Pricing and upper price bounds of relax certificates

Relax certificates are written on multiple underlying stocks. Their payoff depends on a barrier condition and is thus path-dependent. As 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 knock-in claim on the minimum of the stock prices. 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 one or two underlyings only. With the exception of the trivial case of one underlying, the price of the knock-in minimum claim always has to be calculated numerically. Hence, we derive semi-closed form upper price bounds. These bounds are the lowest upper price bounds which can be calculated without the use of numerical methods. In addition, the bounds are especially tight for the vast majority of relax certificates which are traded at a discount relative to the corresponding coupon bond. This is also illustrated with market data.