Professor Ian Giddy

Futures and Options

Introduction to Futures

A futures contract is a legal contract between a buyer and a seller to make a transaction at a future date with terms that are set today . The contract will include: One of the most common mistakes that a small investor makes in futures trading is considering the margin on the futures contract as the investment. Margin is a deposit -- usually 5--10% of the contract's value -- required by the futures exchange from both the buyer and the seller of the contract. Margin helps to ensure that both buyer and seller will perform as specified in the futures contract.

Why Futures?

There has been remarkable growth in the futures market. Most of this growth has come from the financial futures. The leading contracts are the Treasury bond futures (CBOT) and the Eurodollar futures (CME). In addition, there has been significant growth international futures markets. There are a number of reasons for this growth.

Example of Futures Quotes

Commodity Futures Price Quotes for COMEX Gold
Month
Click for chart		 Session
Pr.Day
Open High Low Last Time Sett Chg
Sett OpInt
Jun 02 308.1 309.9 307.1 308.3 10:07 - -1.1
309.4 124152
Aug 02 309 310 308.4 310 09:46 - -0.4
310.4 8593
Oct 02 310 310 310 310 09:21 - -1.3
311.3 4368
Dec 02 311 312.1 310 311.3 10:07 - -0.8
312.1 17134
Feb 03 - - - 315 10:57 - +1.9
313.1 7381
Jun 03 - - - 315.5 08:39 - +0.3
315.2 3050
Dec 03 - - - 322 12:00 - +2.3
319.7 2793
Jun 04 - - - 321.5 11:48 - -3.3
324.8 3788
Dec 04 - - - 329.8 09:33 - -0.6
330.4 2182
Jun 05 - - - 337.5 08:59 - +1
336.5 1209
Dec 05 - - - 305 09:08 - -37.8
342.8 164


What is a Stock Index Futures Contract?

Stock index futures are traded in terms of number of contracts. Each contract is to buy or sell a fixed value of the index. The value of the index is defined as the value of the index multiplied by the specified monetary amount. In the S&P 500 futures contract traded at the Chicago Mercantile Exchange (CME), the contract specification states:

1 Contract = $250 * Value of the S&P 500
If we assume that the S&P 500 is quoting at 1,000, the value of one contract will be equal to $250,000 (250*1,000). The monetary value -- $250 in this case -- is fixed by the exchange where the contract is traded.

Hedging is a way of reducing some of the risk involved in holding an investment. There are many different risks against which one can hedge and many different methods of hedging. When someone mentions hedging, think of insurance. A hedge is just a way of insuring an investment against risk.

Consider a simple case. Much of the risk in holding any particular stock or fund is market risk; i.e. if the market falls sharply, chances are that any particular stock will fall too. So if you own a stock with good prospects but you think the stock market in general is overpriced, you may be well advised to hedge your position. You can hedge by selling financial futures (e.g. the S&P 500 futures). To estimate how many futures contracts to short, one must take into account the market value of the portfolio relative to the market value of an index contract, and the beta of the portfolio relative to the beta of the index contract.

Example:
Assume you own a portfolio with a current market value of $721677. It has a weighted-average beta of 1.4, and the rate you can earn on treasurys is 4.2%. If you want to hedge your portfolio fully for the next 3 months using stock index futures, and if the S&P index futures contract is currently trading at 414, how many contracts would you need to short? The index contact has a multiplier of 250, and has a beta of 1.

Number of contracts=(portfoliovalue*portfoliobeta)/(index*250*indexbeta)
Rounding to nearest number of whole contracts: (721677*1.4)/(414*250*1)=10

Futures and Options

The biggest difference between options and futures is that option purchases have losses limited to the amount paid for the option, whereas futures have no effective limitations on losses. For example, an option to buy 100 shares of IBM stock on December 31, at an exercise price of $90 per share might sell for $500 ($5 per share). If the price per share of IBM is $90, $80, or $70 on December 31, the option expires worthless and the purchaser loses only the $500 price paid for the option. It makes no difference to the option holder how low IBM goes below $90, given the price is below $90. In contrast, the purchaser of a futures contract on Treasury bonds at a price of $90 per $100 face cares a great deal whether the price at expiration is $90, $80 or $70. If the futures contract expires at $90, the purchaser breaks even, if it is $80, she loses $10,000 and if it is $70, she loses $20,000.

Below graphs of the futures and options payoffs are provided. The graphs how that options payoffs are highly nonlinear or skewed relative to the underlying asset's price, whereas the futures payoffs are linear in the underlying asset's price. Of course, this does not mean that options are better or worse than futures. They are just different financial instruments. Due to their limited liability aspect, options are naturally levered and are often 3--10 (or more) times riskier than investing an equivalent amount of money in the underlying stock. With their linear payoffs, futures contracts are much more like buying stocks or bonds (possibly on margin) than they are like options.


 

More on Options

A call option is a contract giving its owner the right to buy a fixed amount of a specified underlying asset at a fixed price at any time or on or before a fixed date. For example, for an equity option, the underlying asset is the common stock. The fixed amount is 100 shares. The fixed price is called the exercise price or the strike price . The fixed date is called the expiration date. On the expiration date, the value of a call on a per share basis will be the larger of the stock price minus the exercise price or zero.

One can think of the buyer of the option paying a premium (price) for the option to buy a specified quantity at a specified price any time prior to the maturity of the option. Consider an example. Suppose you buy an option to buy 1 Treasury bond (coupon is 8%, maturity is 20 years) at a price of $76. The option can be exercised at any time between now and September 19th. The cost of the call is assumed to be $1.50. Let's tabulate the payoffs at expiration.
 
Call Option Payoff



T-bond Price 
on Sept. 19		 Gross Payoff
on Option 		 Net Payoff
on Option



60  0.0 -1.5
70  0.0 -1.5
75  0.0 -1.5
76  0.0 -1.5
77  1.0 -0.5
78  2.0 0.5
79  3.0 1.5
80  4.0 2.5
90  14.0 12.5
100  24.0 22.5

Consider the payoffs diagrammatically. Notice that the payoffs are one to one after the price of the underlying security rises above the exercise price. When the security price is less than the exercice price, the option is referred to as out of the money.

A put option is a contact giving its owner the right to sell a fixed amount of a specified underlying asset at a fixed price at any time on or before a fixed date. On the expiration date, the value of the put on a per share basis will be the larger of the exercise price minus the stock price or zero.

One can think of the buyer of the put option as paying a premium (price) for the option to sell a specified quantity at a specified price any time prior to the maturity of the option. Consider an example of a put on the same Treasury bond. The exercise price is $76. You can exercise the option any time between now and September 19. Suppose that the cost of the put is $2.00.
 
 
Put Option Payoff



T-bond Price 
on Sept. 19		 Gross Payoff
on Option 		 Net Payoff
on Option



60  16.0 14.0
70  6.0 4.0
75  1.0 -1.0
76  0.0 -2.0
77  0.0 -2.0
78  0.0 -2.0
79  0.0 -2.0
80  0.0 -2.0
90  0.0 -2.0
100  0.0 -2.0

The payoff from a put can be illustrated. Notice that the payoffs are one to one when the price of the security is less than the exercise price.

Writing or "shorting" options have the exact opposite payoffs as purchased options.
 

Hedging With Stock Options

If you're concerned about a correction in the stock market, there are steps you can take to help preserve and protect your wealth. You can buy index puts as a hedge against market decline.

Index puts are purchased because they will generally increase in value as the market declines, counteracting the loss in your stock portfolio. Buying puts allows you to participate in the upside of the market while insuring against downward moves. However, if the correction or drop doesn't happen by option expiration, you've got another decision to make: whether to replace the expiring options with options expiring further out in time.

Here's a hypothetical hedging scenario:

You have a $100,000 stock portfolio.
The S&P 100 Index (OEX) stands at 725.
You're concerned about the months of April and May.

Since each OEX option contract represents 100 units, you multiply the index by 100.

725 × 100 = $72,500.

Your portfolio's value divided by the index value will give you the number of puts needed to hedge.

$100,000/$72,500 = 1.37 put option contracts.

Obviously, you can't buy a portion of a contract, so you'll need to choose either more or less protection to fit your needs. If you don't feel you need the entire portfolio hedged, you could buy just one option.

Buy one May 725 strike put option. A May 725 put is currently offered at 18.

Figure your cost this way: 1 × 18 × 100 = $1,800. If you buy two contracts, the cost formula is 2 × 18 × 100 = $3,600.

This hedge will give you the right to collect the amount by which the OEX falls below the 725 strike through May expiration. Should you, however, choose to sell your hedge before expiration, you could probably do so at a substantial premium to the amount that the index is below the strike price.

If the much-anticipated 20% correction occurs, the OEX would drop to 580. Here are the numbers:

100% - 20% loss = 80%
80% of 725 = 580
725 - 580 = 145 points below the strike—or in-the-money, as we call it in the industry.
145 × 1 put × 100 = $14,500—or, if you bought two contracts, 145 × 2 puts × 100 = $29,000.

$14,500 collected from one put
- $1,800 cost of one put contract
=$12,700 to counteract or offset the loss in the stock portfolio.

A 20% decline in a $100,000 portfolio would be a loss of $20,000, which would be offset by the $12,700 collected from the one put. This would bring the total loss to $7,300.

If you bought two option contracts:

$29,000 would be collected
-$ 3,600 cost of two put contracts
=$25,400 to counteract or offset loss in the stock portfolio.

Since the total loss was only $20,000, you'd actually profit from the decline if you purchased two put contracts: $25,400 - $20,000 = $5,400.

Naturally, this is all based on the assumption that your diversified stock portfolio will perform like the S&P 100 Index. Also, remember that taxes and commissions will affect your portfolio.

Determinants of Option Value

Before going into the details of option valuation models, such as the Black-Scholes model, let us identify the key determinants of option prices.

Current Stock Price
As the current stock price goes up, the higher the probability that the call will be in the money. As a result, the call price will increase. The effect will be in the opposite direction for a put. As the stock price goes up, there is a lower probability that the put will be in the money. So the put price will decrease.
Exercise Price
The higher the exercise price, the lower the probability that the call will be in the money. So for call options that have the same maturity, the call with the price that is closest (and greater than) the current price will have the highest value. The call prices will decrease as the exercise prices increase. For the put, the effect runs in the opposite direction. A higher exercise price means that there is higher probability that the put will be in the money. So the put price increases as the exercise price increases.
Volatility
Both the call and put will increase in price as the underlying asset becomes more volatile. The buyer of the option receives full benefit of favorable outcomes but avoids the unfavorable ones (option price value has zero value).
Interest Rates
The higher the interest rate, the lower the present value of the exercise price. As a result, the value of the call will increase. The opposite is true for puts. The decrease in the present value of the exercise price will adversely affect the price of the put option.
Cash Dividends
On ex-dividend dates, the stock price will fall by the amount of the dividend. So the higher the dividends, the lower the value of a call relative to the stock. This effect will work in the opposite direction for puts. As more dividends are paid out, the stock price will jump down on the ex-date which is exactly what you are looking for with a put. (There is also an issue of optimal exercise of the call and put option which will be addressed later.)
Time to Expiration
There are a number of effects involved here. Generally, both calls and puts will benefit from increased time to expiration. The reason is that there is more time for a big move in the stock price. But there are some effects that work in the opposite direction. As the time to expiration increase, the present value of the exercise price decreases. This will increase the value of the call and decrease the value of the put. Also, as the time to expiration increase, there is a greater amount of time for the stock price to be reduced by a cash dividend. This reduces the call value but increases the put value.

Let's summarize these effects.
 
Determinants of Option Value


Effect of Increase on


Call
Option 		 Put 
Option




1.  Current Stock Price Increase Decrease
2.  Exercise Price Decrease Increase
3.  Volatility Increase Increase
4.  Interest Rates Increase Decrease
5.  Dividends Decrease Increase
6.  Time to Expiration Increase Increase

The Black-Scholes Model

The precursor of all modern option pricing models was developed by Fischer Black and Myron Scholes (Black, F. and M. Scholes, "The Pricing of Options and Corporate Liabilities" Journal of Political Economy 81, 1973, 637-654). The main result is an option-pricing formula based on simple and reasonable assumptions in a continuous-time model. The remarkable thing about the result is that it relies on the absence of arbitrage, and part of the proof is a formula that specifies a trading strategy in the underlying stock and the riskless bond that will replicate the payoff of the option at the end. If the option is priced differently in the market, buying or selling the option and following either the trading strategy or the trading strategy in reverse will make money! Using the same sort of analysis gives the trading strategy used by hedge funds to make money taking positions in convertible bonds and the underlying stock. Now we present the Black-Scholes formula for the price of a call option.

In order to understand the model itself, we divide it into two parts. The first part, SN(d1), derives the expected benefit from acquiring a stock outright. This is found by multiplying stock price [S] by the change in the call premium with respect to a change in the underlying stock price [N(d1)]. The second part of the model, Ke(-rt)N(d2), gives the present value of paying the exercise price on the expiration day. The fair market value of the call option is then calculated by taking the difference between these two parts.

[Black and Scholes Model]

Assumptions of the Black-Scholes Model:

1) The stock pays no dividends during the option's life

Most companies pay dividends to their share holders, so this might seem a serious limitation to the model considering the observation that higher dividend yields elicit lower call premiums. A common way of adjusting the model for this situation is to subtract the discounted value of a future dividend from the stock price.

2) European exercise terms are used

European exercise terms dictate that the option can only be exercised on the expiration date. American exercise term allow the option to be exercised at any time during the life of the option, making american options more valuable due to their greater flexibility. This limitation is not a major concern because very few calls are ever exercised before the last few days of their life. This is true because when you exercise a call early, you forfeit the remaining time value on the call and collect the intrinsic value. Towards the end of the life of a call, the remaining time value is very small, but the intrinsic value is the same.

3) Markets are efficient

This assumption suggests that people cannot consistently predict the direction of the market or an individual stock. The market operates continuously with share prices following a continuous Itô process. To understand what a continuous Itô process is, you must first know that a Markov process is "one where the observation in time period t depends only on the preceding observation." An Itô process is simply a Markov process in continuous time. If you were to draw a continuous process you would do so without picking the pen up from the piece of paper.

4) No commissions are charged

Usually market participants do have to pay a commission to buy or sell options. Even floor traders pay some kind of fee, but it is usually very small. The fees that Individual investor's pay is more substantial and can often distort the output of the model.

5) Interest rates remain constant and known

The Black and Scholes model uses the risk-free rate to represent this constant and known rate. In reality there is no such thing as the risk-free rate, but the discount rate on U.S. Government Treasury Bills with 30 days left until maturity is usually used to represent it. During periods of rapidly changing interest rates, these 30 day rates are often subject to change, thereby violating one of the assumptions of the model.

6) Returns are lognormally distributed

This assumption suggests, returns on the underlying stock are normally distributed, which is reasonable for most assets that offer options.

The Greeks

For options traders, it's essential to know how an options position responds to changes in the price of the underlying asset (the Delta), how the Delta changes as the asset's price moves up or down (the Gamma), and how an option's price changes when volatility (Vega), time (Theta) and the interest rate (Rho) change.

Delta:

[Delta]
Delta is a measure of the sensitivity the calculated option value has to small changes in the share price.

Gamma:

[Gamma]
Gamma is a measure of the calculated delta's sensitivity to small changes in share price.

Theta:

[Theta]
Theta measures the calcualted option value's sensitivity to small changes in time till maturity.

Vega:

[Vega]
Vega measures the calculated option value's sensitivity to small changes in volatility.

Rho:

[Rho]
Rho measures the calculated option value's sensitivity to small changes in the interest rate.

Example of Option Quotes

INTERNATIONAL BUSINESS MACHINES (IBM)
 
Symbol Last Time Net Bid Ask Open High Low Close Volume
IBM 84.01 10:12 -0.29 0.00 0.00 83.75 84.80 83.73 84.30 1,205,800

Call Series May 2002 IBM 84.01
Symbol Strike Bid Ask Int.V. Time V. Theo.V. Dif. Vol O.I. Delta Gamma Theta I.Vol. Last
IBMEN 70.00 13.80 14.50 14.01 0.49 14.07 +0.43 0 26 1.00 0.0000 -0.0035 0.0000 12.30
IBMEO 75.00 9.00 9.50 9.01 0.49 9.07 +0.43 10 472 1.00 0.0022 -0.0045 0.3870 9.30
IBMEP 80.00 4.70 5.10 4.01 1.09 4.25 +0.85 197 3,727 0.89 0.0541 -0.0234 0.3567 5.00
IBMEQ 85.00 1.70 1.90 0.00 1.90 0.96 +0.94 238 14,348 0.40 0.1129 -0.0431 0.3163 1.80
IBMER 90.00 0.35 0.45 0.00 0.45 0.07 +0.38 346 21,203 0.05 0.0299 -0.0112 0.3033 0.40
IBMES 95.00 0.00 0.15 0.00 0.15 0.00 +0.15 70 19,631 0.00 0.0014 -0.0005 0.3329 0.10
IBMET 100.00 0.00 0.25 0.00 0.25 0.00 +0.25 0 10,238 0.00 0.0000 -0.0000 0.3957 0.05

Call Series June 2002 IBM 84.01
Symbol Strike Bid Ask Int.V. Time V. Theo.V. Dif. Vol O.I. Delta Gamma Theta I.Vol. Last
IBMFM 65.00 19.20 19.40 19.01 0.39 19.22 +0.18 0 3 1.00 0.0001 -0.0041 0.0000 17.40
IBMFN 70.00 14.50 14.70 14.01 0.69 14.24 +0.46 43 2 1.00 0.0023 -0.0052 0.3281 14.50
IBMFO 75.00 10.10 10.30 9.01 1.29 9.38 +0.92 26 26 0.95 0.0171 -0.0107 0.3974 10.70
IBMFP 80.00 6.30 6.60 4.01 2.59 5.10 +1.50 48 826 0.77 0.0491 -0.0218 0.3198 6.40
IBMFQ 85.00 3.40 3.50 0.00 3.50 2.11 +1.39 74 2,211 0.47 0.0650 -0.0262 0.3061 3.50
IBMFR 90.00 1.55 1.65 0.00 1.65 0.63 +1.02 84 16,036 0.19 0.0448 -0.0174 0.2973 1.65
IBMFS 95.00 0.55 0.65 0.00 0.65 0.14 +0.51 0 8,005 0.05 0.0179 -0.0069 0.2820 0.60
IBMFT 100.00 0.10 0.35 0.00 0.35 0.02 +0.33 0 1,911 0.01 0.0045 -0.0017 0.2786 0.20
Source of these quotes: www.thegumpinvestor.com/options/home.asp
Go to Giddy's Web Portal • Contact Ian Giddy at ian.giddy@nyu.edu