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.
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