M. 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. e way of hedging the risk of On 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 dP 0.1 Consider a portfolio manager who holds hs shares of the stock, and hP shares of the hedging asset, so the value of this part of the portfolio is V(S) hsS +hPPOS). A first order Taylor expansion of the portfolio value around S shows that if the stock price moves to S AS, en the portfolio value moves to approximately V (S AS) hs (S AS) hP (POS) ORAS), so the change in value is approximately (hs OP AS. Technically OP V (S AS) -V(S) has hP AS S C(AS) 2, (1) for some constant C 0 We define the number A OP which is denoted by the "delta" of the hedging asset (with respect to stock price risk). A delta hedge is implemented by purchasing an amount of the hedging asset such that the portfolio is neutral to small movements of the stock price, i.e., such that hs hPAP 0. W e arrive at hs hP 0 for the delta hedge, where positivity follows from the assumption that AP 0, i.e., that the hedging asset's price increases when the stock price decreases. An equivalent formulation of this hedge is of course that hP is chosen such that V'(S) 0