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caperover
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Re: Monte Carlo convergence for LV model

July 6th, 2017, 6:53 pm

-well, it was labeled as local vol surface. I believe it is correct.
-That is right. Backward PDE method has much smaller errors against the input Black vol surface than the Monte Carlo pricer. That makes me wondering why Monte Carlo pricer does not work well for short-dated options.   
 
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Alan
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Re: Monte Carlo convergence for LV model

July 6th, 2017, 7:10 pm

OK. Well accepting they are local vols, what you posted is just a list of discrete values. What is the actual local vol function [$]\sigma(S,t)[$] you use in your backward PDE solver for 0 <= t <= 6/2/2017?
 
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caperover
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Re: Monte Carlo convergence for LV model

July 6th, 2017, 8:17 pm

The local volatility function uses either piecewise linear or piecewise constant interpolation along strike and piecewise constant along the expiry to interpolate the local vol grid, which do not make much difference. That should be fine as long as they are consistent between calibration and validation.
I can expect the forward PDE for calibration and backward PDE for pricing to be consistent because they can be constructed in a similar fashion, which may not be the case for between finite difference and Monte Carlo.
 
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Alan
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Re: Monte Carlo convergence for LV model

July 7th, 2017, 1:19 am

Well, it's your Monte Carlo and your FDM. The most benign choice to me is [$]\sigma(S,t)[$] continuous and bounded --  implying (among the choices you presented) piecewise linear S-interpolation with constant extrapolation for 0 < S < Smin and  Smax < S < infinity. Here Smin and Smax are the min and max strikes of the data you posted. This will mean  [$]\sigma(S,t)[$]  is continuous and bounded at least to 6/2/2017. Then, make sure exactly the same continuous and bounded [$]\sigma(S,t)[$] is used for both the PDE and the Monte Carlo. (I suspect you aren't). With that, I believe both procedures (if decently implemented) will lead to the same option prices through 6/2/2017, which was your concern.

I would also begin to use piecewise linear interpolation in time starting with 6/2/2017 and later. Then,  [$]\sigma(S,t)[$]  will be continuous and bounded for your whole local vol surface.  With that, again I believe both procedures will lead to the same option prices for all strikes, through the furthest expiration of your data. 

Finally, if you want to consider t > tmax, where tmax is the furthest expiration of your data, I would just use constant extrapolation in time. Again this will assure  [$]\sigma(S,t)[$]  will be continuous and bounded -- now for [$](S,t) \in (0,\infty) \times (0,\infty)[$].
 
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Re: Monte Carlo convergence for LV model

July 7th, 2017, 11:06 am

Well, it's your Monte Carlo and your FDM. The most benign choice to me is [$]\sigma(S,t)[$] continuous and bounded --  implying (among the choices you presented) piecewise linear S-interpolation with constant extrapolation for 0 < S < Smin and  Smax < S < infinity. Here Smin and Smax are the min and max strikes of the data you posted. This will mean  [$]\sigma(S,t)[$]  is continuous and bounded at least to 6/2/2017. Then, make sure exactly the same continuous and bounded [$]\sigma(S,t)[$] is used for both the PDE and the Monte Carlo. (I suspect you aren't). With that, I believe both procedures (if decently implemented) will lead to the same option prices through 6/2/2017, which was your concern.

I would also begin to use piecewise linear interpolation in time starting with 6/2/2017 and later. Then,  [$]\sigma(S,t)[$]  will be continuous and bounded for your whole local vol surface.  With that, again I believe both procedures will lead to the same option prices for all strikes, through the furthest expiration of your data. 

Finally, if you want to consider t > tmax, where tmax is the furthest expiration of your data, I would just use constant extrapolation in time. Again this will assure  [$]\sigma(S,t)[$]  will be continuous and bounded -- now for [$](S,t) \in (0,\infty) \times (0,\infty)[$].
It looks as if we are talking about conditions for the SDE to have a L2 bounded unique solution?
In that case, mathematical niceties demand that the coefficients be measurable as well as satisfying Lipschitz and linear growth conditions?

Linear interpolation is not very smooth in general. Is that a problem here?
 
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Alan
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Re: Monte Carlo convergence for LV model

July 7th, 2017, 1:44 pm

Actually, I suspect the calibration procedure requires that [$]\sigma(S,t)[$] be only piecewise constant in time, so even less smooth than linear interpolation -- with jumps (vs. t) at every expiration! But the backward PDE and the MC should tolerate even that. 

Anyway, my suggestion is to first make sure the PDE and MC agree on the fully linearly interpolated  [$]\sigma(S,t)[$]. Then, move on to see if they agree on the less smooth case from the calibration (linearly interpolated  in S, but only piecewise constant in t)   

You can see from the exact solution to BS in the case of piecewise constant (in time) vol that the solution only depends on the integrated variance (integrated in time). So, actually bounded discontinuities for the local vol vs t should be tolerated by a good MC and a good (backwards) PDE, and they should both reproduce this exact solution.  Try it -- say with

[$]\sigma(t) = \sigma_0 1_{\{t \le T_0\}} +  \sigma_1 1_{\{t >T_0\}} [$] with an option expiration at say [$]T = 2 \, T_0[$]. 

In reality we do see evidence of such vol discontinuities, say when a company releases earnings, so the theory (and numerics) needs to accommodate it.
Last edited by Alan on July 7th, 2017, 2:12 pm, edited 1 time in total.
 
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outrun
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Re: Monte Carlo convergence for LV model

July 7th, 2017, 2:11 pm

And the same hold for the S dimension, right?. The only local vol condition I can think of is [$]\sigma(S,t)\ge 0[$]
 
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Alan
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Re: Monte Carlo convergence for LV model

July 7th, 2017, 2:17 pm

Probably you're right -- but not sure. But caperover suggested that linear interp would work for his procedure in that direction.  
 
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caperover
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Re: Monte Carlo convergence for LV model

July 7th, 2017, 3:44 pm

The local volatility surface is bounded, and thus linear interpolation should be fine. My real focus is on why it does not work well for a expiry time shorter than a quarter, even though it works quite well for longer expiry times. I naively think that the "time scale invariant" rule should work, but there can be some fundamental reasons, for example, the distribution is narrower, option price is much smaller, .... 
 
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outrun
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Re: Monte Carlo convergence for LV model

July 7th, 2017, 4:15 pm

Maybe it's that "time scale invariant" rule? What is it you exactly do/assume with that?