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Alan
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Introducing XGBM -- a new stochastic volatility model with some nice properties

September 25th, 2018, 2:22 pm

New research paper, just posted to arXiv:

Exact Solutions for a GBM-type Stochastic Volatility Model having a Stationary Distribution

Abstract:
We find various exact solutions for a new stochastic volatility (SV) model: 
the transition probability density, European-style option values, and (when it exists) the martingale defect. This may represent the first example of an SV model combining exact solutions, GBM-type volatility noise, and a stationary volatility density.
 
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ISayMoo
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Re: Introducing XGBM -- a new stochastic volatility model with some nice properties

September 25th, 2018, 3:26 pm

I don't know why this came to my mind: https://en.wikipedia.org/wiki/Japanese_battleship_Yamato

But, congratulations :)
 
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Alan
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Re: Introducing XGBM -- a new stochastic volatility model with some nice properties

September 25th, 2018, 4:44 pm

Thanks! -- and I kind of like that battleship analogy. 
 
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Costeanu
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Re: Introducing XGBM -- a new stochastic volatility model with some nice properties

September 25th, 2018, 6:02 pm

Congrats Alan. It looks like you put a lot of work into this.  I like that the paper appears to be self-contained, no need for heat kernels on hyperbolic manifolds, geodesic distance and all that. Great job. 

Now a quick question. You are saying it takes roughly ten times longer to price an option in the XGBM model compared to Heston. Do you have some more concrete numbers? 

Thanks,
V
 
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Alan
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Re: Introducing XGBM -- a new stochastic volatility model with some nice properties

September 25th, 2018, 6:35 pm

Thank you! 

Re run-times, later on in Sec. 7, I say that it takes me 0.5 - 2 minutes to compute my (Case 1) option price examples. This is the 'physical case', in the sense that calibrated parameters will fall under Case 1. 

Case 2, which is 'unphysical' (as there is no stationary vol distribution), takes much longer --- that one is for hardcore math finance types only!  :D

Of course, these times are for the exact formulas evaluated in Mathematica. Pure numerics, such as a PDE approach, will be very similar for Heston vs. XGBM, and can be *very* fast.  An expert/reference on very fast PDEs for SV models (including XGBM) is Yiannis Papadopoulos and his project site here

Oct 8, 2018 update: I just added a blog post here , which provides a little more motivation for the model based upon fits of the stationary vol density [$]\psi(\sigma)[$].
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