Trading energy from wind and other forms of stochastic generation in competitive electricity markets is challenging due to the limited predictability of these resources. This paper examines the specific case of single-price balancing markets and derives risk-constrained strategies in a probabilistic framework, going beyond the trivial zero/max solution, which would have participants offer either zero or their maximum energy production based on a prediction of whether the system will be in net up- or down-regulation. The zero/max approach is unacceptable in reality as it exposes the participant to potentially huge imbalance charges, and would violate price taker assumption for a portfolio of significant size. Here, we propose several trading strategies that control risk by hedging against penalising balancing prices in favour of rewarding ones by contracting forecast generation plus some adjustment. These strategies are formulated in a probabilistic framework to address the presence of forecast uncertainty and asymmetric costs in balancing markets. A case study using data from the Great Britain electricity market is presented and it is shown that the proposed strategies are able to simultaneously increase revenue and reduce risk using risk-constrained strategies. Furthermore, the required forecasts of electricity prices and system length are produced using standard tools and widely available explanatory information and are found to have sufficient skill to increase revenue compared to not hedging.