Statistics: Posted by list1 — Yesterday, 6:13 am

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Merton uses no-arbitrage arguments in his proofs. That is fairly compelling logic and the paper is indeed deemed to be a very important one, arguably the best of the early published research efforts in option pricing.

DJN, I agree with importance and your estimate of the value of the paper. On the other hand the proof of the Th2 does not look perfect in the place where Th1 statement related to EO was converted into the same statement for AO. Other part of the proof of the Th2 is also not a sufficiently plausible for understanding.

Statistics: Posted by list1 — April 25th, 2017, 5:19 pm

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I would recommend you check Robert Merton, Theory of Rational Option Pricing, Bell Journal of Economics and Management Science, Vol 4, No 1, Spring 1973, pp 141-183. Theorem 2 deals with your question.

I agree with the reference to Th 2. On the other hand its proof confuse me.

It makes the reference on Th. 1 in which they consider European call but in Th.2 author applies statement of Th.1 to American call, which does not considered in Th.1.

Next, if Amer. call is exercised the exercise time might be at a some future moment prior or equal to maturity. The value of the underlying stock should be random. In proof, they used only the moment [$]\tau[$] that is assigned to the time to maturity corresponding to initial moment.

There is a feeling that this proof does not accurate at least. Though based on importance of the paper I could be wrong too.

Statistics: Posted by list1 — April 25th, 2017, 2:05 am

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I want to understand the logic for why this is:

$$d{S_t} = r{S_t}dt + \sigma {S_t}d{\tilde W_t}$$.

Stock

$$d{S_t} = \mu {S_t}dt + \sigma {S_t}d W_t$$ (1)

is defined on original or 'real' prob space [$] ( \Omega, F , P ) [$]. Then we arrive at parabolic BSE. One can use probabilistic representation of the BSE solution which underlying is

$$d{S_t} = r{S_t}dt + \sigma {S_t}d W_t$$ (2)

Though we can use here any Wiener process. There is an ambiguity we all believe that derivatives take its value from (1) but BS solution has underlying (2). It is a contradiction between experience and logic. It was invented an approach that hides the contradiction. We consider equation (1) on risk neutral world. They call it 'consider stock on risk neutral space [$]( \Omega , F , Q ) [$] . It is formally incorrect as far as stock is defined on [$]( \Omega , F , P ) [$] . More correctly to say that we consider eq. (1) on [$]( \Omega , F , Q ) [$] . Then the finite distributions of the solution eq (1) on risk neutral space are equal to correspondent distribution (2) on real probability space [$]( \Omega , F , P ) [$]

Statistics: Posted by list1 — April 21st, 2017, 10:19 pm

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Statistics: Posted by snufkin — April 21st, 2017, 10:05 pm

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It is possible to understand AAD by a simple 101 example or does one need to have certain background knowledge?

The core idea is as follows:

Dual numbers are an extension of the real numbers, similar to complex numbers, except that instead of an imaginary unit i with the property [$]i^2 = -1[$], we have an infinitesimal unit [$]\varepsilon[$] with the property [$]\varepsilon^2 = 0[$]. The coefficient of [$]\varepsilon[$] is the gradient with respect to [$]x[$]; this is initially 1 since [$]dx/dx\ =\ 1[$]

Since most of the transformations you use in numerical methods are linear, you... get the actual derivative propagated alongside the value — voila.

Moreover, if you redefine the operations to support differentiation, you can work with more complicated models, too: as long as you know what's the derivative of the result of an operation in terms of values and derivatives of the operands, you're fine. E.g. \[ (x + x'\varepsilon) \times (y + y'\varepsilon) = xy + (xy' + x'y)\varepsilon \]

Statistics: Posted by snufkin — April 21st, 2017, 9:23 pm

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$$d{S_t} = r{S_t}dt + \sigma {S_t}d{\tilde W_t}$$

Statistics: Posted by MAYbe — April 21st, 2017, 9:21 pm

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It is possible to understand AAD by a simple 101 example or does one need to have certain background knowledge?

Cuch, for me the eye-opener was this article: https://www.wilmott.com/automatic-for-the-greeks/ — it explains the basic idea and shows the application, which is quite impressive (given how simple the basic idea is!)

Statistics: Posted by snufkin — April 21st, 2017, 9:06 pm

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mtsm wrote:Yes, it is used extremely heavily in ML to perform optimization by gradient descent. There are a lot of ML packages that implement this de facto. Just look at any of the packages released by the big tech firms. It's all about it.

It's also used for various risk calculations in some global IBs.

It is used to compute gradients and Hessian, that kind of area?

It is possible to understand AAD by a simple 101 example or does one need to have certain background knowledge?

Yes, exactly, for gradient. In ML frameworks like tensorflow you specify a graph of computations -like excel does- with some end result that's typically a cost function (lsquares errors, likelihood, entropy) and then it automatically computer the gradient throughout the whole dependency tree and allows you to search for a minimal cost.

Eg

https://stats.stackexchange.com/questio ... tensorflow

Statistics: Posted by outrun — April 21st, 2017, 7:10 pm

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Yes, it is used extremely heavily in ML to perform optimization by gradient descent. There are a lot of ML packages that implement this de facto. Just look at any of the packages released by the big tech firms. It's all about it.

It's also used for various risk calculations in some global IBs.

It is used to compute gradients and Hessian, that kind of area?

It is possible to understand AAD by a simple 101 example or does one need to have certain background knowledge?

Statistics: Posted by Cuchulainn — April 21st, 2017, 5:46 pm

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