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list1
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early exercise American option

August 5th, 2016, 6:13 pm

I check the problem solution at Hull's book and the arguments for non early exercised seem to me non formal. Are there more formal arguments to read for 'non early exercise AOs?
 
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bearish
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Re: early exercise American option

August 5th, 2016, 9:43 pm

How about Merton's "Theory of Rational Option Pricing": http://www.maths.tcd.ie/~dmcgowan/Merton.pdf? Hull's book is not the place to look for mathematical rigor, but it has other redeeming qualities.
 
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Re: early exercise American option

August 5th, 2016, 10:46 pm

How about Merton's "Theory of Rational Option Pricing": http://www.maths.tcd.ie/~dmcgowan/Merton.pdf? Hull's book is not the place to look for mathematical rigor, but it has other redeeming qualities.
Thank you bearish for reference. I will look at it.
Is it a more modern construction available? What I mean is something like following. What is the formula or relationship used for 'optimal' exercising. For example in simplest situation let
[$]dS ( t ) = \mu S ( t ) dt\,+\,\sigma S ( t ) dw ( t )[$] 
with no coupons or dividends. What is the optimal rule to sell stock on finite interval [0 , T ].
Can this rule be applied for the construction a relationship which specifies time for early trading European and early exercise American Option?
 
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Re: early exercise American option

August 6th, 2016, 1:58 pm

A classic rigorous treatment for optimal stopping of a 1D Brownian motion is van Moerbeke (1974).
 (adding a drift or, indeed, optimally stopping a general 1D diffusion is similar).
 
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Re: early exercise American option

August 6th, 2016, 3:22 pm

A classic rigorous treatment for optimal stopping of a 1D Brownian motion is van Moerbeke (1974).
 (adding a drift or, indeed, optimally stopping a general 1D diffusion is similar).
Thank you Alan. I am confused by the general statement about 'do not exercise' American option before maturity and 'the optimal boundary' for optimal stopping time problem. Whether or not early exercise AO and optimal boundary are related or not?
 
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Re: early exercise American option

August 6th, 2016, 4:40 pm

You're welcome. Yes, they are intimately related. The optimal exercise boundary divides the state space, in this case [$]\{(S,t): \, 0 < S < \infty, 0 \le t \le T\}[$], into two regions: the Continuation Region [$]\mathcal{C}[$] and the Stopping Region [$]\mathcal{S}[$]. If, at time t, you find [$](S(t),t) \in \mathcal{C}[$] it is optimal to not exercise; i.e., continue to hold the option.  If, on the other hand, you find [$](S(t),t) \in \mathcal{S}[$] it is optimal to immediately exercise.

In general, the boundary may be degenerate/trivial, lying (at a particular time t) at [$]S=0[$] or [$]S=\infty[$] and these various regions may not be simply connected. 

For the American call option with no dividends and r>0, [$]\mathcal{C}[$] is all of the S-space prior to expiration [$]T[$] and [$]\{S < K\}[$] at [$]t = T[$]. 
For the American put option with no dividends and r>0, [$]\mathcal{C}[$] lies above a non-trivial smooth boundary curve [$]S^*(t)[$] for all [$]t < T[$] which continuously approaches [$]S^*(T)=K[$]. (In both cases, [$]K[$] is the option strike price).
 
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Re: early exercise American option

August 6th, 2016, 7:34 pm

You're welcome. Yes, they are intimately related. The optimal exercise boundary divides the state space, in this case [$]\{(S,t): \, 0 < S < \infty, 0 \le t \le T\}[$], into two regions: the Continuation Region [$]\mathcal{C}[$] and the Stopping Region [$]\mathcal{S}[$]. If, at time t, you find [$](S(t),t) \in \mathcal{C}[$] it is optimal to not exercise; i.e., continue to hold the option.  If, on the other hand, you find [$](S(t),t) \in \mathcal{S}[$] it is optimal to immediately exercise.

In general, the boundary may be degenerate/trivial, lying (at a particular time t) at [$]S=0[$] or [$]S=\infty[$] and these various regions may not be simply connected. 

For the American call option with no dividends and r>0, [$]\mathcal{C}[$] is all of the S-space prior to expiration [$]T[$] and [$]\{S < K\}[$] at [$]t = T[$]. 
For the American put option with no dividends and r>0, [$]\mathcal{C}[$] lies above a non-trivial smooth boundary curve [$]S^*(t)[$] for all [$]t < T[$] which continuously approaches [$]S^*(T)=K[$]. (In both cases, [$]K[$] is the option strike price).
Thanks Alan. Now I understood a connection of the general stopping time an AO early exercised problem. Nevertheless some questions are still remained. In explanations of AO in financial handbooks why it is not reasonable exercise AO early than maturity they explain that say 5% return over q month period  can be not optimal compare with 3% over longer period (q+h) months, h > 0 as far as taking into account that the sum received if AO exercised after 1 month at risk free rate is less than 3% receive exercising AO after  q+h monts. I can not find such possibility in optimal stopping problem. Though such possibility might be included in goal function by multiplying it by the factor[$] B^{ - 1 } ( \tau , T ) [$]. 
Though this is a formally correct approach but it does not specify risk management of the problem, ie ignore some specific of pathwise exercise. Assume that we know max value of the goal function and one sees that say 3 days we reach it while 6 months - 3 days remain to maturity. Whether or not we should exercise AO? To answer we do not have a value of the chance that goal function will reach say 3% more or 1% less. In other words our approach is 'in average' rule for all set of scenarios. It seems in my first paper where only EOs are considered I looked at early sell opportunity of the EO as following. For a given EO price which can be defined by BS model for example we consider the moment at which max of scenario is reached. As far as this random variables defined for each scenario does not markov moment it is difficult do apply the sde theory. Though for one dimensional we can find distribution function of the max GBM. It is perfect exercise as nothing better can not be apply. Nevertheless for any level  Q  we can define markov moment [$]\tau_Q[$] and find statistical characteristics of such moment, its mean and stdv. I did not do that but it seems that we can present probability to reach such level too. Of course if we ignore pathwise risk implied by the standard approach the standard approach looks formally sufficiently better. On the other hand if we pay attention to risks which are attribute of any pricing model the second approach might be look more attractive. 
 
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bearish
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Re: early exercise American option

August 6th, 2016, 9:33 pm

You're welcome. Yes, they are intimately related. The optimal exercise boundary divides the state space, in this case [$]\{(S,t): \, 0 < S < \infty, 0 \le t \le T\}[$], into two regions: the Continuation Region [$]\mathcal{C}[$] and the Stopping Region [$]\mathcal{S}[$]. If, at time t, you find [$](S(t),t) \in \mathcal{C}[$] it is optimal to not exercise; i.e., continue to hold the option.  If, on the other hand, you find [$](S(t),t) \in \mathcal{S}[$] it is optimal to immediately exercise.

In general, the boundary may be degenerate/trivial, lying (at a particular time t) at [$]S=0[$] or [$]S=\infty[$] and these various regions may not be simply connected. 

For the American call option with no dividends and r>0, [$]\mathcal{C}[$] is all of the S-space prior to expiration [$]T[$] and [$]\{S < K\}[$] at [$]t = T[$]. 
For the American put option with no dividends and r>0, [$]\mathcal{C}[$] lies above a non-trivial smooth boundary curve [$]S^*(t)[$] for all [$]t < T[$] which continuously approaches [$]S^*(T)=K[$]. (In both cases, [$]K[$] is the option strike price).
Thanks Alan. Now I understood a connection of the general stopping time an AO early exercised problem. Nevertheless some questions are still remained. In explanations of AO in financial handbooks why it is not reasonable exercise AO early than maturity they explain that say 5% return over q month period  can be not optimal compare with 3% over longer period (q+h) months, h > 0 as far as taking into account that the sum received if AO exercised after 1 month at risk free rate is less than 3% receive exercising AO after  q+h monts. I can not find such possibility in optimal stopping problem. Though such possibility might be included in goal function by multiplying it by the factor[$] B^{ - 1 } ( \tau , T ) [$]. 
Though this is a formally correct approach but it does not specify risk management of the problem, ie ignore some specific of pathwise exercise. Assume that we know max value of the goal function and one sees that say 3 days we reach it while 6 months - 3 days remain to maturity. Whether or not we should exercise AO? To answer we do not have a value of the chance that goal function will reach say 3% more or 1% less. In other words our approach is 'in average' rule for all set of scenarios. It seems in my first paper where only EOs are considered I looked at early sell opportunity of the EO as following. For a given EO price which can be defined by BS model for example we consider the moment at which max of scenario is reached. As far as this random variables defined for each scenario does not markov moment it is difficult do apply the sde theory. Though for one dimensional we can find distribution function of the max GBM. It is perfect exercise as nothing better can not be apply. Nevertheless for any level  Q  we can define markov moment [$]\tau_Q[$] and find statistical characteristics of such moment, its mean and stdv. I did not do that but it seems that we can present probability to reach such level too. Of course if we ignore pathwise risk implied by the standard approach the standard approach looks formally sufficiently better. On the other hand if we pay attention to risks which are attribute of any pricing model the second approach might be look more attractive. 
Just read the f*cking Merton paper already!
 
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Re: early exercise American option

August 6th, 2016, 11:51 pm

You're welcome. Yes, they are intimately related. The optimal exercise boundary divides the state space, in this case [$]\{(S,t): \, 0 < S < \infty, 0 \le t \le T\}[$], into two regions: the Continuation Region [$]\mathcal{C}[$] and the Stopping Region [$]\mathcal{S}[$]. If, at time t, you find [$](S(t),t) \in \mathcal{C}[$] it is optimal to not exercise; i.e., continue to hold the option.  If, on the other hand, you find [$](S(t),t) \in \mathcal{S}[$] it is optimal to immediately exercise.

In general, the boundary may be degenerate/trivial, lying (at a particular time t) at [$]S=0[$] or [$]S=\infty[$] and these various regions may not be simply connected. 

For the American call option with no dividends and r>0, [$]\mathcal{C}[$] is all of the S-space prior to expiration [$]T[$] and [$]\{S < K\}[$] at [$]t = T[$]. 
For the American put option with no dividends and r>0, [$]\mathcal{C}[$] lies above a non-trivial smooth boundary curve [$]S^*(t)[$] for all [$]t < T[$] which continuously approaches [$]S^*(T)=K[$]. (In both cases, [$]K[$] is the option strike price).
Thanks Alan. Now I understood a connection of the general stopping time an AO early exercised problem. Nevertheless some questions are still remained. In explanations of AO in financial handbooks why it is not reasonable exercise AO early than maturity they explain that say 5% return over q month period  can be not optimal compare with 3% over longer period (q+h) months, h > 0 as far as taking into account that the sum received if AO exercised after 1 month at risk free rate is less than 3% receive exercising AO after  q+h monts. I can not find such possibility in optimal stopping problem. Though such possibility might be included in goal function by multiplying it by the factor[$] B^{ - 1 } ( \tau , T ) [$]. 
Though this is a formally correct approach but it does not specify risk management of the problem, ie ignore some specific of pathwise exercise. Assume that we know max value of the goal function and one sees that say 3 days we reach it while 6 months - 3 days remain to maturity. Whether or not we should exercise AO? To answer we do not have a value of the chance that goal function will reach say 3% more or 1% less. In other words our approach is 'in average' rule for all set of scenarios. It seems in my first paper where only EOs are considered I looked at early sell opportunity of the EO as following. For a given EO price which can be defined by BS model for example we consider the moment at which max of scenario is reached. As far as this random variables defined for each scenario does not markov moment it is difficult do apply the sde theory. Though for one dimensional we can find distribution function of the max GBM. It is perfect exercise as nothing better can not be apply. Nevertheless for any level  Q  we can define markov moment [$]\tau_Q[$] and find statistical characteristics of such moment, its mean and stdv. I did not do that but it seems that we can present probability to reach such level too. Of course if we ignore pathwise risk implied by the standard approach the standard approach looks formally sufficiently better. On the other hand if we pay attention to risks which are attribute of any pricing model the second approach might be look more attractive. 
Just read the f*cking Merton paper already!
bearish, I need some time to comprehend the ideas behind Theorems 1,2 of the paper. It looks like there in Th 1,2 AO price is undefined and the underlying distribution does not define and the general ideas suggest not exercise AO earlier then maturity. It can be true but it might be of course that I misunderstand something there.
Actually we can not use AO price f ( S , [$]\tau[$] ; E ) until it is not formally (in math sense) defined. It is nonsense to say we buy AO for f ( S , [$]\tau[$] ; E ) if it is not formally stated what f ( S , [$]\tau[$] ; E ) is it. And [$]\tau[$] looks deterministic though for one scenario it can be one date and for other scenario other date. It might be not a big deal nevertheless it might be good to wait a little bit. 
 
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Re: early exercise American option

August 8th, 2016, 2:14 am

I tried to read Th. and 2 of the Merton paper where early exercise AO is considered. After proof of the Th. 1 I decided not to read proof of Th2 as the formal logic behind proof is somewhat difficult to comprehend. Using modern notations let me present comments to Th.1. It stated> that
[$]C_a ( 0 , S ; t , K ) \ge max\,[ 0 , S - B ( 0 , t ) K ]\, , \,t \, \in\, [ \,0\, ,\, T\, ] [$]                (1)
1. To say that [$]C_e , C_a[$] denote EO or AO prices it still does not mean that we know definitions. These are is only notations. Hence there is no sense in writing formula (1). We can ignore the fact and assume that there exist definitions of [$]C_e , C_a[$] .
About the proof style which was quite popular in old papers. We outline two strategies from which (1) is followed. Is it sufficient to state that there is no other strategy for which (1) does not correct?
2. It was assumed that in t years S ( t ) takes two values [$] S_{up} > K > S_{down}[$] 
Next reasoning do not use real or neutral world distributions. They use dominant notion which is actually equivalent to no arbitrage argument if we talk about rate of return. 
The two investment strategies are:
A) t = 0 : buy call and K shares of bonds 
B) t = 0 : buy stock for S ( 0 )
The statement " unless the current value of A is at least as large as B , A will dominate B" does not fully clear to me. It is clear that one is trying to present no arbitrage argument that rate of return on investment strategy A can not be larger than B. Actually we need to compare rate of return on A and B on [0 , t ]. 
The rate of return on A) is one of  [$]  \frac{ S_{up}} { C_A ( 0 , S ) + K B ( 0 , t ) }\, ,\, \frac{ K }{ C_A ( 0 , S ) + K B ( 0 , t ) }[$] 

The rate of return on B) is one of  [$]  \frac{S_{up}} {S ( 0 )}  , \frac{S_{down}} {S ( 0 )} [$]
Here we applied a simple reduction of the rate of return formula.  It is not clear for me how does we can delete arbitrage. It might comes up later.
 
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bearish
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Re: early exercise American option

August 8th, 2016, 12:34 pm

I tried to read Th. and 2 of the Merton paper where early exercise AO is considered. After proof of the Th. 1 I decided not to read proof of Th2 as the formal logic behind proof is somewhat difficult to comprehend. Using modern notations let me present comments to Th.1. It stated> that
[$]C_a ( 0 , S ; t , K ) \ge max\,[ 0 , S - B ( 0 , t ) K ]\, , \,t \, \in\, [ \,0\, ,\, T\, ] [$]                (1)
1. To say that [$]C_e , C_a[$] denote EO or AO prices it still does not mean that we know definitions. These are is only notations. Hence there is no sense in writing formula (1). We can ignore the fact and assume that there exist definitions of [$]C_e , C_a[$] .
About the proof style which was quite popular in old papers. We outline two strategies from which (1) is followed. Is it sufficient to state that there is no other strategy for which (1) does not correct?
2. It was assumed that in t years S ( t ) takes two values [$] S_{up} > K > S_{down}[$] 
Next reasoning do not use real or neutral world distributions. They use dominant notion which is actually equivalent to no arbitrage argument if we talk about rate of return. 
The two investment strategies are:
A) t = 0 : buy call and K shares of bonds 
B) t = 0 : buy stock for S ( 0 )
The statement " unless the current value of A is at least as large as B , A will dominate B" does not fully clear to me. It is clear that one is trying to present no arbitrage argument that rate of return on investment strategy A can not be larger than B. Actually we need to compare rate of return on A and B on [0 , t ]. 
The rate of return on A) is one of  [$]  \frac{ S_{up}} { C_A ( 0 , S ) + K B ( 0 , t ) }\, ,\, \frac{ K }{ C_A ( 0 , S ) + K B ( 0 , t ) }[$] 

The rate of return on B) is one of  [$]  \frac{S_{up}} {S ( 0 )}  , \frac{S_{down}} {S ( 0 )} [$]
Here we applied a simple reduction of the rate of return formula.  It is not clear for me how does we can delete arbitrage. It might comes up later.
I am sorry, but this should be blindingly obvious, and has nothing to do with rates of return. If (1) is violated then there is a simple, static trading strategy which produces a strictly positive immediate (time zero) cash flow along with non-negative future cash flows for all possible stock price paths. This is the strongest possible definition of an arbitrage opportunity.
 
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Re: early exercise American option

August 8th, 2016, 5:04 pm

I tried to read Th. and 2 of the Merton paper where early exercise AO is considered. After proof of the Th. 1 I decided not to read proof of Th2 as the formal logic behind proof is somewhat difficult to comprehend. Using modern notations let me present comments to Th.1. It stated> that
[$]C_a ( 0 , S ; t , K ) \ge max\,[ 0 , S - B ( 0 , t ) K ]\, , \,t \, \in\, [ \,0\, ,\, T\, ] [$]                (1)
1. To say that [$]C_e , C_a[$] denote EO or AO prices it still does not mean that we know definitions. These are is only notations. Hence there is no sense in writing formula (1). We can ignore the fact and assume that there exist definitions of [$]C_e , C_a[$] .
About the proof style which was quite popular in old papers. We outline two strategies from which (1) is followed. Is it sufficient to state that there is no other strategy for which (1) does not correct?
2. It was assumed that in t years S ( t ) takes two values [$] S_{up} > K > S_{down}[$] 
Next reasoning do not use real or neutral world distributions. They use dominant notion which is actually equivalent to no arbitrage argument if we talk about rate of return. 
The two investment strategies are:
A) t = 0 : buy call and K shares of bonds 
B) t = 0 : buy stock for S ( 0 )
The statement " unless the current value of A is at least as large as B , A will dominate B" does not fully clear to me. It is clear that one is trying to present no arbitrage argument that rate of return on investment strategy A can not be larger than B. Actually we need to compare rate of return on A and B on [0 , t ]. 
The rate of return on A) is one of  [$]  \frac{ S_{up}} { C_A ( 0 , S ) + K B ( 0 , t ) }\, ,\, \frac{ K }{ C_A ( 0 , S ) + K B ( 0 , t ) }[$] 

The rate of return on B) is one of  [$]  \frac{S_{up}} {S ( 0 )}  , \frac{S_{down}} {S ( 0 )} [$]
Here we applied a simple reduction of the rate of return formula.  It is not clear for me how does we can delete arbitrage. It might comes up later.
I am sorry, but this should be blindingly obvious, and has nothing to do with rates of return. If (1) is violated then there is a simple, static trading strategy which produces a strictly positive immediate (time zero) cash flow along with non-negative future cash flows for all possible stock price paths. This is the strongest possible definition of an arbitrage opportunity.
I used rate of return as it was used in definition of the dominant asset. Merton used dominant asset notion to proof the Th1. I do not correct his proof of Th1 and I followed his proof. It might be the statement of Th 1 blindingly obvious from other proof but when such is stated it makes sense to introduce the new proof
 
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Re: early exercise American option

August 8th, 2016, 9:02 pm

Double post
Last edited by list1 on August 8th, 2016, 9:54 pm, edited 1 time in total.
 
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Re: early exercise American option

August 8th, 2016, 9:53 pm


I am sorry, but this should be blindingly obvious, and has nothing to do with rates of return. If (1) is violated then there is a simple, static trading strategy which produces a strictly positive immediate (time zero) cash flow along with non-negative future cash flows for all possible stock price paths. This is the strongest possible definition of an arbitrage opportunity.
I used rate of return as it was used in definition of the dominant asset. Merton used dominant asset notion to proof the Th1. I do not correct his proof of Th1 and I followed his proof. It might be the statement of Th 1 blindingly obvious from other proof but when such is stated it makes sense to introduce the new proof. I doubt if we can proof anything if option price is formally undefined.