Fixed Income Lecture

# Fixed Income Lecture - Debt Instruments and Markets...

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Debt Instruments and Markets Professor Carpenter Financial Engineering 1 Fixed Income Financial Engineering Concepts and Buzzwords From short rates to bond prices The simple Black, Derman, Toy model Calibration to current the term structure Nonnegativity Proportional volatility Lognormal limiting Distribution Independent increments vs. Mean reversion Readings Veronesi, Chapters 10-11 Tuckman, Chapters 11-12

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Debt Instruments and Markets Professor Carpenter Financial Engineering 2 Implementing the No-Arbitrage Derivative Pricing Theory in Practice 1. Start with a model (tree) of one-period rates (short rates) and risk-neutral probabilities. For example, Black-Derman-Toy, Ho and Lee, … 2. Build the tree of bond prices from the tree of short rates using the risk-neutral pricing equation (RNPE) price = discount factor x [p x up payoff + (1-p) x down payoff] 3. Build the tree of derivative prices from the tree of bond prices by pricing by replication. Replication cost can be also represented as price = discount factor x [p x up payoff + (1-p) x down payoff] 4. Calibrate the model parameters (drift, volatility) to make the model match observed bond prices and option prices. Building the Price Tree from the Rate Tree and Risk-Neutral Probabilities (Step 2) Once we have a tree of one-period rates and risk-neutral probabilities, we can price any term structure asset. For example, suppose 0.5-year rates and risk-neutral probabilities are as follows: Time 0.5 Time 1 Time 0 0.5 0.5 0.5 r 0.5 = 5.54% 0.5 r 1 u = 6.004% 0 .5 r 1 d = 4.721% 1 r 1.5 uu = 6.915% 1 r 1.5 ud = 5.437% 1 r 1.5 dd = 4.275%
Debt Instruments and Markets Professor Carpenter Financial Engineering 3 Building the Price Tree from the Rate Tree… Then we have the prices of bonds for maturities 0.5, 1, and 1.5: Time 0 price of the zero maturing at time 0.5 Time 0.5 possible prices of zero maturing at time 1 Time 0 price of the zero maturing at time 1 d 1 = 0.973047 [0.5 x 0.9709 + 0.5x0.9769] = 0.9476 Time 0.5

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