In an uncertain volatility model where only the stock and the money market account are traded, the upper price bound of a European claim is given by the solution of a Black-Scholes-Barenblatt equation. If an additional hedge instrument is available, the price bound can be tightened. This is also true if the set of admissible strategies is restricted to tractable strategies, which are defined as sums of Black-Scholes strategies. We study the structure of both strategies, the general strategies and the tractable strategies, when an additional convex instrument is available. For a call and a bullish vertical spread, we give closed-form solutions for the optimal tractable hedge when the additional instrument is a call option. We show that the position in the additional convex claim as well as the reduction in the price bounds allow to capture the amount of convexity risk a claim is exposed to.