The futures market overcomes the illiquidity and credit risk problems of the forward market. In the forward market, a contract is drawn up between the buyer (B) and seller (S). If one of these parties later changes their minds and wishes to exit the contract, they can’t just transfer their obligation to a third party. In a futures contract, a clearing house (CH) acts as the middleman. The seller S sells to the clearing house CH and then the clearing house sells to the buyer. The intercession of the clearinghouse makes the credit problem go away because each party just deals with the clearing house, usually a large institution such as a bank or large brokerage firm. The credit rating of the clearing house is the concern, not the credit ratings of the counterparties. The contracts are standardized so that they can be traded on large exchanges such as the CBOT, NYMEX and the CME. For example, a futures hedger would call her broker and order one August soybean contract. The contract size, delivery locations, delivery date and grade of soybeans are already set. The only variable is price. Prices are not set but “discovered” in the market.
The clearing house overcomes the credit risk problem observed in the forward market by requiring each counterparty to deposit an initial margin before any trading can occur. This initial margin can consist of cash and/or stocks and other tradable securities, and is monitored daily. When the balance in the margin account dips below some threshold amount, called the maintenance margin, the investor gets a call and is asked to deposit funds sufficient to bring the account back up to the initial margin level. If the investor does not do this, the clearing house will close out the investor’s account, selling off any assets. The sale will not necessarily occur at a price that is in the investor’s favor.
The daily monitoring of the futures position is called marking-to-market. At the end of each day, the positions are evaluated and cash transferred into or out of each investor’s account according to the daily profit/loss position. This process is what lessens the potential for default, and is equivalent to closing out the futures position at the end of each day and establishing a new one at that day’s closing price.
Closing out the futures position
It is amusing to imagine a trader waking up on the day after contract expiration to find a truck pulling up to his house to make delivery of 40,000 pounds of pork bellies. Most market participants never intend to take physical delivery of the underlying, but rather close out their positions prior to expiry by entering a reversing trade on the same underlying with the same maturity. For example, if Tanya has a contract to buy 100 tons of August soybeans, prior to expiry she can call her broker and put in an order to sell 100 tons of August soybeans. (Note that there is no requirement to match this trade with her original counterparty, David. The clearinghouse takes care of the transaction and David may well hold his contract to maturity, or reverse it at some other time with someone else taking the other side.) At expiry, these contracts net each other out.
The question now is: how do we price a futures contract? And, perhaps even more interesting: are futures prices good predictors of future spot prices? That is, does Ft,T = E(ST)?
To begin to answer these questions, consider three categories of underlyings: (i) those that can be stored and have large inventories, (ii) those that can be stored and have moderate inventories and (iii) non-storable commodities such as currencies, stock indices and interest rates (these may pay dividends or interest). We can make the same no-arbitrage argument we did in the forward contract section to preclude arbitrage. For assets having cost of carry only with no dividends or interest payments, the spot-futures parity relationship is:
If we consider cash and carry arbitrage on an underlying that pays a discrete dividend D at time td, we find
Ft,T = (St+s)er(T-t)- der(T-tD)
Assuming that s = 0 for such assets, we have:
This model could easily be extended to account for multiple discrete dividends di paid at times ti. The
general spot-futures parity relationship including discrete dividends is then:
The futures price is reduced by presence of dividends, as it should be. The spot price of the asset would include expectations of future dividends, and when a stock pays a dividend D, its price is immediately reduced by d to reflect this.
For a constant dividend yield at the continuously compounded annual rate d, we have
Transaction t T
Buy spot
-S
te
-d(T-t)S
TBorrow money at rate r
S
te
-d(T-t)-S
te
(r-d)(T-t)Sell futures contract (lock in price)
0
F
t,T-S
TNet Proceeds
0
F
t,T-e
(r-d)(T-t)S
tCash and Carry Arbitrage with Continuous Dividend d ) ( 1 ) ( , rT ti n i i t T r t T t S e e F - = - -
å
= d Transaction t tD T Buy spot -St 0 STBorrow money at rate r St 0 -er(T-t)St
Sell futures contract (lock in price) 0 0 Ft,T-ST
Receive Dividend 0 d 0
Reinvest Dividend at r 0 -d der(T-tD)
Net Proceeds 0 0 Ft,T-er(T-t)St + der(T-tD)
The above relationship implies that dividends act like negative interest rates. In fact, if dividends are paid
at the rate r, the futures price should be exactly the spot price. This leads to an interesting application of
currency. Assuming that an investor in foreign country F can borrow funds at that country’s risk free rate
rF, and a domestic investor can borrow at the domestic country’s risk free rate rD. How would spot-futures
parity work?
Interest rate parity
In the preceding example, we apply the principle of interest rate parity, which relates spot and expected
exchange rates between the two countries to spot and expected interest rates. If E0 is the current exchange
rate between the domestic currency and the foreign currency, and F0 is the forward rate agreed upon today
that will apply at maturity, then E0 and F0 are linked through the ratio of expected interest rates over the
period T via:
Note that if one country pays a higher interest rate than another, it must be balanced by the exchange rates. If interest rate parity did not hold (you guessed it) arbitrage would occur.
As an example, suppose that Pablo is the treasurer of the Mexican cement company Cemex. Most of his supplies come from plants based in Mexico (currency symbol MXN), so he will suffer no exchange rate risk on his Mexican-based costs. But suppose that he also has some factories in Argentina. He is worried about his exposure to anticipated inflation of the Argentinean peso for next year’s manufacturing costs. Since he will have to pay these costs in the local currency, he will need to buy Argentinean pesos (ARS). He can buy them today and invest them at the local rate, or lock in a futures contract today. Suppose the one-year risk-free rate in Mexico is 7% and the one-year rate in Argentina is 10%. If today one Argentinean peso costs 2.98 Mexican pesos, what is the expected exchange rate in one year? If interest rate parity holds, the futures rate that he could lock in today should be
ARS MXN F 1 0 1 0.10 07 . 0 1 98 . 2 ÷ ø ö ç è æ + + = or 2.89
MXN/ARS. The expectation is that the Argentinean currency will weaken relative to the Mexican peso over the next year, so it will cost fewer pesos to buy one Argentinean peso next year. The table of cash flows for cash-and-carry arbitrage looks like the following (based on one unit of foreign currency; the
domestic currency is Mexican peso, the foreign is the Argentinean peso; and St is used for the spot
exchange rate, Ft,T for the future exchange rate):
Thus, Spot-Futures Parity with Continuous Dividend is
(r )( )T t t T t S e F, = -d - T F D r r E F ÷÷ø ö ççè æ + + = 1 1 0 0 simple compounding (rD rF)T e E F0 = 0 - continuous compounding
In the above, Pablo wants to have one unit of Argentinean currency in one year’s time. He will invest this Argentine peso in an Argentine bank to earn the local one year rate of 10%, thus today he must have
purchased 1*e-rf(1) = . e-0.1 ARS = 0.9048 ARS. But he has to pay for this with Mexican pesos, so it will
cost current rate 2.98 MXN/ARS * 0.9048 ARS = 2.6964 MXN. He has to borrow this at the domestic rate r = 7%. This is represented in lines 1, 2 and 3 under time “t” in the above table. At the same time, he sells an FX futures contract for one Argentine Peso at the rate calculated by interest rate parity 2.89 MXN/ARS. Now, what happens after one-year’s time?
His Argentinean currency has grown to 1.0 ARS in the bank. He has to pay back his loan, 2.6964 MXN, at
7%, or 2.6964e0.07(1) = 2.89 MXN. (Note: this is the rate we calculated using interest parity, so it works.)
His futures contract is worth whatever the current exchange rate is, Ft,T. So his position is Ft,T – 2.89 MXN.
He breaks even if the spot exchange rate is 2.89 MXN/ARS. If it is lower, say, 2.80 MXN/ARS, he loses on his futures contract because he has agreed to pay 2.89. His loss is 0.09 MXN/ARS. But, he is paying less for his materials, so his net cost is what he locked in: 2.89 MXN/ARS. If the exchange rate is higher (the Argentinean peso strengthened against the Mexican peso), he wins because he locked in a rate of 2.89,
however, he has to pay more for his supplies. The difference Ft,T-2.89 will help offset these additional
costs. In terms of formulas:
This is interesting because this formula looks exactly like the formula for the futures price with a continuous dividend, except rF is used instead of d. In fact, since Pablo is receiving rF, the foreign interest
rate functions just like a continuously compounded dividend. You can trade on intermarket spreads, borrowing money from one FX futures market and lending in another.
Trading strategy
For all cases, the appropriate trading strategy in the futures market is: if the futures prices exceed the right hand side (expected future value of spot plus cost of carry) then sell futures and buy the spot. If the futures price is lower than the right hand side of the equation, then buy the futures and sell the spot. The rule “buy low, sell high” applies.
(r r )( )T t t
T
t S e F
F, = - -
Line Transaction DOMESTIC FOREIGN DOMESTIC FOREIGN 1 Buy one Unit of Foreign Currency @ rF -Ste-rf(T-t) e
-rf(T-t)
0 0
2 Borrow Domestic Currency Ste-rf(T-t) 0 -Ste(r-rf)(T-t) 0
3 Invest unit of FX at rF 0 -e -rf(T-t)
0 1
4 Sell FX futures contract 0 0 Ft,T -1
5 Net Proceeds 0 0 Ft,T-St e(r-rf)(T-t) 0
t T
Implied interest rate
In any of these formulas, we can invert them to solve for implied interest rates (often called the implied
repo rate), cost of carry, etc. For example, for an asset with cost of carry, we had Taking natural logs of both sides allows us to solve for the implied risk-free rate as
Payoff diagram of futures position
In theory, the payoff graph will look just like the payoff graph for a forward, since payoff = FT-ST, but due
to marking to market and margin requirements, the daily profit/loss of the portfolio will appear different. However, the payoff diagram for a single day should look like the payoff diagram for a forward. Suppose a trader Mike enters a September high-grade copper contract. The contract size for copper is 25,000 lbs and price is quoted in cents per lb. The current spot price (“cash price”) is 72.05 and the September futures price is 73.70. Even though the contract is priced to provide equilibrium between the expected spot price in September and the futures price, so that theoretically the contract should have zero value when entered, there are margin requirements. Suppose that the initial margin requirement is $10,000 and the maintenance margin on this contract is $7,500. Mike would then have to deposit $10,000 in cash or other “marginable securities” into his account. The following table shows a hypothetical table of daily positions over the next five days. The initial futures position is worth $0.7370/lb times the contract size 25,000 lbs = $18,425.
Day Futures Price, $/bu Account Balance Comments
0 0.737 $10,000 Initial margin 1 0.730 $9,825 D =$(0.73-0.737)*25,000 = -$175 2 0.727 $9,750 D =$(0.727-0.730)*25,000 = -$75 3 0.723 $9,650 D =$(0.723-0.727)*25,000 = -$100 4 0.729 $9,800 D =$(0.729-0.723)*25,000 = +$150 … … … 15 0.640 $7,575
16 0.635 $7,450 Margin call – trader must deposit $2,550
17 0.635 $10,000
… … …
30 0.750 $12,875
31 0.760 $13,125 Trader closes out position
Note that when the account balance exceeds the initial margin balance (as in days 30 and 31 above) the excess margin may be withdrawn or kept in the account, where it will generally be invested in interest- earning money market funds.
At first glance, it appears that Mike made a nice profit of $13,125 - $10,000 = $3,125 on the position, but the actual profit/loss over the period is (0.760-0.737)*25,000 = $575 since he had to inject an additional cash flow of $2,550 into the account (note that $13,125 - $10,000 - $2,550 = $575). This illustrates one of
(
-)
ççèæ + ÷÷øö = s S F t T r t T t, ln 1(
)
r( )T t t T t S se F, = + -the fundamental differences between futures and forwards, since, had he purchased a forward instead, he would have had a net gain of only $575. There is no marking to market on forward contracts.
Swaps
Swaps are agreements to exchange assets. There are interest rate swaps, currency swaps, equity swaps and commodity swaps. The swap market is huge. According to the Bank for International Settlements, U.S.- dollar denominated interest rate swaps accounted for almost $33 billion dollars in notional, approximately 50% of the total notional amount for all OTC derivatives in the first half of 2001. Outright forwards and forex (foreign exchange) swaps were the second largest, at almost $15 billion, and currency swaps accounted for about $2.3 billion.
Swaps, of whatever flavor, may be used to transform a company’s liabilities to better match their assets, to borrow more cheaply in a different market, or to gain exposure to a desired market. As an example, a finance company such as Ford Credit or GE Capital makes fixed-rate loans to customers purchasing vehicles (cars, trucks, ambulances, fire trucks.) These loans are the company’s assets and are funded by borrowing at a floating rate such as LIBOR and/or, perhaps, the commercial paper rate. They may also float huge bond issues that must be paid back at the agreed-upon fixed rate. These loans are the company’s liabilities. If the funding cost of the bonds is lower than the interest income they receive, they profit; if not, they lose money. This spread between funding costs and interest income on customer loans is what drives companies’ income and profitability.
Imagine that the finance company has a five-year bond issue that they are legally obliged to repay at 5%. Imagine also that they have a portfolio of car loans yielding 6%. The income from these loans can finance the debt payment on their borrowings. There are at least two potential risks to profitability here. First, there is a mismatch between the maturities of the assets and liabilities. The liability is for five years but the average car loan may be for three years or less, because even if people borrow for five years, there are always prepayments or trade-ins on new vehicles that reduce the average loan life to about two and a half years. And even if no new loans were issued at all for the first two and a half years, when new loans are made, they will likely be made at a different rate. If interest rates have fallen over this period, say, to 4%, then the company is in trouble. They still have the fixed 5% liability, but now there is a loss due to the gap between asset and liability. Of course, new loans are made continuously and new debt is incurred as well. The second risk is called basis risk. This occurs when the liabilities are funded at one benchmark and the assets at another. For example, if a finance company borrows at three-year LIBOR plus some spread, but lends at the three year treasury rate plus some spread, and if these rates do not move identically over time, then the spreads between borrowing and lending will vary – again, potentially impacting profitability. To solve these problems a company might enter into an interest-rate swap to trade their loan income to some counterparty for a fixed payment with a maturity matching their liabilities. This is called a fixed-for-
fixed or fixed-fixed type of swap. Or, they might swap out their debt obligation for one that floats with some benchmark rate such as LIBOR plus a spread. This is called pay fixed receive floating interest rate
swap. In fact, GE Capital and Ford Credit are the two largest users of interest rate derivatives. (Pricing of
Options
An option is a derivative on an underlying asset such as a stock, index, Eurodollar futures contract, commodity, or interest rate. In some ways options are similar to forwards and futures, but in other ways they’re significantly different. In contrast to futures and forwards, options cost more money upfront to buy. This cost is called the option premium. The buyer of the option has the right but not the obligation to exercise the option, while the seller of the option is obligated to perform. This is unlike the forward contract where, once entering the contract, both parties are obligated to deliver.
Options are traded on most major exchanges. They are used for hedging purposes, for speculation and by arbitrageurs. A speculator is an investor who wishes to take a position on a particular asset without actually owning the asset. Since options on an underlying can be purchased at a fraction of the cost it would take to acquire a long position in the underlying, options provide leverage. Leverage magnifies the returns, both positive and negative, which is the reason that some people say that options are very risky. Options range from the simple, “plain vanilla” flavors such as simple puts and calls on a stock, to the exotic such as binary options, barrier options, Asian options and compound options. The value of such exotic options depends not just on the value of the underlying at expiry, but also on the price path over the entire time period. There are options on currencies, options on interest rates, options on futures, options on swaps and even options on options. (While you’re reading this, someone is probably engineering an option on an option to acquire an option.)
The work that we have already done to understand forwards and futures provides us with a good foundation to begin our discussion of options. Pricing options is far more complex than pricing either a forward or a future. In forwards and futures, we were able to develop a price due to an appeal to futures-spot parity (or forward-spot parity) and the no-arbitrage arguments. The prices we developed depended only on the risk- free rate, assumed to remain constant over the period, and on any dividends or costs of carry. We didn’t concern ourselves with the actual path that the spot took over time, we locked in a price that we wanted and, at expiry, our profit was determined by the difference of the futures price with the spot at some point in time, whether positive or negative.
In contrast, with options we pre-select a price or level in advance (the strike price) as well as an expiration time. In order to model the behavior of the option, we need a model of the dynamics of the underlying. The model for the movement of the underlying asset is fairly complicated and is known as the Black-
Scholes partial differential equation, which will be developed and solved in the following. We can apply the model, with modifications, to interest rates, stock indices and other options as well.