dorsal/arxiv
View SchemaLinear vs. Nonlinear Diffusion and Martingale Option Pricing
| Authors | J. L. McCauley, G. H. Gunaratne, K. E. Bassler |
|---|---|
| Categories | |
| ArXiv ID | physics/0606035 |
| URL | https://arxiv.org/abs/physics/0606035 |
Abstract
First, classes of Markov processes that scale exactly with a Hurst exponent H are derived in closed form. A special case of one class is the Tsallis density, advertised elsewhere as nonlinear diffusion or diffusion with nonlinear feedback. But the Tsallis model is only one of a very large class of linear diffusion with a student-t like density. Second, we show by stochastic calculus that our generalization of the Black-Scholes partial differential equation (pde) for variable diffusion coefficients is equivalent to a Martingale in the risk neutral discounted stock price. Previously, this was proven for the restricted case of Gaussian logarithmic returns by Harrison and Kreps, but we prove it here for large classes of empirically useful and theoretically interesting returns models where the diffusion coefficient D(x,t) depends on both logarithmic returns x and time t. Finally, we prove that option prices blow up if fat tails in returns x are included in the market distribution.
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"abstract": "First, classes of Markov processes that scale exactly with a Hurst exponent H\nare derived in closed form. A special case of one class is the Tsallis density,\nadvertised elsewhere as nonlinear diffusion or diffusion with nonlinear\nfeedback. But the Tsallis model is only one of a very large class of linear\ndiffusion with a student-t like density. Second, we show by stochastic calculus\nthat our generalization of the Black-Scholes partial differential equation\n(pde) for variable diffusion coefficients is equivalent to a Martingale in the\nrisk neutral discounted stock price. Previously, this was proven for the\nrestricted case of Gaussian logarithmic returns by Harrison and Kreps, but we\nprove it here for large classes of empirically useful and theoretically\ninteresting returns models where the diffusion coefficient D(x,t) depends on\nboth logarithmic returns x and time t. Finally, we prove that option prices\nblow up if fat tails in returns x are included in the market distribution.",
"arxiv_id": "physics/0606035",
"authors": [
"J. L. McCauley",
"G. H. Gunaratne",
"K. E. Bassler"
],
"categories": [
"physics.soc-ph",
"q-fin.PR"
],
"title": "Linear vs. Nonlinear Diffusion and Martingale Option Pricing",
"url": "https://arxiv.org/abs/physics/0606035"
},
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