Financial Derivatives-PORTFOLIO INSURANCE-Notes-Financial Management, Study notes of Financial Management

Download Financial Derivatives-PORTFOLIO INSURANCE-Notes-Financial Management and more Study notes Financial Management in PDF only on Docsity! PORTFOLIO INSURANCE Portfolio managers holding a well-diversified stock portfolio are sometimes interested in insuring themselves against the value of the portfolio dropping below a certain level. One way of doing this is by holding, in conjunction with the stock portfolio, put options on a stock index. This strategy was discussed in earlier units. Consider, for example, a fund manager with a Rs. 30 million portfolio whose value mirrors the value of the S&P 500. Suppose that the S&P 500 is standing at 300 and the manager wishes to insure against the value of the portfolio dropping below Rs. 29 million in the next six months. One approach is to buy 1,000 six-month put option contracts on the S&P 500 with a strike price of 290 and a maturity in six months. If the index drops below 290, the put options will become in the money and provide the manager with compensation for the decline in the value of the portfolio. Suppose, for example, that the index drops to 270 at the end of 6 months. The value of the manager's stock portfolio is likely to be about Rs. 27 million. Since each option contract is on 100 times the index, the total value of the put options is Rs. 2 million. This brings the' value of the entire holding back up to Rs. 29 million. Of course, insurance is not free. In this example the put options could cost the portfolio manager as much as Rs. 1 million Creating Options Synthetically An alternative approach open to the portfolio manager involves creating the put options synthetically. This involves taking a position in the underlying asset (or futures on the underlying asset) so that the delta of the position is maintained equal 240 to the delta of the required option. If more accuracy is required, the next step is to use traded options to match the gamma and vega of the required option. The position necessary to create an option synthetically is the reverse of that nec essary to hedge it. This is a reflection of the fact that a procedure for hedging an option involves the creation of an equal and opposite option synthetically. There are two reasons why it may be more attractive for the portfolio man- ager to create the required put option synthetically than to buy it in the market. The first is that options markets do not always have the liquidity to absorb the trades that managers of large funds would like to carry out. The second is that fund managers often require strike prices and exercise dates that are different from those available in traded options markets. The synthetic option can be created from trades in stocks themselves or from trades in index futures contracts. We first examine the creation of a put option by trades in the stocks themselves. Consider again the fund manager with a well-- diversified portfolio worth Rs. 30 million who wishes to buy a European put on the portfolio with a' strike price of Rs. 29 million and an exercise date in six months. Recall that the delta of a European put on an index is given by earlier would keep the Rs. 30 million stock portfolios intact and short index futures contracts. From equations (4.1) and (4.4), the amount of futures contracts shorted as a proportion of the value of the portfolio should be e-q(T-t) e- (r - q) ( T* - t) )[1- N(d1) ] = e-q ( T* - T) e-r (T* - t) [1- N(d1) ] Where T* is the maturity date of the futures contract. If the portfolio is worth. K1 times the index and each index futures contract is on K2 times the index, this means that the number of futures contracts shorted at any given time should be e-q ( T* - T) e-r (T* - t) [1- N(d ) ] k1 k 2 Example 4.11 In the example given at the beginning of this section, suppose that the volatility of the market is 25% per annum, the risk-free interest rate is 9% per annum, and the dividend yield on the market is 3% per annum. In this case, S = 300, X = 290, r = 0.09, q = 0,03, σ = 0.25, and T - t = 0.5. The delta of the option that is required is e-q(T-t) )[ N(d1) - 1 ] = - 0.322 Hence, if trades in the portfolio are used to create the option, 32.2% of the portfolio should be sold initially. If nine-month futures contracts on the S&P 500 are used, T* - T = 0.25, T' - t = 0.75, K1 = 100,000, K2 = 500, so that the number of futures contracts shorted should be 1 e-q ( T* - T) e-r (T* - t) [1- N(d1) ] k1 = 61.6 k 2 An important issue when put options are created synthetically for portfo lio insurance is the frequency with which the portfolio manager's position should be adjusted or rebalanced. With no transaction costs, continuous rebalancing is 243 ,optimal. However, as transactions costs increase, the optimal frequency of rebal- ancing declines. This issue is discussed by Leland F 0 7 E . Up to now we have assumed that the portfolio mirrors the index. As dis- cussed in Chapter 12, the hedging scheme can be adjusted to deal with other situations. Tile strike price for the options used should be the expected level of the market index when the portfolio's value reaches its insured value. The number of index options used should be β times the number of options that would be required if the portfolio had a beta of 1.0. Example 4.12 Suppose that the risk-free rate of interest is 5% per annum, the S&P 500 stands at 500, and the value of a portfolio with a beta of 2.0 is Rs. 10 million. Suppose that the dividend yield on the S&P 500 is 3%, the dividend yield on the portfolio is 2%, and that the portfolio manager wishes to insure against a decline in the value of the portfolio to below Rs. 9.3 million in the next year. If the value of the portfolio declines to Rs. 9.3 million at the end of the year, the total return (after taking account of the 2% dividend yield) is approximately -5% per annum. This is 10% per annum less than the risk-free rate. We expect the market to perform 5% worse than the risk-free rate (i.e., .to provide zero return) in these circumstances. Hence, we expect a 3% decline in the S&P 500 since this index does not take any account of dividends. The correct strike price for the put options that are created is therefore 485. The number of put options required is beta times the value of the portfolio I divided by the value of the index, or 40,000 (i.e., 400 contracts). F 0 7 E 15See H. E. Leland, "Option Pricing and Replication with Transactions Costs," Journal of Finance, 40 (December 1985), 1283-1301. 244