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Options on Stock Indices, Currencies, and Futures Chapter 13. STOCK INDEX OPTIONS ONE CONTRACT VALUE = (INDEX VALUE)($MULTIPLIER) One contract = (I)($m) ACCOUNTS ARE SETTLED BY CASH. STOCK INDEX OPTIONS FOR PORTFOLIO INSURANCE Problems: How many puts to buy?
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ONE CONTRACT VALUE =
One contract = (I)($m)
ACCOUNTS ARE SETTLED BY CASH
The protective put consists of holding the portfolio and purchasing n puts.
THE CAPITAL ASSET PRICING MODEL.
For any security i,the expected excess return on the security and the expected excess return on the market portfolio are linearly related by their beta:
THE INDEX TO BE USED IN THE STRATEGY, IS TAKEN TO BE A PROXY FOR THE MARKET PORTFOLIO, M. FIRST, REWRITE THE ABOVE EQUATION FOR THE INDEX I AND ANY PORTFOLIO P :
In a more refined way, using V and I for the portfolio and index market values, respectively:
NEXT, use the ratio Dp/V0 as the portfolio’s annual dividend payout ratio qP and DI/I0 the index annual dividend payout ratio, qI.
The ratio V1/ V0 indicates the portfolio required protection ratio.
The manager wants V1, to be down to no
more than 90% of the initial portfolio
market value, V0: V1 = V0(.9).
We denote this desired level by (V1/ V0)*.
This is the decision variable.
2.The exercise price, K, is determined by substituting I1 = K and the required level, (V1/ V0)* into the equation:
and solving for K:
We are now ready to calculate the floor level of the portfolio: V1+n($m)(K- I1)
Min portfolio value = V1+n($m)(K- I1)
This is the lowest level that the portfolio value can attain. If the index falls below the exercise price and the portfolio value declines too, the protective puts will be exercised and the money gained may be invested in the portfolio and bring it to the value of:
The desired level of protection is made at time 0. This determines the exercise price and management can also calculate the minimum portfolio value.
EXAMPLE: A portfolio manager expects the market to fall by 25% in the next six months. The current portfolio value is $25M. The manager decides on a 90% hedge by purchasing 6-month puts on the S&P500 index. The portfolio’s beta with the S&P500 index is 2.4. The S&P500index stands at a level of 1,250 points and its dollar multiplier is $100. The annual risk-free rate is 10%, while the portfolio and the index annual dividend payout ratios are 5% and 6%, respectively. The data is summarized below:
Solution: Purchase n = 480 six-months puts with exercise price K = 1,210.
Holding the portfolio and purchasing 480, 6-months protective puts on the S&P500 index, with the exercise price K = 1210, guarantees that the portfolio value, currently $25M, will not fall below $22,505,000 in six months.
Solution: Purchase n = 10 six-months puts with exercise price K = 960.
Holding the portfolio and purchasing 10, 3-months protective puts on the S&P500 index, with the exercise price K = 960, guarantees that the portfolio value, currently $500,000 will not fall below $450,000 in three months.
βp = 1 and qP =qI, then:
Standardized Options Currencies Traded: The PHLX lists six dollar-based standardized currency option contracts, which settle in the actual physical currency. These are the most heavily traded currencies in the global inter bank market.
Contract Size: The amounts of currency controlled by the various currency options contracts are geared to the needs of the widest possible range of participants.
Exercise Style: American- or European
options available for physically settled
contracts; Long-term options are
With the Canadian dollar spot price currently at a level of USD.6556/CD, strike prices would be listed in half-cent intervals ranging from 60 to 70. i.e., 60, 60.5, 61, …, 69, 69.5, 70. If the Canadian dollar spot rate should move to say USD.7060/CD, additional strikes would be listed. E.G, 70, 70.5, 71, …, 75.Standardized Options
Exercise PricesExercise prices are expressed in terms of U.S. cents per unit of foreign currency. Thus, a call option on EUR with an exercise price of 82 would give the option buyer the right to buy Euros at 82 cents per EUR.
Strike price intervals vary for the different expiration time frames. They are narrower for the near-term and wider for the long-term options.
A premium of 1.00 for a given EUR option is one cent (USD.01) per EUR.
Since each option is for 62,500 EURs, the total option premium would be
[62,500DM][USD.01/EUR] = USD625.
Customized currency options can be traded on any combination of the currencies currently available for trading. Currently, AUD must be denominated in U.S. dollars. AUD premiums must be denominated in USD.
In the case of an option on the USD in CD, the underlying currency is U.S. dollars. The strike prices and premiums are quoted in Canadian dollars. E.G, a call option on the USD with a strike price of 1.542 gives the buyer the right to purchase 50,000 USD at CD1.542/USD.
Base CurrencyThe base currency is that currency in which terms the underlying is being quoted, i.e. strike price.
Expiration/Last Trading DayExpirations can be established for any business day up to two years from the trade date. Customized option contracts expire at 10:15 a.m., Eastern Time on the expiration day in contrast with standardized options which expire at 11:59 p.m., Eastern Time on the expiration day.
The contract size for customized currency options is:
Australian dollar 50,000AUD
British Pound 31,250GBP
Canadian dollar 50,000CD
Japanese Yen 6,250,000JPY
Mexican Peso 250,000MXP
Spanish Peseta 5,000,000ESP
Swiss Frank 62,500CHF
Exercise-style European-style only.
Minimum Transaction SizeSince customized currency options were designed for the institutional market, there is a minimum opening transaction size which equals or exceeds 50 contracts.
A call option on the EUR quoted in American terms would have a strike price expressed in USD. For example, USD.8484 per EUR.
A similar option expressed in European terms would be a put option on USD with a strike price expressed in EUR. For example, 1.1787 EUR per USD.
The premium for an option on the EUR with a strike price in USD (EUR is the underlying currency and the USD is the base currency) quoted in cents per EUR(premium of 2.50) would be calculated as follows: the aggregate premium for each contract =
USD1562.50(USD.025 x 62,500EUR per contract ).Similarly, if this option were quoted in percentage of underlying and the premium was 2.5%, the premium amount for each contract = 1562.5 EUR (.025 x 62,500 EUR per contract).
If the put option costs USD.01/EUR, the floor price can be roughly approximated as:
USD.80/EUR - USD.O1/EUR = USD.79/EUR.
That is, if the option is used, the put holder will be able to sell the EUR for the USD.80/EUR strike price, but in the meantime, have paid a premium of USD.01/EUR. Deducting the cost of the premium leaves USD.79/EUR as the floor price established by the purchase of the put. This calculation ignores fees and interest rate adjustments.
USD.80/EUR + USD.02/EUR = USD.82/EUR
or the strike price plus the premium.
1- Foreign currency put options on spot can be used as insurance to establish a floorprice on the domestic currency value of foreign exchange. This floor price is approximately
Floor price = exercise price of put
- put premium
2- Foreign currency call options on spot can be used as insurance to establish a ceiling price on the domestic currency cost of foreign exchange. This ceiling price is approximately
Ceiling price = exercise price of call
+ call premium.
8(£3l,250)($.0220/£) + 8($4) = $5,532.
$5,532/£250,000 + $1.50/£= $.02213/£ + $1.50/£= USD1.52213/GBP.
Notice here that the add factor USD.02213/GBP is larger than the option premium of USD.0220/GBP by USD.00013/GBP, which represents the dollar brokerage cost per pound. The number USD1.52213/GBP is the importer's ceiling price. The importer is assured he will not pay more than this, but he could pay less. The price the importer will actually pay will depend on the spot price on the November payment date. To illustrate this, we can consider two scenarios for the spot rate.
- (sale value of options- $32)/£250,000.
The USD.02213/GBP that was the original cost of the premium and brokerage fee turned out in this case to be an unnecessary expense.
[Now, to be strictly correct, a further adjustment to the calculation should be made. Namely, the $1.46 and $.02213 represent cash flows at two different times. Thus, if R is the amount of interest paid per dollar over the September 16 to November time period, the proper calculation is
- (sale value of options-$32)/250,000.]
(USD.055/GBP))( £250,000) + 8($4)
This amount is:
$373,782/£250,000 = USD1.49513/GBP.
USD1.49513/GBP + USD.02213(l + R)/GBP
If the importer chooses instead to exercise the option, the calculations will be similar except that the brokerage fee will be replaced by an exercise fee.
This concludes Example 1.
Suppose the company buys from its bank an American put on USD50M with a strike price of JPY130/USD.
JPY130/USD - JPY4/USD = JPY126/USD.
Again, we can consider two scenarios, one in which the yen falls in value to JPY145/USD and the other in which the yen rises in value to JPY115/USD.
The yen falls to JPY145/USD. In this case the company will not exercise the option to sell dollars for yen at JPY13O/USD, since the company can do better than this in the exchange market. The company will have obtained a net value of
JPY145/USD - JPY4/USD = JPY141/USD.
The JPY rises to JPY115/USD. The company will exercise the put and sell each U.S. dollar for JPY130/USD. The company will obtain, net,
JPY130/USD - JPY4/USD = JPY126/USD.
This is JPY11 better than would have been available in the FX market and reflects a case where the “insurance” paid off. This concludes Example 2.
General considerations. The writer of a foreign currency option on spot or futures is in a different position from the buyer of these options. The buyer pays the premium up front and then can choose to exercise the option or not. The buyer is not a source of credit risk once the premium has been paid. The writer isa source of credit risk, however, because the writer has promised either to sell or to buy foreign currency if the buyer exercises his option. The writer could default on the promise to sell foreign currency if the writer did not have sufficient foreign currency available, or could default on the promise to buy foreign currency if the writer did not have sufficient domestic currency available.
The following example will illustrate:
the risk/return trade-off for the case of an oil company with an exchange rate risk, that chooses to become an option writer.
Clearly, the options might expire worthless or they might be exercised. In either case, however, Iris walks away with the full amount of the options premium:
To illustrate Iris' strategy, consider an anticipated cash flow of CD300M over the next 180 days. With hedge ratio of 1:1, Iris sells CD300,000,000/CD50,000 = 6,000 PHLX calls.
F = USD.7447/CD.
For the current level of spot rate, logical strike price choices for the calls might be X = USD.74, or USD.75, or USD.76.
For the illustration, assume that Iris’ brokerage fee is USD4 per written call and let the hypothetical market values of the options be those listed in the following:
c(K = SUD.75/CD) = USD.00650;
c(K = USD.76/CD) = USD.00313.
Actually, the above table consolidates three profit profile tables, each corresponding to one of the three strike prices under consideration.
The three tables are as follows:
As illustrated by the consolidated table and the three separate profit profile tables, the lower the strike price chosen, the better the protection against a depreciating CD. On the other hand, a lower strike price limits the corresponding profitability of the strategy if the CD happens to appreciate against the USD in six months. The optimal decision of which strategy to take is a function of the spot exchange rate at expiration. One possible comparison is to evaluate the options strategy vis-à-vis the immediate forward exchange.
options strategy the forward
exchange rate is
F = USD.7447/CD.
Thus, a future break-even spot
rate can be calculated for every
corresponding exercise price
IF: K =.74,
S(CD300M) + 4,113,000 = USD223,410,000
SBE = USD.7310/CD.
IF K= .75,
S(CD300M) + 1,926,000 = USD223,410,000
SBE = USD.7383/CD.
IF K= .76,
S(CD300M) + 915,000 = USD223,410,000
SBE = USD.7416/CD.
Writing the calls will protect Iris’ flow
in USD six months from now better
than an immediate forward exchange,
for all spot rates (in six months) that
are above the corresponding break-
even exchange rates.
Iris either maintains an open long position of CD300M un hedged. Alternatively, Iris writes 6,000 PHLX calls with 180-day expiration period. Possible strike prices are USD.76/CD, USD.75/CD, USD.74/CD. Current spot and forward exchange rates are USD.75/CD and USD.7447/CD, respectively.
Terminal Spot rateOptimal Decision
S >.76235 Hold long currency only
.75267 < S< .76235 Write options with K = .76
.74477 < S< .75267 Write options with K = .75
S < .74477 Write options with K = .74
Because of the large OCC fee of USD35 per exercised call assumed in the example, it might be less expensive for Iris to buy back the calls and pay the brokerage fee of USD4 per call in the event the options were in dancer of being exercised. In addition, it is assumed that Iris will have the CD300M on hand if the options are exercised. This would not be the case if actual Canadian dollar revenues were less than anticipated. In that event, the options would need to be repurchased prior to expiration.
will have a different payoff, depending on
the movement in the exchange rate. But
Iris' expectation regarding the exchange
rate is not the only relevant criterion for
choosing a risk-management strategy.
The possible variation in the underlying
position should also be considered.
payoffs for each of the call-writing
choices, compared to the un hedged
position and a forward market hedge:
Position None. Zero.
forward: USD223,410,000 USD223,410,000
.76 call: USD228,705,000 Unhedged minimum + USD915,000.
.75 call USD226,716,000 Unhedged minimum + USD1,926,000.
.74 call USD225,903,000 Unhedged minimum + USD4,113,000.
When a call futures option is exercised the holder acquires
1. A long position in the futures
2. A cash amount equal to the excess of
the most recent settlement futures price over the strike price
When a put futures option is exercised the holder acquires
1. A short position in the futures
2. A cash amount equal to the excess of
the strike price over the most recent settlement futures price
If the futures position is closed out immediately:
Payoff from call = F0 – K
Payoff from put = K – F0
where F0 is futures price at time of exercise
Consider the following two portfolios:
1. European call plus Ke-rT of cash
2. European put plus long futures plus cash equal to F0e-rT
They must be worth the same at time T so that
Set S0 = current futures price (F0)
Set q = domestic risk-free rate (r )