14.1.1 US Stock Options

We use the example of US stock options and stock index options to illustrate some of the administrative procedures which operate when these are traded on an exchange.

14.1.2 Contract Size

Options are traded on CBOT (in Chicago) and options on individual stocks are usually for delivery of 100 stocks. Options on stock indexes are also available and are cash settled. The exchange sets the dollar value of an index point (e.g. $100). For example, the cash payoff to a long call option on a stock index (S&P 500) for and is , which is paid by an (assigned) writer (seller) of the option (who has not closed out). The cash payment is usually based on the stock index quote (on the NYSE) at the end of the day on which the option is exercised (rather than when the exercise order is made).

14.1.3 Expiration Dates

These are fixed by the exchange and stock options are traded for expiration up to 4.30 p.m. (Central Time) on the third Friday of the expiry month. The holder of a long call on a stock can instruct his broker to exercise the option up until 4.30 p.m. on that Friday. The broker then has until 10.59 p.m. the next day (Saturday) to notify the exchange that exercise will happen.

Expiration dates for options on individual stocks usually extend to about 9 months – with some exceptions. For example, LEAPS (Long Term Equity Anticipation Securities), which are primarily options on around 500 individual (different) stocks (but some are also on stock indices), have expiration dates of about 3 years ahead. Similarly FLEX (short for ‘flexible’) options on stocks and stock indices can have any expiration date of up to 5 years ahead and in addition they permit the purchaser to set any exercise price and can be European or American.

14.1.4 Strike/Exercise Prices

These, for example, might be set at $2.50 intervals when the underlying stock price is less than $25, at $5 intervals when the stock price is between $25 and $200 and at $10 intervals for a stock price over $200. Strike prices are set either side of the current stock price and as the stock price moves up or down, options with new strike prices are added by the exchange.

14.1.5 Trading

Over 95% of orders on the CBOE are via an electronic platform. But trading still takes place on the floor of the CBOE. An individual who has purchased (or rents) a seat on the CBOE may be a ‘market maker’ who stands ready to quote both bid and ask (offer) prices on the option. The ‘bid’ price is the price the market maker is prepared to buy the option and the ‘ask’ price is the price at which he is prepared to sell. Prices are quoted by open outcry on the floor of the exchange. Market makers must stand ready to trade with investors. An investor who has purchased an option can close out her position by selling (writing) the same option (i.e. with the same strike and expiration date).

14.1.6 Options Clearing Corporation (OCC)

For US options markets (except those trading futures options) the Options Clearing Corporation (OCC) standardises contracts and acts as an intermediary, effectively creating two separate contracts.

For example, if a trader buys a call option the OCC guarantees that the writer will honour the contract. An option writer represents a credit risk to the OCC since she may not have funds to purchase the underlying asset (e.g. stock) in the spot market to facilitate delivery. The OCC therefore requires the writer to post a margin payment (usually in cash and equal to at least 30% of the value of the stocks underlying the option contract plus the call premium). There is also a maintenance margin which might be set at a minimum of 15–25% of the value of the stocks underlying the option contract. An option buyer has no margin requirement with the OCC since the most she can lose is the option premium, which is paid in full at the outset of the contract.

Initial margins vary depending on whether one has a naked (uncovered) position (i.e. no offsetting holding in the underlying stocks) or a covered/hedged position. The latter is less risky and involves less initial margin. Margin positions on strategies such as straddles, spreads etc. are governed by special rules described in publications on the CBOE website.

Take, for example, a written call on a stock, where the initial margin is 30% of the value of the stock, plus the option premium. However, if the call is out-of-the-money then the margin is reduced by . If , (i.e. out-of-the-money) and , and each call is written on 100 stocks, then the margin payment would be . If the call had been in-the-money or at-the-money the deduction of would not be allowable. A similar rule applies to written puts.

Option prices on a stock index are less volatile and hence (uncovered) written index options would have a margin requirement based on, say, 15% of the dollar value of the stock-index, rather than the 30% for individual stock options.

14.1.7 Orders

For example, suppose in January Ms Long wants to buy a December-call option on AT&T stocks with a strike of . Her broker would instruct (electronically) a market maker with a seat on the exchange. Trading is conducted in a pit. The buyer shouts out the bid (e.g. $4) and accepts the lowest offers ‘shouted’ to him by the market maker (e.g. $4.1), giving a contract price of $410 (for delivery of 100 stocks per contract). The OCC keeps track of all trades on the exchange.

Next Ms Long deposits $410 with her broker, who passes this on to his OCC ‘clearing firm-ABC’ by the next (business) morning. This money is credited to the option writer's clearing firm-XYZ. The option writer's clearing firm-XYZ also receives a margin payment from the writer of the option. The clearing firms ABC and XYZ then aggregate all the transactions of their customers and deposit funds with the OCC as surety on the net contracts outstanding, according to a prearranged formula. The OCC is therefore the ultimate guarantor for Ms Long, the purchaser of the call option. Because the clearing firms also hold some default risk, the OCC imposes minimum capital requirements on them.

14.1.8 Offsetting Order

If in January Ms Long originally purchased a December-90 call option on AT&T at $4.10 (total cost = $410) then she can sell this contract in June (say) by placing an offsetting order. If Ms Long sells at $4.50 then $450 will be passed to her clearing firm ABC (and then on to Ms Long) and the OCC will cancel Ms Long's position in this contract. The trader who buys the call option from Ms Long will usually not be the person from whom Ms Long initially purchased the contract. If the purchaser of Ms Long's sale of her option is establishing a new long position in the contract then the ‘open interest’ would remain the same, with the new ‘owner’ noted by the OCC. If both traders are closing existing positions, the open interest falls by one contract.

14.1.9 Exercising an Option

If the buyer (Ms Long) of a December-90 AT&T call option holds the option to maturity and exercises the option, her broker notifies the OCC member (ABC) that clears its trades. ABC then places an exercise order with the OCC who ‘assigns’ a member (XYZ) with an outstanding short position in the same option (i.e. with same strike, same maturity, same underlying asset). This may be a random assignment or a ‘first-in, first-out’ rule may be used. The OCC member (XYZ) then selects a specific investor (Ms Short) who has an existing short position – using a specific procedure, for example random allocation. Ms Short having then been assigned, must fulfil the delivery terms – she must deliver 100 stocks of AT&T and she will receive per stock. A cash amount equal to the strike price is passed from Ms Long's broker via her clearing firm-ABC to the assigned writer's clearing firm-XYZ. Hence the ‘assigned writer’ is unlikely to be the original trader who initially sold Ms Long the call option.

Suppose Ms Long holds a December-90 put option on a stock to its expiration date. Then a trader holding a short position must take delivery of 100 stocks of AT&T from Ms Long and pay her per stock. When an option is exercised the open interest falls by one. All at-the-money options should be exercised (if the payoff at maturity is greater than any transactions costs) and some brokers will do this automatically for their clients. A very rough estimate is that about 10% of calls and puts on the CBOE are exercised and about 30–40% expire out-of-the money.

14.1.10 Commissions

Market makers trade their own book but are obliged to ‘fill’ all public orders. They make profits (losses) on their own book and also earn the bid–ask spread on a buy-sell ‘round trip’. Market makers buy at the bid price and sell the same contract at a higher offer (ask) price. The exchange sets upper limits for the bid-offer spread – for example, $0.75 for options prices between $10 and $20.

Commissions for retail investors are usually a fixed cost plus a proportion of the value of the trade – and discount brokers charge less than full-service brokers. For example, a dollar trade of between $3,000 and $10,000 via a discount broker might involve commission of ‘$30 +1% of the dollar amount’ in the trade. Larger trades generally involve larger fixed costs but lower variable cost. When offsetting an existing position in an option, the commission must be paid again. When exercising a stock option, the commission will probably be the same as for a buy or sell order and may be around 1% of the stock's dollar value.

14.1.11 Position Limits and Exercise Limits

The exchange (often on the instructions of the regulator) also specifies position limits, namely the maximum number of contracts that an investor can hold on one side of the market. For individual stock options (CBOE) with high market capitalisation and trading volume, the position (and exercise) limit might be set at 250,000 contracts (on the same side of the market). Long calls and short puts are on the same side of the market because each contract's value increases (decreases) as the underlying price increases (decreases). Put differently, the premia on long calls and short puts are positively correlated. Short calls and long puts are also on one side of the market.

Investors are also limited in the number of contracts that can be exercised in any five consecutive business days (i.e. exercise limit). The figure for the exercise limit is usually the same as the position limit. The purpose of position and exercise limits is to prevent single traders or groups of traders acting together and having a significant influence on the market price. However, theses limits may reduce liquidity and may drive some business into the OTC market.

14.1.12 Newspaper Quotes

Illustrative price quotes on 2 November for call and put options on stock-XYZ are shown in Table 14.1. The expiration dates are in the first column. The current price (on the NYSE) of the underlying stock is $26.80. The strike prices in the second column are set (by the exchange) above and below the current stock price. The columns labelled ‘Price’ denote the closing price for the call or put option (for the 3 p.m. trade) and these columns are followed by the daily trade volume as well as the open interest (i.e. number of long or short contracts outstanding). Note that the quoted option price for the ‘3pm trade’ in Chicago might not be based on the recorded price for the underlying stock on the NYSE because the two prices might not be taken at exactly the same time, especially if the option is rather illiquid and hence infrequently traded.

From Table 14.1 you can see that the alternative exercise (strike) prices range from 20.00 to 30.00 (there are other strike prices available which are not reported here). Call prices (premiums) for the strike for expiration months November, December, January and April are 4.29, 4.40, 4.55, and 5.04 respectively – so the call premium increases with the maturity date of the option. The quoted option premium assumes delivery of one stock but 100 stocks must be delivered for each contract, so the invoice price for the option = ‘100 × quoted price’.

By looking at the November contracts with strike prices , 22.50, 25.00, 27.50, and 30.00 you can see that the call premia fall as the strike price increases. The converse applies to the put premia which are positively related to the strike price. By looking at the put premia for contracts that have expiration dates of 25 November and 25 April (of the next year), you can see that put premia increase with the time to maturity of the contract.

14.2.1 Positions in Options

As we see below there are always two parties to any options trades and these are classified as follows:

14.2.1.1 Long Call: Insurance

Consider, for example, the purchase of a (European) call option on stock-XYZ on 15 July, when the current price of stock-XYZ on the NYSE is . One contract is for delivery of 100 stocks and if the quoted call premium , the contract will cost $300. Assume the strike price is (i.e. an ATM option) and the expiry date T is in about 3 months' time on 25 October (Figure 14.1).

First consider how the purchases of a call option can provide insurance. Suppose that in July a pension fund knows it will receive an inflow of funds in October and wants to invest in stocks-XYZ. If in July, the pension fund purchases an October-call option (in Chicago), it will have set a maximum purchase price of , if it decides to exercise the contract in October – this is a form of insurance.

Clearly, if on 28 October then the pension fund will exercise the option in Chicago, take delivery and pay $80 per stock – which is cheaper than purchasing stocks-XYZ on the NYSE at $88 (Figure 14.1). A trader who has an outstanding short position in the October-call is legally bound to deliver the stock to the pension fund and receives $80.

Alternatively, the long call option can be ‘cash settled’ by the pension fund which receives in Chicago (a trader with an outstanding short position in the October call, provides the $8). The pension fund can then buy stock-XYZ on the NYSE for , which when offset against the receipt of $8 from ‘cash settled’ call, gives an effective cost for the stocks-XYZ of $80 – the same as the strike price in the call option contract. Hence, (ignoring transactions costs) ‘delivery’ or ‘cash settlement’ results in the same value to the holder of the long call.

On the other hand, if the stock price on 25 October is (i.e. below the strike price of ), then the pension fund will not exercise the call, since it can purchase the stocks-XYZ at lower cost on the NYSE.

Hence, the call option provides insurance in the form of a maximum price payable of (if ) but also allows the pension fund to not exercise the call option (if ) and then the pension fund buys stock-XYZ at the lower price on the NYSE. The cost of this insurance is the call premium of which is paid in July.

One further thing to note is that the maximum price the pension fund ‘locks in’ is the strike price in the options contract and not the stock price (on the NYSE) when the pension fund purchases the call option in July. For example, suppose the actual stock price on the NYSE on 15 July, is . This price of $78 is not the maximum purchase price the pension fund will pay on 25 October when it exercises the call option. The maximum price the pension fund will pay in October when exercising the call option is the strike price .

14.2.1.2 Long Call: Speculation

Now consider purchasing an October-call option on 15 July for , in order to speculate on the future price of stock-XYZ over the next 3 months (Table 14.2). If the actual price of stock-XYZ on 25 October turns out to be , then the holder of the call option can take delivery at and sell each stock-XYZ on the NYSE at $88 – which implies a payoff of $8. Alternatively the speculator can cash settle the long call and receive $8 (from the options clearing house). In either case the ‘payoff’ from exercising the call on 15 October is $8 per stock:

Current stock price, S 0 = $78

Traders' desk (today, 15 July)    Strike price,  Call premium (price),

Outcome (3 months later on 15 October, time T)  Stock price at expiry,  Cash payoff at expiration:  Profit (net of call price):

Payoff: long call = (+1)[max(0,S_T − K)] ¹          if S_T > K exercise the option.

The ‘+1’ implies we are long one call option. In October, the ‘profit’ (after deduction of the call premium ) is $5 per stock (Figure 14.2).²

(14.1)

The breakeven stock price occurs when :

(14.2)

Each call option contract is for ‘delivery’ of 100 stocks; hence, if the speculator makes an overall profit of $500. On the other hand, if at expiration (< K) the speculator would not exercise the option. The call option expires worthless but the speculator has only lost the call premium of $300, which was paid when she initially purchased the option in July.

Hence a speculator who holds (i.e. is long) a call option can at most lose the call premium. However, the speculator can benefit from an unlimited upside potential, if the stock price in October ends up well above the strike price. The return to the speculator from a long call is asymmetric.

If the stock price at expiration is $82 then the profit to the speculator from the long call is –, that is, a loss of $1. However, in this case it is still worth exercising the long call, since if the speculator does not do so, her net loss is the premium of , which is larger than the $1 loss from exercising the call. In fact, if the stock price exceeds the strike price at the expiration date, it always pays to exercise the call option (since this positive payoff will contribute to reducing the effective cost of the call premium).

It is useful to designate the payoff profile from options in terms of direction vectors – this will be especially useful when we look at options strategies in Chapter 17. The payoff from a long call is represented by the direction vector . When , the payoff to the long call is zero – hence the use of ‘0’ in the payoff vector. When , the long call earns a positive payoff which increases dollar-for-dollar with the stock price – this is a positive relationship, hence the use of ‘+1’ in the direction vector (see inset in Figures 14.1 and 14.2). For a light-hearted view of how call options can be used, see Finance Blog 14.1.

14.2.1.3 Write (Sell) a Call

One simple way to find out what happens to the profit at maturity for the writer of a call is to work out the profit to the long call (+$5) and simply reverse the sign. If the long call makes a cash profit of $5 then the person who wrote (sold) the call makes a cash loss of $5. Let us look at this in more detail.

If , on 25 October then the writer of the call has to purchase the stock in the cash market (NYSE) at but receives only when she delivers the stock (in Chicago) to the holder of the long call (Figure 14.3). The writer of the call has a payoff of –$8 (= –().

Alternatively, if the long call is cash settled then the trader with an outstanding short positon (i.e. the writer) makes a cash payment of $8 to the holder of the call but on 15 July she had received when she sold the call option, so her profit is , that is, a loss. The writer makes a total loss on one contract of $500 which is the mirror image of the $500 gain, made by the long call.

Alternatively, if the actual stock price on 25 October is (say) (i.e. below the strike price of ), then the holder of the call option (‘the long’) will not exercise it (in Chicago), since she can purchase the stocks at lower price on the NYSE. Hence, when , the writer of the call option makes a profit of $3, which is the option premium she received in July. The payoff at maturity from one written call is:

The ‘–1’ implies you initially wrote (sold) one call option. The profit at maturity for the writer of the call is:

(14.3)

14.3.1 Long Put + Stock: Insurance

First let us see how the purchase of a put option (i.e. you go long the put) can be used to set a minimum value you will receive in the future, for stocks that you already own. The stock + put is like an insurance contract which sets a minimum future selling price for the stock but allows most of the ‘upside capture’ should stock prices rise in the future.

Suppose on 15 July Ms Prudence, a pension fund manager, holds 100 stocks of XYZ, with a price of on the NYSE. Ms Prudence has to pay out someone's lump-sum pension in 3 months’ time (on 25 October) and she is worried that the stock price will fall below $70. But she also wants to take advantage of any rise in stock prices, should this occur. Ms Prudence can eliminate most of the downside risk by purchasing a put option, with a strike price and expiration date on 25 October. Each put contract is for delivery of 100 stocks. Ms Prudence must pay the put premium today.

By purchasing the put, Ms Prudence guarantees that she can, if she wishes, sell her stocks for each, by exercising the put contract in Chicago on 25 October – even if the price of the stocks on the NYSE is much lower, say per stock. The options market in Chicago must honour Ms Prudence's decision to exercise the put, by delivering her stocks to Chicago, for which she receives the strike price in the put contract. The paid to Ms Prudence, will be provided by a Chicago options trader (Mr Short) who has a short position in the October-70 put, on the 25 October (i.e. Mr Short has previously sold an October-put with a strike of and has not closed out his position before 25 October). Mr Short's $70 payment to Ms Prudence will be paid via the options clearing house.

Alternatively, the long put can be ‘cash settled’ by Ms Prudence for receipt of cash (provided by Mr Short via the options clearing house). Ms Prudence then sells the stock (owned by the pension fund) on the NYSE for , giving an ‘effective’ cash value for the put+stock position of Ms Prudence of $70 – the same as the strike price in the put contract.

On the other hand, if in October, then Ms Prudence can ‘walk away’ from the put contract (i.e. not exercise it in Chicago) but instead sell stocks-XYZ on the NYSE at the high price of $75 per stock.

This means that whatever happens to the stock price on the NYSE over the next 3 months, Ms Prudence can either exercise the put and sell stocks-XYZ at a minimum price of (in Chicago), or if the stock price rises above $70, she can ‘throw away’ the put option (i.e. not exercise it in Chicago) and sell her stocks-XYZ at the higher spot price on the NYSE. Hence, if you already own some stocks, buying a put option today provides insurance in the form of a guaranteed minimum price when you sell your stocks (on the expiration date of the put), whilst also allowing you to benefit from any ‘upside potential’, should the stock prices rise on the NYSE.

For a light-hearted analysis of the payoffs when holding a ‘spot market asset’ and using puts to insure a minimum selling price, see Finance Blog 14.2.

14.3.2 Long Put: Speculation

There are also opportunities for speculation with put options. On 15 July, suppose a speculator Ms Doom, expects the price of stock-XYZ to fall substantially between July and October (say). To benefit from a future stock price fall, Ms Doom buys an October-put option contract on 15 July, with , exercise date 25 October and pays the put premium on 15 July. (Note that here Ms Doom only holds a put option, she does not hold any stocks).

Suppose the price of stock-XYZ on the NYSE on 25 October is – Figure 14.4. Then on 28 October, Ms Doom could buy 100 stocks-XYZ on the NYSE for per stock, and then exercise the put option by delivering these to Chicago for which she receives per stock from the options clearing house. Ms Doom has a positive payoff of $5 per stock³ and a net profit of $3 after paying the put premium of (Figure 14.5). If one put contract is for delivery of 100 stocks then the outcome is given in Table 14.3.

Current stock price,

Traders' desk (today, 15 July)     Strike price,   Put premium (price),

Outcome (3 months later: 25 October, time T)  Stock price at expiry,  Cash payoff at expiration:  Profit (net of call price):

Hence the payoff from holding one long put is:

Payoff: ⁴ exercise the put if

The profit from the long put is:

(14.4)

If then Ms Doom does not exercise the put (which expires worthless) but the most she loses is the put premium, . The breakeven stock price occurs when :

(14.5)

The payoff profile for a long put (Figure 14.5) is represented by the direction vector {–1, 0}. When , the payoff to the long put is zero – hence the use of ‘0’ in the payoff vector. When , the long put earns a positive payoff which increases dollar-for-dollar as the stock price falls – this is a negative relationship, hence the use of ‘–1’ in the direction vector.

14.3.3 Write (Sell) a Put

The payoff to the seller of a put is the opposite of that of the buyer of a put – there are two sides to every trade – if one side ‘wins’ the other ‘loses’. Let us look at this in more detail.

Suppose Ms Doom already has a long position in the October-put. If , then Ms Doom will exercise the put and receive a payoff of $5. Hence, Ms Writer, who has previously sold (written) a put, will have a negative payoff if the (long) put is exercised (i.e. ). Ms Writer is legally obliged to ‘receive’ the stock from Ms Doom in Chicago and pay Ms Doom . But Ms Writer can only sell the stock for on the NYSE (Figure 14.6) – a loss of $5 to Ms Writer. But Ms Writer when she initially sold (written) the October-put in July received the put premium , so her net loss is $3. Hence the payoff to Ms Writer (at expiration) is:

Payoff: Short put = (−1)[max(0,K − S_T)]      (–1, implies Ms Writer has sold a put).

The profit at maturity for Ms Writer is:

(14.6)

The payoff to the writer of the put option is just the ‘mirror image’ of the payoff for Ms Doom who is long the put (see Equation 14.4).

14.4.1 In, Out, and At-the-Money

For a long position in a call option the option is said to be:

• In-the-Money (ITM): if Current Spot Price > Strike Price ()
• At-the-Money (ATM): if Current Spot Price = Strike Price ()
• Out-of-the-Money (OTM): if Current Spot Price < Strike Price ()

Strictly speaking an option is ‘in-the-money’ if exceeds the present value of the strike price (i.e. ) but the simpler definitions above are usually used.

14.4.2 Newspaper Quotes: Calls

On 1 July, the 360-October calls (Table 14.4) must be worth at least $16 because the long has the right to exercise the American call option and buy stock-A at (in Chicago) and immediately sell the stocks for on the NYSE. The cash payout from immediately exercising a call option is the ‘intrinsic value’ of the option.

(14.7)

The October-390 call has a call premium of $36.5, which is greater than the intrinsic value of $16. The reason for this is that there is a chance that between 1 July and the October expiration date, the stock price will increase even further (thus increasing the value of the call). Hence, the option has an additional source of value known as the ‘time value’ of the option.

(14.8)

Now consider the October-390 call. This has an intrinsic value of zero since the holder of this call would not wish to take immediate delivery in the option contract and pay and then immediately sell the stock on the NYSE for . However, there is a chance that on or before the maturity date of the option, the stock price will rise above . For the October-390 call, the long is willing to pay $21.5 on 1 July, for that chance.

Notice that for either of the calls the long is willing to pay a higher call premium, the longer the time to expiry – hence these options have time value that increases with time to maturity (look along the rows in Table 14.4). This is because there is a longer time over which the stock price might rise above (or well above) the strike price. In summary we have:

14.4.3 Newspaper Quotes: Puts

Let us undertake a similar analysis of put options on stock-A (Table 14.5). The October-390 puts have an intrinsic value of $14 . This is because the holder of the put option could buy the stocks on the NYSE for and sell (deliver) them immediately for (in Chicago) by immediately exercising the American put option.

Strike price	Expiry month
	October	January	April
360	16 (0 16)	25 (0 25)	27.5 (0 27.5)
390	31 (14 17)	40 (14 26)	43.5 (14 29.5)
Stock price (NYSE, 1 July):

Notes: Prices denoted in dollars. (..) = (intrinsic time value)

The 390-puts have an intrinsic value of , but the put premium for the October expiry is . The time value is therefore . This reflects the possibility that between 1 July and the October maturity date of the option, the price of stock-A might fall futher, thus increasing the payoff to holding the put option.

Put premia increase as the time to expiry increases (look along the rows in Table 14.5), since over a long horizon there is an increased chance that the stock price will end up below the strike price and the put payoff will be positive.

14.4.4 In/Out-of-the-Money

Note that the 360-calls (Table 14.4) are in-the-money (ITM) (i.e. ) while the 390-calls are out-of-the-money (OTM). Hence the 360-calls have relatively high intrinsic values and relatively low time values. The converse applies for the 390-calls which are OTM and hence have zero intrinsic value and relatively large time values. Similar considerations apply to the puts.