Does this work for you?
library(plyr)
vol1<-ddply(mydata,.(DATE), transform, max.name=NAME[which.max(quantile(VOLATILITY))])
DATE NAME VOLATILITY max.name
1 01/01/2011 VENTOS DEAD - 06/06/384 0.0538 VENTOS DEAD - 06/06/384
2 01/01/2012 ELECTRORAIL DEAD - DELIST 21/02/387 0.0730 ELECTRORAIL DEAD - DELIST 21/02/387
3 01/01/2012 BLYSTEIN FL.1384 0.0730 ELECTRORAIL DEAD - DELIST 21/02/387
4 01/02/2010 LOUVAIN-LA-NEUVE 0.0494 LOUVAIN-LA-NEUVE
5 01/06/2010 VILENZO DEAD - 25/11/370 0.0370 VILENZO DEAD - 25/11/370
6 01/07/2010 ECONOSTO KON. DEAD - 07/07/374 0.0386 ECONOSTO KON. DEAD - 07/07/374
7 01/08/2010 BBL DEAD - 30/06/465 0.0945 BBL DEAD - 30/06/465
8 01/08/2012 STORK DEAD - TAKEOVER 905099 0.1083 STORK DEAD - TAKEOVER 905099
9 01/09/2010 BRAINE-LE-COMTE SUSP 14/02/460 0.0357 BRAINE-LE-COMTE SUSP 14/02/460
10 01/09/2011 NOORD-EUR.HOUTH. 0.1361 <NA>
11 01/09/2011 BRUGEFI DEAD - 31/07/475 0.0687 <NA>
12 01/09/2012 GENERALE BANQUE DEAD - DEL. 30/12/490 0.1032 GENERALE BANQUE DEAD - DEL. 30/12/490
13 01/10/2010 BRILL (KON.) 0.0460 BRILL (KON.)
14 01/10/2011 HOEK'S MACHINE DEAD - DELIST. 0.3383 <NA>
15 01/10/2011 GENK LOGISTICS 0.0465 <NA>
16 01/10/2011 PALTHE DEAD - 4/2/475 0.2183 <NA>
17 01/10/2011 GIB NEW 0.3308 <NA>
18 01/10/2012 WORLD SCOPE (CADB TEST STOCK) 0.0840 WORLD SCOPE (CADB TEST STOCK)
19 01/11/2011 OBOURG (CIMENTS) 0.0393 OBOURG (CIMENTS)
20 01/12/2011 GROENIJK.YLCBN. DEAD - DELIST.31/05/479 0.2008 GROENIJK.YLCBN. DEAD - DELIST.31/05/479
Updated solution:
library(plyr)
vol2<-ddply(x,.(DATE), transform,quantile=ifelse(VOLATILITY<quantile(VOLATILITY,p=0.25),1,
ifelse(((VOLATILITY>quantile(VOLATILITY,p=0.25))& (VOLATILITY<quantile(VOLATILITY,p=0.5))),2,ifelse(((VOLATILITY>quantile(VOLATILITY,p=0.5))& VOLATILITY<quantile(VOLATILITY,p=0.75)),3,4))))
DATE NAME VOLATILITY quantile
1 01/01/2011 VENTOS DEAD - 06/06/384 0.0538 4
2 01/01/2012 ELECTRORAIL DEAD - DELIST 21/02/387 0.0730 4
3 01/01/2012 BLYSTEIN FL.1384 0.0730 4
4 01/02/2010 LOUVAIN-LA-NEUVE 0.0494 4
5 01/06/2010 VILENZO DEAD - 25/11/370 0.0370 4
6 01/07/2010 ECONOSTO KON. DEAD - 07/07/374 0.0386 4
7 01/08/2010 BBL DEAD - 30/06/465 0.0945 4
8 01/08/2012 STORK DEAD - TAKEOVER 905099 0.1083 4
9 01/09/2010 BRAINE-LE-COMTE SUSP 14/02/460 0.0357 4
10 01/09/2011 NOORD-EUR.HOUTH. 0.1361 4
11 01/09/2011 BRUGEFI DEAD - 31/07/475 0.0687 1
12 01/09/2012 GENERALE BANQUE DEAD - DEL. 30/12/490 0.1032 4
13 01/10/2010 BRILL (KON.) 0.0460 4
14 01/10/2011 HOEK'S MACHINE DEAD - DELIST. 0.3383 4
15 01/10/2011 GENK LOGISTICS 0.0465 1
16 01/10/2011 PALTHE DEAD - 4/2/475 0.2183 2
17 01/10/2011 GIB NEW 0.3308 3
18 01/10/2012 WORLD SCOPE (CADB TEST STOCK) 0.0840 4
19 01/11/2011 OBOURG (CIMENTS) 0.0393 4
20 01/12/2011 GROENIJK.YLCBN. DEAD - DELIST.31/05/479 0.2008 4