The QWebFrame.evaluateJavaScript will do it, but it doesn't work that well. It doesn't always return the right type, and I've found that it always returns None on function calls. One way of getting the result is to set the returned item to a variable you have access to then call evaluateJavaScript again on that variable.
frame.evaluateJavaScript("myVariable = getResult()")
result = frame.evaluateJavaScript("myVariable")
I just looked at the "DukascopyApplet", and its not a typical data type. Only standard data types like strings, ints, float, bools ... can be transferred or returned to python. You will probably have to go through the "DukascopyApplet" and find the specific data you want. Another thing that may help is to attach a Python object to the JavaScript. You can then call that python object's slot methods inside of your JavaScript.
class MyCLass(object):
@QtCore.Slot(str)
def doSomething(self, info):
# do something with the string info here
frame.addToJavaScriptWindowObject("varName", MyClass)
frame.evaluateJavaScript("varName.doSomething(DukascopyApplet.params.height)")
web.settings().setAttribute(QtWebKit.QWebSettings.WebAttribute.DeveloperExtrasEnabled, True)
inspector = QtWebKit.QWebInspector()
inspector.setPage(web.page())
After looking through the DukascopyApplet it doesn't look like there is any useful stored information there. Finding that data may be difficult.