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Kernel-Correlated Lévy Field Driven Forward Rate and Application to Derivative Pricing

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Abstract

We propose a term structure of forward rates driven by a kernel-correlated Lévy random field under the HJM framework. The kernel-correlated Lévy random field is composed of a kernel-correlated Gaussian random field and a centered Poisson random measure. We shall give a criterion to preclude arbitrage under the risk-neutral pricing measure. As applications, an interest rate derivative with general payoff functional is priced under this pricing measure.

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Notes

  1. The kernel-correlated Gaussian field is well known in the field of stochastic partial differential equations (SPDEs). In fact its formal (or generalized) derivative is the so-called colored (or spatial-correlated) noise, which have been studied by many authors (see, e.g., Dalang [12], Mytnik, Perkins and Sturm [42] and Dalang and Mueller [13]).

  2. \({\mathcal{D}}(\mathbb{R}^{d+1})\) denotes the topological vector space of functions \(\phi\in C_{0}^{\infty}(\mathbb{R}^{d+1})\) with the topology corresponding to the following: ϕ n →ϕ if and only if: (i) there exists a compact set K of ℝd+1 such that the support \({\rm supp}(\phi_{n}-\phi)\subset K\) for all n, and (ii) D α ϕ n →D α ϕ uniformly on K for each multi-index α=(α 1,…,α d ).

  3. We generalize the model in [6] by replacing the Wiener process with a kernel-correlated Gaussian random field.

  4. We assume that μ(t,T), σ(ξ,t,T), γ(ξ,t,T) and f 0(T) are “smooth” enough to guarantee that f(t,T) is continuous in the second variable and cadlag in the first one.

  5. This condition is well defined when \(\sigma(\cdot,\cdot,\cdot),\psi(\cdot,\cdot)\in{\mathcal{P}}\) (see (2.2)) and γ(⋅,⋅,⋅),φ(⋅,⋅)∈Ψ (see (2.6)).

  6. For general theory on arbitrage pricing, one may refer to [4, Chap. 15], [15, Chap. 6] or [27].

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Acknowledgements

The authors would like to thank the two reviewers for their valuable comments and suggestions that greatly improve the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, Xidian University, Xi’an, 710071, China

    Lijun Bo

  2. School of Business, Nankai University, Tianjin, 300071, China

    Yongjin Wang

  3. School of Management and Engineering, Nanjing University, Nanjing, 210093, China

    Xuewei Yang

Authors
  1. Lijun Bo
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  2. Yongjin Wang
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  3. Xuewei Yang
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Corresponding author

Correspondence to Xuewei Yang.

Additional information

This work was partially supported by the NSF of China (Nos. 11001213, 71201074, 70932003). The research of Bo was also supported by NCET-12-0914. The research of Yang was also supported by the Fundamental Research Funds for the Central Universities (No. 1127011812).

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Bo, L., Wang, Y. & Yang, X. Kernel-Correlated Lévy Field Driven Forward Rate and Application to Derivative Pricing. Appl Math Optim 68, 21–41 (2013). https://doi.org/10.1007/s00245-013-9196-2

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