Steven R. Dunbar
Department of Mathematics
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University of Nebraska-Lincoln
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Math 489/Math 889
Stochastic Processes and
Advanced Mathematical Finance
Dunbar, Fall 2010

Speculation and Hedging

Rating

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Question of the Day

Discuss examples in your experience of speculation. (Example: think of “scalping tickets”.) A hedge is an investment that is taken out specifically to reduce or cancel out risk. Discuss examples in your experience of hedges.

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Key Concepts

Vocabulary

Mathematical Ideas

Options have two primary uses, speculation and hedging. Consider speculation first.

Example: Speculation on a stock with calls

An investor who believes that a particular stock, say XYZ, is going to rise may purchase some shares in the company. If she is correct, she makes money, if she is wrong she loses money. The investor is speculating. Suppose the price of the stock goes from $2.50 to $2.70, then the investor makes $0.20 on each $2.50 investment, or a gain of 8%. If the price falls to $2.30, then the investor loses $0.20 on each $2.50 share, for a loss of 8%. These are both standard calculations.

Alternatively, suppose the investor thinks that the share price is going to rise within the next couple of months, and that the investor buys a call option with exercise price of $2.50 and expiry date in three months’ time.

Now assume that it costs $0.10 to purchase a European call option on stock XYZ with expiration date in three months and strike price $2.50. That means in three months time, the investor could, if the investor chooses to, purchase a share of XYZ at price $2.50 per share no matter what the current price of XYZ stock is! Note that the price of $0.10 for this option may or may not be an appropriate price for the option, I use $0.10 simply because it is easy to calculate with. However, 3-month option prices are often about 5% of the stock price, so this is reasonable. In three months time if the XYZ stock price is $2.70, then the holder of the option may purchase the stock for $2.50. This action is called exercising the option. It yields an immediate profit of $0.20. That is, the option holder can buy the share for $2.50 and immediately sell it in the market for $2.70. On the other hand if in three months time, the XYZ share price is only $2.30, then it would not be sensible to exercise the option. The holder lets the option expire. Now observe carefully: By purchasing an option for $0.10, the holder can derive a net profit of $0.10 ($0.20 revenue less $0.10 cost) or a loss of $0.10 (no revenue less $0.10 cost.) The profit or loss is magnified to 100% with the same probability of change. Investors usually buy options in quantities of hundreds, thousands, even tens of thousands so the absolute dollar amounts can be quite large. Compared to stocks, options offer a great deal of leverage, that is, large relative changes in value for the same investment. Options expose a portfolio to a large amount of risk cheaply. Sometimes a large degree of risk is desirable. This is the use of options and derivatives for speculation.

Example: Speculation on a stock with calls

Consider the profit and loss of a investor who buys 100 call options on XYZ stock with a strike price of $140. Suppose the current stock price is $138, the expiration date of the option is two months, and the option price is $5. Since the options are European, the investor can exercise only on the expiration date. If the stock price on this date is less than $140, the investor will clearly choose not to exercise the option since buying a stock at $140 that has a market value less than $140 is not sensible. In these circumstances the investor loses the whole of the initial investment of $500. If the stock price is above $140 on the expiration date, the options will be exercised. Suppose for example,the stock price is $155. By exercising the options, the investor is able to buy 100 shares for $140 per share. If the shares are sold immediately, then the investor makes a gain of $15 per share, or $1500 ignoring transaction costs. When the initial cost of the option is taken into account, the net profit to the investor is $10 per option, or $1000 on an initial investment of $500. This calculation ignores the time value of money.

Example: Speculation on a stock with puts

Consider an investor who buys 100 European put options on XYZ with a strike price of $90. Suppose the current stock price is $86, the expiration date of the option is in 3 months and the option price is $7. Since the options are European, they will be exercised only if the stock price is below $90 at the expiration date. Suppose the stock price is $65 on this date. The investor can buy 100 shares for $65 per share, and under the terms of the put option, sell the same stock for $90 to realize a gain of $25 per share, or $2500. Again, transaction costs are ignored. When the initial cost of the option is taken into account, the investor’s net profit is $18 per option, or $1800. This is a profit of 257% even though the stock has only changed price $25 from an initial of $90, or 28%. Of course, if the final price is above $90, the put option expires worthless, and the investor loses $7 per option, or $700.

Example: Hedging with calls on foreign exchange rates

Suppose that a U.S. company knows that it is due to pay 1 million pounds to a British supplier in 90 days. The company has significant foreign exchange risk. The cost in U.S. dollars of making the payment depends on the exchange rate in 90 days. The company instead can buy a call option contract to acquire 1 million pounds at a certain exchange rate, say

1.7

in 90 days. If the actual exchange rate in 90 days proves to be above

1.7

, the company exercises the option and buys the British pounds it requires for $1,700,000. If the actual exchange rate proves to be below

1.7

, the company buys the pounds in the market in the usual way. This option strategy allows the company to insure itself against adverse exchange rate increases while still benefiting from favorable decreases. Of course this insurance comes at the relatively small cost of buying the option on the foreign exchange rate.

Example: Hedging with a portfolio with puts and calls

Since the value of a call option rises when an asset price rises, what happens to the value of a portfolio containing both shares of stock of XYZ and a negative position in call options on XYZ stock? If the stock price is rising, the call option value will also rise, the negative position in calls will become greater, and the net portfolio should remain approximately constant if the positions are held in the right ratio. If the stock price is falling then the call option value price is also falling. The negative position in calls will become smaller. If held in the proper amounts, the total value of the portfolio should remain constant! The risk (or more precisely, the variation) in the portfolio is reduced! The reduction of risk by taking advantage of such correlations is called hedging. Used carefully, options are an indipensable tool of risk management.

Consider a stock currently selling at $100 and having a standard deviation in its price fluctuations of 10%. We can use the Black-Scholes formula derived later in the course to show that a call option with a strike price of $100 and a time to expiration of one year would sell for $11.84. A 1 percent rise in the stock from $100 to $101 would drive the option price to $12.73.

Suppose a trader has an original portfolio comprised of 8944 shares of stock selling at $100 per share. (The unusual number of 8944 shares will be calculated later from the Black-Scholes formula as a hedge ratio. ) Assume also that a trader short sells call options on 10,000 shares at the current price of $11.84. That is, the short seller borrows the options from another trader and must later repay it, creating a negative position in the option value. Once the option is borrowed, the short seller sells it and takes the money from the sale. The transaction is called short selling because the trader sells a good he or she does not actually own and must later pay it back. In the table this short position in the option is indicated by a minus sign. The entire portfolio of shares and options has a net value of $776,000.

Now consider the effect of a 1 percent change in the price of the stock. If the stock increases 1 percent,the shares will be worth $903,344. The option price will increase from $11.84 to $12.73. But since the portfolio also involves a short position in 10,000 options, this creates a loss of $8,900. This is the additional value of what the borrowed options are now worth, so it must additionally be paid back! After these two effects are taken into account, the value of the portfolio will be $776,044. This is virtually identical to the original value. The slight discrepancy of $44 is rounding error due to the fact that the number of stock shares calculated from the hedge ratio is rounded to an integer number of shares, and the change in option value is rounded to the nearest penny.

On the other hand of the stock price falls by 1 percent, there will be a loss in the stock of $8944. The price on this option will fall from $11.84 to $10.95 and this means that the entire drop in the price of the 10,000 options will be $8900. Taking both of these effects into account, the portfolio will then be worth $776,956. The overall value of the portfolio will not change regardless of what happens to the stock price. If the stock price increases, there is an offsetting loss on the option, if the stock price falls, there is an offsetting gain on the option.

Original Portfolio	$S = 100$ , $C = $ 11.84$

8,944 shares of stock	$894,400
Short position on 10,000 options	-$118,400
Total value	$776,000


Stock Price rises 1%	$S = 101$ , $C = $ 12.73$

8,944 shares of stock	$903,344
Short position on 10,000 options	-$127,300
Total value	$776,044


Stock price falls 1%	$S = 99$ , $C = $ 10.95$

8,944 shares of stock	$885,456
Short position on options	-$109,500
Total value	$775,956

This example is not intended to illustrate a prudent investment strategy. If an investor desired to maintain a constant amount of money, instead putting the sum of money invested in shares into the bank or in Treasury bills would safeguard the sum and even pay a modest amount of interest. If the investor wished to maximize the investment, then investing in stocks solely and enduring a probable 10% loss in value would still leave a larger total investment.

This example is a first example of short selling. It is also an illustration of how holding an asset and short selling a related asset in carefully calibrated ratios can hold a total investment constant. The technique of holding and short-selling to hold a portfolio constant will later be an important component in deriving the Black-Scholes formula.

Sources

The ideas in this section are adapted from Options, Futures and other Derivative Securities by J. C. Hull, Prentice-Hall, Englewood Cliffs, New Jersey, 1993 and The Mathematics of Financial Derivatives by P. Wilmott, S. Howison, J. Dewynne, Cambridge University Press, 1995, Section 1.4, “What are options for?”, Page 13, and Financial Derivatives by Robert Kolb, New York Institute of Finance, New York, 1994, page 110.

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Problems to Work for Understanding

Reading Suggestion:

References

[1] John C. Hull. Options, Futures, and other Derivative Securities. Prentice-Hall, second edition edition, 1993. economics, finance, HG 6024 A3H85.

[2] Robert W. Kolb. Financial Derivatives. Institute of Finance, 1993.

[3] Paul Wilmott, S. Howison, and J. Dewynne. The Mathematics of Financial Derivatives. Cambridge University Press, 1995.

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