Chapter 7: Common Contracts 313
Call option is an option to buy the underlying asset for a
specified price, the strike or exercise price, at (European)
or before (American) a specified data, the expiry or expira-
tion. The underlying can be any security. They are bought to
benefit from upward moves in the underlying, or if volatility
is believed to be higher than implied. In the latter case the
buyer would delta hedge the option to eliminate exposure
to direction. Calls are written for the opposite reasons, of
course. Also a holder of the underlying stock might write a
call to gain some premium in a market where the stock is
not moving much. This is called covered call writing. Simulta-
neous buying of the stock and writing a call is a buy-write
strategy. For calls on lognormal underlyings in constant
or time-dependent volatility worlds there are closed-form
expressions for prices. With more complicated underlyings or
volatility models these contracts can be priced by Monte Carlo
or finite difference, the latter being more suitable if there is
early exercise.
Other contracts may have call features or an embedded call.
For example, a bond may have a call provision allowing the
issuer to buy it back under certain conditions at specified
times. If the issuer has this extra right then it may decrease
the value of the contract, so it might be less than an equiva-
lent contract without the call feature. Sometimes the addition
of a call feature does not affect the value of a contract,
this would happen when it is theoretically never optimal
to exercise the call option. The simplest example of this is
an American versus a European call on a stock without any
dividends. These both have the same theoretical value since
it is never optimal to exercise early.
Cap is a fixed-income option in which the holder receives a
payment when the underlying interest rate exceeds a specified
level, the strike. This payment is the interest rate less the
strike. These payments happen regularly, monthly, or quarterly
etc., as specified in the contract, and the underlying interest
rate will usually be of the same tenor as this interval. The
life of the cap will be several years. They are bought for
protection against rises in interest rates. Market practice is to
quote prices for caps using the Black ’76 model. A contract
with a single payment as above is called a caplet.