In a simple form, an option is a contract between two parties that describes a right, but not an obligation, held by one of those parties over the other to:
• buy (in the case of a CALL OPTION) or sell (in the case of a PUT OPTION)
• an underlying asset (the UNDERLYING)
• at a specified price (the STRIKE PRICE)
• in a specified amount of the underlying (the NOMINAL SIZE)
• with the EXPIRY DATE of the option contract specified
• with the right to exercise the option up to and including the expiry date (in the case of an AMERICAN STYLE option) or only on the expiry date (in the case of a EUROPEAN STYLE option)
• the purchase price paid for the option being termed the PREMIUM
• the buyer of an option is the option BUYER
• the buyer of an option may in due course be referred to as the option HOLDER
• the seller of an option is the option WRITER
Some reasons for trading options
• the buyer of a put option can 'lock-in' a specified sale price for an asset that he owns, and the buyer of a call option can lock-in a specified purchase price for an asset that he may wish to buy; options thereby allow price risk to be HEDGED
• an option can be used to achieve high degrees of LEVERAGE, in that the percentage returns on the premium invested by an option buyer can be very large relative to the percentage changes in underlying price; meanwhile, the option writer may not be required to commit any funds whatsoever, although the option writer's credit rating will be closely scrutinised by the buyer in such circumstances
• option buyers can trade market direction and RESTRICT MAXIMUM LOSSES to the value of the premium paid
• traders may use options to speculate on the amount of VOLATILITY encountered in the underlying price during the option period
At-, In-, and Out-of-the-money options
• an option whose strike price is the same as the current spot price of the underlying is said to be AT THE MONEY SPOT
• an option whose strike price is the same as the current forward price of the underlying (for the expiry date) is said to be AT THE MONEY FORWARD
• an option whose strike gives the holder a more advantageous trading price than the current spot price is said to be IN THE MONEY SPOT
• an option whose strike gives the holder a more advantageous trading price than the forward price (for the expiry date) is said to be IN THE MONEY FORWARD
• an option whose strike gives the holder a less advantageous trading price than the current spot price is said to be OUT OF THE MONEY SPOT
• an option whose strike gives the holder a less advantageous trading price than the forward price (for the expiry date) is said to be OUT OF THE MONEY FORWARD
• the amount by which an option is in-the-money-spot is termed INTRINSIC VALUE
• the difference between intrinsic value and current option value is often termed TIME VALUE
Trading and valuing options
When an option is traded, the various contractual conditions that define the option are confirmed between the counterparties. These conditions are sometimes referred to as the option TERMS. In addition, the names of the counterparties, their bank accounts and contact information, and of course the premium that is being paid in respect of the option, will be confirmed between the parties and any intermediary to the deal. The precise nature of the trading process will depend upon which type of market the trade is conducted in, for example exchange traded options are traded differently to over-the-counter options.
The process of calculating a fair value for the option premium is called option VALUATION and is carried out by sophisticated mathematical programmes which operate by modelling the possible behaviour of the underlying price during the life of the option. Such models are called option MODELS. The process of analysing the sensitivity to market variables of the value of a portfolio of options and underlying assets is a task that is usually carried out by a RISK MANAGEMENT SYSTEM. The valuation model and the risk management system are often combined in one software package, allowing a trader to speedily estimate the impact upon portfolio performance of any one given option.
Each underlying has different features and option terms will vary accordingly. For example, an option on 100 ounces of gold will have a strike price since gold is quoted in price terms, whilst an option on a swap will have a strike rate since swaps are quoted in rate terms. An option on Soya beans will be traded in a nominal size of a certain number of bushels of Soya, whilst an option on a bond will be traded in a certain face value of bonds.
Option premiums can be quoted in a variety of ways. Three such ways are to quote the premium:
• as a value per unit of the physical amount of underlying (e.g. a gold option)
• as a percentage of the nominal value of the underlying traded (e.g. a swap option)
• as a percentage of the market value of the underlying traded (e.g. an index option)
An option on gold
$450 Gold Call in 100 ounces, expiry 15/09/95 European, premium $8.50 at $9.50 per ounce.
This option can only be exercised on the 15th September 1995 since it is a European option. The option gives the holder the right to buy 100 ounces of gold (the nominal size) at $450 per ounce (the strike price) on that date. The price quoted for this option by the option market maker is $8.50 at $9.50 per ounce of nominal. Thus the total value that must be paid for the option (on 100 ounces) is $950. Alternatively a counterparty may sell this option to the market maker for a total amount of $850.
An option on a swap
6% annual versus 3's Payers option (a call on the fixed swap rate) in DM50 million on a 5 year interest rate swap, expiry 15/09/95 American style, physical settlement, premium 0.20% at 0.30% of nominal.
This option can be exercised at any time from the moment of purchase up to and including the expiry date of 15th September 1995 since it is an American option. It gives the holder the right to pay 6% annual and receive 3 month floating interest on a 50 million 5 year Deutschmark swap that commences at the moment of exercise. It is thus a physically settled option. The premium on the option is quoted by the market maker as a percentage of the nominal amount, DM50 million. Therefore a counterparty can pay DM150,000 for the option or sell it to the market maker for DM100,000.
An option on a share index
2150 CALL on the DAX 30 German share index in DM10,000 per point, expiry 15/09/95 European style, cash settlement, premium 5% at 6% of index value.
This option gives the holder the right to buy the DAX index at expiry at a level of 2150, for cash settlement, in a nominal amount of DM10,000 per point. Hence, upon exercise of the option, the writer will pay the holder an amount of DM10,000 for each point that the index level exceeds 2150. The premium quoted by the market maker is 5% at 6% of index value. If the index at the time of trading the option stands at 2200, then the index value for this nominal size totals 2200 * 10000, i.e. DM22,000,000. The premium charged would be DM1,320,000 whilst the premium received would be DM1,100,000.
Gain on exercise
Option traders like to forecast how much profit or loss may be made on an option position. One type of forecast involves an examination of the various possible market prices for the underlying asset on the exercise date and the subsequent calculation, for each different price, of the amount of profit or loss made. For example, the holder of a $450 call on gold might exercise the call on the expiry date with the gold market price at $473 (per ounce). For each ounce of nominal size the trader makes a $23 GAIN ON EXERCISE because the gold can be bought at $450 from the option writer and simultaneously sold into the market at $473 per ounce. However the $23 gain is not the same as the profit made since the trader will have had to pay a certain amount of premium to purchase the option initially. If the premium were $9.50 per ounce, the profit on this option trade would be $13.50 per ounce.
Of course the profit of $13.50 per ounce of nominal only applies to one market price (here $473) at the time of exercise. Other market prices will give rise to different gains on exercise and hence different profits or losses. Thus, if the option were exercised with a market price of $459.50 per ounce, the gain on exercise would equal the premium paid and the payoff would be zero. $459.50 is therefore the break even point. By plotting a graph of the profit or loss that would arise for each underlying price at exercise, we produce what is known as an option PAYOFF DIAGRAM.
An option can be valued at any time after it is first created in an option trade. The valuing of an option according to current market conditions does not of course change the original premium amount paid, but simply calculates a current market value for it. Perhaps the original option buyer wishes to sell the option at a higher price than was originally paid for it. Note that the option holder does not have to exercise the option in order to make a profit. Such buying and selling of existing options (or options with identical terms to those held or written by the trader) forms a large part of option market turnover. Naturally then, traders are keen to know in what way changes in the market environment affect the premium value of an option over the course of its life. The major variables that can influence option value are introduced below.
Time to expiry
As an option approaches its expiry date the expected range of the underlying price or rate distribution at expiry will narrow. The implication of a shorter time horizon is therefore smaller gains on exercise relative to longer dated options. Given this fact, options with short times to expiry will generally have lower premiums than those with longer times to expiry. However, there is an important exception to this general rule with regard to European options and forward prices to be discussed later.
Imagine valuing an out-of-the-money put or call on a currency that operates within a fixed exchange rate system. The value would be nil (or at least very low) since there is little chance that the option will ever have intrinsic value. The theoretical value of exactly the same option under a floating exchange rate system would be greater since there would be a greater chance of making a gain on exercise. Hence, the higher the estimate of future volatility in the underlying price, the higher the option premium.
Strike price and Market price
Given that gains on exercise are determined by reference to both the strike and the underlying market prices, it should be clear that the relationship between the market price and strike price is of great importance. Changes in this relationship, in other words changes in the amount by which an option is in or out-of-the-money, will therefore affect premium value. As an option moves 'toward' or 'into' the money, its premium increases. Of course strike prices themselves do not change (for most types of option) during the option life, rather it is the level of the market that effectively determines the strike/market price relationship. Thus when an option is said to be 'moving into the money', it is in fact the market price that is 'moving', not the option strike.
Yield on asset and Money market rate
As money market interest rates increase in relation to the asset yield, so too do forward prices. Given that option models incorporate some form of forward price element into their workings, it is therefore the case for many types of option that an increase in money market interest rates relative to yield will lead to calls becoming further in-the-money forward and puts becoming further out-of-the-money forward. The effect of this upon option premiums is for an increase in money market rates to cause an increase in call values and a decrease in put values, but again there are exceptions to be dealt with later.
Option payoff diagrams show the trader what potential returns are available from single or combined option positions at exercise. Literature is available that describes the many possible payoffs that can be achieved from a combination of two or more individual options, but the reader must bear in mind that such payoff diagrams merely help to identify at what level the underlying price must trade at exercise in order for the option to achieve a given payoff.
We now turn to the subject of option valuation, namely the process by which one may arrive at a fair premium value for a given option. Here, a trader will rely upon an option model that employs mathematical formula of some complexity and which is therefore usually computer based. Traders will make differing assumptions in the construction of their models, hence one trader's calculation of fair premium value may not be the same as anothers. It is due to differences in the nature of such assumptions, and the best way of employing mathematical formula to represent them, that option professionals will differ over which approach is best for any given valuation.
Option models do not of course pretend to predict the future. The mathematics does however aim to depict in some way the statistical nature of the behaviour of the underlying price over time. This enables a representation to be made of the possible levels of the underlying price at various points in time. With this information, a model can proceed to define the possible range and size of gains on exercise that may result. By incorporating a factor that represents probability into its calculation, an option model can 'probability?weight' these possible gains on exercise so as to provide a fair premium value for a given option.
All of the information required in order to define an option will be immediately apparent simply by examining the contract on which the option terms are written (i.e. type of underlying, the expiry date, the strike price, the nominal size transacted, whether the option is American or European, put or call, etc.). An option model will however require further information (for example, on the current market price of the underlying and its yield during the option period, the money market rate of interest, and the estimated volatility of the underlying price for the option period) since these factors also influence option value despite being outside the scope of the option contract itself. Except for estimated volatility, all of this information will be directly observable in the marketplace, through the normal quote vendors for example. Hence, the means of arriving at an appropriate value for estimated volatility often takes centre stage in discussions relating to option valuation.
COMPARISON OF AMERICAN AND EUROPEAN OPTIONS
American option values can differ from European option values, even where the terms of both types of option are otherwise identical. Here is an intuitive explanation of these differences.
European options can only be exercised at the expiry date. An American option in contrast can be exercised at any time until the expiry date, so it offers the holder all of the exercise possibilities of the European option plus the possibility of early exercise. An American option must therefore always be at least as valuable as a European option.
Since the European option can only be exercised on the expiry date, its value will depend largely upon the forward price for that date. However, the possibility of early exercise for an American option brings into consideration possible events that may occur before the expiry date. One such factor is that of immediate exercise. Therefore the current spot price or rate of the underlying is of importance. For example, a one year European call on a bond, strike 110%, with the market spot price at 115% and the forward price at 110%, will be at-the-money forward and in-the-money spot. There will however be no possibility of exercising this European option immediately after purchase in order to achieve a gain on exercise of 5%. Such a gain is however available to the holder of an otherwise identical American option. Thus the American call must have a premium of at least 5% in order to avoid arbitrage possibilities, whereas the European call has no such arbitrage limit acting upon its premium value. European calls may therefore trade more cheaply than American options in such an environment. A similar effect can be derived for American puts when forward prices are above spot prices.
Let us take an example of a share option in order to further demonstrate the possible consequences of the early exercise facility of the American style option. An owner of an American call option on a share may wish to exercise the option before an ex-dividend date in order to own the share and thereby receive the dividend. Where a large dividend payment is due, the American call gives its holder the opportunity to receive that dividend (should one occur during the option's life). This flexibility, not available to the holder of a European call of otherwise identical terms, can add to the value of American call options on dividend paying stocks. It also adds to the value of American put options on dividend paying stocks since immediately after a dividend payment a share price will fall by the amount of the dividend. Again, the European put holder does not have the flexibility to exercise early in such cases.
Where the asset has no yield at all, it will never be optimal to exercise an American call option early. The possibility of early exercise of an American call to catch a dividend payment will not arise. In this case, little advantage is derived from holding an American option in preference to a European one and there will therefore be no difference between their respective premiums. The early exercise of an American call option on a non-dividend paying stock simply results in the wastage of its time value, since only the intrinsic value of the option will be received as the gain on exercise by the option holder. However, it may still be worthwhile exercising an American put early under these circumstances if it is deep in the money. To see this, imagine an underlying spot price that is zero. Exercising the American put at this time will result in the maximum possible gain on exercise, which can then be invested at the prevailing money market rate of interest. Delaying exercise will simply reduce the amount of interest that can be earned on the gain on exercise, but will not offer any opportunity for a larger gain on exercise than is currently available.
OVERVIEW OF 'EXOTIC' OPTIONS
So called EXOTIC options involve more complex contractual conditions than for simpler put and call options. The following are a few examples :
A compound option is an option on an option. For example, an option that conferred the right to purchase a call on gold at a specific premium would be a type of compound option. The strike price of a compound option refers to the premium that must be paid for the underlying option in the event of exercise.
Path dependent options
A path dependent option is one whose gain on exercise is determined by the course followed by the underlying during the life of the option. The following are both types of path dependent option:
a) A LOOK-BACK option is exercised for an amount equalling the largest gain on exercise that could have been achieved with the option's strike price. The option counterparties simply look back over the path followed by the underlying during the period and thus identify the largest in-the-money value achieved. This amount then becomes the settlement amount; and
b) an AVERAGE RATE option is settled against the average value of the underlying price or rate experienced during the life of the option.
A barrier option is one whose validity depends upon specified movements in the price or rate of the underlying. The two basic forms of barrier option are KNOCK-IN options and KNOCK-OUT options.
a) A Knock-in option requires the underlying price or rate to move above or below a specified threshold before the option contract becomes valid (referred to as UP-AND-IN and DOWN-AND-IN respectively).
b) An example of a Knock-in option is a down-and-in gold call option, strike $400 per ounce, knock-in at $350 per ounce, which becomes valid once the price of gold drops below $350 per ounce.
A Knock-out option is one in which a movement of the underlying price or rate above or below a specified threshold will invalidate the option contract (referred to as UP-AND-OUT and DOWN-AND-OUT respectively).
An example of a Knock-out option is an up-and-out gold put option, strike $400 per ounce, knock-out at $450 per ounce, which becomes invalid if the price of gold rises above $450 per ounce.
A note on reference prices
With both path dependent and barrier options, a suitable independent reference price must be chosen, a fixing or future market settlement for instance, with which the correct gain on exercise can be determined or by which the option can be said to have been knocked-in or knocked out.