Models of Yield Curve and Yield Curve Dynamics

1
Models of Yield Curve andYield Curve Dynamics
2
Outline

• We are going to focus on just on a few of the
  models
• Random walk model of interest rates
• Mean reverting interest rates
• The Cox Ingersoll Ross Model
• The BDT Model

3
Binomial Model of Bond Prices

• It is tempting and commonly done to
  extrapolate the binomial stock price model to
  bond prices. This can create some problems,
  because it ignores certain aspects of bond
  prices, namely
• Bonds are not infinite lived
• Bonds must converge to par (100) at their
  maturity
• Bonds can go to 0

4

• If we work with a typical Cox, Ross, Rubinstein
  model
• Where u e s ((T-t)/N)1/2 e s (dt)1/2 1/d
• and the up jump probability is q 0.5 0.5
  u/s (dt)1/2
• This results in bond price has a distribution
  dP/P u dt d2
• which means that bond prices are lognormally
  distributed, even at maturity.
• As a result, the CRR type of model is not widely
  used

5
Binomial Model of Interest Rates

• Another approach, of course, would be to simply
  model the underlying interest rate as a binomial
  model. This, in fact, is one of the basic
  techniques used in term structure modeling today.

6
Binomial Model of Interest Rates

• Let q be the probability of an up jump and (1-q)
  the probability of a down jump
• In reality you would like for the probability to
  adjust depending on the level of rates, but for
  starters we will ignore that possibility.
• Consider how a two discount bond will evolve

7
Binomial Model of Interest Rates

• We now invoke the notion of the local
  expectations Hypothesis.
• This says, essentially, that the expected return
  over one period on any (default free) bond should
  be the same as the one-period rate prevailing at
  that time.
• Under LEH, therefore, at time t1, this must
  hold
• 1 / bu (1,2) 1 ur (t) and 1 / bd (1,2) 1
  dr (t)
• Recall that ur(t) and dr(t) are rates, not
  products necessarily

8
Binomial Model of Interest Rates

• We can rearrange these to get a pricing method
• We can now work backwards to get a price at time
  0

9
Binomial Model of Interest Rates

• We begin with the LEH condition
• This is really nothing more than a variant of how
  we have been pricing options all along
• This whole business of the LEH is really just to
  justify the following if you are at a node in
  lattice where the rate is r, then all bonds
  should be discounted at that rate

q . bu (1,2) (1-q) bd (1,2)
or
10
Example 17 - 1

• The initial one-period rate is 10, the up factor
  (u) is 1.25 and d0.80. We define the probability
  of an up (or down) move as 0.50. Based on this,
  rates evolve as

11
Binomial Model of Interest Rates

• Given this structure we can really start to
  calculate some interesting values
• What is a one-period zero-coupon bond worth?
• To answer this we can examine just the first two
  levels of the lattice. The bond pays 1 with
  certainty at the end of the period

Now the first thing to notice is that the only
relevant rate is the one at the central node,
since it is the correct discount rate for the
time period 0-1. In essence r 1,112.5 and r 1,0
8.0 become effective at the start of period 1,
while the 1 cash flows at paid at the end of
period 0.
12
Binomial Model of Interest Rates

• Within the lattice we tend to draw the end of 0
  and the beginning of 1 as occurring at the same
  point, but they are actually different. In fact,
  some people draw these as

13
Binomial Model of Interest Rates

• So to get the 1 period bond price you do the
  following
• We can easily extend this to a two period example

1/1.10
b (0,1)


0.9091
1
0.8889
1
0.9259
1.08
1(0.88) 1(0.92)
0,2

b 0,0

0.8249
1.10
1
14
Binomial Model of Interest Rates

• So the overall lattice looks like
• Frankly the superscript is frequently suppressed
  people assume you know which bond you are
  using.
• Notice that we now know the price of a zero
  coupon bond for periods 1 and 2, i.e 90.91 and
  82.49.

15
Binomial Model of Interest Rates

• From this we can determine the yields for one and
  two period zero coupon bonds

1/0.9091 1 y1
y1 10
16
Binomial Model of Interest Rates

• We can now show the evolution of the two period
  bond prices
• We can now apply the same basic methodology to a
  three period bond

17
Binomial Model of Interest Rates
18
Binomial Model of Interest Rates
Notice a few things here. First, we can solve for
the three period yield
Y3 (1/0.7475)1/3 1 10.1888
Y1 10 Y2 10.10 Y3 10.188
Thus we now know the three period zero coupon
yield curve
19
Binomial Model of Interest Rates

• We can also talk about the evolution of the
  two-period zero coupon yield over time. We know
  that at time 0 the two period zero coupon bond
  was worth 0.8249, and has a yield of 10.10.
• Based on our results we know that a two year bond
  at node (1,0) is worth 0.8560 and a two year bond
  at node (1,1) is worth 0.7884.
• From this we got the evolution of two year,
  constant maturity bond prices

20
Binomial Model of Interest Rates

• Now the two-year zero coupon rate at node 1,0 is
  given by
• y (2)1,0 (1/0.8560)1/2 1 0.80844
• And at 1,1
• y (2)1,1 (1/0.8560)1/2 1 0.80844
• So the two year constant maturity zero coupon
  yield evolves as

21
Binomial Model of Interest Rates

• Thus you can see that the two year
  constant-maturity rate evolves, just as the one
  period rate evolves
• The yield curve is, therefore, evolving over
  time, but the only factor driving this yield
  curve is the evolution at the one period interest
  rate.

Notice that the yield curve is not moving in
parallel, however. It is steeper at node 1,1 than
at 0,0 and it is flatter at 1,0 then at 0,0
22
Binomial Model of Interest Rates

• If we extend our model out more periods we can
  observe the evolution if more distant points on
  the yield curve for example we could see how
  the three year constant maturity rate evolved.
• In fact, if we wanted to we could describe the
  evolution of the entire yield curve as being
  driven by this single factor.
• Note that you have to pay attention to some
  issues here. You can discuss (as the book largely
  does) how a single bond evolves its price through
  time. For example you could see how a 4 period
  zero coupon starts at a given price and then in
  period 1 has a new price (or set of prices) and
  then get another set of prices in times 2 3.

23
Binomial Model of Interest Rates

• But you are observing a 4 year bond in time 0, a
  3 year bond in time 1, etc. You also want to pay
  attention to how the four year (or 3 year, etc)
  rate varies from period to period and from node
  to node. To do that you have to look at different
  bonds through time.
• I can also extract implied forward rates and see
  how they evolve over time, based on how the yield
  curve evolves which again is based on the
  evolution of the one period spot rate (i.e. one
  period yield).
• To see this, lets return to our lattice
  describing how the yield curve evolves over the
  first two periods

24
Binomial Model of Interest Rates

• Recall that the forward rate is the rate to which
  you can contract today to borrow in a future
  period. So at any point the one period forward
  rate is given by
• f(1,2) ((1y2)2 / (1y)) 1
• So at node 0,0 f(1,2)0,0 ((1.1010)2 / (1.10))
  110.2
• Note that this is not the expected spot rate in
  one period

25
Binomial Model of Interest Rates

• At node 1,0 that forward rate is given by
• f(2,3)1,0 ((1.080844)2 / (1.08)) 18.1688
• And at node 1,1 it is given by
• f(2,3)1,1 ((1.12662)2 / (1.125)) 112.824
• So we get an evolution of the one period forward
  curve as

So the one period spot rate evolution drives not
only the yield curve evolution, but also the
evolution of the forward rate. In fact the
forward rate and spot rates roles are sometimes
reversed. In the HJM model it is the forward rate
which is allowed to evolve and the spot rate
evolution is implied by that evolution.
26
Binomial Model of Interest Rates

• Now there are a couple of points we can make here
  about using this lattice structure. It is very
  good for pricing options, interest rates and
  other contingent claims, and you discount
  everything at the one period rate.
• Lets look at a couple of examples

27
Basic Bond Option

• Lets say that you hold a call option which gives
  you the right, at the beginning of period 3 (so
  the option expires at the end of period 2) to
  purchase a one period bond at price 90. What is
  this option worth today?
• Lets begin by recalling what our evolution of
  rates and prices looks like

28
Basic Bond Option

• So at nodes (2,0), (2,1), and (2,2) we get one
  period zero coupon bond prices of 0.9398, 0.9091
  and 0.8649 respectively. Since our strike is
  0.90, our payoffs to the option are given by
• C2,0 max (0, 0.9398 0.90) 0.0398
• C2,1 max (0, 0.9091 0.90) 0.0091
• C2,2 max (0, 0.8649 0.90) 0
• We can now move backwards through the lattice to
  determine the price of the option

C 2,20
C 1,1 0.0040
0.125
C 0,0
0.10
C 2,1 0.0091
C 1,0 0.0226
0.08
C 2,0 0.0398
29
Basic Bond Option
Or 1.212 on a 100 bond
30
Interest Rate Cap

• Lets say that you are a bank. Your average
  depositor puts money into 2 year CDs.
• So you have significant exposure to the two year
  zero coupon bond rate (realize that in any year ½
  of your depositors roll into new CDs at the rate
  for that year).
• Since you wish to manage your exposure you elect
  to use an interest rate cap on the two year zero
  coupon bond rate.
• Lets say that we have 100 million in total
  deposits, so you hedge 100 million notional.
  Your payoff is equal to two years interest on the
  notional amount.

31
Interest Rate Cap

• Recall that today the two year zero coupon bond
  rate is 10.10, and lets assume that this is the
  strike rate. The payoff to the cap would be
• Cap t 100 . Max (0, zero 2,t cap-rate)
• Recall earlier that we noted that the yield curve
  was evolving according to

32
Interest Rate Cap

• I have extended this another period using the
  rates found on page 601.
• P(1)3,3 1/1.19531 0.8366
• P(1)3,2 1/1.125 0.8888
• P(1)2,1 1/1.108 0.9259
• P(1)2,0 1/1.10512 0.95129

y(2)2,2((1/0.7461)1/2) -115.7706
y(2)2,1((1/0.82486)1/2) -110.10
y(2)2,0((1/0.88213)1/2) -16.471
33
Interest Rate Cap

• So I can now convert this into cash flows. Recall
  that the cap pays off annually based on the 100
  million notional. I am simplifying by assuming
  the cap pays immediately. It normally would not.

34
Interest Rate Cap

• CF 2,2 100 . Max(0,y2 cap-rate)
• 100 . Max(0, 0.157706-0.1010) 100m X 0.0567
• 5.67 million
• CF 2,1 0
• CF 2,0 0
• CF 1,1 100 . Max(0, 0.12662-0.1010) 2.562 m
• CF 0, 0 0

35
Interest Rate Cap

• Lets say that the initial futures price is 100,
  and it can raise or fall by 10, farther, the
  risk fee rate is 5, and the strike is 100.
• Assume we construct a portfolio consisting of
• 1 short call
• ? futures contracts

36
Interest Rate Cap

• And we select ? such that the portfolio has the
  same value in all states of the world at time 4
• - 10 ?10 -9.09 ?
• 19.09 ? 10
• ? 10/19.09 5.238
• Thus the portfolio value is

Thus the portfolio will be worth 4.762/1.05
-4.534 today
The time 0 futures contract is worth 0, so the
portfolio value is given by -4.534 0 (1)4.534
So the call value is 4.534
37
Interest Rate Cap

• We can generalize this
• ? (Cu Cd) / (Hu Hd) Cu Cd / (u- d) H
• The value at the end of the time period, the is
  always
• (Hu H) ? - Cu Since this is by
  construction the top part if the lattice / you
  could do this with the bottom as well)
• Which has present value
• (Hu H) ? - Cu e rT
• And we know that the value of the futures
  contract today must be zero, so the portfolio
  value at time 0 only comes from the time 0 option
  value
• - c (Hu H) ? - Cu e rT

38
Interest Rate Cap

• Substituting for ? and simplifying will result
  in
• Note It may be optimal to exercise an American
  call on a bond futures contract early, although
  it generally is not optimal to do so on a call on
  the bond itself
• To see this lets consider the following

39
Interest Rate Cap

• Assume that the call will be in the money
  regardless of the final state, i.e. both Cu and
  Cd gt 0. So Cu Hu k and Cd Hd k.
• Consider the time 0 price for the option
• Substitute (1-d)/(u-d) for p

C max H k, (h k)/(1r), so early
exercise is indeed optimal
40
Properties of Rates

• Now, so far, we have ignored the question of how
  we develop the interest rate lattice.
• Clearly we want to build it in a way that is
  consistent with basic interest rate processes
  that we have observed over time.
• Before we can begin with a specific lattice
  model, therefore, we want to fully understand
  what we want in an interest rate model.
• We want the interest rate model to be stochastic.
• We want the model to exhibit mean-reversion.
• Why? Because in reality rate exhibit this
  property!

41
Property of Rates
42
A Simple Model with Mean Reversion

• Sundaresan presents a simple model that
  incorporates mean reversion.
• This model is not really that great for pricing
  assets, but it is good for illustrating a few
  properties of mean reversion.
• The model works as follows.
• The one-period interest rate is assumed to be
  bound by an upper limit of 2µ and a lower limit
  of 0.
• Over one step in the binomial lattice, rates can
  rise of fall by an amount d, which is selected
  based on the time-step of the lattice.
• The probability of an up-jump from a given node
  in the lattice (t,j), is given by
• This generates a model that looks like this

43
A Simple Model with Mean Reversion
Upper limit r2µ
Lower limit r0
44
A Simple Model with Mean Reversion

• Notice that for any node (t,j), the rate can be
  represented as
• rt,j r0,0jd-(t-j)d
• Thus, r1,1 r0,0 (1) d-(1-1) d r00 d
• Similarly, r3,2 r0,0(2) d-(3-2) d r0,02 d-1
  d r0,0 d
• And r3,0r0,0 (0) d (3-0) d r0,0-3 d
• Example
• Assume r0,010, d1, and µ12. What would the
  lattice and the transition probabilities look
  like?

45
A Simple Model with Mean Reversion
Upper limit r2µ2(12)24
Lower limit r0
46
A Simple Model with Mean Reversion

• Notice that as the lattice grows toward 2µ or 0,
  that the probabilities force you to move back to
  the center of the lattice.
• At the two boundaries, you get the following
  probabilities of up and down jumps
• If r2µ, then q0, so (1-q)1, meaning that a
  down jump is guaranteed.
• If r0, then q1, and (1-q)0, meaning that an
  up jump is guaranteed.
• You can then price bonds, options, swaps, etc.
  backwards through the lattice as we did earlier
  in the model.

47
More Advanced Models

• Now, one has to be a little bit careful when
  talking about term structure models as to what is
  being meant.
• Sometimes people refer to a specific incarnation
  of a lattice as being the model. What they mean
  is both the underlying distribution of interest
  rates and the numerical method (the lattice) that
  insures they price bonds using that distribution.
• In other contexts people are referring only to
  the underlying distribution of rates.
• This is really the more general way of thinking
  of the term structure model, since there may be
  more than one numerical procedure that can
  determine prices under that distribution.
• We are going to first examine a model known as
  the Vasicek model and one of its variants known
  as the Cox, Ingersoll, and Ross (CIR) model.

48
More Advanced Models

• Theorists really like the Vasicek and CIR models
  because they are General Equilibrium models.
• This means that they are consistent within an
  entire economy that is defined by just a few
  parameters.
• They are also well-liked because you can get
  closed-formed solutions for zero coupon bonds.
• The difficulty with these models is that they are
  notoriously difficult to calibrate.
• This means that the input parameters they require
  are very difficult to estimate econometrically to
  within the level of accuracy needed for pricing
  real-world assets.
• There are many numerical models that people use
  that are consistent with Vasicek and CIR, but we
  will primarily focus on their use within Monte
  Carlo.

49
Vasicek Model

• The Vasicek class of models assume that only one
  fact, the one-period interest rate (r),
  determines the entire term structure. The process
  that r follow is
• (note this differs from Sundaresan) where
• r the spot rate
• ? a speed of adjustment factor
• µ the long-run mean of the interest rate
  process
• s the volatility of the short-run rate
• dz a white noise process.

50
Vasicek Model

• The model admits closed form solutions for a zero
  coupon bond

51
Vasicek Model

• The model does not, however, admit closed form
  solutions for a lot of other assets, including
  path-dependent assets.
• As a result, many times people either have to
  break down more complex assets such as
  coupon-bearing bonds into constituent zero
  coupon bonds, or they use Monte Carlo to model
  the evolution of the interest rate through time.
• Lets price some assets both ways to see how this
  process works.
• One issue you do have to be careful with is the
  you have to make your time-step small enough so
  that you dont inadvertently draw negative rates.

52
Vasicek Model

• Lets begin with pricing two bonds, one a zero
  coupon bond and one a coupon-bearing bond.
• The zero coupon bond matures in 5 years, and the
  coupon bearing bond matures in 5 years as well.
  Assume that the coupon bearing bond pays 12 but
  with semi-annual payments.
• The parameters for the model are
• r .10
• ? .10
• µ .12
• s 0.02
• So we have an upward sloping term structure.
• Lets price the zero coupon bond first.

53
Vasicek Model

• The model admits closed form solutions for a zero
  coupon bond

54
Vasicek Model

• From the five year zero coupon bond price we can
  also determine the five year yield
• A nice feature of the Vasicek model is that you
  can use it to determine the entire yield curve at
  a point in time. For example, using the
  parameters in this example, the yield curve is
  given by

55
Vasicek Model
56
Vasicek Model

• But what about the 12 coupon-bearing bond?
• We price it as a collection of zeros. That
  is
• Or, in this case 105.50.
• Notice that this is a Bond-Equivalent Yield of
  10.556, which is equivalent to a
  continously-compounded yield of 10.2869.
• Recall that the 5 year zero coupon yield was
  10.03096

57
Vasicek Model

• So this is fine if you want to analyze a bond at
  a specific point in time t.
• What if you want to model the evolution of the
  rate r over time.
• First, realize that this is equivalent to
  modeling the entire yield curve over time!
• There are many ways people do this some lattice
  methods, more frequently finite difference
  methods, and, of course, Monte Carlo. We will
  focus on Monte Carlo.
• Our goal is to develop a model that we can
  evaluate a bond, option, etc., over time, and to
  do this we can directly model the evolution of r
• To do this, we just have to use standard Monte
  Carlo methods.

58
Monte Carlo Methods

• A Monte Carlo model works by simulating the
  random process to generate potential interest
  rate paths, prices the mortgage (or other asset)
  under each of those paths, and then treats the
  average price found for all of the paths as the
  true price of the mortgage.
• This is a valid procedure (assuming that you have
  correctly modeled the interest rate process)
  because of the statistical laws known as the
  Central Limit Theorem and the Law of Large
  Numbers.
• Basically these laws state that if you draw from
  a random distribution (in this case from the
  simulated interest rates and the resultant
  mortgage prices), then you can be certain that
  the average of your sample population will
  approach the true average at a rate that is
  proportional to the square root of the number of
  simulations you are using.

59
Monte Carlo Simulation

• What this means is that as you increase the
  number of interest rate paths the average price
  that you get will become a better and better
  approximation of the real price.
• Operationally, a Monte-Carlo process is
  relatively easy to implement.
• Let N denote the number of interest rate paths
  that you are going to use, and let Pave be the
  average price you calculate from the simulation.
• The following flow chart demonstrates the general
  process

60
Monte Carlo Simulation
Set N to number of Iterations
Simulate the ith interest rate path
Add the Credit Spread to each rate along the
interest rate path
Determine the asset cash flowsgiven the ith rate
path
Increment i
Discount the cash flows to determine Pi
Update Pave Pave Pi/N
No
iN?
Yes
Report Pave as price of asset
61
Monte Carlo Simulation

• In reality what Monte Carlo returns is an
  estimate of the real price of the asset, and the
  Central Limit Theorem basically tells you the
  rate at which that estimate will become well
  behaved in a statistical sense.
• Since we are working with a estimate, we would
  like a measure of how good the estimate is. One
  such measure is the estimated standard deviation,
  of the distribution.

62
Monte Carlo Simulation

• Really, however, the sample standard deviation is
  only an estimate of the standard deviation for
  the population. What we want is a measure of how
  much deviation there is in our estimate of the
  mean. For this we want to use the standard error.
  (If you are having trouble recalling the
  differences between a standard error and a
  standard deviation check out the online
  statistics text hyperstat (http//davidmlane.com/h
  yperstat/) it has a very nice review of this
  stuff)

63
Monte Carlo Simulation

• Once we have Standard Errors, we can construct
  confidence intervals. What they do is tell us the
  upper and lower bounds within which the real
  price of the asset will fall (1-p) times, where p
  is the level of confidence we want. That is if
  p1, the upper and lower confidence interval
  will tell us the range in which the real price of
  the asset will fall into 99 of the time.
• The confidence interval narrows as the number of
  iterations increases.

64
Vasicek Model

• The Vasicek model assumes that the one-period
  interest rate evolves according to
• Lets assume that once again we have the same
  base parameters as before
• r0.10, µ0.12, ?0.10, s0.02
• The key, of course is to be able to simulate the
  randomness from a standard normal (i.e. from dz).
  This is relatively easy to do using a computer.
  Its a three step process
• Draw a value from the uniform distribution over
  the range 0-1, inclusive.
• Assume this draw represents the value of the
  cumulative density function (i.e. area under the
  curve to the left) at our point x.
• Invert the cumulative density function to
  determine the point x, treat x as our draw from
  dz!

65
Vasicek Model

• Lets say we did this
• Let x be the draw from the cumulative uniform,
  and say that value came out to be .20, so x0.20.
• First, assume that our draw will correspond to
  that value that has 20 of the area to the left
  in the cumulative density function.

66
Vasicek Model

• Lets say we did this
• We can now use the inverse of the normal density
  to determine what number has exactly 20 of the
  area of a standard normal to its left. We can use
  the Excel function norminv(x,mean,stdev) to do
  this.
• norminv(0.20,0,1)
• The value is -0.8416, and we assume that dz is
  (for this draw),
• -0.8416
• So what does this mean for our simulation? Well,
  since we know r0, we can use it to determine r1
• dr .10(.12-.10)(1).02(-0.8416)
• that is, dr 0.001667-0.016832 -0.016665
• r1r0dr 0.10 0.01665 0.083333
• Of course we have to scale by the time step if dt
  is not 1.
• So what does a simulation look like?

67
Vasicek Model
68
Vasicek Model

• Remember, however, that in the Vasicek model, rt
  determines the entire yield curve for time t.
• This means that as t evolves over time, the yield
  curve evolves over time.
• Thus, when we observe a line chart like the one
  we just created, we are really only looking at
  half of the story.
• We also want to get a sense for how the yield
  curve itself is evolving over time.
• This will be easiest to do by examining a surface
  area plot.

69
Vasicek Model
70
Vasicek Model

• It may be easier to see the evolution of the
  yield curve by examining the curve at specific
  points in time.
• Lets look at the curve at times 0, 12 months, 2
  years, 5 years, and 10 years

71
Vasicek Model
72
Vasicek Model
73
Vasicek Model
74
Vasicek Model
75
Vasicek Model
76
Vasicek Model

• So we have demonstrated how to generate the
  interest rates needed for the Vasicek model, but
  how would we use it to price a specific asset?
• Well, it is relatively straightforward. Once we
  have laid out the evolution of the spot rate, rt
  using the model, we then determine the cash flows
  given each specific interest rate path and
  discount them back along that path.
• We then take the average price over all N
  interest rate paths, and that is value of our
  asset.

77
Vasicek Model

• As an illustration, lets begin with a really
  simple, the value of a 5 year zero coupon bond.
• Now, we know that the Vasicek has a closed-form
  solution for the value of a zero coupon bond, but
  we are going to use Monte Carlo anyway, just to
  illustrate the process. We will then move on
  toward a swap to see how a more complex
  instrument would be valued.
• Using the same parameters as before (r0.10,
  µ0.12, ?0.10, s0.02), the value of a five year
  zero is 59.72, and is yielding 10.3096.
• So, lets examine how we would price it under
  Monte Carlo.

78
Vasicek Model

• We first have to decide how many iterations to
  use.
• Why? Because we are relying on the Central Limit
  Theorem and the Law of Large numbers to give us
  the correct solution, on average.
• Although each individual iteration is unbiased,
  they are draws from a sample distribution, and
  will only converge to the actual distribution
  over several trials.
• Lets go through a couple of cycles to illustrate
  this point.
• Recall the basic process

79
Monte Carlo Simulation
Set N to number of Iterations
Simulate the ith interest rate path
Add the Credit Spread to each rate along the
interest rate path
Determine the asset cash flowsgiven the ith rate
path
Increment i
Discount the cash flows to determine Pi
Update Pave Pave Pi/N
No
iN?
Yes
Report Pave as price of asset
80
Vasicek Model

• For the current sample, let us set N2 (we will
  quickly up this.)
• First, we begin with r0 and create r1 through r59
  using the formula
• The chart on the following page illustrates this
  sample path.
• Obviously the cash flows are straightforward in
  this case since they do not depend upon r.
• CF00, CF10, CF20,,CF580, CF59100
• Notice that I am using base 0 notation.
• We can then discount the cash flows back, on a
  monthly basis to get a time zero value

81
Vasicek Model
82
Vasicek Model

• In this case that turns out to generate a value
  of 54.5396.
• We then repeat the process for a second
  iteration.
• We again start with r0 (which is always the
  same!), and then generate 60 new interest rates
  using
• This is illustrated in the figure on the next
  page.
• We then apply the same basic valuation formula
• Which in this case yields a value of 77.83.
• So our average value is (54.5477.83)/2 66.18.
  Which is pretty far afield from our true value of
  59.72.

83
Vasicek Model
84
Vasicek Model

• Obviously we want to increase the number of
  iterations that we use to value this asset.
• How close to the correct value we get is a
  function of how many iterations we use.

85
Vasicek Model

• First, we should probably be struck by how
  inefficient Monte Carlo is. This is why generally
  we prefer to have closed-form solutions. Monte
  Carlo is nice because it will work for most types
  of assets.
• One issue to pay attention to is the way in which
  the confidence interval evolves.

86
Vasicek Model
87
Vasicek Model

• Keep in mind, of course, that the zero coupon
  bond is just about the easiest thing we could
  ever want to price.
• Lets price something more difficult, a type of
  swap.
• Lets assume that you have a 5 year swap with
  annual payments. The swap payments are made at
  the end of the year. This means that there will
  be cash flows occurring at the end of months 11,
  23, 35, 47, and 59, with the swap payments
  determined by the rate environments at times 0,
  11, 23, 35, and 47.
• Assume that the swap is the 1 year risk-free rate
  on the floating side and 10 on the fixed side.
  Assume you are paying fixed.
• Note that we still discount based on r following
• but have to determine the 1 year zero coupon
  rate for the payments.

88
Vasicek Model

• Keep in mind, of course, that the zero coupon
  bond is just about the easiest thing we could
  ever want to price.
• Lets price something more difficult, a swap.
• Lets assume that you have a 5 year swap with
  annual payments. The swap payments are made at
  the end of the year. This means that there will
  be cash flows occurring at the end of months 11,
  23, 35, 47, and 59, with the swap payments
  determined by the rate environments at times 0,
  11, 23, 35, and 47.
• Assume that the swap is the 1 year risk-free rate
  on the floating side and 10 on the fixed side.
  Assume you are paying fixed.
• Note that we still discount based on r following
• but have to determine the 1 year zero coupon
  rate for the payments.

89
Vasicek Model

• We will solve for the one-year zero coupon yields
  using the closed form solution that the Vasicek
  model generates.
• Lets go through a sample iteration to see what
  this looks like.
• Using the same sample path that we used earlier
  (next screen), the value of r and the one-year
  yield at times 0, 11, 23, 35 and 47 months are

90
Vasicek Model

• Assuming a million dollar notional amount, and
  that you are paying fixed, your cash flows will
  beWe can then discount the bonds back
  using the closed-form value of a zero coupon
  bond. We wind up with a value of 53,576 for the
  swap.

91
Vasicek Model
92
Vasicek Model

• Of course, that is just for one path. We again
  have to replicate this many times to get a real
  value for the swap.
• I should point out that there are closed-form
  solutions for swaps. What is more important here
  is seeing how the Monte Carlo works as opposed to
  the actual pricing of the swap.
• If we price the swap using Monte Carlo, however,
  we can once again generate prices and confidence
  intervals, as the next page demonstrates.

93
Vasicek Model

• The following chart demonstrates what happens as
  we increase the number of iterations.

94
Vasicek Model

• Again, we can examine the confidence interval and
  what happens to it as we increase the number of
  iterations.

95
Vasicek Model

• It is also interesting to see what happens to a
  histogram of the swap prices as you increase the
  iterations.
• Watch how we build the distribution.

96
Vasicek Model
97
Vasicek Model
98
Vasicek Model
99
The CIR Model

• Cox, Ingersoll, and Ross proposed a term
  structure model that is very similar to Vasiceks
  model, but with one very important difference
  they scale the stochastic component of the short
  rate process to be proportional to the square
  root of the interest rate process.
• The process they specify, therefore is
• The effect of including the square root of r is
  relatively straightforward to see as r
  approaches zero, the mean-reverting portion of
  the process becomes much more important.

100
The CIR Model

• We can see the effects of this easily enough with
  a few examples.
• First, let us set r0.10, ?.10, ?.10, and
  s.02, and then set up an example where dz is
  forced to be -1 for 20 straight times.
• Second, let us use the same parameters as above,
  but with randomly drawn dz values, and with
  comparisons of the Vasicek and CIR models.
• Third, it is frequently the case that when
  comparing Vasicek and CIR, that researchers will
  set the variance of the CIR model such that at
  time 0 the following relationship holdsIn the
  example above, this means that sCIR would be set
  to 0.6324.

101
The CIR Model
102
The CIR Model
103
The CIR Model

• Like Vasicek, the CIR model does admit closed for
  solutions for bonds, but with some modifications
  to the formulas for A and B.

104
Partial Equilibrium Models

• Both Vasicek and CIR work as full-equilibrium
  models, which means that all yield curve
  dynamics, including the time 0 yield curve, are
  endogenous to the model.
• Researchers have not had a lot of success in
  selecting parameters of these models such that
  they will correctly price time 0 bonds.
• A second approach to term structure modeling has
  been to develop so-called partial equilibrium
  models.
• These are models that take the time 0 term
  structure as an input and then build a
  self-consistent model given that term structure.
• Examples include the models of Ho and Lee, Hull
  and White, Black and Karasinski, and Black,
  Derman and Toy.

105
Partial Equilibrium Models

• General idea
• The general idea behind a partial equilibrium
  model is this you have an asset, x, which you
  wish to price or just wish to model as evolving
  through time.
• You want to insure that the pricing model you are
  using is correct given todays market.
• To insure this, you take a basket of other
  instruments that are in the market and you search
  for the parameters of the model that will cause
  the model to generate the correct prices for
  those other instruments. Denote the other
  instruments as y.
• You can then price the asset x with the same
  lattice and feel reasonably confident that you
  are getting prices that are consistent with y.

106
Partial Equilibrium Models

• Now, keep in mind several things
• First, we may not be as interested in the price
  of x as we are in its future evolution, for
  example, we may want to use the model to
  calculate hedge parameters such as delta.
• Second, the basket of other instruments that we
  have to use may vary from model to model and
  potentially from instrument to instrument.
• Frequently firms will maintain separate Treasure
  and LIBOR based models.
• Third, you will normally need at least two assets
  at each time step one to determine the level
  of rates at that time step, and one to determine
  the volatility (dispersion) or rates at that time
  step. Typically you will use zero coupon bond
  prices for the first asset, and something (such
  as caps) that is sensitive to volatility as the
  second.

107
Black, Derman, and Toy

• The Black, Derman, and Toy model (BDT), we first
  proposed in 1990. It was the first of these
  models to gain widespread acceptance in the
  industry.
• The model is somewhat dated now, but is still the
  standard starting point for students interested
  in learning about these types of models.
• Its also nice to use because it is a binomial
  model.
• In this binomial model, we are modeling the
  evolution of the spot (one-period) interest rate,
  rt,j.
• The probability of an up or a down jump (p and
  1-p) are defined to be 50.

108
Black, Derman, and Toy

• The Black, Derman, and Toy model (BDT), we first
  proposed in 1990. It was the first of these
  models to gain widespread acceptance in the
  industry.
• Lets begin with a discussion of how to construct
  the lattice, and then we will work through an
  example of pricing an asset.
• Unlike the binomial models that we have built
  before, there is not a simple construction
  method. You have to build the lattice via a
  search algorithm.
• We begin with market data on yields and
  volatilities

109
BDT

• We can convert the yields to prices
• So, working with a 1 year time step, we know that
  the initial one year rate must be 10, since
  90.91 100/1.1
• Thus, we say that r0,0.10.
• So we can start to set up our binomial lattice

110
BDT
r1,1
r0,0.10
r1,0

• So our next step is to figure out r1,1 and r1,0.
• We need to select value for r1,0 and r1,1 that
  will get P281.16 and s219.
• As show by equation (17.4) in Sundaresan, the
  volatility is defined as

111
BDT

• We know from the table, that s20.19, so this
  means that
• So as soon as we determine r1,0, we can calculate
  r1,1. So in one sense this now becomes a guessing
  game. We literally begin we a guessed value for
  r1,0. To see how this would work, lets say we
  start with an initial r1,0 guess of 9. This
  would make r1,1 13.16

112
BDT

• We then plug this back into the lattice we have
  already created and then calculate bond prices.

113
BDT

• Well, clearly 81.86 is more than the 81.16 that
  we expect to find.
• If we want to decrease the price, we have to
  increase r, so we raise it to a new level, say to
  9.792.

114
BDT

• Now, at this point we need to think very
  carefully about what we mean by volatility.
• There are really two potential ways we can define
  volatility
• The spread in the spot rate along a given time
  step,
• The spread in the yields over one period.
• It turns out that what we really want to focus on
  in this context is the yield volatility.
• The yield volatility of an N period bond is
  defined to be half of the natural log of the
  ratio of the yields of the bond at node (1,0) and
  (1,1).
• For the bond in the previous time step, that was
  easy to find since r1,1 and r1,0 are the yields
  for the bond at time 1.
• In our table, we see that the yield volatility of
  a bond that matures at time 2 is 18.

115
BDT

• This means that we have to build a three-step
  lattice

116
BDT

• We know that the price of a bond maturing at the
  end of period 2 will be 71.17.
• We also know that the yield volatility should be
  18.
• Denote the yield of a bond at node t,j that
  matures at the end of period I as yt,j(i).
• The one-period yield volatility of that bond is
  given by
• In contrast, the short-rate volatility at time
  step 2 would be denoted as s2r and is given by

117
BDT

• Since we know that the yield volatility is 18,
  we also know that
• So we have to once again search for the values of
  r2,0, r2,1, and r2,2 that will generate the
  correct P2 and s2 values.
• The book demonstrates that the values are 9.76,
  13.7669, and 19.4187, respectively.
• The next slide demonstrates the calculations

118
BDT
119
BDT

• So we get the right price, but what about the
  yield volatility?
• First, we want to calculate the yields on the
  zero coupon bonds that mature at the end of step
  2 at nodes 1,0 and 1,1.
• We can then calculate the yield volatility

120
BDT

• So we have verified that this is the yield
  volatility, but what about the short-rate
  volatility?
• Well, we can calculate it in two ways, using
  either r2,0 and r2,1, or using r2,1 and r2,2.
• Just to be clear, lets denote the period 2 rate
  volatility as s2r.
• This raises an interesting point about the BDT
  lattice by construction, at any time step t, the
  difference between the log of any two adjacent
  nodes is equal to two times the rate volatility
  for that time step. That is,

121
BDT

• Indeed, a lot of people find it easier to specify
  the evolution of the rate volatility instead of
  the yield volatility when working with the BDT
  model.
• Once you specify one, however, you have really
  specified the other.
• The BDT model does not really allow you to
  control simultaneously the yield and rate
  volatility.
• If you specify a flat rate volatility (say
  constant at 10), then this will eventually force
  the yield volatility toward zero.
• What process does the spot rate follow in the BDT
  model?
• Its is a log-normal process specified as follows

122
BDT

• In this specification, ?(t) is a time-varying
  long-run average yield, s(t) is the time-varying
  spot rate volatility, and s(t) is the derivative
  of that volatility, and dz has it normal meaning.
• In practice you almost never see it written out
  like this, instead, you solve for the parameters
  implicitly by solving for the prices and either
  yield or spot rate volatility.
• A closely related model to the Black, Derman, and
  Toy model is the Black, Karasinksi model. In that
  model you have a specification that is given by
• By relaxing the specification of a(t), they allow
  you to specify both a yield and spot rate
  volatility.

123
BDT

• Of course in reality we do not simply guess at
  the values of rt,j that will cause the model to
  generate the appropriate yields and yield/spot
  rate volatilities.
• What you normally do is to solve for those rates
  using a two-dimensional Newtons method. That is
  you simultaneously solve for the price and the
  volatility.