Fixed Income #44 - The Arbitrage-Free Valuation Framework

Question 1

arbitrage-free valuation of a fixed-income securitiy

Answer 1

LOS 44.a

arbitrage-free valuation - method of valuing security such that no market participant can earn risk-free “arbitrage” profits using that security, i.e.:

• no initial cash outlay
• positive riskless profit in the future
• basic principle of “law of one price” in freely functioning markets

Question 2

two types of arbitrage opportunities

Answer 2

LOS 44.a

arbitrage-free framework:

• upholds value additivity principle - value of whole differs from the sum of parts
  
  • stripping: V_bond < strips ⇒ buy whole bond and sell its parts
  • reconstitution: V_bond > strips ⇒ buy strips and sell the whole bond\
• does not allow for dominance - otherwise indentical securitiies having different market prices

Question 3

binomial interest rate model

Answer 3

LOS 44.c

binomial interest rate model - system for building interest rate models

• binomial tree with equal probability up/down interest rate paths
• lognormal random walk based on assumed volatility in interest rates (non-negative rates and higher volatility at higher rates

Question 4

binomial interest rate tree

Answer 4

LOS 44.c

i₀ → use spot rate s₁

nodal values → use single period forward rate

​i_2,LU = i_2,LLe^2σ

​i_2,LU = i_2,UL

​i_2,UU = i_2,LLe^4σ

FYI - know how to use the tree; won’t be building the tree on exam

Question 5

backward induction methodology

Answer 5

LOS 44.d

backward induction methodology - value a bond by moving backward from last period to time zero. Things to know:

• value at maturity is known
• value at any node is the average PV of the two possible values from next period
• discount rate used is the foreward rate of that node

Question 6

backward induction example

Answer 6

LOS 44.d

Question 7

calibrating a binomial interest rate tree to match a specific term structure

Answer 7

LOS 44.e

binomial interest rate tree must be calibrated to conform to 3 rules:

1. arbitrage-free interest rate values
2. adjacent forward rates (in the same time period) are 2σ apart
3. middle forward rate (or mid-point for even number of rates) in a time period is ≈ implied (from benchmark spot rate curve) one-period forward rate for that period.

think “Ho-Lee” model