Exchange-traded and over-the-counter options
Options are traded both on organized exchanges and over-the-counter. The two
modes of trading are quite different and lead to important differences in market
conventions. The over-the-counter currency and interest rate option markets have
become much more liquid in recent years. Many option market participants prefer
the over-the-counter markets because of the ease with which option contracts tailored
to a particular need can be acquired. The exchanges attract market participants who
prefer or are required to minimize the credit risk of derivatives transactions, or who
are required to transact in markets with publicly posted prices.
Most money-center commercial banks and many securities firms quote over-thecounter
currency, interest rate, equity and commodity option prices to customers. A
smaller number participates in the interbank core of over-the-counter option trading,
making two-way prices to one another. The Bank for International Settlements (BIS)
compiles data on the size and liquidity of the derivatives markets from national
surveys of dealers and exchanges. The most recent survey, for 1995, reveals that over-the-counter markets dominate trading in foreign exchange derivatives and a
substantial portion of the interest rate, equity, and commodity derivatives markets.
We can summarize the key differences between exchange-traded and over-thecounter
option contracts as follows:
Ω Exchange-traded options have standard contract sizes, while over-the-counter
options may have any notional underlying amount.
Ω Most exchange-traded options are written on futures contracts traded on the
same exchange. Their expiration dates do not necessarily coincide with those of
the futures contracts, but are generally fixed dates, say, the third Wednesday of
the month, so that prices on successive days pertain to options of decreasing
maturity. Over-the-counter options, in contrast, may have any maturity date.
Ω Exchange-traded option contracts have fixed exercise prices. As the spot price
changes, such an option contract may switch from out-of-the-money to in-themoney,
or become deeper or less deep in- or out-of-the-money. It is rarely exactly
at-the-money. Thus prices on succesive days pertain to options with different
moneyness.
Ω Mostly American options are traded on the exchanges, while primarily European
options, which are simpler to evaluate, are traded over-the-counter.
Prices of exchange traded options are expressed in currency units. The lumpiness
of the tick size is not a major issue with futures prices, but can be quite important
for option prices, particularly prices of deep out-of-the-money options with prices
close to zero. The price of such an option, if rounded off to the nearest basis point
or 1
32 may be zero, close to half, or close to double its true market value. This in turn
can violate no-arbitrage conditions on option prices. For example, if two options with
adjacent exercise prices both have the same price, the convexity requirement is
violated. It can also lead to absurdly high or low, or even undefined, implied volatilities
and greeks.
In spite of their flexibility, there is a good deal of standardization of over-thecounter
option contracts, particularly with respect to maturity and exercise prices:
Ω The typical maturities correspond to those of forwards: overnight, one week, one,
two, three, six, and nine months, and one year. Interest rate options tend to have
longer maturities, with five- or ten-year common. A fresh option for standard
maturities can be purchased daily, so a series of prices on successive days of
options of like maturity can be constructed.
Ω Many over-the-counter options are initiated at-the-money forward, meaning
their exercise prices are set equal to the current forward rate, or have fixed deltas,
so a series of prices on successive days of options of like moneyness can be
constructed.
Fixed income options
The prices, payoffs, and exercise prices of interest rate options can be expressed in
terms of bond prices or interest rates, and the convention differs for different
instruments. The terms and conditions of all exchange-traded interest rate options
and some over-the-counter interest rate options are expressed as prices rather than
rates. The terms and conditions of certain types of over-the-counter interest rate
options are expressed as rates. A call expressed in terms of interest rates is identical
to a put expressed in terms of prices.
Caplets and floorlets are over-the-counter calls and puts on interbank deposit
rates. The exercise price, called the cap rate or floor rate, is expressed as an interest
rates rather than a security price. The payoff is thus a number of basis points rather
than a currency amount.
Ω In the case of a caplet, the payoff is equal to the cap rate minus the prevailing
rate on the maturity date of the caplet, or zero, which ever is larger. For example,
a three-month caplet on six-month US dollar Libor with a cap rate of 5.00% has
a payoff of 50 basis points if the six-month Libor rate six months hence ends up
at 5.50%, and a payoff of zero if the six-month Libor rate ends up at 4.50%
Ω In the case of a floorlet, the payoff is equal to the prevailing rate on the maturity
date of the cap minus the floor rate, or zero, which ever is larger. For example, a
three-month floorlet on six-month US dollar Libor with a floor rate of 5.00% has
a payoff of 50 basis points if the six-month Libor rate six months hence ends up
at 4.50%, and a payoff of zero if the six-month Libor rate ends up at 5.50%
Figure 1.8 compares the payoffs of caplets and floorlets with that of a FRA.
4.6 4.8 5 5.2 5.4
ST
40
20
0
20
40
payoff bp
payoff on FRA
payoff on caplet
payoff on floorlet
Figure 1.8 FRAs, caps and floors.
A caplet or a floorlet also specifies a notional principal amount. The obligation of
the writer to the option owner is equal to the notional principal amount times the
payoff times the term of the underlying interest rate. For example, for a caplet or
floorlet on six-month Libor with a payoff of 50 basis points and a notional principal
amount of USD1 000 000, the obligation of the option writer to the owner is
USD0.0050 · 1
2 · 1 000 000ó2500.
To see the equivalence between a caplet and a put on a bond price, consider a
caplet on six-month Libor struck at 5%. This is equivalent to a put option on a sixmonth
zero coupon security with an exercise price of 97.50% of par. Similarly. A
floor rate of 5% would be equivalent to a call on a six-month zero coupon security
with an exercise price of 97.50.
A contract containing a series of caplets or floorlets with increasing maturities is
called a cap or floor. A collar is a combination of a long cap and a short floor. It
protects the owner against rising short-term rates at a lower cost than a cap, since
the premium is reduced by approximately the value of the short floor, but limits the
extent to which he benefits from falling short-term rates.
Swaptions are options on interest rate swaps. The exercise prices of swaptions,
like those of caps and floors, are expressed as interest rates. Every swaption obliges
the writer to enter into a swap at the initiative of the swaption owner. The owner will
exercise the swaption by initiating the swap if the swap rate at the maturity of the
swaption is in his favor. A receiver swaption gives the owner the right to initiate a
swap in which he receives the fixed rate, while a payer swaption gives the owner the
right to initiate a swap in which he pays the fixed rate.
There are two maturities involved in any fixed-income option, the maturity of the
option and the maturity of the underlying instrument. To avoid confusion, traders in
the cap, floor and swaption markets will describe, say, a six-month option on a twoyear
swap as a ‘six-month into two year’ swaption, since the six-month option is
exercised ‘into’ a two-year swap (if exercised).
There are highly liquid over-the-counter and futures options on actively traded
government bond and bond futures of industrialized countries. There are also liquid
markets in over-the-counter options on Brady bonds.
13 Şubat 2011 Pazar
Put-call parity
Calls can be combined with forwards or with positions in the underlying asset and
the money market to construct synthetic puts with the same exercise price (and vice
versa). In the special case of European at-the-money forward options:
Ω The value of an at-the-money forward European call is equal to the value of
an at-the-money forward European put.
The reason is that, at maturity, the forward payoff equals the call payoff minus the
put payoff. In other words, you can create a synthetic long forward by going long one
ATM forward call and short one ATM forward put. The construction is illustrated in
Figure 1.7.
the money market to construct synthetic puts with the same exercise price (and vice
versa). In the special case of European at-the-money forward options:
Ω The value of an at-the-money forward European call is equal to the value of
an at-the-money forward European put.
The reason is that, at maturity, the forward payoff equals the call payoff minus the
put payoff. In other words, you can create a synthetic long forward by going long one
ATM forward call and short one ATM forward put. The construction is illustrated in
Figure 1.7.
Time value
Time value is defined as the option value minus intrinsic value and is rarely negative,
since option value is usually greater than intrinsic value. Time value is greatest for
at-the-money-options and declines at a declining rate as the option goes in- or outof-
the-money.
The following restriction pertains to sets of American options which are identical
in every respect – exercise prices, underlying asset – except their times to maturity.
Ω A plain-vanilla American call or put option must be worth more than a similar
option with a shorter time to maturity.
This restriction does not necessarily hold for European options, but usually does.
since option value is usually greater than intrinsic value. Time value is greatest for
at-the-money-options and declines at a declining rate as the option goes in- or outof-
the-money.
The following restriction pertains to sets of American options which are identical
in every respect – exercise prices, underlying asset – except their times to maturity.
Ω A plain-vanilla American call or put option must be worth more than a similar
option with a shorter time to maturity.
This restriction does not necessarily hold for European options, but usually does.
Option basics Option terminology
A call option is a contract giving the owner the right, but not the obligation, to
purchase, at expiration, an amount of an asset at a specified price called the strike
or exercise price. A put option is a contract giving the owner the right, but not the
obligation, to sell, at expiration, an amount of an asset at the exercise price. The
amount of the underlying asset is called the notional principal or underlying
amount. The price of the option contract is called the option premium.
The issuer of the option contract is called the writer and is said to have the short
position. The owner of the option is said to be long. Figure 1.5 illustrates the payoff
profile at maturity of a long position in a European call on one pound sterling against
the dollar with an exercise price of USD1.60.
There are thus several ways to be long an asset:
Ω long the spot asset
Ω long a forward on the asset
Ω long a call on the asset
Ω short a put on the asset
There are many types of options. A European option can be exercised only at
expiration. An American option can be exercised at any time between initiation of
the contract and expiration.
A standard or plain vanilla option has no additional contractual features. An exotic option has additional features affecting the payoff. Some examples of
exotics are
Ω Barrier options, in which the option contract is initiated or cancelled if the asset’s
cash price reaches a specified level.
Ω Average rate options, for which the option payoff is based on the average spot
price over the duration of the option contract rather than spot price at the time
of exercise.
Ω Binary options, which have a lump sum option payoff if the spot price is above
(call) or below (put) the exercise price at maturity.
Currency options have an added twist: a domestic currency put is also a foreign currency
put. For example, if I give you the right to buy one pound sterling for USD1.60
in three months, I also give you the right to sell USD1.60 at £0.625 per dollar.
Intrinsic value, moneyness and exercise
The intrinsic value of a call option is the larger of the exercise price minus the
current asset price or zero. The intrinsic value of a put is the larger of the current
asset price minus the exercise or zero. Denoting the exercise price by X, the intrinsic
value of a call is StñX and that of a put is XñSt .
Intrinsic value can also be thought of as the value of an option if it were expiring
or exercised today. By definition, intrinsic value is always greater than or equal to
zero. For this reason, the owner of an option is said to enjoy limited liability, meaning
that the worst-case outcome for the owner of the option is to throw it away valueless
and unexercised.
The intrinsic value of an option is often described by its moneyness:
Ω If intrinsic value is positive, the option is said to be in-the-money.
Ω If the exchange rate is below the exchange rate, a call option is said to be out-ofthe-
money.
Ω If the intrinsic value is zero, the option is said to be at-the-money.
If intrinsic value is positive at maturity, the owner of the option will exercise it, that
is, call the underlying away from the writer. Figure 1.6 illustrates these definitions
for a European sterling call with an exercise price of USD1.60.
1.50 1.55 1.60 1.65 1.70
Spot
0.05
0.10
Intrinsic value
out of the money
at the money
in the money
Figure 1.6 Moneyness.
Owning a call option or selling a put option on an asset is like being long the asset.
Owning a deep in-the-money call option on the dollar is like being long an amount
of the asset that is close to the notional underlying value of the option. Owning a
deep out-of-the-money call option on the dollar is like being long an amount of the
asset that is much smaller than the notional underlying value of the option.
Valuation basics
Distribution- and preference-free restrictions on plain-vanilla option prices
Options have an asymmetric payoff profile at maturity: a change in the exchange
rate at expiration may or may not translate into an equal change in option value.
The difficulty in valuing options and managing option risks arises from the asymmetry
in the option payoff. Options have an asymmetric payoff profile at maturity: a change
in the exchange rate at expiration may or may not translate into an equal change in
option value. In contrast, the payoff on a forward increases one-for-one with the
exchange rate.
In this section, we study some of the many true statements about option prices
that do not depend on a model. These facts, sometimes called distribution- and
preference-free restrictions on option prices, meaning that they don’t depend on
assumptions about the probability distribution of the exchange rate or about market
participants’ positions or risk appetites. They are also called arbitrage restrictions to
signal the reliance of these propositions on no-arbitrage arguments.
Here is one of the simplest examples of such a proposition:
Ω No plain vanilla option European or American put or call, can have a negative
value: Of course not: the owner enjoys limited liability.
Another pair of ‘obvious’ restrictions is:
Ω A plain vanilla European or American call option cannot be worth more than
the current cash price of the asset. The exercise price can be no lower than
zero, so the benefit of exercising can be no greater than the cash price.
Ω A plain vanilla European or American put option cannot be worth more than
the exercise price. The cash price can be no lower than zero, so the benefit of
exercising can be no greater than the exercise price.
Buying a deep out-of-the-money call is often likened to buying a lottery ticket. The
call has a potentially unlimited payoff if the asset appreciates significantly. On the
other hand, the call is cheap, so if the asset fails to appreciate significantly, the loss
is relatively small. This helps us to understand the strategy of a famous investor who
in mid-1995 bought deep out-of-the-money calls on a large dollar amount against
the Japanese yen (yen puts) at very low cost and with very little price risk. The dollar
subsequently appreciated sharply against the yen, so the option position was then
equivalent to having a long cash position in nearly the full notional underlying
amount of dollars.
The following restrictions pertain to sets of options which are identical in every
respect – time to maturity, underlying currency pair, European or American style –
except their exercise prices:
Ω A plain-vanilla European or American call option must be worth more than a
similar option with a lower exercise price.
Ω A plain-vanilla European or American put option must be worth more than a
similar option with a higher exercise price.
We will state a less obvious, but very important, restriction:
Ω A plain-vanilla European put or call option is a convex function of the
exercise price.
To understand this restriction, think about two European calls with different
exercise prices. Now introduce a third call option with an exercise price midway
between the exercise prices of the first two calls. The market value of this third option
cannot be greater than the average value of the first two.
Current value of an option
Prior to expiration, an option is usually worth at least its intrinsic value. As an
example, consider an at-the-money option. Assume a 50% probability the exchange
rate rises USD0.01 and a 50% probability that the rate falls USD0.01 by the
expiration date. The expected value of changes in the exchange rate is
0.5 · 0.01ò0.5 · (ñ1.01)ó0. The expected value of changes in the option’s value is
0.5 · 0.01ò0.5 · 0óñ0.005. Because of the asymmetry of option payoff, only the
possibility of a rising rate affects a call option’s value.
Analogous arguments hold for in- and out-of-the-money options. ‘But suppose the
call is in-the-money. Wouldn’t you rather have the underlying, since the option might
go back out-of-the-money? And shouldn’t the option then be worth less than its
intrinsic value?’ The answer is, ‘almost never’. To be precise:
Ω A European call must be worth at least as much as the present value of the
forward price minus the exercise price.
This restriction states that no matter how high or low the underlying price is, an
option is always worth at least its ‘intrinsic present value’.
We can express this restriction algebraically. Denote by C(X,t,T ) the current (time
t) market value of a European call with an exercise price X, expiring at time T. The
proposition states that C(X,t,T )P[1òrt,T (Tñt)]ñ1(Ft,TñX). In other words, the call
must be worth at least its discounted ‘forward intrinsic value’.
Let us prove this using a no-arbitrage argument. A no-arbitrage argument is based
on the impossibility of a set of contracts that involve no cash outlay now and give
you the possibility of a positive cash flow later with no possibility of a negative cash
flow later. The set of contracts is
Ω Buy a European call on one dollar at a cost of C(X,t,T ).
Ω Finance the call purchase by borrowing.
Ω Sell one dollar forward at a rate Ft,T.
The option, the loan, and the forward all have the same maturity. The net cash
flow now is zero. At expiry of the loan, option and forward, you have to repay
[1òrt,T(Tñt)]C(X,t,T ), the borrowed option price with interest. You deliver one dollar
and receive Ft,T to settle the forward contract. There are now two cases to examine:
Case (i): If the option expires in-the-money (ST[X), exercise it to get the dollar to
deliver into the forward contract. The dollar then costs K and your net proceeds from
settling all the contracts at maturity are Ft,TñXñ[1òrt,T(Tñt)]C(K,t,T ).
Case (ii): If the option expires out-of-the-money (STOX), buy a dollar at the spot rate
ST to deliver into the forward contract. The dollar then costs ST and your net proceeds
from settling all the contracts at maturity is Ft,TñSTñ[1òrt,T(Tñt)]C(X,t,T ).
For arbitrage to be impossible, these net proceeds must be non-positive, regardless
of the value of ST.
Case (i): If the option expires in-the-money, the impossibility of arbitrage implies
Ft,TñXñ[1òrt,T(Tñt)]C(X,t,T )O0.
Case (ii): If the option expires out-of-the-money, the impossibility of arbitrage implies
Ft,TñSTñ[1òrt,T(Tñt)]C(X,t,T )O0,
which in turn implies Ft,TñXñ[1òrt,T(Tñt)]C(X,t,T )O0.
This proves the restriction.
purchase, at expiration, an amount of an asset at a specified price called the strike
or exercise price. A put option is a contract giving the owner the right, but not the
obligation, to sell, at expiration, an amount of an asset at the exercise price. The
amount of the underlying asset is called the notional principal or underlying
amount. The price of the option contract is called the option premium.
The issuer of the option contract is called the writer and is said to have the short
position. The owner of the option is said to be long. Figure 1.5 illustrates the payoff
profile at maturity of a long position in a European call on one pound sterling against
the dollar with an exercise price of USD1.60.
There are thus several ways to be long an asset:
Ω long the spot asset
Ω long a forward on the asset
Ω long a call on the asset
Ω short a put on the asset
There are many types of options. A European option can be exercised only at
expiration. An American option can be exercised at any time between initiation of
the contract and expiration.
A standard or plain vanilla option has no additional contractual features. An exotic option has additional features affecting the payoff. Some examples of
exotics are
Ω Barrier options, in which the option contract is initiated or cancelled if the asset’s
cash price reaches a specified level.
Ω Average rate options, for which the option payoff is based on the average spot
price over the duration of the option contract rather than spot price at the time
of exercise.
Ω Binary options, which have a lump sum option payoff if the spot price is above
(call) or below (put) the exercise price at maturity.
Currency options have an added twist: a domestic currency put is also a foreign currency
put. For example, if I give you the right to buy one pound sterling for USD1.60
in three months, I also give you the right to sell USD1.60 at £0.625 per dollar.
Intrinsic value, moneyness and exercise
The intrinsic value of a call option is the larger of the exercise price minus the
current asset price or zero. The intrinsic value of a put is the larger of the current
asset price minus the exercise or zero. Denoting the exercise price by X, the intrinsic
value of a call is StñX and that of a put is XñSt .
Intrinsic value can also be thought of as the value of an option if it were expiring
or exercised today. By definition, intrinsic value is always greater than or equal to
zero. For this reason, the owner of an option is said to enjoy limited liability, meaning
that the worst-case outcome for the owner of the option is to throw it away valueless
and unexercised.
The intrinsic value of an option is often described by its moneyness:
Ω If intrinsic value is positive, the option is said to be in-the-money.
Ω If the exchange rate is below the exchange rate, a call option is said to be out-ofthe-
money.
Ω If the intrinsic value is zero, the option is said to be at-the-money.
If intrinsic value is positive at maturity, the owner of the option will exercise it, that
is, call the underlying away from the writer. Figure 1.6 illustrates these definitions
for a European sterling call with an exercise price of USD1.60.
1.50 1.55 1.60 1.65 1.70
Spot
0.05
0.10
Intrinsic value
out of the money
at the money
in the money
Figure 1.6 Moneyness.
Owning a call option or selling a put option on an asset is like being long the asset.
Owning a deep in-the-money call option on the dollar is like being long an amount
of the asset that is close to the notional underlying value of the option. Owning a
deep out-of-the-money call option on the dollar is like being long an amount of the
asset that is much smaller than the notional underlying value of the option.
Valuation basics
Distribution- and preference-free restrictions on plain-vanilla option prices
Options have an asymmetric payoff profile at maturity: a change in the exchange
rate at expiration may or may not translate into an equal change in option value.
The difficulty in valuing options and managing option risks arises from the asymmetry
in the option payoff. Options have an asymmetric payoff profile at maturity: a change
in the exchange rate at expiration may or may not translate into an equal change in
option value. In contrast, the payoff on a forward increases one-for-one with the
exchange rate.
In this section, we study some of the many true statements about option prices
that do not depend on a model. These facts, sometimes called distribution- and
preference-free restrictions on option prices, meaning that they don’t depend on
assumptions about the probability distribution of the exchange rate or about market
participants’ positions or risk appetites. They are also called arbitrage restrictions to
signal the reliance of these propositions on no-arbitrage arguments.
Here is one of the simplest examples of such a proposition:
Ω No plain vanilla option European or American put or call, can have a negative
value: Of course not: the owner enjoys limited liability.
Another pair of ‘obvious’ restrictions is:
Ω A plain vanilla European or American call option cannot be worth more than
the current cash price of the asset. The exercise price can be no lower than
zero, so the benefit of exercising can be no greater than the cash price.
Ω A plain vanilla European or American put option cannot be worth more than
the exercise price. The cash price can be no lower than zero, so the benefit of
exercising can be no greater than the exercise price.
Buying a deep out-of-the-money call is often likened to buying a lottery ticket. The
call has a potentially unlimited payoff if the asset appreciates significantly. On the
other hand, the call is cheap, so if the asset fails to appreciate significantly, the loss
is relatively small. This helps us to understand the strategy of a famous investor who
in mid-1995 bought deep out-of-the-money calls on a large dollar amount against
the Japanese yen (yen puts) at very low cost and with very little price risk. The dollar
subsequently appreciated sharply against the yen, so the option position was then
equivalent to having a long cash position in nearly the full notional underlying
amount of dollars.
The following restrictions pertain to sets of options which are identical in every
respect – time to maturity, underlying currency pair, European or American style –
except their exercise prices:
Ω A plain-vanilla European or American call option must be worth more than a
similar option with a lower exercise price.
Ω A plain-vanilla European or American put option must be worth more than a
similar option with a higher exercise price.
We will state a less obvious, but very important, restriction:
Ω A plain-vanilla European put or call option is a convex function of the
exercise price.
To understand this restriction, think about two European calls with different
exercise prices. Now introduce a third call option with an exercise price midway
between the exercise prices of the first two calls. The market value of this third option
cannot be greater than the average value of the first two.
Current value of an option
Prior to expiration, an option is usually worth at least its intrinsic value. As an
example, consider an at-the-money option. Assume a 50% probability the exchange
rate rises USD0.01 and a 50% probability that the rate falls USD0.01 by the
expiration date. The expected value of changes in the exchange rate is
0.5 · 0.01ò0.5 · (ñ1.01)ó0. The expected value of changes in the option’s value is
0.5 · 0.01ò0.5 · 0óñ0.005. Because of the asymmetry of option payoff, only the
possibility of a rising rate affects a call option’s value.
Analogous arguments hold for in- and out-of-the-money options. ‘But suppose the
call is in-the-money. Wouldn’t you rather have the underlying, since the option might
go back out-of-the-money? And shouldn’t the option then be worth less than its
intrinsic value?’ The answer is, ‘almost never’. To be precise:
Ω A European call must be worth at least as much as the present value of the
forward price minus the exercise price.
This restriction states that no matter how high or low the underlying price is, an
option is always worth at least its ‘intrinsic present value’.
We can express this restriction algebraically. Denote by C(X,t,T ) the current (time
t) market value of a European call with an exercise price X, expiring at time T. The
proposition states that C(X,t,T )P[1òrt,T (Tñt)]ñ1(Ft,TñX). In other words, the call
must be worth at least its discounted ‘forward intrinsic value’.
Let us prove this using a no-arbitrage argument. A no-arbitrage argument is based
on the impossibility of a set of contracts that involve no cash outlay now and give
you the possibility of a positive cash flow later with no possibility of a negative cash
flow later. The set of contracts is
Ω Buy a European call on one dollar at a cost of C(X,t,T ).
Ω Finance the call purchase by borrowing.
Ω Sell one dollar forward at a rate Ft,T.
The option, the loan, and the forward all have the same maturity. The net cash
flow now is zero. At expiry of the loan, option and forward, you have to repay
[1òrt,T(Tñt)]C(X,t,T ), the borrowed option price with interest. You deliver one dollar
and receive Ft,T to settle the forward contract. There are now two cases to examine:
Case (i): If the option expires in-the-money (ST[X), exercise it to get the dollar to
deliver into the forward contract. The dollar then costs K and your net proceeds from
settling all the contracts at maturity are Ft,TñXñ[1òrt,T(Tñt)]C(K,t,T ).
Case (ii): If the option expires out-of-the-money (STOX), buy a dollar at the spot rate
ST to deliver into the forward contract. The dollar then costs ST and your net proceeds
from settling all the contracts at maturity is Ft,TñSTñ[1òrt,T(Tñt)]C(X,t,T ).
For arbitrage to be impossible, these net proceeds must be non-positive, regardless
of the value of ST.
Case (i): If the option expires in-the-money, the impossibility of arbitrage implies
Ft,TñXñ[1òrt,T(Tñt)]C(X,t,T )O0.
Case (ii): If the option expires out-of-the-money, the impossibility of arbitrage implies
Ft,TñSTñ[1òrt,T(Tñt)]C(X,t,T )O0,
which in turn implies Ft,TñXñ[1òrt,T(Tñt)]C(X,t,T )O0.
This proves the restriction.
The expectations hypothesis of the term structure
In fixed-income markets, the efficient markets hypothesis is called the expectations
hypothesis of the term structure. As is the case for efficient markets models of other
asset prices, the expectations hypothesis can be readily formulated in terms of the
forward interest rate, the price at which a future interest rate exposure can be locked in. Forward interest rates are often interpreted as a forecast of the future spot interest
rate. Equivalently, the term premium or the slope of the term structure – the spread
of a long-term rate over a short-term rate – can be interpreted as a forecast of changes
in future short-term rates. The interpretation of forward rates as forecasts implies
that an increase in the spread between long- and short-term rates predicts a rise in
both short- and long-term rates.
The forecasting performance of forward rates with respect to short-term rates has
generally been better than that for long-term rates. Central banks in industrialized
countries generally adopt a short-term interest rate as an intermediate target, but
they also attempt to reduce short-term fluctuations in interest rates. Rather than
immediately raising or lowering interest rates quickly by a large amount to adjust
them to changes in economic conditions, they change them in small increments over
a long period of time. This practice, called interest rate smoothing, results in
protracted periods in which the direction and likelihood, but not the precise timing,
of the next change in the target interest rate can be guessed with some accuracy,
reducing the error in market predictions of short-term rates generally.
Central banks’ interest rate smoothing improves the forecasting power of shortterm
interest rate futures and forwards at short forecasting horizons. This is in
contrast to forwards on foreign exchange, which tend to predict better at long
horizons. At longer horizons, the ability of forward interest rates to predict future
short-term deteriorates. Forward rates have less ability to predict turning points in
central banks’ monetary stance than to predict the direction of the next move in an
already established stance.
hypothesis of the term structure. As is the case for efficient markets models of other
asset prices, the expectations hypothesis can be readily formulated in terms of the
forward interest rate, the price at which a future interest rate exposure can be locked in. Forward interest rates are often interpreted as a forecast of the future spot interest
rate. Equivalently, the term premium or the slope of the term structure – the spread
of a long-term rate over a short-term rate – can be interpreted as a forecast of changes
in future short-term rates. The interpretation of forward rates as forecasts implies
that an increase in the spread between long- and short-term rates predicts a rise in
both short- and long-term rates.
The forecasting performance of forward rates with respect to short-term rates has
generally been better than that for long-term rates. Central banks in industrialized
countries generally adopt a short-term interest rate as an intermediate target, but
they also attempt to reduce short-term fluctuations in interest rates. Rather than
immediately raising or lowering interest rates quickly by a large amount to adjust
them to changes in economic conditions, they change them in small increments over
a long period of time. This practice, called interest rate smoothing, results in
protracted periods in which the direction and likelihood, but not the precise timing,
of the next change in the target interest rate can be guessed with some accuracy,
reducing the error in market predictions of short-term rates generally.
Central banks’ interest rate smoothing improves the forecasting power of shortterm
interest rate futures and forwards at short forecasting horizons. This is in
contrast to forwards on foreign exchange, which tend to predict better at long
horizons. At longer horizons, the ability of forward interest rates to predict future
short-term deteriorates. Forward rates have less ability to predict turning points in
central banks’ monetary stance than to predict the direction of the next move in an
already established stance.
Swaps and forward swaps
A plain vanilla interest rate swap is an agreement between two counterparties to
exchange a stream of fixed interest rate payments for a stream of floating interest
rate payments. Both streams are denominated in the same currency and are based
on a notional principal amount. The notional principal is not exchanged. The design
of a swap has three features that determine its price: the maturity of the swap, the
maturity of the floating rate, and the frequency of payments. We will assume for
expository purposes that the latter two features coincide, e.g. if the swap design is
fixed against six-month Libor, then payments are exchanged semiannually.
At initiation, the price of a plain-vanilla swap is set so its current value – the net
value of the two interest payment streams, fixed and floating – is zero. The swap can
be seen as a portfolio which, from the point of view of the payer of fixed interest
(called the ‘payer’ in market parlance) is long a fixed-rate bond and short a floatingrate
bond, both in the amount of the notional principal. The payer of floating-rate
interest (called the ‘receiver’ in market parlance) is long the floater and short the
fixed-rate bond.
The price of a swap is usually quoted as the swap rate, that is, as the yield to
maturity on a notional par bond. What determines this rate? A floating-rate bond
always trades at par at the time it is issued. The fixed-rate bond, which represents
the payer’s commitment in the swap, must then also trade at par if the swap is to
have an initial value of zero. In other words, the swap rate is the market-adjusted
yield to maturity on a par bond.
Swap rates are also often quoted as a spread over the government bond with a
maturity closest to that of the swap. This spread, called the swap-Treasury spread,
is almost invariably positive, but varies widely in response to factors such as liquidity
and risk appetites in the fixed-income markets.
A forward swap is an agreement between two counterparties to commence a swap
at some future settlement date. As in the case of a cash swap, the forward swap rate
is the market-adjusted par rate on a coupon bond issued at the settlement date. The
rate on a forward swap can be calculated from forward rates or spot rates.
exchange a stream of fixed interest rate payments for a stream of floating interest
rate payments. Both streams are denominated in the same currency and are based
on a notional principal amount. The notional principal is not exchanged. The design
of a swap has three features that determine its price: the maturity of the swap, the
maturity of the floating rate, and the frequency of payments. We will assume for
expository purposes that the latter two features coincide, e.g. if the swap design is
fixed against six-month Libor, then payments are exchanged semiannually.
At initiation, the price of a plain-vanilla swap is set so its current value – the net
value of the two interest payment streams, fixed and floating – is zero. The swap can
be seen as a portfolio which, from the point of view of the payer of fixed interest
(called the ‘payer’ in market parlance) is long a fixed-rate bond and short a floatingrate
bond, both in the amount of the notional principal. The payer of floating-rate
interest (called the ‘receiver’ in market parlance) is long the floater and short the
fixed-rate bond.
The price of a swap is usually quoted as the swap rate, that is, as the yield to
maturity on a notional par bond. What determines this rate? A floating-rate bond
always trades at par at the time it is issued. The fixed-rate bond, which represents
the payer’s commitment in the swap, must then also trade at par if the swap is to
have an initial value of zero. In other words, the swap rate is the market-adjusted
yield to maturity on a par bond.
Swap rates are also often quoted as a spread over the government bond with a
maturity closest to that of the swap. This spread, called the swap-Treasury spread,
is almost invariably positive, but varies widely in response to factors such as liquidity
and risk appetites in the fixed-income markets.
A forward swap is an agreement between two counterparties to commence a swap
at some future settlement date. As in the case of a cash swap, the forward swap rate
is the market-adjusted par rate on a coupon bond issued at the settlement date. The
rate on a forward swap can be calculated from forward rates or spot rates.
Forward rate agreements
Forward rate agreements (FRAs) are forwards on time deposits. In a FRA, one party
agrees to pay a specific interest rate on a Eurodeposit of a specified currency,
maturity, and amount, beginning at a specified date in the future. FRA prices are
defined as the spot rate the buyer agrees to pay on a notional deposit of a given
maturity on a given settlement date. Usually, the reference rate is Libor. For example,
a 3î6 (spoken ‘3 by 6’) Japanese yen FRA on óY 100 000 000 can be thought of as a
commitment by one counterparty to pay another the difference between the contracted
FRA rate and the realized level of the reference rate on a óY 100 000 000
deposit.
Suppose the three-month and six-month Swiss franc Libor rates are respectively 3.55% and 3.45%. Say Bank A takes the long side and Bank B takes the short side
of a DM10 000 000 3î6 FRA on 1 January at a rate of 3.30%, and suppose threemonth
DM Libor is 3.50%. If the FRA were settled by delivery, Bank A would place a
three-month deposit with Bank B at a rate of 3.30%. It could then close out its
position by taking a deposit at the going rate of 3.50%, gaining 0.002î 90
360î
10 000 000ó5000 marks when the deposits mature on 1 June.
FRAs are generally cash-settled by the difference between the amount the notional
deposit would earn at the FRA rate and the amount it would earn at the realized
Libor or other reference rate, discounted back to the settlement date. The FRA is
cash-settled by Bank B paying Bank A the present value of DM5000 on 1 March.
With a discount factor of 1.045î 90
360ó1.01125, that comes to DM4944.38.
agrees to pay a specific interest rate on a Eurodeposit of a specified currency,
maturity, and amount, beginning at a specified date in the future. FRA prices are
defined as the spot rate the buyer agrees to pay on a notional deposit of a given
maturity on a given settlement date. Usually, the reference rate is Libor. For example,
a 3î6 (spoken ‘3 by 6’) Japanese yen FRA on óY 100 000 000 can be thought of as a
commitment by one counterparty to pay another the difference between the contracted
FRA rate and the realized level of the reference rate on a óY 100 000 000
deposit.
Suppose the three-month and six-month Swiss franc Libor rates are respectively 3.55% and 3.45%. Say Bank A takes the long side and Bank B takes the short side
of a DM10 000 000 3î6 FRA on 1 January at a rate of 3.30%, and suppose threemonth
DM Libor is 3.50%. If the FRA were settled by delivery, Bank A would place a
three-month deposit with Bank B at a rate of 3.30%. It could then close out its
position by taking a deposit at the going rate of 3.50%, gaining 0.002î 90
360î
10 000 000ó5000 marks when the deposits mature on 1 June.
FRAs are generally cash-settled by the difference between the amount the notional
deposit would earn at the FRA rate and the amount it would earn at the realized
Libor or other reference rate, discounted back to the settlement date. The FRA is
cash-settled by Bank B paying Bank A the present value of DM5000 on 1 March.
With a discount factor of 1.045î 90
360ó1.01125, that comes to DM4944.38.
Kaydol:
Kayıtlar (Atom)