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MAYbe
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Why changing measure is necessary?

April 21st, 2017, 9:21 pm

I want to understand the logic for why this is:
We have our model for the stock price behaviour:
$$d{S_t} = \mu {S_t}dt + \sigma {S_t}d{\tilde W_t}$$
It describes the development of a stock price over time using the risk-adjusted expected return $\mu$ and the real uncertainty in the stochastic term.
We want to change the probability measure in such a way that the stochastic process remains a Brownian motion but with a drift of r instead of $\mu$. .... To repeat the manner of speaking, we want the process to change gear from an instantaneous increase of $\mu$ to r and leave the rest as before.

$$d{S_t} = r{S_t}dt + \sigma {S_t}d{\tilde W_t}$$
 
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snufkin
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Re: Why changing measure is necessary?

April 21st, 2017, 9:49 pm

Could you please provide any context? Usually the change of measure allows for simpler math; without context, it's really hard to say whether it's the case here.
 
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snufkin
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Re: Why changing measure is necessary?

April 21st, 2017, 10:05 pm

Looks like a quote from "Finance: A Quantitative Introduction" by Nico van der Wijst. Surprisingly, the paragraph is titled "Why the change of measure is necessary" — so what exactly puzzles you there?
 
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list1
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Re: Why changing measure is necessary?

April 21st, 2017, 10:19 pm

I want to understand the logic for why this is:
We have our model for the stock price behaviour:
$$d{S_t} = \mu {S_t}dt + \sigma {S_t}d{\tilde W_t}$$
It describes the development of a stock price over time using the risk-adjusted expected return $\mu$ and the real uncertainty in the stochastic term.
We want to change the probability measure in such a way that the stochastic process remains a Brownian motion but with a drift of r instead of [$]\mu[$]. .... To repeat the manner of speaking, we want the process to change gear from an instantaneous increase of [$]\mu[$] to r and leave the rest as before.

$$d{S_t} = r{S_t}dt + \sigma {S_t}d{\tilde W_t}$$. 
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 [$]( \Omega , F , P ) [$] 
 
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Orbit
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Re: Why changing measure is necessary?

May 16th, 2017, 12:00 am

I want to understand the logic for why this is:
We have our model for the stock price behaviour:
$$d{S_t} = \mu {S_t}dt + \sigma {S_t}d{\tilde W_t}$$
It describes the development of a stock price over time using the risk-adjusted expected return $\mu$ and the real uncertainty in the stochastic term.
We want to change the probability measure in such a way that the stochastic process remains a Brownian motion but with a drift of r instead of $\mu$. .... To repeat the manner of speaking, we want the process to change gear from an instantaneous increase of $\mu$ to r and leave the rest as before.

$$d{S_t} = r{S_t}dt + \sigma {S_t}d{\tilde W_t}$$
The average return "mu" goes with a probability measure that isn't suitable for pricing (i.e. the so-called historical or "objective" measure). So switch to a measure that goes with "r." This measure is known as the Risk-Neutral measure. Now switch the numeraire to the bank-account, and you've got a martingale. When your process is a martingale, the math is quite straightforward on how to build an option model. This is the motivation.
 
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list1
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Re: Why changing measure is necessary?

May 16th, 2017, 4:04 pm

we do not switch measure. original correct derivation did not use it and mention about probability measure. B&S showed that no arbitrage pricing should use sde which corresponds to ( r , [$]\sigma^2[$] ) heuristic random process which could not be associated with any sock. Switching measure is mathematical trick which attempts to hide embarrassment of such curious phenomena.  
 
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Orbit
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Re: Why changing measure is necessary?

May 17th, 2017, 12:26 am

we do not switch measure. original correct derivation did not use it and mention about probability measure. B&S showed that no arbitrage pricing should use sde which corresponds to ( r , [$]\sigma^2[$] ) heuristic random process which could not be associated with any sock. Switching measure is mathematical trick which attempts to hide embarrassment of such curious phenomena.  
"We do not switch probability measure." Yes, we do. We do it all the time.
"original correct derivation did not use it and mention about probability measure." That's true if you mean the original Black-Scholes approach. They solved a PDE. But that's not what the OP asked about.
"B&S showed that no arbitrage pricing should use sde which corresponds to ( r , [$]\sigma^2[$] ) heuristic random process which could not be associated with any sock." Yes, B&S used a geometric Brownian motion; I'm not sure why you state it cannot be associated with any stock. Or did you mean "sock?" Anyways their formula is used on stocks all the time.
"Switching measure is mathematical trick which attempts to hide embarrassment of such curious phenomena." Switching measure is not a "trick." It's a powerful, subtle and profound aspect of mathematical finance. As for "hiding embarrassment of such curious phenomena," I'm not sure what to say, other than you seem to be interjecting totally irrelevant nonsense.
The OP asked why his text was switching measure. I explained it. Everything I wrote was true and correct. If you'd like further elaboration, I suggest you ask politely. 
 
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list1
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Re: Why changing measure is necessary?

May 17th, 2017, 2:52 am

In original post there is no words like option pricing though question is related to it. In option pricing we assume that there exists [$]( \mu , \sigma )[$] stock, risk free rate r and definition of the European call option. These are form market. Pricing does not involve any other assets.  The BSE solution is assigned to be the price of the Call. Using probabilistic representation one discovers that BSE solution has underlying [$]( r , \sigma )[$]. This random process does not represent an asset on our market. This is complete option pricing approach. We do not need anything else to add. 
Why does the change measure is a trick? There is actually no needs in risk neutral space because BS option pricing is already perfectly defined. As far as we already defined option price we can observe that there is a big difference between benchmark statement that option is a derivative and it takes its value from the underlying stock and the fact that BS price is defined by heuristic [$]( r , \sigma )[$] process. Trick is that we place real stock on risk neutral prob space which image on the real space is risk neutral process. In such trick it looks like that underlying of the option is real stock.
 
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Orbit
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Re: Why changing measure is necessary?

May 17th, 2017, 8:31 pm

list1, the OP question was "I want to understand the logic for why this is." Then he quoted a passage.
I answered him thoroughly and correctly. It's to do with measure change.
Your follow-on posts are very close to nonsensical.
It's true the BSE solution originally did not explicitly use a measure change. But the OP did not ask about that.
Do you understand?
 
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list1
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Re: Why changing measure is necessary?

May 17th, 2017, 9:22 pm

list1, the OP question was "I want to understand the logic for why this is." Then he quoted a passage.
I answered him thoroughly and correctly. It's to do with measure change.
Your follow-on posts are very close to nonsensical.
It's true the BSE solution originally did not explicitly use a measure change. But the OP did not ask about that.
Do you understand?
As you noted that the original solution did not used measure change then measure change was not used to get BSE solution. This is primary logic. Secondary logic what does measure change used for? In the upper reply it was explained that measure changed is used in order to present connection of the sock equation on risk neutral world and risk neutral process on the original (real) probability space. Remarkably that stock price eq is defined on the real prob space hence to consider this eq with respect to measure Q is a kind of math trick. There is nothing sensational or nonsensantial in such explanation. 
 
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Orbit
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Re: Why changing measure is necessary?

May 18th, 2017, 1:49 am

As you noted that the original solution did not used measure change then measure change was not used to get BSE solution. This is primary logic.
Look, I hate to be the one to break it to you, but the original BSE has a measure change in it, and it happens whether you like it or not. It's the reason why there is a probability term N[d1] and N[d2]. Note they are not the same. They are different measures.
Secondary logic what does measure change used for?
It's used explicitly in martingale theory.
...it was explained that measure changed is used in order to present connection of the sock equation on risk neutral world and risk neutral process on the original (real) probability space.
Correct. We define a connection between these two things.
Remarkably that stock price eq is defined on the real prob space hence to consider this eq with respect to measure Q is a kind of math trick. There is nothing sensational or nonsensantial in such explanation.
It's no more a "trick" than for any other mathematical transformation.
By the way I think you're kind of a nut case. Nothing personal.
 
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list1
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Re: Why changing measure is necessary?

May 18th, 2017, 2:07 am

As you noted that the original solution did not used measure change then measure change was not used to get BSE solution. This is primary logic.
Look, I hate to be the one to break it to you, but the original BSE has a measure change in it, and it happens whether you like it or not. It's the reason why there is a probability term N[d1] and N[d2]. Note they are not the same. They are different measures.

////  BSE does not have any measure and measure change. It is a parabolic equation with initial condition given at T. N [d i ] is a notation.

Secondary logic what does measure change used for?
It's used explicitly in martingale theory.

////To define solution of the BSE we do not need to know about martingales or measures. If we know something more than we need we of course can discover additional connections between martingales or measures with solution of the BSE. It is similar to write English without knowing Latin. But if we know Latin we can find something new in English sense and grammar constructions.
...it was explained that measure changed is used in order to present connection of the sock equation on risk neutral world and risk neutral process on the original (real) probability space.
Correct. We define a connection between these two things.
Remarkably that stock price eq is defined on the real prob space hence to consider this eq with respect to measure Q is a kind of math trick. There is nothing sensational or nonsensantial in such explanation.
It's no more a "trick" than for any other mathematical transformation.
By the way I think you're kind of a nut case. Nothing personal.
////It is trick because we can safely forget about measure change at all and BS pricing will not lost its importance and sense.  
 
frolloos
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Re: Why changing measure is necessary?

May 18th, 2017, 7:07 am

By the way I think you're kind of a nut case. Nothing personal.
Is this your baptism of fire? :-D Most of us have been there..
 
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list1
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Re: Why changing measure is necessary?

May 18th, 2017, 9:07 am

By the way I think you're kind of a nut case. Nothing personal.
Is this your baptism of fire? :-D Most of us have been there..
It is indeed nothing personal, we can use measure change techniques for BSE derivation but we also can use original B&S derivation without any lost of financial ideas. Then one can ask about what does measure change used for? for math or finance?
 
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Orbit
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Re: Why changing measure is necessary?

May 18th, 2017, 3:55 pm

By the way I think you're kind of a nut case. Nothing personal.
Is this your baptism of fire? :-D Most of us have been there..
LOL. I'm done trying to get him to have a "light bulb moment." I now regard his stuff as nothing other than humor. Sometimes it's pretty good, actually