dorsal/arxiv
View SchemaA Closed-Form Approximation of Likelihood Functions for Discretely Sampled Diffusions: the Exponent Expansion
| Authors | Luca Capriotti |
|---|---|
| Categories | |
| ArXiv ID | physics/0703180 |
| URL | https://arxiv.org/abs/physics/0703180 |
Abstract
In this paper we discuss a closed-form approximation of the likelihood functions of an arbitrary diffusion process. The approximation is based on an exponential ansatz of the transition probability for a finite time step $\Delta t$, and a series expansion of the deviation of its logarithm from that of a Gaussian distribution. Through this procedure, dubbed {\em exponent expansion}, the transition probability is obtained as a power series in $\Delta t$. This becomes asymptotically exact if an increasing number of terms is included, and provides remarkably accurate results even when truncated to the first few (say 3) terms. The coefficients of such expansion can be determined straightforwardly through a recursion, and involve simple one-dimensional integrals. We present several examples of financial interest, and we compare our results with the state-of-the-art approximation of discretely sampled diffusions [A\"it-Sahalia, {\it Journal of Finance} {\bf 54}, 1361 (1999)]. We find that the exponent expansion provides a similar accuracy in most of the cases, but a better behavior in the low-volatility regime. Furthermore the implementation of the present approach turns out to be simpler. Within the functional integration framework the exponent expansion allows one to obtain remarkably good approximations of the pricing kernels of financial derivatives. This is illustrated with the application to simple path-dependent interest rate derivatives. Finally we discuss how these results can also be used to increase the efficiency of numerical (both deterministic and stochastic) approaches to derivative pricing.
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"abstract": "In this paper we discuss a closed-form approximation of the likelihood\nfunctions of an arbitrary diffusion process. The approximation is based on an\nexponential ansatz of the transition probability for a finite time step $\\Delta\nt$, and a series expansion of the deviation of its logarithm from that of a\nGaussian distribution. Through this procedure, dubbed {\\em exponent expansion},\nthe transition probability is obtained as a power series in $\\Delta t$. This\nbecomes asymptotically exact if an increasing number of terms is included, and\nprovides remarkably accurate results even when truncated to the first few (say\n3) terms. The coefficients of such expansion can be determined\nstraightforwardly through a recursion, and involve simple one-dimensional\nintegrals.\n We present several examples of financial interest, and we compare our results\nwith the state-of-the-art approximation of discretely sampled diffusions\n[A\\\"it-Sahalia, {\\it Journal of Finance} {\\bf 54}, 1361 (1999)]. We find that\nthe exponent expansion provides a similar accuracy in most of the cases, but a\nbetter behavior in the low-volatility regime. Furthermore the implementation of\nthe present approach turns out to be simpler.\n Within the functional integration framework the exponent expansion allows one\nto obtain remarkably good approximations of the pricing kernels of financial\nderivatives. This is illustrated with the application to simple path-dependent\ninterest rate derivatives. Finally we discuss how these results can also be\nused to increase the efficiency of numerical (both deterministic and\nstochastic) approaches to derivative pricing.",
"arxiv_id": "physics/0703180",
"authors": [
"Luca Capriotti"
],
"categories": [
"physics.soc-ph",
"cond-mat.other",
"math.ST",
"physics.comp-ph",
"q-fin.ST",
"stat.TH"
],
"title": "A Closed-Form Approximation of Likelihood Functions for Discretely Sampled Diffusions: the Exponent Expansion",
"url": "https://arxiv.org/abs/physics/0703180"
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