Trigger Authcache Reload on Field Change for Drupal Commerce Products
Authcache comes with an integration module for the Cache Expiration project. See also the Authcache 2 tuning guide - integrate with Cache Expiration module.

It might get tricky to find all pages where a particular product/price is displayed due to the separation of products and display nodes in commerce. Depending on how frequent prices are changed, you might consider nuking the whole cache, e.g. using a combination of Rules and the Cache Actions module.