An Option Model for Hedging Investment Risk

1
Example 2.4

• An Option Model for Hedging Investment Risk

2
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 Dells 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 Harrys 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?

3
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

4
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.

5
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.

6
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.

7
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.

8
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 CurrPriceNumPuts
  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, NumPutsIF(FutPricegtExerPr
  ice,0,ExerPrice-FutPrice) and
  SUM(B14B15) in cells B14, B15 and B16.

9
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.

10
(No Transcript)
11
Return as a Function of Future Price
12
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.

13
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 Dells price
  decreases his return by 1/94.25 1.06. If he
  owns a put, then any decrease in Dells 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
  Dells stock price below 80.

14
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 Harrys 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.

15
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 Harrys attitude toward risk.
  Does he mind taking risks, or does he avoid them
  whenever possible?

16
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.