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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 Chapter 11.

Consider, for example, a fund manager with a $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 $29 million in the next 6 months. One approach is to buy 1,000 6-month put option contracts on the S&P 500 with an exercise price of 290 and a maturity in 6 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 $27 million. Since each option contract is on 100 times the index, the total value of the put options is $2 million. This brings the value of the entire holding back up to $29 million. Of course, insurance is not free. In this example the put options could cost the portfolio manager as much as $1 million.

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 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 necessary to hedge it. This is simply 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 manager 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 will 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 $30 million who wishes to buy a European put on the portfolio with a strike price of $29 million and an exercise date in 6 months. Recall that the delta of a European put on an index is given by

where, with the usual notation,

Since, in this case, the fund manager's portfolio mirrors the index, this is also the delta of a put on the portfolio when it is regarded as a single security. The delta is negative. Accordingly, in order to create the put option synthetically, the fund manager should ensure that at any given time a proportion e-o(T-Of! _ N(fh)~\

of the stocks in the original $30 million portfolio have been sold and the proceeds invested in riskless assets. As the value of the original portfolio declines, the delta of the put becomes more negative and the proportion of the portfolio sold must be increased. As the value of the original portfolio increases, the delta of the put becomes less negative and the proportion of the portfolio sold must be decreased (i.e., some of the original portfolio must be repurchased).

Using this strategy to create portfolio insurance means that at any given time funds are divided between the stock portfolio on which insurance is required and riskless assets. As the value of the stock portfolio increases, riskless assets are sold and the position in the stock portfolio is increased. As the value of the stock portfolio declines, the position in the stock portfolio is decreased and riskless assets are purchased. The cost of the insurance arises from the fact that the portfolio manager is always selling after a decline in the market and buying after a rise in the market.

Using index futures to create portfolio insurance can be preferable to using the underlying stocks, as the transactions costs associated with trades in index futures are generally less than those associated with the corresponding trades in the underlying stocks. The portfolio manager considered earlier would keep the $30 million stock portfolio intact and short index futures contracts. The dollar amount of futures contracts shorted as a proportion of the value of the portfolio should from equations (13.2) and (13.5) be e-q(T-t)e-(r-q)(T>-,){l _ N{di)] = eq(T--T)e-riT'-t)[l _ N ^

where T* is the maturity date of the futures contract. If the portfolio is worth K\ 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 t*<r-r>e-r<r-t)[l _ Nidl)]£L

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