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 <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+hpP(S). A first order Taylor expansion of the portfolio value around S shows that if the stock price moves to SAS, then the portfolio value moves to approximately V(S+AS) ≈hs(S+ AS) + hp (P(S)+AS), so the change in value is approximately (hs + hp05) AS. Technically, |V(S+ for some constant C > 0. +AS) V(S) - -(" ӘР - ӘР as as hs+hp AS C(AS)², (1) We define the number Ap = 0, 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. We arrive at hp = hs ΔΡ > 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. (a) The classical Black-Scholes formula for the price, PK, of a so-called put option on the stock with strike price K is -σντ PK(S) = Ke˜³T [1 – N(x — o√√T)] – S [1 – N(x)], - where x = In(S/K)+(r+02/2)T οντ Here, is the instantaneous discount rate, σ is the stock's return volatility, and T is the time to maturity, all of which, together with K, are treated as constants so that PK is viewed as a function of a single variable, S. The function N(x) is the so-called cumulative distribution function of the standard normal distribution, a strictly positive increasing function with range between zero and one, with the important property that 1 N'(x) = 2π e We will discuss options and the cumulative normal distribution in some further detail later in the course, but this is sufficient for the current exercise. Finally, In(x) is the natural logarithm of x, eln(x) = x. Derive the following formula for the delta of the put option: APK = N(x) - 1. (b) As we know from class, an even better approximation of a function around a point is obtained by a second order Taylor expansion. The portfolio management equivalence of this principle is the use of a delta-gamma hedge. Technically, the holdings in the assets are chosen such that V'(S) = 0, and V"(S) = 0, leading to an error of the order (AS)³ for small AS. Define the gamma of an asset Ip = 32 and note, of course, that the stock's gamma is Ts = 0. N'(x) So√T Derive the following formula for the gamma of the put option: TPK = (c) To create a delta-gamma hedge, two put options are needed. Consider a portfolio with hs shares, h₁ put options with strike price K₁, and h₂ options with strike price K2 (and the same r, σ, and T). Use Taylor's formula to derive expressions for h₁ and h2 in the delta-gamma neutral hedge, as a function of APK₁, APK₂ TPKI' and TPK2