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Fixed Income

Fixed Income. Chris Lamoureux. 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.

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Fixed Income

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  1. Fixed Income Chris Lamoureux Fixed Income

  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. Fixed Income

  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.) Fixed Income

  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. Fixed Income

  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). Fixed Income

  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 Fixed Income

  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 Fixed Income

  8. BDT 6 As mentioned in the introductory remarks, the use of such models is often to evaluate derivatives. Let’s 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. Fixed Income

  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. Fixed Income

  10. Exercise Consider the following scenario: Fixed Income

  11. Exercise (Cont’d.) 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). Fixed Income

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