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Bond price modelling using continuous-time Markov chains
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Bond prices are based on the expected accumulation of a stochastic interest rate, the spot rate. Well-known spot rate models include Vasicek and CIR, which are based on stochastic differential equations. In this talk, we present an alternative description of the spot rate in terms of a finite state-space Markov process, where the spot rates are piecewise deterministic (or even constant) in the different states. In this so-called Markovian interest model, we show that the bond price coincides with the survival function of a phase-type distribution. This striking coincidence allows for calibrating a Markovian interest rate model (using maximum likelihood) to observed data (prices) or other theoretical models.

In life insurance models, discounting by stochastic interest rates is usually dealt with by assuming independence and converting the stochastic rates into deterministic forward rates. A problem with this approach is that the computation of reserves and premiums becomes much more involved since Thiele's differential equation no longer remains valid. We discuss how the Markovian interest model can be used as an alternative in multi-state life insurance models and how the spot rates naturally integrate into formulas for reserves, moments of future payments and equivalence premiums.

HEC-DSA

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