HJM Framework Lecture

Introduction

• Overview of the HJM framework
• Focus on understanding the dynamics of an instantaneous forward rate with fixed maturity
• Framework developed using a generic maturity T

Dynamics of the Instantaneous Forward

• Similar to other stochastic differential equations: consists of a drift term and volatility term
• Conditions for the drift term must avoid arbitrage
• Derivation of conditions under the risk-neutral measure and forward measure
• Understanding the drift conditions is crucial

Volatility Term

• Volatility dictates the stochastic process of the instantaneous forward
• Simplifications of volatility for a tractable model: Gaussian and Markovian assumptions
• Assumption of randomness from one Brownian motion; can be extended to multiple Brownian motions
• Volatility as a function of the instantaneous forward

Deriving Dynamics Under Risk-Neutral Measure

• If volatility of an asset process is known, its dynamics under the risk-neutral measure can be written
• Instantaneous forward is not a traded asset but zero-coupon bond of the same maturity is
• Dynamics of instantaneous forward inferred from zero-coupon bond
• Use Ito's lemma for the differential of logarithm of bond price
• Resulting dynamics for the instantaneous forward under the risk-neutral measure
• Conditions on volatility terms and drift terms

Change of Measure

• Change of numeraire approach to switch probability measures
• Use price of zero-coupon bond as numeraire
• Derive dynamics under forward measure from those under risk-neutral measure
• Radon-Nikodým derivative and Girsanov theorem used to change measure
• Instantaneous forward as a martingale under the forward measure

Practical Modeling Considerations

• Typically model multiple forwards; choose longest maturity zero-coupon bond as numeraire
• Analysis valid for entire period since numeraire is alive throughout
• Compare valuation under risk-neutral and forward measures
• Formulae adjusted for different maturities of bonds

Volatility and Dynamics

• Volatility is crucial for driving dynamics of instantaneous forwards
• Explore deterministic function of volatility for Gaussian process
• Discuss Markov process and its advantages in modeling
• Show how separable volatility function makes a process Markovian
• Explain why lognormal type specification won’t work in continuous settings due to possible arbitrage
• In discrete settings, issues with lognormal specification are resolved

Conclusion

• Covered key topics of HJM framework
• Next steps: Deriving short rate models like Hull-White extended Vasicek model
• Homework on CIR model