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A heat kernel approach to interest rate models. (English) Zbl 1304.35313

The paper is meant as a presentation of the heat kernel approach in the construction of state price densities. The idea of the approach is the following: For a Markov process \((X_t)_t\), assume that one can easily compute expectations \(\mathbb E [f(X_t)]\) and one explicitly knows one additional function \(p\) depending on time and the state variables; then one can construct explicit formulae for bond prices, and analytically tractable formulae for caps, swaptions and other derivatives: by this it is meant that these formulae can be calculated by one numerical integration with respect to the law of the underlying Markov process. This approach can be easily calibrated to market data, and is efficient in the modelling of interest rates with jumps, but not only. The basic concepts are presented with great detail, as well as many illuminating examples

MSC:

35K08 Heat kernel
60J25 Continuous-time Markov processes on general state spaces
35R60 PDEs with randomness, stochastic partial differential equations
35Q91 PDEs in connection with game theory, economics, social and behavioral sciences
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