What is a natural monopoly. Give an example of an industry that is a natural monopoly.
You work in the hedging department of LMN, a company selling variable annuities with a GMWB rider. Your manager is interested in understanding more about volatility management strategies and asks you to prepare a memo. [a] {1.5 points] Describe principai objectives of volatility management strategies of equity-based guarantee products from the perspectives of manufacturer and client respectively. [MN currentlyr uses a capped volatility strategy to manage volatility risks. [MN targets a 60% allocation to the 5&P5DD index. The trigger level for the company is SEES. The sum of squared daily returns of the portfolio from the last 21 business days is now D.031 [i.e., 21 2 ij j; r E = = D3031, wherej r = daily return of the portfolio from a;- business days ago]. Assume that there are 252 business days in a year and lD'E': is the maximum equity allocation. [b] [1.5 points} Determine the equity allocation of the portfolio after any changes driven by the capped volatility strategy. [c] {(1.5 points} Describe actions, if any, to take to achieve the changes in equity allocation in part [b]. 1I"our co-worker mentions that he had just read about VIK indexed volatility management strategies. He stated that the VIEindexed fee rider enables the company to charge cliehts for all of the hedging costs of the company as they occur and that this in turn makes it a good strategy for dealing with spikes in volatility. {d} [1 point} Critique your co coworker' s thoughts on VIEindexed volatility manaoomant errata-ips (7 points) Your company uses Black's model with caplets' forward volatilities to price caps and is interested in offering more general options. You have chosen the LIBOR market model for pricing these options. Given: . f, (1, T,T) is the n times compounded annual forward rate as seen at time for the period [r, 7]. . . (7,I) is the n times compounded annual LIBOR spot rate at time r for the period [z, 7]. (a) (/ point) Describe the distribution of r (7.7) in the LIBOR market model. (b) (J point) Define caplet forward volatilities, 5, "(T., ), 1=0, 1, ..., and identify their advantages in pricing caps. Consider a call option on the "square root" of the 6-month LIBOR rate with payoff P( r, T) = N max( vr (r, T) - K,O), given that: N is the notional amount and it is set at $1 million. 1 = 0.5. . f. (0, 0.5, 1) = 3%. The option strike price K is set at 0.2. The one-year forward caplet volatility of" (1) is 0.2. The one-year discount factor Z (0, 1) is 0.95. (c) (2.5 points) Calculate the value of the call option. Suppose that the payoff function in the above is changed to P(r, I ) = Ne Kin)-41. (d) (/ point) Outline an algorithm to calculate the option value using the Monte Carlo method.You plan to set the forward rate volatility of (?) of f (1, T. T, ,) as given below: S, I
