Example 2.4

An Option Model for Hedging Investment Risk

Background Information On June 30, 1998 Harry Rockefeller purchases 1 share of Dell

Computer for $94.25.

However, Harry is worried about the possibility of Dell’s price falling, so he decides to buy some European puts(expiring on November 22, 1998) with an exercise price of $80. Each put is priced at $5.25.

As a function of the number of puts purchases, construct a worst-case and best-case scenario for the return on Harry’s portfolio between June 30, 1998 and November 22, 1998, assuming he sells his stock on the latter date.

How does the analysis change if Harry purchases 100 shares rather than 1 share?

Definitions

Before beginning to answer the questions posed in this example you need to know the following definitions.

First the return from any portfolio of investments is given by the formula

investedAmount

investedAmountreceivedAmounteturnR

Definitions – continued

Second, a European option on a stock is a contract that gives the owner of the option the right to buy (if the option is a call option) or the right to sell (if the option is a put option) 1 share of the stock for a particular price (called the exercise price)on a particular date in the future (called the exercise date).

Finally, we note that an analysis of a business situation often calls for computation of best-case and worst- case scenarios.

Understanding Options Since you might not have encountered options before,

we provide some intuition. The key is why Harry is considering put options.

He believes the price of the stock is going to fall. Therefore, he stands to lose on the stock he owns, and he wants to hedge this loss.

Now consider what would happen if he purchases puts on the stock.

– If the stock price falls from $80 to $75, then he can sell a share of stock in November for $80, buy it back for $75, and make $5.

The Model To model Harry's problem, we must make some

assumptions about the possible price of Dell stock on November 22, 1998.

– We assume that the price on this date will be between $40 and $150.

Next, we need to determine the value of the expiration. Recall that a put gives Harry the right to see a share of stock for $80 on November 22, 1998.

– If the price on that date is $80 or more, no value can be obtained by selling, so Harry will let his option expire.

– If the price on that date is less than $80, Harry will exercise his put.

The Model -- continued After entering the appropriate inputs, naming the

ranges, entering any trial values for the number of puts purchase and the future stock price, the spreadsheet model can be developed as shown here.

The Model -- continued The following steps need to occur to develop the shown

model.

– Amount invested. The total amount invested is the June price of 1 share of stock plus the cost of the puts, so enter the formula =CurrPrice+NumPuts*PutPrice in the AmtINvested cell.

– Amount received. In November, Harry will sell his 1 share of the stock at the going price, and he will exercise his option if the November price is below $80. Therefore, enter the formulas

=FutPrice, =NumPuts*IF(FutPrice>ExerPrice,0,ExerPrice-FutPrice)

and =SUM(B14:B15) in cells B14, B15 and B16.

The Model -- continued

The model shows a positive 9.09% return, but this is obviously a function of the number of puts Harry purchases and the future price of the stock.

To examine this dependency more closely, we use a two-way data table to determine the portfolio return for each stock price between $40 and $150 and each number of puts from 0 to 5.

The first few rows of this table along with a line chart created from the table are shown on the next slides.

Return as a Function of Future Price

The Model -- continued

The line chart clearly shows how puts shield Harry from risk if the price of the stock falls precipitously.

In fact, the more puts he buys, the more he stands to gain if the price falls significantly.

If the price stays about the same or increases, he loses slightly by buying more puts, but the difference is fairly minor.

This is illustrated in rows 21 and 22 where we use the MIN and MAX functions to find the worst-case scenario and the best-case returns.

The Model -- continued

Note that if Harry buys no puts, his worst case is a 57.56% loss, whereas if he buys 1 put, his worst case is only a 19.6% loss. Why?

With no puts, each $1 decrease in Dell’s price decreases his return by 1/94.25 = 1.06%. If he owns a put, then any decrease in Dell’s price below $80 costs him $1, but this is offset by a $1 increase in the cash flow from the put. Thus purchasing 1 put, makes him immune to decrease in Dell’s stock price below $80.

The Model -- continued Of course by purchasing a put, he caps his maximum

return at 50.75%. If he buys no puts, his maximum return is 59.15%.

In effect, the put is portfolio insurance that hedges Harry’s downside risk but limits his upside benefits.

It is interesting to note that purchasing more puts actually makes Harry's worst case inferior to purchasing 1 put.

The reason is that if Dell drops to $80, Harry loses money on this stock, and his investment in 5 puts is a complete washout.

Solution

Nevertheless, it is still not clear how many puts Harry should purchase.

This depends on the probability distribution of the future stock price.

It also depends on Harry’s attitude toward risk. Does he mind taking risks, or does he avoid them whenever possible?

Solution -- continued

Finally, it is easy to scale this model. If Harry purchases 100 shares of stock rather than 1, we simply multiply the appropriate quantities in the model by 100.

Specifically, his purchase amount for the stock and then amount he receives by selling the stock both increase by a factor of 100.

Of course, if he decides to buy, say 3 puts to hedge the risk from owning 1 share of stock, he will probably buy 300 puts if he owns 100 shares.