In general, yes, but for the model at hand, r=0.

For clarity, I didn't mean use the chain rule to verify Breeden-Litzenberger, but to use it with the Black-Scholes formula on the r.h.s. of the first line I wrote.

A small side step how sensitive is Breeden-Litzenberger to approximations? Even for Black-Scholes where we get out Log-Normal from Breeden-Litzenberger (second partial derivative with respect to strike), how sensitive when we are very very far out in tails, because many (all?) Cumulative normal distributions functions are approximations, typically far outside practical relevant levels, but I just wonder. I plotted it far out without issues, but at some point it should run into problems, likely only of theoretical interest...

Statistics: Posted by Collector — Yesterday, 7:21 pm

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Statistics: Posted by Cuchulainn — April 22nd, 2018, 6:36 pm

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mtsm wrote:

I recently sent a number of questions (mostly numerics and feasibility) to the authors after having read this and related articles a few times. Hopefully get some feedback soon.

If you ever write a self-contained comment or a critique of applications of ML to PDE solving, I'd really like to see it.

Statistics: Posted by ISayMoo — April 21st, 2018, 10:35 pm

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'Concrete' can be at many 'levels' e.,g. objects and types.

Just think about metrics for NNs, input representation and the mappings (hopefurrry bijective) between metric spaces etc.

Statistics: Posted by Cuchulainn — April 21st, 2018, 5:07 pm

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I find trying to grasp all these articles a bit painful. Lots of theory/theorems etc. but where is the example? e.g. that you can check against.

I wonder what percentage of academic math papers have examples?

Statistics: Posted by Traden4Alpha — April 21st, 2018, 4:37 pm

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Statistics: Posted by Cuchulainn — April 21st, 2018, 12:57 pm

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I recently sent a number of questions (mostly numerics and feasibility) to the authors after having read this and related articles a few times. Hopefully get some feedback soon.

Statistics: Posted by Cuchulainn — April 17th, 2018, 4:41 pm

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From the perspective of financial theory, one might worry that the sum of squared differences of [$] y_i - g ( u_i ) [$] in (2) may not be the right measure of loss, since an investor is only interested in relative prices. This concern can be addressed by using the underlying asset price as numéraire. By setting [$]w_i = S^{−2}_{t}[$] and switching to a spot moneyness space [$]\tilde u = u/S_t[$], one can conduct the minimization on relative option prices after some obvious adjustments to the no-arbitrage constraints in (19). The resulting curve estimate [$](\mathbf{\tilde g ^\top }, \tilde \gamma ^\top )[$] can be inflated again via [$]g_i(u) = S_t \tilde g_i(\tilde u)[$] and [$]\gamma_i(u) = \tilde \gamma _i(\tilde u)/S_t[$], which yields a natural cubic spline as can be verified from (12). Seemingly this approach comes at the additional cost of a homogeneity assumption. However, as can be observed from (15), in choosing as smoothing parameter [$] \tilde \lambda = \lambda S_{t}^{−3}[$] the program in relative prices is equivalent to the former one in absolute prices (up to the aforementioned scales).

Before putting into the optimization software, I convert all my strikes (knots [$]g_i[$]) to moneyness space (ensuring my [$]h[$] are now correctly spaced). Then my weights [$]w_i = S^{−2}_{t}[$] only impact the [$]\mathbf{y}[$] vector and the matrix [$]\mathbf{B}[$].

The optimization solves without the no-arb constraints - and is a good fit but still has arb opportunities in the IVS (as it should, the constraints have not been added). I believe I'm having a problem with the "obvious" adjustments to the no-arb constraints. Can we be specific about what they are? Also, can we adjust them for puts, as well?

The optimization without the no-arb constraints should require no modifications for put prices (because it is a generic method for fitting a spline to observed data - it's from Green & Silverman - Nonparametric Regression and Generalized Linear Models).

$$ \begin{align} \frac{g_2 - g_1}{h_1} - \frac{h1}{6}\gamma_2 & \geq & -e^{-\int_{t_m}^{T} r_s ds} & \\

- \frac{g_n - g_{n-1}}{h_{n-1}} - \frac{h_{n-1}}{6}\gamma_{n-1} & \geq & 0 & \\

g_1 & \leq & e^{- \int_{t_m}^{T} \delta_s ds} S_t \\

g_1 & \geq & e^{- \int_{t_m}^{T} \delta_s ds} S_t - e^{- \int_{t_m}^{T} r_s ds} u_1

\end{align}$$

The original optimization problem also includes [$]g_n \geq 0[$] and [$]\lambda_i \geq 0[$] which I've excluded because they won't change for puts or for moneyness space. What are the changes for puts and the obvious changes for moneyness space?

Statistics: Posted by jayjo — April 17th, 2018, 12:41 am

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ISayMoo wrote:Or that it's a FSB controlled account and the shift handover procedures failed.

Spoke too hastily, my friend, yet again.

Has anyone ever compared you to a certain crustacean from a certain cartoon?

Statistics: Posted by ISayMoo — April 15th, 2018, 4:11 pm

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I am happy when the qualitative behaviour (existence results etc.) of ODE/SDE have been finalised and then move to numerics. So, both are needed. You can't have one without the other.

Statistics: Posted by Cuchulainn — April 15th, 2018, 7:12 am

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I am happy when the qualitative behaviour (existence results etc.) of ODE/SDE have been finalised and then move to numerics. So, both are needed.

Statistics: Posted by Cuchulainn — April 15th, 2018, 7:11 am

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lovenatalya wrote:While communism and its brethren socialism are abhorrences, I usually cut artists some slacks, so long as they do not try to impose those utopian visions on others.

I agree. Quite magnanimous of you. Utopian ideals should not get into the wrong hand. We have a government for that.

BTW if you know your history, Lennon beat the System.

Yes, absolutely, Cuchulainn. By the way, when I said "that would be marvelous" I was referring to what ISayMoo's line "Wilmotters talking maths in peace" rather than the "imagine no possessions" part of the lyrics. I agree with sensible things anybody says.

By "beat the System" are you referring to Lennon's battle with the US immigration authority and the court battle against the FBI? Maybe his political activism as well?

Statistics: Posted by lovenatalya — April 15th, 2018, 6:41 am

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Or that it's a FSB controlled account and the shift handover procedures failed.

Spoke too hastily, my friend, yet again.

Statistics: Posted by lovenatalya — April 15th, 2018, 6:17 am

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