the thief is to be cut off for one quarter of a dinar, which (in that case) would be twenty-five dirhams. 

Dirhams are no more than items, like garments or sheep. If one quarter of a dinar, or something equal in value to one quarter of a dinar, or something equal in value to ten sheep, is stolen, then the hand is to be cut off for one quarter of a dinar, which is equivalent to ten sheep. 

Similarly, if a thief steals something that is equivalent to one quarter of a dinar, and it is one quarter of a sheep, then his hand is cut off for the equivalent of one quarter of a dinar. 

The basic measure is the dinar, so the dirham is like trade goods, and no attention is to be paid to whether its value is low or high [at the time of the theft]. 

End quote from al-Umm (6/159) 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: 

The words “one quarter of a dinar”, and the dinar is the mithqaal [a unit of weight], and the mithqaal weighs 4.25 grams. So one quarter of a dinar weighs one and one sixthteenth grams. So if a person steals gold that weighs one gram plus one sixtheenth of a gram, his hand is to be cut off, because of the hadith of ‘Aa’ishah (may Allah be pleased with her), according to which the Prophet (blessings and peace of Allah be upon him) said: “The hand of the thief is not to be cut off except for one quarter of a dinar or more.” This is narrated in as-Saheehayn. Based on that, this hadith is specifying the general meaning of the verse in which Allah, may He be exalted, says (interpretation of the meaning): “And (as for) the male thief and the female thief, cut off (from the wrist joint) their (right) hands… ” [al-Maa’idah 5:38]. So for anything below this threshold, the hand is not to be cut off. 

With regard to the other hadith, which says that the Messenger of Allah (blessings and peace of Allah be upon him) cut off a thief’s hand for stealing a shield the value of which was three dirhams, this is to be interpreted as indicating that three dirhams were equivalent to one quarter of a dinar at that time, and a dinar was equivalent to twelve dirhams of silver.

End quote from ash-Sharh al-Mumti‘ (14/334) 

As the questioner mentioned, a dinar weighs 4.25 grams of gold, and one quarter of a dinar is equivalent to one and one sixteenth grams, which as the questioner says is 1.0625 grams. 

As for defining the price of that in terms of currency, it is clear from the answer that the price varies from one day to another, according to the price of gold. So it is the price on the day of the theft that should be worked out.