4. Portfolio hedging: An important role of a portfolio manager is to control for risk. For example, a portfolio may contain a high number of shares in an individual firm and therefore be highly exposed to movements of the stock price, S, of that firm. One way of hedging the risk of S changing is, of course, to sell some of the shares, but for various reasons this may not be feasible. Another approach is then to purchase another asset that increases in value if the stock price decreases. Mathematically, if the price of this hedging asset is P, this is expressed by requiring that