1. BS (and Merton - often forgotten) is correct if its assumptions are correct. BSM assumptions are not correct, but still a beautiful idea. Now used as a quoting mechanism, the BS formula that is, not the BS model!

2. Yes, even a hedgehog knows that, nothing new.

3. What do you mean?

4. Not true. Multiple connections.

In 3. I meant that

In Hull book 8ed p 607 in Implied Volatility section is written:

Financial institutions sometimes want to go one stage further and use a model that provides an exact fit to the prices of these options.12 In 1994 Derman and Kani, Dupire, and Rubinstein developed a model that is designed to do this. It has become known as the implied volatility function (IVF) model or the implied tree model.13 It provides an exact fit to the European option prices observed on any given day, regardless of the shape of the volatility surface.

The risk-neutral process for the asset price in the model has the form

dS = ( r – q ) S dt + σ ( t , S ) S dz

where r ( t ) is the instantaneous forward interest rate for a contract maturing at time t and q ( t ) is the dividend yield as a function of time. The volatility σ ( S , t ) is a function of both S and t and is chosen so that the model prices all European options consistently

with the market. It is shown both by Dupire and by Andersen and Brotherton-Ratcliffe that σ ( S , t ) can be calculated analytically:

[$]\sigma ^2 ( K , T ) = 2 .... [$]

where c ( K , T ) is the market price of a European call option with strike price K and maturity T.

In this reading one can see that LV [$]\sigma ( K , T ) [$] is mixed with Implied Volatility phenomenon.

Do you mean that the Hull's description of the Implied Volatility connection between [$]\sigma ( S , t )\, and \, \sigma ( K , T ) [$] is one of the yours " 4. Not true. Multiple connections" or you know other one?

Statistics: Posted by list1 — Yesterday, 5:38 pm

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3. What do you mean?

4. Not true. Multiple connections.

Statistics: Posted by frolloos — Yesterday, 4:28 pm

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Sorry, but where are you trying to go with this (monologue)?

1. BS pricing is a correct approach of a formal pricing.

2. Implied volatility for different options even with the same T , K are different and therefore it is illustrative example that market price of the options 'could be' or 'almost sure' different than it is represented by BS formulas.

3 Implied Volatility could not be used to justify transition from a GBM to SDE with nonlinear volatility term.

4. Implied Volatility does not have any connection to Local Volatility.

Statistics: Posted by list1 — Yesterday, 4:23 pm

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In above message I made a few corrections which specified my point on connection between IV and LV. The connection and difference between are expressed in (1), (2). When people write that bearing in mind that IV depends on K they consider underlying stock process in the form

[$] dS = \mu S dt + \sigma_{iv} ( t , S ) S dw [$] (3)

it looks incorrect. The right point is expressed in formulas (1) , (2) in above message. The implied volatility based on (1) is the same volatility as it is represented in original stock eq regardless whether it is going use for options or not. When we note that implied volatility depends on K it means that we consider options with fixed t , S ( t ) for different K and might be for different T. Then implied volatility is related to eq (2). Hence, it might be no problems with concept of LV except the moment when one attempts to rename variables ( T , K ) by ( t , S ) as it shown for example in (3).

I read a section in the book Jarrow & Turnbull, Derivatives Securities, about IV. They considered Call and Put IVs with the same K, T,. They figured out that different options on the same stock, IVs are different. They highlighted reasons why BS model is an appropriate. These are bid-ask spread, non-simultaneous observations, and options and stock prices are presented in discrete time format while BS model uses continuous time.

Obviously such arguments if they are a related to paradox can cover the case of call options with different Ks. Then the smile paradox was not a paradox.

Of course if we calculate IV based on BS we should get stock volatility, if we calculate IV based on Dupire eq we should arrive at Dupire's eq volatility of the Dupire's equation heuristic underlying. It is heuristic because underlying of the Dupire equation does not observable random process. The fact that IVs of the some calls and puts with equal K, T shows that market prices such options not in the BS's way. There are other factors that affect pricing. Jarrow& Turnbull remarks can be apply to any trading instruments not only for options themselves.

Statistics: Posted by list1 — Yesterday, 10:10 am

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Thanks a lot

Statistics: Posted by burnrate00 — Yesterday, 8:35 am

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NOW I AM GETTING TO MY ACTUAL QUESTION. A claim has come up in some of the interviews I made that mortgage-style amortization is illegal in Western Europe and/or Northern America and is only being used in Eastern European countries such as Croatia where regulation of the banking industry is less stringent and banks generally get away with more. From what I have read, however, it seems that mortgage-style amortization is very far from being illegal in the US at least and to the contrary is widely used there.

it would be interesting to find what the source of that was

it is quite common for auto loans in north america to have equal payments, i.e. amortizing loan.

the exact opposite to what you were told actually appears to be true: consumer loans that are not fully amortized are sometimes regarded as "predatory" because the borrower owes more at maturity than he borrowed initially and finds he can never repay it

the term they use is "negative amortization" which was prohibited in the US under the 1994 Home Ownership and Equity Protection Act (HOEPA). I believe it was carried over into the newer Dodd-Frank legislation

Statistics: Posted by ppauper — June 19th, 2017, 9:29 pm

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it looks incorrect. The right point is expressed in formulas (1) , (2) in above message. The implied volatility based on (1) is the same volatility as it is represented in original stock eq regardless whether it is going use for options or not. When we note that implied volatility depends on K it means that we consider options with fixed t , S ( t ) for different K and might be for different T. Then implied volatility is related to eq (2). Hence, it might be no problems with concept of LV except the moment when one attempts to rename variables ( T , K ) by ( t , S ) as it shown for example in (3).

Statistics: Posted by list1 — June 18th, 2017, 2:56 pm

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Practice show that [$]\sigma= \sigma ( K ) [$]. One way to explain phenomenon is that the market pricing of the options does not correspond to basic of B&S assumption that [$]C_{BS} ( t , x ; T , K ; \sigma ) \neq E f ( S ( T ; t , x )) [$] where [$] S ( t ) = x [$].

Other more constructive approach applies Loc Vol concept. In BS construction option [$]C ( t , S ; T , K ; \sigma )[$] is considered as a function ( t , S ) [$] t \le T [$] when T , K is fixed. The LV consider the same function [$]C ( t , S ( t ) ; T , K ; \sigma )[$] as a function of ( T , K ) , where [$] t \le T , K [$]. There ar two ways of the representation of the the same call option by using probabilistic representation of the stock SDE and the random process that corresponds Dupire equation

[$]C ( t , x ; T , K ; \sigma_{BS} ) = E \,max ( S_r ( T ; t , x ) - K , 0 ) [$] (1)

[$]C ( t , x ; T , K ; \sigma_{D} ) = E \, max ( S - k ( t ; T , K ) , 0 ) [$] (2)

Here k ( t ; T , x ) is the solution of the backward SDE , which represents underlying of the Dupire equation with boundary condition given at T = t ,

max ( S - K , 0 ).

Hence, dealing with implied volatility which shows its dependence on K we should pay attention to underlying that is used for sigma calculations. It can be the process S or to process k. In the second case, we deal with local volatility which is the function of T , K. we should proof again that C is also a monotonic function in Dupire's [$]\sigma[$] . Then if [$] C_D ( t , S ; T , K ; \sigma_D ( T , K ))[$] is monotonic in [$]\sigma[$] then we again can find implied volatility by applying Dupire eq solution [$] C_D[$]. Note that implied volatility implied by BSE and Dupire equations are diferent functions.

Thus we should be first focus on underlying that is used for practical calculation of the implied volatility. If implied volatility calculations use underlying S but not k then it is most likely to assume that market prices options not as presented by BSE.

Statistics: Posted by list1 — June 17th, 2017, 12:42 pm

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Statistics: Posted by list1 — June 17th, 2017, 12:32 pm

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https://search.yahoo.com/search;_ylc=X3 ... s&ei=UTF-8

I was a bit confused by Fig.1 The upper title is : Implied Volatility Vs Moneyness for a fixed maturity,

the low title is : Implied Volatility Vs Strike Price

Moneyness and strike Price are completely different notions. In particular moneyness by definition depends on K differently than K itself and next calculations shows that they deal with moneyness which is some kind of - ln K in contrast to linear K.

Statistics: Posted by list1 — June 16th, 2017, 9:07 pm

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The problems you describe seem to be due to a variable interest rate and changing principal. It indeed seems like a terrible idea to combine all three elements -- simply from a complexity viewpoint. Perhaps it's that combination that has been made illegal in some jurisdictions. But, consider the alternative, which would have probably been interest-only payments with all the principal due at the end. In that case, given the same interest rate and principal changes, the borrowers would now be even worse off -- now owing even more! (If you are bad at math, find a colleague to confirm this for you).

So, the source of their problems seems not so much the amortization, but the other two variables, not to mention poor consumer choices and likely poor disclosure.

My two cents.

Statistics: Posted by Alan — June 16th, 2017, 7:02 pm

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list1 wrote:Paul wrote:Get some data and look at implied vol vs your measure of drift.

Paul, the scheme presented above covers stock drift and volatility estimates. When you talk about implied vol it somewhat relates to option(s). I am a little confused about a task.

??? I'm just following what you said. You had

[$] \frac{ \partial C ( t , S ( t ) ; T , K ) }{ \partial \mu} = 0[$]

You were talking about option sensitivity to drift!

I do not have historical data for options. It should be for options and its underlying stock. ie open, max-min , close prices of the day for different K when t , T are fixed when T - t is say 30 or 60 days. Though one can try use only open prices for options with many K and its underlying for t , T fixed.

Statistics: Posted by list1 — June 15th, 2017, 10:16 am

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