Avalising
It's an even more recent word in English. The earliest I found (not counting French-English dictionaries) is from 1975.
Avali[s|z]e[|d|r|ing] barely registers with Google Ngrams Viewer.
A snippet of a what appears to be a 1975 edition of Euromoney says:
Often, notes carry an "aval" (unconditional commitment to pay by an avalizing bank) or a separate payment guarantee. It is essential that in the case of guarantees separate from the underlying trade obligations, the guarantees are irrevocable, unconditional and transferable.
Aval
However, the financial term aval is a bit older in English, although often written in italics or quote marks that suggests it's uncommon or a loanword.
In the following 1843 edition of The Merchants' Magazine and Commercial Review, they are clearly discussing the French system and writing Aval and pour Aval in italics as foreign words:
It's defined as a French declaration in the The dictionary of trade products, manufacturing, and technical terms (1858):
And it's also defined as part of Canadian law in A Law Dictionary, Adapted to the Constitution and Laws of the United States of America (1868):
The French law is discussed in 1811, 1814, 1839, 1854, 1864 and 1868, and of course in many French-English dictionaries.
It seems as these laws and contracts became more common in English-speaking countries, so the term was used more. One interesting excerpt can be found in The Law Quarterly Review of 1888. In a review of an English translation of The French Code of Commerce on the top of page 102, they imply the French word was in common use and well understood:
On the other
hand, the translation of 'aval' by 'surety' (art. 141) is calculated to mislead.
The word 'aval' belongs to the universal phraseology of the law merchant.
Ḥawāla
The French aval has Persian roots. A snippet from Roman Ghirshman's 1951 Iran from the Earliest Times to the Islamic Conquest (and repeated in full here) seems to suggest:
The bill has been known since the second millenium BC but its use had been limited, and in effect was no more than the recognition of a debt fixing ther date of repayment. In the Sassanian period [224-637 AD] it became a legally recognised title-deed. The banks of the empire run by Iranians or Jews employed a highly developed system of monetary exchange by writing. How many financiers and bankers know, for example, that the word 'cheque' or the term avaliser come from the Pahlavi language and were invented by the Iranian banking institutions of this remote age?
From Wikipedia:
Hawala has its origins in classical Islamic law and is mentioned in texts of Islamic jurisprudence as early as the 8th century. Hawala itself later influenced the development of the agency in common law and in civil laws, such as the aval in French law and the avallo in Italian law. The words aval and avallo were themselves derived from hawala.
