The key here is that jquantlib requires slf4j-api version 1.4.0 in the jar manifest:
(from MANIFEST.MF in jquantlib.jar)
Implementation-Vendor-Id: org.jquantlib Extension-List: slf4j-api slf4j-api-Extension-Name: slf4j-api slf4j-api-Implementation-Version: 1.4.0
Tomcat will try to use the version of slf4j-api that you're including in your app, but it's likely that what you've included isn't actually compatible. See http://grepcode.com/file/repo1.maven.org/maven2/org.apache.tomcat/tomcat-catalina/7.0.34/org/apache/catalina/util/Extension.java#Extension.isCompatibleWith%28org.apache.catalina.util.Extension%29 for details.
Bottom line - try to use slf4j-api version 1.4.0 in your application.