Fixed-Income Products
¾ Simple fixed-income products
¾ Some concepts
¾ Vasicek model
Stats243 Summer 2007
Simple fixed-income instruments
• Zero-coupon bond
– It is a contract paying a known fixed amount (principal) at a given
date in the future (maturity date).
• Coupon-bearing bond
– It pays both coupons and the principal.
• Money market account
– The bank account (checking, saving, certificate of deposit, …)
• Floating rate bonds
– The interest rate you get varies from time to time.
– A common measure of interest is London Interbank Offer Rate
(LIBOR), which is the rate of interest offered between Eurocurrency
banks for fixed-term deposits.
Stats243 Summer 2007
Simple fixed-income instruments
• Forward rate agreement
– It is an agreement between two parties that a prescribed interest rate
will apply to a prescribed principal over some specified period in the
future.
• Repos
– It is a repurchase agreement. Party A sells some security to party B,
and will buy it back at a fixed date and for a fixed amount.
……
Stats243 Summer 2007
Some Concepts
• Accrued interest
– It is the amount of interest that has built up since the last coupon
payment.
– The market price of bonds are clean prices (without accrued
interest).
– Dirty price = clean price + accrued interest
• The yield to maturity (Internal rate of return)
– It is the value y that we need in the following equation to make the
present value of the bond at time t (V) equal to the market price of
the bond.
Stats243 Summer 2007
Some Concepts
• Yield curve
– The plot of yield to maturity against time to maturity.
– The higher the yield, the higher the discount factor, the lower the
price of a bond.
Humped
Stats243 Summer 2007
Some Concepts
• Duration
– It is a measure of the average life of the bond.
– The Macaulay duration is defined as
Stats243 Summer 2007
Some Concepts
• Convexity
– It is a measure of the curvature in the price/yield relationship.
– The convexity is defined as
Stats243 Summer 2007
Duration-based hedging strategies
Stats243 Summer 2007
Fixed-Income Products
• Swap
– It is an agreement between two parties to exchange future cash
flows. Most popular swaps are currency swaps and interest rate
swaps.
– In the interest rate swap, the two parties exchange cash flows that
are represented by the interest on a notional principal. Typically, one
party agrees to pay the other a fixed interest rate and the cash flow in
the opposite side is a floating rate.
B: floating rate
t
A: fixed rate
Stats243 Summer 2007
Fixed-Income Products
Stats243 Summer 2007
One-factor interest rate modeling
• It models the spot interest rate (the interest rate received by
the shortest possible deposit) as a random variable by
using a single source of randomness.
• Suppose that the interest rate r is governed by another
stochastic differential equation of the form
dr = u(r,t) dt + w(r,t) dWt,
where u and w are functions of r and t, and Wt follows the
Brownian motion.
Stats243 Summer 2007
One-factor interest rate modeling
• Vasicek model
–
–
–
–
dr = (η - γ r) dt + β1/2 dWt
The model is mean reverting to a constant level.
The modeled interest rate can be negative.
There are explicit formulas for many interest rate derivatives. For
example, the value of a zero-coupon bond is
Stats243 Summer 2007