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list1
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Re: Exercise boundaries for american call and put

August 2nd, 2016, 10:09 pm

No, for the put (under constant interest rates), the fair value is [$]\max_{t \le u \le T} E[e^{-r (u - t)} (K - S_u)^+ | S_t][$], where [$]u[$] is actually a stopping-time policy.  The optimal exercise policy is to stop at the first time after [$]t[$] such that [$]S_u \le S_c(u)[$]. For the case ppauper was asking about, [$]S_t < S_c(t)[$], and so the optimal policy rule is to stop immediately, namely at [$]u = t[$].  
If u is actually a stopping-time policy then it looks like it is a random variable and therefore [$]\max_{t \le u \le T}[$] before the expectation sign  E should be at least explained,ie if u is a random variable on [ t , T ] then after integration E we do not have u.
 
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Kamil90
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Joined: February 15th, 2012, 2:02 pm

Re: Exercise boundaries for american call and put

August 3rd, 2016, 5:49 pm

No, for the put (under constant interest rates), the fair value is [$]\max_{t \le u \le T} E[e^{-r (u - t)} (K - S_u)^+ | S_t][$], where [$]u[$] is actually a stopping-time policy.  The optimal exercise policy is to stop at the first time after [$]t[$] such that [$]S_u \le S_c(u)[$]. For the case ppauper was asking about, [$]S_t < S_c(t)[$], and so the optimal policy rule is to stop immediately, namely at [$]u = t[$].  
I am still confused about the exercise boundary and how I can use that result. Assume I solved the pricing equation for an America put numerically and obtained the exercise boundary S_c(t). Now what? Does it imply as time passes by and I own this call, the first time S goes below S_c I exercise? Is this boundary now a constant?  What is the use of it after I priced my put and bought it. Does it tell me about when is it optimal to exercise it?
Yes, as long as the model parameters stay constant (i.e., in the theoretical setting), once you have solved for the optimal exercise boundary it stays solved. You exercise the option optimally at the first point in time and (price) space that you hit the boundary. 
ok, so in real life, rate levels change, volatility changes, meaning I have no use of that boundary. Would I need it then in practice as time passes by this is not optimal anymore? If I don't, why would I keep it?
 
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bearish
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Re: Exercise boundaries for american call and put

August 3rd, 2016, 8:30 pm

There are a couple of different ways to think about this. The general approach is applicable in a world of stochastic rates and volatility, you just have to parameterize their processes and solve the optimal stopping time problem in a richer space, and my comment remains correct, mutatis mutandis. Of course, then you will tell me that the parameters of the rate and/or vol dynamics will change, and we are back to where we started. The more pragmatic answer is that you estimate the original model boundary in order to determine the option value and whether it is *currently* optimal to exercise it. If it is not, then you kind of forget about it and repeat the process until you reach the earlier of maturity or a point where you decide that early exercise is indeed optimal. As usual, model recalibration introduces a kind of temporal inconsistency of behavior, but you get used to that after a while (for better or worse).  
 
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list1
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Joined: July 22nd, 2015, 2:12 pm

Re: Exercise boundaries for american call and put

August 3rd, 2016, 9:03 pm

We often use optimal notion talking about American options. It makes sense to define formally the sense of optimality. If we define optimum within markov stopping times we indeed can consider [$] sup\, E \,exp - ( \tau ( \omega ) - t ) OP ( T )[$] , were OP ( T ) is an option payoff function and sup is taking over all possible distribution of[$]\tau[$].I afraid that such problem have not studied yet. More difficult and more interesting consider optimality in the sense of 
E sup exp - ([$] \tau ( \omega )[$] - t ) OP ( T )  
Here sup is taking over all markov stopping times [$]\tau[$]. If u in above formula is nonrandom then it is quite simple or oversimplified case. It means that we are looking optimal exercise time which does not depend on the path of the underlying stock whether it goes up or down we use the same moment to exercise option.
Though I might confuse something here too.
 
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Kamil90
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Joined: February 15th, 2012, 2:02 pm

Re: Exercise boundaries for american call and put

August 4th, 2016, 12:35 pm

There are a couple of different ways to think about this. The general approach is applicable in a world of stochastic rates and volatility, you just have to parameterize their processes and solve the optimal stopping time problem in a richer space, and my comment remains correct, mutatis mutandis. Of course, then you will tell me that the parameters of the rate and/or vol dynamics will change, and we are back to where we started. The more pragmatic answer is that you estimate the original model boundary in order to determine the option value and whether it is *currently* optimal to exercise it. If it is not, then you kind of forget about it and repeat the process until you reach the earlier of maturity or a point where you decide that early exercise is indeed optimal. As usual, model recalibration introduces a kind of temporal inconsistency of behavior, but you get used to that after a while (for better or worse).  
I am looking at the practical application and wonder if I need to keep the exercise boundary out of my solver at all. Market most often evolves not exactly as implied by the model and therefore at any instance I need to reprice an option and as a result recalibrate my boundary. Assume I am holding an American put and as I am at the next day, now I ask myself again, what is the price and is it optimal to exercise today? I assume intrinsic value is always less then the price, as otherwise I would exercise immediately. So, do I get any use from yesterday's numerical boundary I got from the numerical solver?
 
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bearish
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Re: Exercise boundaries for american call and put

August 4th, 2016, 1:28 pm

It sounds like you may indeed be able to leave the exercise boundary as an internal matter for your solver and not include it in your output (which is the typical design). There are two practical use cases (that I can think of off the top of my head) where you may want access to the boundary. The first is as a starting point in estimating the boundary next time around, assuming that you are using an iterative algorithm for the boundary estimation. The second, and probably more widespread, is for generating forward looking scenarios/paths for risk management calculations, including counterparty exposure. 

As an aside, I am slightly puzzled by your statement that you "assume intrinsic value is always less then the price".  If with "price" you mean model value, then that should be an output and not an input/assumption, and if you actually refer to an observable market price, then that is what it is.
 
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Kamil90
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Joined: February 15th, 2012, 2:02 pm

Re: Exercise boundaries for american call and put

August 4th, 2016, 1:37 pm

It sounds like you may indeed be able to leave the exercise boundary as an internal matter for your solver and not include it in your output (which is the typical design). There are two practical use cases (that I can think of off the top of my head) where you may want access to the boundary. The first is as a starting point in estimating the boundary next time around, assuming that you are using an iterative algorithm for the boundary estimation. The second, and probably more widespread, is for generating forward looking scenarios/paths for risk management calculations, including counterparty exposure. 

As an aside, I am slightly puzzled by your statement that you "assume intrinsic value is always less then the price".  If with "price" you mean model value, then that should be an output and not an input/assumption, and if you actually refer to an observable market price, then that is what it is.
yes, by price I mean my model output, as if the relation doesn't hold for my model, then my model produces arbitrage. But I get your comments and it makes sense now, thanks.
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