We study the destabilising effect of dynamic hedging strategies on the price of the underlying in the presence of sunk costs of transaction. Once sunk costs of transaction are taken into account, continuous portfolio rehedging is no longer an optimal strategy. Using a non-optimising (local in time) strategy for portfolio rebalancing, explicit dynamics for the price of the underlying are derived, focusing in particular on the excess volatility and feedback effects of these portfolio insurance strategies. Further, we show how these latter depend on the heterogeneity of the insured payoffs. Finally, conditions are derived under which it may still be reasonable, from a practical viewpoint, to implement Black - Scholes strategies.