Chris Lamoureux

1
Fixed Income

• Chris Lamoureux

2
Motivation

• On Wall Street, fixed income analysis usually
  starts by fitting a model to the observed yield
  curve. (Arbitrage-free models.)
• This model is then used to price instruments
  which are derivative to the yield curve.
• The Black-Derman-Toy model is both a simple
  example, and a model that is widely used on the
  Street.

3
BDT 1

• In BDT, we make an assumption about the behavior
  of short-term rates. (In the example, the
  shortest horizon is one year).
• The primitives in the model are the observed spot
  rates on T-Year zero coupon Treasury securities,
  and the volatility of the short rate at each
  date. (These may be implied volatilities from
  the swap market, or estimated from historical
  data.)

4
BDT 2

• Using these primitives, we imply a binomial tree
  for short rates.
• Key formulas for this process include
• st ½ ln(ru rd) (assumption is that sigma
  depends on t but not r so this works from any
  node).
• For the zeros, S 100 / (1 y)N where y is the
  yield (-to-maturity) on the zero.

5
BDT 3

• We can fill in the rate tree as demonstrated in
  the companion spreadsheet.
• Exercise for next class
• Continue the process in the companion
  spreadsheet to develop the tree out to years 3
  and 4 (as in Figure F).

6
BDT 4

• Once we generate the implied tree for the short
  rate process, we can price more complicated
  securities.
• As an example, a coupon bond is a portfolio of
  zeros.
• As an example, a 10 coupon, 3-Year Bond is
  identical to a portfolio containing
• A 1-Yr 10 Zero
• A 2-Yr 10 Zero
• A 3-Yr 110 Zero

7
BDT 5

• Now we can use the implied tree on the companion
  spreadsheet to evaluate these three zeros or
  more simply, the yields on the different terms
• 10 / (1.1) (9.09)
• 10 / (1.11)2 (8.12)
• 110 / (1.12)3 (78.30)
• PV 95.51

8
BDT 6

• As mentioned in the introductory remarks, the use
  of such models is often to evaluate derivatives.
  Lets look at a European call and put on this
  coupon bond - both options have a 2-Year term
  and a strike of 95.
• The bond price tree is used to determine the
  option values upon expiration. These are then
  discounted back to get their current values, as
  shown in the companion spreadsheet.

9
BDT 7

• The options hedge ratios are of equal importance
  to traders and investment banks as their values.
• Next, we use our model to derive the options
  hedge ratios.

10
Exercise

• Consider the following scenario

Time Yield Volatility
1 .038 .15
2 .043 .165
3 .047 .18
4 .049 .17
5 .052 .16
6 .055 .15
7 .058 .14
8 .059 .135
11
Exercise (Contd.)

• Use the information in the table and the BDT
  model to
• Evaluate an 8-Year 6 coupon bond.
• Evaluate a call provision in the bond that allows
  the issuer to retire the bond at par after 5
  years (and anytime thereafter).