Abstract
This article presents a numerically efficient approach for constructing an interest rate lattice for multi-state variable multi-factor term structure models in the Makovian HJM [Econometrica 70 (1992) 77] framework based on Monte Carlo simulation and an advanced extension to the Markov Chain Approximation technique. The proposed method is a mix of Monte Carlo and lattice-based methods and combines the best from both of them. It provides significant computational advantages and flexibility with respect to many existing multi-factor model implementations for interest rates derivatives valuation and hedging in the HJM framework.
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Notes
For practical applications h ij can be selected as a function of the short rate r, for example \(h_{ij} =c_{ij} r^{\gamma_j},\) where c ij and γ j are some constant parameters.
dτ is an infinitely small increment of time, and dz j (t) is the Wiener process for the jth factor.
t k is the time of k-lattice layer.
In practice, the lattice, built on \(K=3,000\) or more paths, provides pretty good convergence.
All other time layers are built analogously.
In the presented example the one-factor model is considered for demonstration purposes; in general, the equal probabilities are as follows: \(p=\frac{1}{2M},\) where M is the number of stochastic factors.
The average state of a set of states is defined as the state with state variables equal to averages of corresponding state variables of set states.
This (external) set is, in general, different from the (internal) set of MC paths used for the lattice building. This allows one to compute cash flows on a few (three, for example) MC paths while the lattice (and the corresponding exercise boundary) is built using a different, bigger set of, for example, 5,000 paths.
The most important difference between the LS (2001) approach and the LB approach is the implicit exercise boundary.
This can be achieved with given accuracy by increasing the number of MC paths used to build the lattice.
For the LS approach it is represented by the exercise time \(T_n\) for every path (see Eq. 7).
It should be noted here that the lattice could be viewed as a set of special MC paths with the number of paths equal to the number of possible scenario combination within the lattice structure. And this number is usually is much larger than the number of MC paths used to construct the lattice.
In terms of pricing the LM approach does not provide additional advantages over the LB approach. The former is based on the latter and is more computationally expensive.
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Acknowledgement
The authors are grateful to Vladimir Novakovsky for implementation of numerous numerical algorithms and many stimulating discussions.
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Kramin, M.V., Nandi, S. & Shulman, A.L. A multi-factor Markovian HJM model for pricing American interest rate derivatives. Rev Quant Finan Acc 31, 359–378 (2008). https://doi.org/10.1007/s11156-007-0078-z
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DOI: https://doi.org/10.1007/s11156-007-0078-z
Keywords
- Monte Carlo simulation
- Lattice
- Recombining tree
- American derivatives
- Markovian HJM framework
- Multi-state variable multi-factor model
- Interest rate options
- Computational efficiency