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CFA® Level II · Fixed Income · free lesson
The Arbitrage-Free Valuation Framework
Every bond, from a plain government note to a callable corporate or a mortgage-backed security, is valued on the same principle: its price must leave no room for risk-free profit. This module applies that principle with spot rates, builds and calibrates binomial interest rate trees, values bonds by backward induction and by pathwise valuation, explains when Monte Carlo simulation is needed, and compares the equilibrium and arbitrage-free models that generate interest rate paths. It is the toolkit for the next module on bonds with embedded options.
What you will learn
1. Explain arbitrage-free valuation, identify arbitrage opportunities, and price an option-free bond with spot rates.
- Arbitrage requires no net investment, no risk and a positive profit.
- Value additivity: a coupon bond equals the sum of its zero-coupon parts.
- Bond below the sum of its parts: buy and strip; above: buy the zeros and reconstitute.
- Option-free bond value = sum of cash flows discounted at their spot rates.
- Interest-rate-contingent cash flows need a rate model calibrated to benchmark prices.
2. Build and calibrate a binomial interest rate tree to the benchmark curve and value a bond by backward induction.
- Adjacent rates at a date differ by the factor e^(2 sigma) for annual steps; up and down moves are equally likely.
- Backward induction: V = 0.5[(V_H + C) + (V_L + C)]/(1 + r) at each node.
- Calibrate date by date so the tree reprices benchmark par bonds at 100.
- A calibrated tree values option-free bonds exactly as spot rates do.
- Volatility, often implied from caps, floors and swaptions, matters for bonds with options.
3. Value a bond by pathwise valuation and explain how Monte Carlo simulation of interest rate paths is used.
- Pathwise valuation: value = average of PVs along all 2^(N-1) equally likely paths in an annual tree.
- Pathwise and backward induction give the same value for a calibrated tree.
- Path-dependent cash flows, such as MBS prepayments, need pathwise or Monte Carlo methods.
- Monte Carlo adds a drift adjustment so simulated paths reprice benchmarks; mean reversion keeps rates realistic.
- More paths reduce sampling error but not model error.
4. Compare equilibrium and arbitrage-free term structure models and describe the features of the main one-factor models.
- Equilibrium models (CIR, Vasicek) use few parameters and economic assumptions; they need not fit today's curve.
- CIR: mean-reverting, volatility proportional to sqrt(r), no negative rates; Vasicek: mean-reverting, constant volatility, negative rates possible.
- Arbitrage-free models (Ho-Lee, KWF, BDT) are calibrated to current market prices via a time-dependent drift.
- Ho-Lee is normal (negative rates possible); KWF is lognormal (positive rates).
- Arbitrage-free models are the standard for valuing embedded options and derivatives.
Module at a glance
- Arbitrage-free value = sum of cash flows discounted at their spot rates; mispricing invites stripping or reconstitution.
- Lognormal tree: adjacent rates differ by e^(2 sigma); up and down moves are equally likely.
- Backward induction: V = 0.5[(V_H + C) + (V_L + C)]/(1 + r); remember the coupon at every node.
- Calibrate the tree date by date to reprice benchmark par bonds at 100.
- Pathwise value = average PV over 2^(N-1) paths; Monte Carlo handles path-dependent cash flows such as MBS.
- Equilibrium models (CIR, Vasicek) use economic parameters; arbitrage-free models (Ho-Lee, KWF) fit today's prices.
This free module has 4 lessons, 16 practice questions with full explanations.