Transcript No Slide Title

18 Extensions
A major advantage of the term structure model
presented in this textbook is that it is easily
extended to incorporate additional term
structures.
The introduction of additional term structures is
the generalization needed to price and hedge
foreign-currency derivatives, credit derivatives,
and commodity options.
1
A Foreign-Currency Derivatives
To price and hedge foreign-currency derivatives,
one needs a spot exchange rate of foreign into
domestic currency and two zero-coupon bond
price curves: (i) one for the domestic currency and
(ii) one for the foreign currency.
The method for building an arbitrage-free
evolution of these term structures proceeds in a
fashion identical to that given in Chapters 3-9.
The only complication is that in constructing the
tree, two price vectors are included at each node.
2
One vector is for the domestic currency zero-coupon curve (just as
before), and one vector is for the foreign currency zero-coupon
curve with the spot exchange rate appended.
The arbitrage-free conditions correspond to the existence of pseudo
probabilities, which make all dollar-denominated and dollartranslated securities (foreign zero-coupon bonds) martingales after
normalization by the domestic money market account.
Market completeness corresponds to the uniqueness of these pseudo
probabilities. Pricing and hedging is done via the risk-neutral
valuation procedure.
The only difficulty in applying these extensions in practice is that
the computation time increases as more term structures are
introduced into the model.
Efficient numerical procedures become an important issue.
3
B Credit Derivatives and Counterparty Risk
An important extension of the default-free term
structure model to multiple term structures is
when one includes securities with different levels
of bankruptcy risk.
The pricing and hedging of corporate debt and the
pricing and hedging of swaps with counterparty
risk are two prime examples.
The easiest way to analyze this pricing problem is
to transform it into a foreign-currency derivative
problem and then to use the methods for pricing
and hedging foreign currency derivatives (with
obvious modifications).
4
To see the foreign currency analogy, consider two
term structures of zero-coupon bonds: (i) the
default-free term structure and (ii) the term
structure for a risky firm.
Call the risky firm XYZ. XYZ's zero-coupon
bonds provide only a promised dollar payoff at
future dates. The promised dollar is paid only if
XYZ is not bankrupt at the payoff date.
One can think of XYZ zero-coupon debt
differently. Consider XYZ zero-coupon bonds as
first paying off in a hypothetical (foreign)
currency, called XYZ dollars. That is, each XYZ
zero-coupon bond pays one XYZ dollar for sure at
its maturity. In XYZ dollars, XYZ debt can be
considered default-free.
5
But XYZ dollars need to be converted into actual
dollars for analysis. The conversion rate (or spot
exchange rate from XYZ dollars to dollars) is the
payoff ratio at the zero-coupon bond's maturity.
If XYZ is not bankrupt, the payoff ratio is unity.
If it is bankrupt, less than the promised dollar is
received.
Given this foreign currency analogy, the pricing
and hedging problem for foreign currency
derivatives can now be applied.
This analogy also applies to counterparty risk as
well.
A counterparty to a contract only promises to
make a payment, and when the contract
provisions come due, payment is made only if the
counterparty is not in default.
6
C Commodity Derivatives
The final extension studied is the pricing of
commodity derivatives.
Examples include oil futures, options on oil
futures, precious metal futures, and options on
precious metals.
This pricing and hedging problem has two term
structures.
The first is the same as that already studied, the
term structure of default-free zero-coupon bonds.
The second term structure is the term structure of
commodity futures prices for future delivery.
7
Given these two term structures, the analysis
proceeds in a fashion similar to that of Chapters 3
- 9. The only difference is that in constructing the
tree, two price vectors are included at each node.
One is for the default-free zero-coupon bond
prices, and the second is for the commodity
futures prices for future delivery.
The arbitrage-free conditions correspond to the
existence of pseudo probabilities, which make the
zero-coupon bond prices normalized by the money
market account martingales and make the futures
prices martingales (see Chapter 12 for the
motivation of this last condition).
Market completeness corresponds to the
uniqueness of these pseudo probabilities. Pricing
and hedging is done using the risk-neutral
valuation procedure
8