Gold leasing

Market participants can borrow and lend gold in the gold lease market. Typical
lenders of gold in the lease market are entities with large stocks of physical gold on
which they wish to earn a rate of return, such as central banks. Typical borrowers
of gold are gold dealing desks.
Suppose you are a bank intermediating in the gold market. Let the spot gold price
(in US dollars) be Stó275.00, let the US dollar 6-month deposit rate be 5.6% and let
the 6-month gold lease rate be 2% per annum. The 6-month forward gold price must
then be
Ft,Tó 1ò0.056ñ0.02
2 · 275ó279.95
The gold lease rate plays the role of the dividend rate in our framework.
A mining company sells 1000 ounces gold forward for delivery in 6 months at the
market price of USD279.95. You now have a long forward position to hedge, which
you can do in several ways. You can use the futures market, but perhaps the delivery
dates do not coincide with the forward. Alternatively, you can lease gold from a
central bank for 6 months and sell it immediately in the spot market at a price of
USD275.00, investing the proceeds (USD 275 000) in a 6-month deposit at 5.6%.
Note that there is no net cash flow now.
In 6 months, these contracts are settled. First, you take delivery of forward gold
from the miner and immediately return it to the central bank along with a wire
transfer of USD2750. You redeem the deposit, now grown to USD282 700, from the
bank and pay USD279 950 to the miner.
As noted above, it is often difficult to take short positions in physical commodities.
The role of the lease market is to create the possibility of shorting gold. Borrowing
gold creates a ‘temporary long’ for the hedger, an obligation to divest himself of gold
6 months hence, which can be used to construct the synthetic short forward needed
to offset the customer business.
Futures
Futures are similar to forwards in all except two important and related respects.
First, futures trade on organized commodity exchanges. Forwards, in contrast, trade over-the-counter, that is, as simple bilateral transactions, conducted as a rule by
telephone, without posted prices. Second, a forward contract involves only one cash
flow, at the maturity of the contract, while futures contracts generally require interim
cash flows prior to maturity.
The most important consequence of the restriction of futures contracts to organized
exchanges is the radical reduction of credit risk by introducing a clearinghouse as
the counterparty to each contract. The clearinghouse, composed of exchange members,
becomes the counterparty to each contract and provides a guarantee of performance:
in practice, default on exchange-traded futures and options is exceedingly
rare. Over-the-counter contracts are between two individual counterparties and have
as much or as little credit risk as those counterparties.
Clearinghouses bring other advantages as well, such as consolidating payment
and delivery obligations of participants with positions in many different contracts.
In order to preserve these advantages, exchanges offer only a limited number of
contract types and maturities. For example, contracts expire on fixed dates that may
or may not coincide precisely with the needs of participants. While there is much
standardization in over-the-counter markets, it is possible in principle to enter into
obligations with any maturity date. It is always possible to unwind a futures position
via an offsetting transaction, while over-the-counter contracts can be offset at a
reasonable price only if there is a liquid market in the offsetting transaction. Settlement
of futures contracts may be by net cash amounts or by delivery of the
underlying.
In order to guarantee performance while limiting risk to exchange members, the
clearinghouse requires performance bond from each counterparty. At the initiation
of a contract, both counterparties put up initial or original margin to cover potential
default losses. Both parties put up margin because at the time a contract is initiated,
it is not known whether the terminal spot price will favor the long or the short. Each
day, at that day’s closing price, one counterparty will have gained and the other will
have lost a precisely offsetting amount. The loser for that day is obliged to increase
his margin account and the gainer is permitted to reduce his margin account by an
amount, called variation margin, determined by the exchange on the basis of the
change in the futures price. Both counterparties earn a short-term rate of interest
on their margin accounts.
Margining introduces an importance difference between the structure of futures
and forwards. If the contract declines in value, the long will be putting larger and
larger amounts into an account that earns essentially the overnight rate, while the
short will progressively reduce his money market position. Thus the value of the
futures, in contrast to that of a forward, will depend not only on the expected future
price of the underlying asset, but also on expected future short-term interest rates
and on their correlation with future prices of the underlying asset. The price of a
futures contract may therefore be higher or lower than the price of a congruent
forward contract. In practice, however, the differences are very small.
Futures prices are expressed in currency units, with a minimum price movement
called a tick size. In other words, futures prices cannot be any positive number, but
must be rounded off to the nearest tick. For example, the underlying for the
Eurodollar futures contract on the Chicago Mercantile Exchange (CME) is a threemonth
USD1 000 000 deposit at Libor. Prices are expressed as 100 minus the Libor
rate at futures contract expiry, so a price of 95.00 corresponds to a terminal Libor
rate of 5%. The tick size is one basis point (0.01). The value of one tick is the increment in simple interest resulting from a rise of one basis point: USD1 000 000 · 0.0001·
90
360ó25. Another example is the Chicago Board of Trade (CBOT) US Treasury bond
futures contract. The underlying is a T-bond with a face value of USD100 000 and a
minimum remaining maturity of 15 years. Prices are in percent of par, and the tick
size is 1
32of a percentage point of par.
The difference between a futures price and the cash price of the commodity is
called the basis and basis risk is the risk that the basis will change unpredictably.
The qualification ‘unpredictably’ is important: futures and cash prices converge as
the expiry date nears, so part of the change in basis is predictable. For market
participants using futures to manage exposures in the cash markets, basis risk is
the risk that their hedges will offset only a smaller part of losses in the underlying
asset.
At expiration, counterparties with a short position are obliged to make delivery to
the exchange, while the exchange is obliged to make delivery to the longs. The
deliverable commodities, that is, the assets which the short can deliver to the long
to settle the futures contract, are carefully defined. Squeezes occur when a large
part of the supply of a deliverable commodity is concentrated in a few hands. The
shorts can then be forced to pay a high price for the deliverable in order to avoid
defaulting on the futures contract.
In most futures markets, a futures contract will be cash settled by having the
short or long make a cash payment based on the difference between the futures price
at which the contract was initiated and the cash price at expiry. In practice, margining
will have seen to it that the contract is already largely cash settled by the expiration
date, so only a relatively small cash payment must be made on the expiration date
itself.
The CBOT bond futures contract has a number of complicating features that
make it difficult to understand and have provided opportunities for a generation of
traders:
Ω In order to make many different bonds deliverable and thus avoid squeezes, the
contract permits a large class of long-term US Treasury bonds to be delivered into
the futures. To make these bonds at least remotely equally attractive to deliver,
the exchange establishes conversion factors for each deliverable bond and each
futures contract. The futures settlement is then based on the invoice price,
which is equal to the futures price times the conversion factor of the bond being
delivered (plus accrued interest, if any, attached to the delivered bond).
Ω Invoice prices can be calculated prior to expiry using current futures prices. On
any trading day, the cash flows generated by buying a deliverable bond in the
cash market, selling a futures contract and delivering the purchased bond into
the contract can be calculated. This set of transactions is called a long basis
position. Of course, delivery will not be made until contract maturity, but the
bond that maximizes the return on a long basis position, called the implied repo
rate, given today’s futures and bond prices, is called the cheapest-to-deliver.
Ω Additional complications arise from the T-bond contract’s delivery schedule. A
short can deliver throughout the contract’s expiry month, even though the contract
does not expire until the third week of the month. Delivery is a three-day
procedure: the short first declares to the exchange her intent to deliver, specifies
on the next day which bond she will deliver, and actually delivers the bond on the
next day. 14

Foreign exchange

Forward foreign exchange is foreign currency deliverable in the future. Its price is
called forward exchange rate or the forward outright rate, and the differential of
the forward minus the spot exchange rate is called the swap rate (not to be confused
with the rate on plain-vanilla interest rate swaps).
To apply the general mechanics of a forward transaction described above to this
case, let rt,T and r*t ,T represent the domestic and foreign money-market interest rates.
The spot and forward outright exchange rates are St and Ft,T, expressed in domestic
currency units per foreign currency unit.
To create a synthetic long forward,
Ω Borrow St/(1òr*t,Tq) domestic currency units at rate rt,T and buy 1/(1òr*t,Tq)
foreign currency units. Deposit the foreign currency proceeds at rate r*t ,T. There is
no net cash outlay now.
Ω At time T, the foreign currency deposit has grown to one foreign currency unit,
and you must repay the borrowed
St(1òrt,T q)
1òr*t ,T q
including interest. This implies that the forward rate is
Ft,Tó1òrt,T q
1òr*t ,T q
St
Here is a numerical example of this relationship. Suppose the Euro-dollar spot exchange rate today is USD1.02 per Euro. Note that we are treating the US dollar as
the domestic and the Euro as the foreign currency. Suppose further that the US
1-year deposit rate is 5.75% and that the 1-year Euro deposit rate is 3.0%. The
1-year forward outright rate must then be
1.0575
1.03
1.02ó1.0472
Typically, forward foreign exchange rates are quoted not as outright rates but in
terms of forward points. The points are a positive or negative quantity which is
added to the spot rate to arrive at the forward outright rate, usually after dividing by
a standard factor such as 10 000. In our example, the 1-year points amount to
10 000 · (1.0472ñ1.02)ó272. If Euro deposit rates were above rather than below US
rates, the points would be negative.

Cost-of-carry with a known dividend

If the commodity pays dividends or a return dT (expressed as a percent per period of
the commodity price, discretely compounded), which is known in advance, the
analysis becomes slightly more complicated. You can think of dt,T as the dividend
rate per ‘share’ of the asset: a share of IBM receives a dividend, an equity index unit
receives a basket of dividends, $100 of par value of a bond receives a coupon, etc.
The dT may be negative for some assets: you receive a bill for storage and insurance
costs, not a dividend check, on your 100 ounces of platinum. The amount of dividends
received over Tñt years in currency units is dt,TSt (Tñt).
The synthetic long forward position is still constructed the same way, but in this
case the accrued dividend will be received at time T in addition to the commodity price.
The net cash flow is STòdt,TSt (Tñt)ñ[1òrt,T(Tñt)]St . The no-arbitrage condition is
now
STñFt,TóSTñ[1ò(rt,Tñdt,T)(Tñt)]St .
The forward price will be lower, the higher the dividends paid:
Ft,Tó[1ò(rt,Tñdt,T)(Tñt)St .
The forward price may be greater than, less than or equal to than the spot price if
there is a dividend. The long’s implied financing cost is reduced by the dividend
received.

Forwards, futures and swaps

Forwards and forward prices
In a forward contract, one party agrees to deliver a specified amount of a specified
commodity – the underlying asset – to the other at a specified date in the future (the
maturity date of the contract) at a specified price (the forward price). The commodity
may be a commodity in the narrow sense, e.g. gold or wheat, or a financial asset, e.g.
foreign exchange or shares. The price of the underlying asset for immediate (rather
than future) delivery is called the cash or spot price.
The party obliged to deliver the commodity is said to have a short position and
the party obliged to take delivery of the commodity and pay the forward price for it
is said to have a long position.
A party with no obligation offsetting the forward contract is said to have an open
position. A party with an open position is sometimes called a speculator. A party
with an obligation offsetting the forward contract is said to have a covered position.
A party with a closed position is sometimes called a hedger.
The market sets forward prices so there are no cash flows – no money changes
hands – until maturity. The payoff at maturity is the difference between forward
price, which is set contractually in the market at initiation, and the future cash
price, which is learned at maturity. Thus the long position gets STñFt,T and the short
gets Ft,TñST, where Tñt is the maturity, in years, of the forward contract (for
example, Tó1/12 for a one-month forward), ST is the price of the underlying asset
on the maturity date, and Ft,T is the forward price agreed at time t for delivery at time
T. Figure 1.4 illustrates with a dollar forward against sterling, initiated at a forward
outright rate (see below) of USD1.60. Note that the payoff is linearly related to the
terminal value ST of the underlying exchange rate, that is, it is a constant multiple,
in this case unity, of ST.
1.45 1.50 1.55 1.60
ST
0.05
0.10
0.05
0.10
payoff
forward rate
Figure 1.4 Payoff on a long forward.
No-arbitrage conditions for forward prices
One condition for markets to be termed efficient is the absence of arbitrage. The
term ‘arbitrage’ has been used in two very different senses which it is important to
distinguish:
Ω To carry out arbitrage in the first sense, one would simultaneously execute a set
of transactions which have zero net cash flow now, but have a non-zero probability
of a positive payoff without risk, i.e. with a zero probability of a negative payoff in
the future.
Ω Arbitrage in the second sense is related to a model of how asset prices behave. To
perform arbitrage in this sense, one carries out a set of transactions with a zero
net cash flow now and a positive expected value at some date in the future.
Derivative assets, e.g. forwards, can often be constructed from combinations of
underlying assets. Such constructed assets are called synthetic assets.
Covered parity or cost-of-carry relations are relations are between the prices of
forward and underlying assets. These relations are enforced by arbitrage and tell us
how to determine arbitrage-based forward asset prices.
Throughout this discussion, we will assume that there are no transactions costs
or taxes, that markets are in session around the clock, that nominal interest rates
are positive, and that unlimited short sales are possible. These assumptions are
fairly innocuous: in the international financial markets, transactions costs typically
are quite low for most standard financial instruments, and most of the instruments
discussed here are not taxed, since they are conducted in the Euromarkets or on
organized exchanges.
Cost-of-carry with no dividends
The mechanics of covered parity are somewhat different in different markets,
depending on what instruments are most actively traded. The simplest case is that
of a fictitious commodity which has no convenience value, no storage and insurance
cost, and pays out no interest, dividends, or other cash flows. The only cost of holding
the commodity is then the opportunity cost of funding the position.
Imagine creating a long forward payoff synthetically. It might be needed by a dealer
hedging a short forward position:
Ω Buy the commodity with borrowed funds, paying St for one unit of the commodity
borrowed at rt,T, the Tñt-year annually compounded spot interest rate at time t.
Like a forward, this set of transactions has a net cash flow of zero.
Ω At time T, repay the loan and sell the commodity. The net cash flow is
STñ[1òrt,T(Tñt)]St .
This strategy is called a synthetic long forward.
Similarly, in a synthetic short forward, you borrow the commodity and sell it,
lending the funds at rate rt,T,: the net cash flow now is zero. At time T, buy the
commodity at price ST and return it: the net cash flow is [1òrt,T(Tñt)]StñST.
The payoff on this synthetic long or short forward must equal that of a forward
contract: STñ[1òrt,T (Tñt)]STóSTñFt,T. If it were greater (smaller), one could make
a riskless profit by taking a short (long) forward position and creating a synthetic
long (short) forward. This implies that the forward price is equal to the future value
of the current spot price, i.e. the long must commit to paying the financing cost of
the position: Ft,Tó[1òrt,T(Tñt)]St .
Two things are noteworthy about this cost-of-carry formula. First, the unknown
future commodity price is irrelevant to the determination of the forward price and
has dropped out. Second, the forward price must be higher than the spot price, since
the interest rate rt,T is positive.
Short positions can be readily taken in most financial asset markets. However, in
some commodity markets, short positions cannot be taken and thus synthetic short
forwards cannot be constructed in sufficient volume to eliminate arbitrage entirely.
Even, in that case, arbitrage is only possible in one direction, and the no-arbitrage
condition becomes an inequality: Ft,TO[1òrt,T(Tñt)]St .

Autocorrelation of returns

The distribution of many asset returns is not only kurtotic and skewed. The return
distribution may also change over time and successive returns may not be independent
of one another. These phenomena will be reflected in the serial correlation or
autocorrelation of returns. Table 1.1 displays evidence that asset returns are not
typically independently and identically distributed. The rightmost column displays a
statistic which measures the likelihood that there is serial correlation between
returns on a given day and returns on the same asset during the prior five trading
days. High values of this statistic indicate a high likelihood that returns are
autocorrelated.
Table 1.1 Statistical properties of selected daily asset returns
Standard
Asset deviation Skewness Kurtosis Autocorrelation
Dollar–Swiss franc 0.0069 0.3472.485 6.0
Dollar–yen 0.0078 0.660 6.181 8.0
Dollar–Mexican peso 0.0132 ñ3.015 65.94756.7
Dollar–Thai baht 0.0080 ñ0.461 25.879 87.4
Crude oil 0.0204 0.249 4.681 41.1
Gold 0.0065 ñ0.165 4.983 21.3
Nikkei 225 average 0.0138 0.213 3.131 26.3
S&P 500 average 0.0087 ñ0.578 8.391 25.6

Skewness

The skewness of a distribution is a measure of the frequency with which large returns
in a particular direction occur. An asset which displays large negative returns more
frequently than large positive returns is said to have a return distribution skewed to
the left or to have a ‘fat left tail’. An asset which displays large positive returns more
frequently than large negative returns is said to have a return distribution skewed
to the right or to have a ‘fat right tail’. The normal distribution is symmetrical, that
is, its coefficient of skewness is exactly zero. Thus a significantly positive or negative
skewness coefficient is inconsistent with the assumption that returns are normal.
Figure 1.3 compares a skewed, but non-kurtotic, distribution with a normal
distribution with the same variance. Table 1.1 presents estimates of the kurtosis and
skewness of some widely traded assets. All the assets displayed have significant
positive or negative skewness, and most also have a coefficient of kurtosis significantly
greater than 3.0.
The exchange rates of the Mexican peso and Thai baht vis-a` -vis the dollar have
the largest coefficients of kurtosis. They are examples of intermittently fixed exchange
rates, which are kept within very narrow fluctuation limits by the monetary authorities.
Typically, fixed exchange rates are a temporary phenomenon, lasting decades in
rare cases, but only a few years in most. When a fixed exchange rate can no longer
be sustained, the rate is either adjusted to new fixed level (for example, the European
Monetary System in the 1980s and 1990s and the Bretton Woods system until 1971)
or permitted to ‘float’, that is, find a free-market price (for example, most emerging
market currencies). In either case, the return pattern of the currency is one of
extremely low returns during the fixed-rate period and extremely large positive or
negative returns when the fixed rate is abandoned, leading to extremely high kurtosis.
The return patterns of intermittently pegged exchange rates also diminishes the
forecasting power of forward exchange rates for these currencies, a phenomenon
known as regime-switching or the peso problem. The term ‘peso problem’ has its
origin in experience with spot and forward rates on the Mexican peso in the 1970s.
Observers were puzzled by the fact that forward rates for years ‘predicted’ a significant
short-term depreciation of the peso vis-a` -vis the US dollar, although the peso–dollar
exchange rate was fixed. One proposed solution was that the exchange rate peg was
not perfectly credible, so market participants expected a switch to a new, lower value
of the peso with a positive probability. In the event, the peso has in fact been
periodically permitted to float, invariably depreciating sharply.

Kurtosis

The kurtosis or leptokurtosis (literally, ‘fat tails’) of a distribution is a measure of
the frequency of large positive or negative asset returns. Specifically, it measures the
frequency of large squared deviations from the mean. The distribution of asset
returns will show high kurtosis if asset returns which are far above or below the
mean occur relatively often, regardless of whether they are mostly above, mostly
below, or both above and below the mean return.
Kurtosis is measured in comparison with the normal distribution, which has a
coefficient of kurtosis of exactly 3. If the kurtosis of an asset return distribution is
significantly higher than 3, it indicates that large-magnitude returns occur more
frequently than in a normal distribution. In other words, a coefficient of kurtosis well
over 3 is inconsistent with the assumption that returns are normal. Figure 1.2
compares a kurtotic distribution with a normal distribution with the same variance.

Behavior of asset prices

Efficient markets hypothesis
The efficient market approach to explaining asset prices views them as the present
values of the income streams they generate. Efficient market theory implies that all
available information regarding future asset prices is impounded in current asset
prices. It provides a useful starting point for analyzing derivatives.
One implication of market efficiency is that asset returns follow a random walk.
The motion of the asset price has two parts, a drift rate, that is, a deterministic rate
at which the asset price is expected to change over time, and a variance rate, that
is, a random change in the asset price, also proportional to the time elapsed, and
also unobservable. The variance rate has a mean of zero and a per-period variance
equal to a parameter p, called the volatility. This assumption implies that the
percent changes in the asset price are normally distributed with a mean equal to the
drift rate and a variance equal to p2.
The random walk hypothesis is widely used in financial modeling and has several
implications:
Ω The percent change in the asset price over the next time interval is independent
of both the percent change over the last time interval and the level of the asset
price. The random walk is sometimes described as ‘memoryless’ for this reason.
There is no tendency for an up move to be followed by another up move, or by a
down move. That means that the asset price can only have a non-stochastic trend
equal to the drift rate, and does not revert to the historical mean or other ‘correct’
level. If the assumption were true, technical analysis would be irrelevant.
Ω Precisely because of this lack of memory, the asset price tends over time to wander
further and further from any starting point. The proportional distance the asset
price can be expected to wander randomly over a discrete time interval q is the
volatility times the square root of the time interval, p q.
Ω Asset prices are continuous; they move in small steps, but do not jump. Over a
given time interval, they may wander quite a distance from where they started,
but they do it by moving a little each day.
Ω Asset returns are normally distributed with a mean equal to the drift rate and a
standard deviation equal to the volatility. The return distribution is the same each
period.
The Black–Scholes model assumes that volatility can be different for different asset
prices, but is a constant for a particular asset. That implies that asset prices are
homoskedastic, showing no tendency towards ‘volatility bunching’. A wild day in
the markets is as likely to be followed by a quiet day as by another wild day.
An asset price following geometric Brownian motion can be thought of as having
an urge to wander away from any starting point, but not in any particular direction.
The volatility parameter can be thought of as a scaling factor for that urge to wander.
Figure 1.1 illustrates its properties with six possible time paths over a year of an
asset price, the sterling–dollar exchange rate, with a starting value of USD1.60, an
annual volatility of 12%, and an expected rate of return of zero.
Empirical research on asset price behavior
While the random walk is a perfectly serviceable first approximation to the behavior
of asset prices, in reality, it is only an approximation. Even though most widely
traded cash asset returns are close to normal, they display small but important ‘nonnormalities’.
In particular, the frequency and direction of large moves in asset prices,
which are very important in risk management, can be quite different in real-life
markets than the random walk model predicts. Moreover, a few cash assets behave
very differently from a random walk.
The random walk hypothesis on which the Black–Scholes model is based is a good
first approximation to the behavior of most asset prices most of the time. However,
even nominal asset returns that are quite close to normally distributed display small
but important deviations from normality. The option price patterns discussed below
reveal how market participants perceive the distribution of future asset prices.
Empirical studies of the stochastic properties of nominal returns focus on the
behavior of realized asset prices. The two approaches largely agree.