21 Futures, synthetics and put–call parity 223On the other hand, the holder of the long futures position forgoes the dividends payable for the next six weeks, and therefore the value of
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Frequently, however, on a rally the skew can remain in place, and the implieds of all strikes are unchanged Effectively, the implied volatility decreases because the focal point of the skew moves to the new at-the-money strike The solid line of Figure 20.9 illustrates this: XYZ rallies from
100 to 105, and the new ATM implied, now at the 105 strike, is less than that of the former 100 strike
This situation often occurs with skews in stock indexes as they rally to former levels The options market is unfazed by the upside retracement This also occurs in commodities that have negative put skews as the commodities retrace from a rally; there the graph is the mirror image of Figure 20.9.Another possibility is that on a break, the skew can remain in place Effectively, the implied volatility increases because the focal point of the skew moves to the new at-the-money strike The dotted line of Figure 20.9 illustrates this: XYZ breaks from 105 to 100, and the new ATM implied, now at the 100 strike, is greater than that of the former 105 strike
This latter situation often occurs with skews in stock indexes as they break The options market is fearful that this is the big one When it really is the big one, then the entire skew will shift vertically upward, and the put wing will become more positive
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A note on market sentiment
In all cases where a straight long or short option is chosen for a directional strategy, skew risk can be minimised by trading the long or short call or put spread
Volatility skews are indicators of market
senti-ment Positive skews indicate fear, while negative
skews indicate complacence Sentiment, as we
know, can often be wrong, but it cannot be
ignored
Volatility skews are indicators of market sentiment
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Basic non-essentials
Trang 8Synthetic positions are used primarily by professional market-makers to simplify the view of their options inventory in order to manage risk better They are of little practical use to traders who take options positions based
on market outlooks, but they can be studied in order to understand how options markets work
In order to understand synthetics, it is best if you understand why they exist Like all options positions, they are based on a relation to an under-lying contract, which may be a cash investment or a futures contract
If we briefly take this subject step by step, then we will avoid future disorientation
What a futures contract is
A futures contract is simply an agreement to trade a commodity, stock,
bond or currency at a specified price at a specified future date Because no cash is exchanged for the time being, the future buyer is said to have a
long position, and the future seller is said to have a short position As a
result, the holder of the long position profits as the market moves up and takes a loss as the market moves down The holder of the short position has the opposite profit/loss
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If short selling were not possible, investors would only be able to buy from those who wanted to sell physical holdings; liquidity would suffer and market volatility would increase Most exchanges require a security deposit
in order to open a futures contract, and this deposit is known as initial margin The value of the contract as traded on the exchange invariably
fluctuates, and so results in a profit to one party and a loss to the other The party who has a loss is then required to deposit the amount of the loss, and this additional deposit is known as variation margin Margin
may be in the form of cash, or it may be in the form of liquid securities such as treasury bills or gilts, for which the depositor still collects interest Meanwhile the party who has the profit is credited with variation margin, and he receives interest on the balance
Futures contracts have traditionally been used in commodities markets
in order to hedge supply shortages and surpluses They are now used in stocks, stock indexes, bonds and currencies Many excellent books describe how these forms of futures contracts operate
An example of a futures contract
Consider the following example of a closing price of the S&P 500 index with the settlement price of the December futures contract and the settle-ment prices of the at-the-money call and put on the futures contract.S&P index: 1133.68
Trang 1021 Futures, synthetics and put–call parity 223
On the other hand, the holder of the long futures position forgoes the dividends payable for the next six weeks, and therefore the value of the December future is decreased by that amount The formula for the value of the futures contract is approximated as follows:
Futures contract = cash value of index + interest or cost of carry on index until expiration – dividends payable until expiration
In practice, the formula is more complicated because annualised rates of carry and dividend yields are used Here, we are simply concerned with why the above future trades above or below the cash
Until recently short-term interest rates paid more than dividend yields, and so stock index futures traded at a premium to their underlying indexes The situation is now reversed, and it is
similar to the 1950s, where dividend yields paid
more than short-term interest rates in order to
compensate for the risk of owning stock This was
a holdover from the crash of 1929, when many
stock-holders’ investments were wiped out The
reason now, however, is that after the recent
bank-ing crisis, the central banks are trybank-ing to maintain
liquidity by keeping interest rates low
Occasionally, shortly before expiration, there may be a large amount of dividends payable in a stock or stock index Then the dividend outweighs the interest amount and the future trades at a discount to the index Once the dividend or dividends are paid, then the future trades above the cash
In any event, the futures contract and the cash index converge at tion because then there is no remaining differential between cost of carry and payable dividends The futures contract simply expires to the current cash value of the index
expira-There, the holder of the long futures contract pays the cash value of all the stocks in the index The holder of the short futures contract receives the cash value of all the stocks in the index The ultimate amount exchanged is deter-mined by the value of the index at expiration times the contract multiplier
In the case of a physical commodity such as corn or crude oil, the futures contract is deliverable to the quantity of the commodity specified in the contract at the settlement price
The futures contract and the cash index converge at expiration because then there is
no remaining differential between cost of carry and payable dividends
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Synthetic futures contract
As we already know, a long XYZ 100 call, by virtue of its right to buy, equals a long XYZ position when XYZ is above 100 at expiration We also know that a short XYZ 100 put, by virtue of its obligation to buy, equals a long XYZ position when XYZ is below 100 at expiration The sum of these two options positions, therefore, equals a synthetic long XYZ position with a strike price of 100 This is a result of the combined right and obliga-tion Consider the example in Figure 21.1
We also know that a short XYZ 100 call, by virtue of its obligation to sell, equals a short XYZ position when XYZ is above 100 at expiration A long XYZ 100 put, by virtue of its right to sell, equals a short XYZ position when XYZ is below 100 at expiration The sum of these options positions, therefore, equals a synthetic short position with a strike price of 100 This again is a result of the combined right and obligation Consider the exam-ple in Figure 21.2
Assuming that interest rates will eventually rise, then the S&P 500 ple above is typical of the modern era A long December 1140 call plus a short December 1140 put equals a synthetic long futures contract valued
exam-at 1140 If you pay 34.40 for the call, and sell the put exam-at 33.70, then you have paid a net 0.70 for the synthetic at 1140 In other words, you have paid 0.70 to go long the future at 1140 You have paid 1140.70 for the synthetic long future
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Note that the actual December future is valued at 1140.70 Your synthetic options position is valued the same, and always will be, as a futures contract
If, on the other hand, you sell the call at 34.40 and pay 33.70 for the put, then you have sold the synthetic future at 1140.70 Here, you have the obligation to sell the future above 1140, and the right to sell the future below 1140
The profit/loss of the two synthetics is graphed in Figure 21.3
Long December 1140 put
Short December 1140 put
Long December 1140 call
Short December 1140 call
Figure 21.3 Synthetic long December SPZ futures contract + synthetic short December SPZ futures contract
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Synthetics on individual stocks
In the case of individual stocks, there are also a synthetic futures tion, because the holder of a long call plus short put position at any strike controls a long stock position without having to pay for the stock The situation is the same as with the S&P example above, but often there is no underlying future for comparison Still,the synthetic future exists In the stock options the synthetic future is often spoken of simply as the syn-thetic, or occasionally, the combo
posi-Synthetic long call position
When a long XYZ 100 put is combined with a long underlying position, the profit/loss’s of the put and the underlying cancel each other below
100, leaving the upside, profit-making leg of the underlying The sum equals a synthetic long call For the purpose of illustration, let’s assume that the call was purchased for free At expiration, the synthetic position would be as shown in Figure 21.4
Now let’s return to the example based on the S&P 500 futures and options
on the futures:
XYZP/L
100Long XYZ 100 put
Long XYZ
Long XYZ
Figure 21.4 Synthetic long 100 call
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S&P 500 December future: 1140.70
December 1140 call: 34.40
December 1140 put: 33.70
Suppose you take a long position in the futures contract at 1140.70 and
at the same time you pay 33.70 for the December 1140 put You know that below 1140 the profit/loss of the put and the futures contract offset each other because below 1140 you have the right to sell what you own
at the price at which it was purchased less the cost of the put Above 1140 you are simply long the futures contract Being net long a futures contract above 1140 is the same as owning a December 1140 call The cost of your synthetic call breaks down as follows
The futures contract costs 1140.70, and the right to sell it at 1140 costs 33.70 With your futures contract you have paid 0.70 more for what you own than for your potential selling price With your put your total cost is 0.70 + 33.70
= 34.40, or the price of the December 1140 call Compare the profit/loss tables for the 1140 call (Table 21.1) and the 1140 synthetic call (Table 21.2)
Table 21.1 Profit/loss of SPZ December 1140 call at expiration
Profit/loss of synthetic long
call
–34.40 –34.40 0.00 25.60
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Synthetic short call position
If instead XYZ is sold at 100, and at the same time a 100 put is sold, a thetic short 100 call results Below 100 the profit on the short underlying position and the loss on the short put offset each other Above 100, a loss
syn-is taken on the short underlying position Let’s assume that the put was sold for free The graph at expiration would be as shown in Figure 21.5
Returning to our SPZ example, suppose the above December 1140 put
is sold for 33.70 and a short position is taken in the futures contract at 1140.70, the result is a synthetic short call The profit/loss is the opposite
to the above long synthetic long call (see Table 21.3)
Table 21.3 Profit/loss of synthetic short SPZ December 1140 call
XYZ
P/L
100Short XYZ
Short XYZ 100 put Short XYZ
Figure 21.5 Synthetic short XYZ call
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Synthetic long put position
When a long XYZ 100 call is combined with a short underlying position, the profit/loss of the call and the underlying cancel each other above 100, leaving the downside, profit-making leg of the underlying The sum equals
a synthetic long put We’ll assume that the put is traded for free At tion, the profit/loss graph is shown in Figure 21.6
expira-Returning to our SPZ example, suppose the December 1140 call is chased for 34.40, and a short position in the futures contract is taken at 1140.70 The result is a synthetic long put purchased for 33.70 Tables 21.4 and 21.5 show a comparison of the profit/loss of the synthetic and the straight put
pur-Table 21.4 Profit/loss of long SPZ December 1140 put at
Short XYZLong XYZ 100 call
Figure 21.6 Synthetic long XYZ put
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Table 21.5 Profit/loss of long SPZ December 1140 synthetic put
put
26.30 0.00 –33.70 –33.70
Synthetic short put position
When a short XYZ 100 call is combined with a long underlying position, the profit/loss of the call and the underlying cancel each other above 100, leaving the downside, loss-taking leg of the underlying The sum equals a synthetic short put Again, we’ll assume that the put is traded for free At expiration, the profit/loss graph is shown in Figure 21.7
Returning to our SPZ example, if the December 1140 call is sold at 34.40, and a long position is taken in the underlying at 1140.70, the result is a synthetic short put sold at 33.70 The profit/loss is the opposite of the above long synthetic put (see Table 21.6)
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Table 21.6 Profit/loss of short SPZ December 1140 synthetic
The complex problem of put–call parity
The above are illustrations of put–call parity, which tells us that by
know-ing the value of the underlyknow-ing, the strike price, and either the call or put, the price of the unknown call or put can be determined The formulas for determining the value of a corresponding call or put at a particular strike are as follows
Call – put = futures – strike price (34.40 – 33.70 = 1140.70 – 1140),
therefore
Call = futures – strike price + put (34.40 = 1140.75 – 1140 + 33.70), or
Put = call – futures + strike price (33.70 = 34.40 – 1140.70 + 1140)
This equation can also be solved for the other two variables
Futures = call – put + strike price (1140.70 = 34.40 – 33.70 + 1140), and
Strike price = futures + put – call (1140 = 1140.70 + 33.70 – 34.40)
All this really tells us is that a call and a put at the same strike have the same amount of time premium, or volatility coverage If you’ve read this book with open eyes, you’ve already arrived at the same conclusion, at least intuitively The mysterious and complex world of put–call parity is now exposed as a trifle You, the intelligent reader, have more important things to think about, such as choosing your socks in the morning