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Alan
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Looking for a simpler proof -- Moment explosions

July 5th, 2016, 4:33 pm

There is a relatively old, well-known, result that for a GBM process, [$]dX_t = m X_t \, dt + \sigma X_t \, dB_t[$],we have [$]E[e^{\alpha \int_0^T X_t \, dt}] = +\infty[$] for every [$]T > 0[$] with any [$]\alpha > 0[$].(The parameter [$]m[$] is an arbitrary real, and any [$]\sigma > 0[$] will do).For example, the result is more or less the same as the proven explosion of the expected value of the money market account inlognormal one-factor interest rate models: see Hogan and Weintraub (1998)The result also implies the instantaneous explosion of all moments greater than one in the uncorrelated lognomal SABR model: [$]E[S_T^p] = \infty[$] for every [$]T > 0[$] with any [$]p > 1[$]. ([$]dS_t = \sigma_t S_t dW_t, \quad d \sigma_t = \nu \, \sigma_t \, dB_t[$], assuming any [$]\nu > 0[$]).A look at the Hogan and Weintraub proof (see the link) shows it is rather involved and relies on an integral representation of a Bessel function.Does anyone know or see a more direct (but rigorous) proof?Thanks! ==============================================================p.s. Let me add that I am aware of a heuristic hand-wavy proof:Suppose [$]T > 0[$] but very small. Then[$]E[e^{\alpha \int_0^T X_t \, dt} \approx E[e^{\alpha \frac{T}{2}(X_0 + X_T)}] = +\infty[$] by an elementary application of the normal density for [$]\log X_T[$]; i.e. [$]E[e^{\alpha \, e^z}]=\infty[$], [$]z \sim[$] Normal.While this captures the essence of what is going on, it is not rigorous. Can the argument be cleaned up?
Last edited by Alan on July 4th, 2016, 10:00 pm, edited 1 time in total.
 
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frolloos
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Looking for a simpler proof -- Moment explosions

July 7th, 2016, 4:29 pm

I am feeling overconfident today, so dont take this seriously, but starting with binomial model calculate the expectation, and then take the continuous limit?
Last edited by frolloos on July 6th, 2016, 10:00 pm, edited 1 time in total.
 
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LocalVolatility
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Looking for a simpler proof -- Moment explosions

July 7th, 2016, 4:52 pm

See attached for some old notes that I once took while studying for a term structure modelling class. They deal with the bank account case you mentioned.(Please note that this is not my own work but reproduced from some other source that unfortunately I do not have anymore.)
 
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Alan
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Looking for a simpler proof -- Moment explosions

July 8th, 2016, 12:22 am

QuoteOriginally posted by: LocalVolatilitySee attached for some old notes that I once took while studying for a term structure modelling class. They deal with the bank account case you mentioned.(Please note that this is not my own work but reproduced from some other source that unfortunately I do not have anymore.)Perfect -- thank you!
 
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Alan
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Looking for a simpler proof -- Moment explosions

July 8th, 2016, 1:44 pm

QuoteOriginally posted by: frolloosI am feeling overconfident today, so dont take this seriously, but starting with binomial model calculate the expectation, and then take the continuous limit?Interesting idea -- can you carry this through?
 
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frolloos
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Looking for a simpler proof -- Moment explosions

July 9th, 2016, 2:23 pm

QuoteOriginally posted by: AlanQuoteOriginally posted by: frolloosI am feeling overconfident today, so dont take this seriously, but starting with binomial model calculate the expectation, and then take the continuous limit?Interesting idea -- can you carry this through?Just saw your reply, thanks. Yes, I could try this out once I have a bit more time. Still need to spend time on the stoch ir-equity vol mixing from the other thread plus some other things.
 
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Antonio
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Re: Looking for a simpler proof -- Moment explosions

October 6th, 2016, 7:20 am

Hello Alan,
If you just write out the integrals, you have something of the form
$$
\int_{\mathbb{R}}\exp\left\{\alpha\int_{0}^{T}e^{z\sqrt{t}}dt\right\}\exp\left\{-\frac{z^2}{2}\right\}dz.
$$
As z tends to infinity, the integrand is positive and does not tend zero, for any $\alpha>0$.
 
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Alan
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Re: Looking for a simpler proof -- Moment explosions

October 7th, 2016, 1:03 am

Thanks, although I have to say I don't see the 'writing out the integrals' step.
(There was a nice attachment that Local Volatility provided, which had a more involved but rigorous argument).
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