MSc IT

1
Chapter 23 Interest Rate Models
2
Introduction

• We will begin by seeing how the Black model can
  be used to price bond and interest rate options.
• The Black model is a version of the Black-Scholes
  model for which the underlying asset is a futures
  contract.
• Next, we see how the Black-Scholes approach to
  option pricing applies to bonds.
• Finally, we examine binomial interest rate
  models, in particular the Black-Derman-Toy model.

3
Bond Options, Caps, and the Black Model

• If we assume that the bond forward price is
  lognormally distributed with constant volatility
  ?, we obtain the Black formula for a bond option
• (23.3)
• where
• and F is an abbreviation for the bond forward
  price F0,T P(T,T s), where Pt(T,T s)
  denotes the time-t price of a zero-coupon bond
  purchased at T and paying 1 at time T s.

4
Bond Options, Caps, and the Black Model

• One way for a borrower to hedge interest rate
  risk is by entering into a call option on a
  forward rate agreement (FRA), with strike price
  KR.
• This option, called a caplet, at time T s pays
• Payoff to caplet max 0, RT(T, T s) KR
  (23.5)
• An interest rate cap is a collection of caplets.
• Suppose a borrower has a floating rate loan with
  interest payments at times ti, i 1, . . ., n. A
  cap would make the series of payments
• Cap payment at time ti1 max 0, Rti(ti, ti1)
  KR (23.8)

5
Bond Pricing

• Some difficulties of bond pricing are
• A casually specified model may give rise to
  arbitrage opportunities (for example, if the
  yield curve is assumed to be flat).
• In general, hedging a bond portfolio based on
  duration does not result in a perfect hedge.

6
An equilibrium equation for bonds

• When the short-term interest rate is the only
  source of uncertainty, the following partial
  differential equation must be satisfied by any
  zero-coupon bond
• (23.26)
• where r is the short-term interest rate, which
  follows the Itô process dr a(r)dt ?(r)dZ, and
  ?(r,t) is the Sharp ratio for dZ.
• This equation characterizes claims that are a
  function of the interest rate. The risk-neutral
  process for the interest rate is obtained by
  subtracting the risk premium from the drift
• dr a(r) ? ?(r)?(r,t)dt ?(r)dZ (23.27)

7
An equilibrium equation for bonds

• Given a zero-coupon bond, Cox et al. (1985) show
  that the solution to equation (23.26) is
• (23.28)
• where E represents the expectation taken with
  respect to risk-neutral probabilities and R(t,T)
  is the random variable representing the
  cumulative interest rate over time
• (23.29)
• Thus, to value a zero-coupon bond, we take the
  expectation over all the discount factors implied
  by these paths.

8
An equilibrium equation for bonds

• In summary, an approach to modeling bond prices
  is exactly the same procedure used to price
  options on stock
• We begin with a model of the interest rate and
  then use equation (23.26) to obtain a partial
  differential equation that describes the bond
  price.
• Next, using the PDE together with boundary
  conditions, we can determine the price of the
  bond.

9
Delta-Gamma Approximations for Bonds

• Equation (23.26) implies that
• (23.20)
• Using the risk-neutral distribution, bonds are
  priced to earn the risk-free rate.
• Thus, the delta-gamma-theta approximation for the
  change in a bond price holds exactly if the
  interest rate moves one standard deviation.
• However, the Greeks for a bond are not exactly
  the same as duration and convexity.

10
Equilibrium Short-Rate Bond Price Models

• We discuss three bond pricing models based on
  equation (23.26), in which all bond prices are
  driven by the short-term interest rate, r.
• the Rendleman-Bartter model,
• the Vasicek model,
• the Cox-Ingersoll-Ross model.
• They differ in their specification of ?(r), ?(r),
  ?(r).

11
Equilibrium Short-Rate Bond Price Models

• The simplest models of the short-term interest
  rate are those in which the interest rate follows
  arithmetic or geometric Brownian motion. For
  example,
• dr adt ?dZ (23.31)
• In this specification, the short-rate is
  normally distributed with mean r0 ar and
  variance ?2t.
• There are several objections to this model
• The short-rate can be negative.
• The drift in the short-rate is constant.
• The volatility of the short-rate is the same
  whether the rate is high or low.

12
The Rendleman-Bartter model

• The Rendleman-Bartter model assumes that the
  short-rate follows geometric Brownian motion
• dr ardt ?rdz (23.32)
• An objection to this model is that interest rates
  can be arbitrarily high. In practice, we would
  expect rates to exhibit mean reversion.

13
The Vasicek model

• The Vasicek model incorporates mean reversion
• dr a(b ? r)dt ?dz (23.33)
• This is an Ornstein-Uhlenbeck process. The a(b ?
  r)dt term induces mean reversion.
• It is possible for interest rates to become
  negative and the variability of interest rates is
  independent of the level of rates.

14
The Cox-Ingersoll-Ross model

• The Cox-Ingersoll-Ross (CIR) model assumes a
  short-term interest rate model of the form
• dr a(b ? r)dt ??r dz (23.35)
• The model satisfies all the objections to the
  earlier models
• It is impossible for interest rates to be
  negative.
• As the short-rate rises, the volatility of the
  short-rate also rises.
• The short-rate exhibits mean reversion.

15
Comparing Vasicek and CIR

• Yield curves generated by the Vasicek and CIR
  models

16
A Binomial Interest Rate Model

• Binomial interest rate models permit the interest
  rate to move randomly over time.
• Notation
• h is the length of the binomial period. Here a
  period is 1 year, i.e., h1.
• rt0(t, T) is the forward interest rate at time t0
  for time t to time T.
• rt0(t, T j) is the interest rate prevailing from
  t to T, where the rate is quoted at time t0 lt t
  and the state is j.
• At any point in time t0, there is a set of both
  spot and implied forward zero-coupon bond prices,
  Pt0(t, T j).
• p is the risk-neutral probability of an up move.

17
A Binomial Interest Rate Model

• Three-period interest rate tree of the 1-year
  rate

18
Zero-coupon bond prices

• Because the tree can be used at any node to value
  zero-coupon bonds of any maturity, the tree also
  generates implied forward interest rates of all
  maturities, as well as volatilities of implied
  forward rates.
• Thus, we can equivalently specify a binomial
  interest rate tree in terms of
• interest rates,
• zero-coupon bond prices, or
• volatilities of implied forward interest rates.

19
Zero-coupon bond prices

• We can value a bond by considering separately
  each path the interest rate can take.
• Each path implies a realized discount factor.
• We then compute the expected discount factor,
  using risk-neutral probabilities.
• All bond valuation models implicitly calculate
  the following equation
• (23.44)
• where E is the expected discount factor and ri
  represents the time-i rate.

20
Zero-coupon bond prices

• The volatility of the bond price is different
  from the behavior of a stock.
• With a stock, uncertainty about the future stock
  price increases with horizon.
• With a bond, volatility of the bond price
  initially grows with time. However, as the bond
  approaches maturity, volatility declines because
  of the boundary condition that the bond price
  approached 1.

21
Yields and expected interest rates

• Uncertainty causes bond yields to be lower than
  the expected average interest rate.
• The discrepancy between yields and average
  interest rates increases with volatility.
• Therefore, we cannot price a bond by using the
  expected interest rate.

22
Option pricing

• Using the binomial tree to price a bond option
  works the same way as bond pricing.
• Suppose we have a call option with strike price K
  on a (T ? t)-year zero-coupon bond, with the
  option expiring in t ? t0 periods.
• The expiration value of the option is
• O (t, j) max 0, Pt(t, T j) K (23.46)
• To price the option we can work recursively
  backward through the tree using risk-neutral
  pricing, as with an option on a stock..
• The value one period earlier at the node j is
• O (t ? h, j) Pt ? h (t ? h, t j)
• ? p ? O (t, 2 ? j 1) (1 ? p) ? O (m, 2 ?
  j) (23.47)

23
Option pricing

• In the same way, we can value an option on a
  yield, or an option on any instrument that is a
  function of the interest rate.
• Delta-hedging works for the bond option just as
  for a stock option.
• The underlying asset is a zero-coupon bond
  maturing at T, since that will be a (T ?
  t)-period bond in period t.
• Each period, the delta-hedged portfolio of the
  option and underlying asset is financed by the
  short-term bond, paying whatever one-period
  interest rate prevails at that node.

24
The Black-Derman-Toy Model

• The models we have examined are arbitrage-free in
  a world consistent with their assumptions. In the
  real world, however, they will generate apparent
  arbitrage opportunities, i.e., observed prices
  will not match theoretical prices.
• Matching a model to fit the data is called
  calibration.
• This calibration ensures that it matches observed
  yields and volatilities, but not necessarily the
  evolution of the yield curve.
• The Black-Derman-Toy (BDT) tree is a binomial
  interest rate tree calibrated to match
  zero-coupon yields and a particular set of
  volatilities.

25
The Black-Derman-Toy Model

• The basic idea of the BDT model is to compute a
  binomial tree of short-term interest rates, with
  a flexible enough structure to match the data.
• Consider market information about bonds that we
  would like to match

26
The Black-Derman-Toy Model

• Black, Derman, and Toy describe their tree as
  driven by the short-term rate, which they assume
  is lognormally distributed.

27
The Black-Derman-Toy Model

• For each period in the tree, there are two
  parameters
• Rih can be though of as a rate level parameter at
  a given time and
• ?i can be though of as a volatility parameter.
• These parameters can be used to match the tree
  with the data.

28
The Black-Derman-Toy Model

• The volatilities are measured in the tree as
  follows
• Let the time-h price of a zero-coupon bond
  maturing at T when the time-t short-term rate is
  r(h) be Ph, T, r(h).
• The yield of the bond is
• yh, T, r(h) Ph, T, r(h)?1/(T ? h) ? 1
• At time h, the short-term rate can take on the
  two values ru and rd. The annualized lognormal
  yield volatility is then
• (23.48)

29
The Black-Derman-Toy Model

• The BDT tree, which depicts 1-year effective
  annual rates, is constructed using the above
  market data

30
The Black-Derman-Toy Model

• The tree behaves differently from binomial trees.
  
• Unlike a stock-price tree, the nodes are not
  necessarily centered on the previous periods
  nodes.
• This is because the tree is constructed to match
  the data.

31
The Black-Derman-Toy Model

• Verifying that the tree is consistent with the
  data
• To verify that the tree matches the yield curve,
  we need to compute the prices of zero-coupon
  bonds with maturities of 1, 2, 3, and 4 years.
• To verify the volatilities, we need to compute
  the prices of 1-, 2-, and 3-year zero-coupon
  bonds at year 1, and then compute the yield
  volatilities of those bonds.

32
The Black-Derman-Toy Model

• The tree of 1-year bond prices implied by the BDT
  interest rate tree

33
The Black-Derman-Toy Model

• Verifying yields

34
The Black-Derman-Toy Model

• Verifying volatilities
• For each bond, we need to compute implied bond
  yields in year 1 and then compute the volatility.
• Using equation (23.48), the yield volatility
• for the 2-year bond (1-year bond in year 1) is
• for the 3-year bond in year 1 is

35
Constructing a Black-Derman-Toy tree

• Given the data, how do we generate the tree?
• We start at early nodes and work to the later
  nodes, building the tree outward.
• The first node is given by the prevailing 1-year
  rate. The 1-year bond price is
• 0.9091 1 / (1R0) (23.49)
• Thus, R0 0.10.

36
Constructing a Black-Derman-Toy tree

• For the second node, the year-1 price of a 1-year
  bond is P(1, 2, ru) or P(1, 2, rd). We require
  that two conditions be satisfied
• (23.50)
• (23.51)
• The second equation gives us ? 0.1, which
  enables us to solve the first equation to obtain
  R1 0.1082.
• In the same way, it is possible to solve for the
  parameters for each subsequent period.

37
Black-Derman-Toy examples

• Caplets and caps
• An interest rate cap pays the difference between
  the realized interest rate in a period and the
  interest cap rate, if the difference is positive.
• The payments in each node in a tree are the
  present value of the cap payments for the
  interest rate at that node.

38
Black-Derman-Toy examples

• Forward rate agreements (FRA)
• The standard FRA calls for settlement at maturity
  of the loan, when the interest payment is made.
• If r(3, 4) is the 1-year rate in year 3, the
  payoff to a standard FRA 4 years from today is
• (23.54)
• We can compute by taking the discounted
  expectation along a binomial tree of r(3, 4) paid
  in year 4 and dividing by P(0, 4).

39
Black-Derman-Toy examples

• Forward rate agreements (FRA)
• Eurodollar-style settlement, calls for payment at
  the time the loan is made.
• If a FRA settles on the borrowing date in year
  3, then
• (23.55)
• We can compute by taking the discounted
  expectation along a binomial tree of r(3, 4) paid
  in year 3, and dividing by P(0, 3).
• There is no marking-to-market prior to
  settlement, but the timing of settlement is
  mismatched with the timing of interest payments.
• Thus, the two settlement procedures generate
  different fair forward interest rates.