Yeah, it's read only:
var multiplier: CGFloat { get } 
You can only specify the multiplier at creation time.  In contrast, you can change the non-constant constant property at run-time:
var constant: CGFloat
Edit:
It takes a little effort, but to change immutable properties, you have to create a new constraint and copy all but the property that you want to change.  I keep playing around with different techniques for this, but am currently exploring this form:
let constraint = self.aspectRatioConstraint.with() {
    (inout c:NSLayoutConstraint.Model) in
    c.multiplier = self.aspectRatio }
or used in a more realistic context:
var aspectRatio:CGFloat = 1.0 { didSet {
    // remove, update, and add constraint
    self.removeConstraint(self.aspectRatioConstraint)
    self.aspectRatioConstraint = self.aspectRatioConstraint.with() {
        (inout c:NSLayoutConstraint.Model) in
        c.multiplier = self.aspectRatio }
    self.addConstraint(self.aspectRatioConstraint)
    self.setNeedsLayout()
}}
Where the backing code is:
extension NSLayoutConstraint {
    class Model {
        init(item view1: UIView, attribute attr1: NSLayoutAttribute,
            relatedBy relation: NSLayoutRelation,
            toItem view2: UIView?, attribute attr2: NSLayoutAttribute,
            multiplier: CGFloat, constant c: CGFloat,
            priority:UILayoutPriority = 1000) {
                self.firstItem = view1
                self.firstAttribute = attr1
                self.relation = relation
                self.secondItem = view2
                self.secondAttribute = attr2
                self.multiplier = multiplier
                self.constant = c
                self.priority = priority
        }
        var firstItem:UIView
        var firstAttribute:NSLayoutAttribute
        var relation:NSLayoutRelation
        var secondItem:UIView?
        var secondAttribute:NSLayoutAttribute
        var multiplier:CGFloat = 1.0
        var constant:CGFloat = 0
        var priority:UILayoutPriority = 1000
    }
    func priority(priority:UILayoutPriority) -> NSLayoutConstraint {
        self.priority = priority;
        return self
    }
    func with(configure:(inout Model)->()) -> NSLayoutConstraint {
        // build and configure model
        var m = Model(
            item: self.firstItem as! UIView, attribute: self.firstAttribute,
            relatedBy: self.relation,
            toItem: self.secondItem as? UIView, attribute: self.secondAttribute,
            multiplier: self.multiplier, constant: self.constant)
        configure(&m)
        // build and return contraint from model
        var constraint = NSLayoutConstraint(
            item: m.firstItem, attribute: m.firstAttribute,
            relatedBy: m.relation,
            toItem: m.secondItem, attribute: m.secondAttribute,
            multiplier: m.multiplier, constant: m.constant)
        return constraint
    }
}