Pricing and Upper Price Bounds of Relax Certificates

Relax certificates are written on multiple underlying stocks. The payoff depends on a barrier condition such that it is path–dependent. So long as none of the underlying assets crosses a lower barrier, the investor receives the payoff of a coupon bond. Otherwise, there is a cash settlement at maturity which depends on the lowest stock return. Thus, the products consist of a knock-out coupon bond and a knockin minimum option. In a Black–Scholes model setup, the price of the knock–out part can be given in closed (or semi–closed) form in the case of two underlyings, but not for more than two. Without considering the trivial case of one underlying, the price of the knock-in minimum option is to be calculated numerically. Alternatively to exact pricing, we derive semi–closed form upper price bounds. These bounds are the lowest upper price bounds which can be derived without the usage of numerical methods. In addition, the bounds are especially tight with respect to the market convention that the certificates are traded at a discount of the corresponding coupon bond. This is also illustrated with market data.