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Kamil90
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Exercise boundaries for american call and put

July 12th, 2016, 12:59 pm

Hi,
As I am learning how to solve for American type of exercise, I see how I can easily incorporate that in the FDM scheme with just one line of code. Now, how should I understand my results on the exercise boundary?
Call: it is a known fact, it is never optimal to exercise the option early, i.e. at every node (S-K)^+ should always be less than the value of the node function, i.e. there is no exercise boundary at all, so basically this is equivalent to pricing European option and I should not even bother adding the feature?
Put: this one is more interesting: so I have the exercise boundary, I.e. at each time level t_i I am likely to have S_J below which it is better to exercise. But how do I use it in practice, is there a single time I can pass back to the user along with the price saying this is when you will be exercising?  
 
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Alan
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Re: Exercise boundaries for american call and put

July 12th, 2016, 1:43 pm

For the put, the buyer should exercise at the first time when [$]S(t) < S_c(t)[$], a critical boundary function which you can compute (approximately, numerically). In practice, some put solvers simply report a fair value. But, it is interesting to calculate and look at [$]S_c(t)[$].
 
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Kamil90
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Re: Exercise boundaries for american call and put

July 12th, 2016, 2:43 pm

For the put, the buyer should exercise at the first time when [$]S(t) < S_c(t)[$], a critical boundary function which you can compute (approximately, numerically). In practice, some put solvers simply report a fair value. But, it is interesting to calculate and look at [$]S_c(t)[$].
Thanks for your answer. By the fair value you mean the price itself at time t=0? But then why would I omit the [$]S_c(t)[$] out of my solver? would not it be a very important piece of info if I am a holder of American put once I bought it?
And related question: for the callable bond, is there a similar argument why I would/would not call the bond? Or because it pays coupons on a regular basis and the stock example I describe above is dividend free, there is nothing like that and for both callable and puttable bonds there is a exercise(call/put) boundary?
 
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Alan
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Re: Exercise boundaries for american call and put

July 12th, 2016, 3:46 pm

Yes, fair value = time 0 value. Agree, you want to know the critical boundary, but many solvers don't report it explicitly (although it plays a role in the solution). In principle, if you are only told the fair value of your put, if it's at parity, you know you should exercise. Don't know on the bond questions, beyond the fact that there should be an optimal strategy for both issuer and buyer. 

p.s. A little googling turns up this:  
ftp://ftp.repec.org/opt/ReDIF/RePEc/ibf ... 2009-5.pdf
Looks like a good example to work out and display the critical call boundary (call by the issuer).
 
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bearish
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Re: Exercise boundaries for american call and put

July 12th, 2016, 6:05 pm

Going back to the original question, it may be worth recalling that puts and calls are the same thing: they both represent the option to exchange one asset for another. Any qualitative difference in their behavior must therefore arise from payouts arising from holding a position (long or short) in each asset, such as dividends, interest and borrowing cost. The old Merton proof that you will not rationally exercise an American call on a non-dividend paying stock is not generally valid. In particular, it fails in the presence of negative interest rates, which may not be an entirely negligible case these days. 
 
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Re: Exercise boundaries for american call and put

July 13th, 2016, 10:39 am

You can use the well-known front-fixing method for PDE moving boundaries to explicitly compute V and B(t).
 
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ppauper
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Re: Exercise boundaries for american call and put

July 16th, 2016, 10:02 am

In principle, if you are only told the fair value of your put, if it's at parity, you know you should exercise.
the value is  parity on the exercise boundary
In the area where the put should already have been exercised, would the scheme report a value below parity?
 
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list1
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Re: Exercise boundaries for american call and put

July 16th, 2016, 1:46 pm

Hi,
As I am learning how to solve for American type of exercise, I see how I can easily incorporate that in the FDM scheme with just one line of code. Now, how should I understand my results on the exercise boundary?
Call: it is a known fact, it is never optimal to exercise the option early, i.e. at every node (S-K)^+ should always be less than the value of the node function, i.e. there is no exercise boundary at all, so basically this is equivalent to pricing European option and I should not even bother adding the feature?
Put: this one is more interesting: so I have the exercise boundary, I.e. at each time level t_i I am likely to have S_J below which it is better to exercise. But how do I use it in practice, is there a single time I can pass back to the user along with the price saying this is when you will be exercising?  
Let t be a spot moment and t = [$]t_0[$] < [$]t_1[$] < ... [$]t_n[$] = T . Then say call option dynamics on [ t , [$]t_1[$] ) is similar to delta shares of underlying stocks on this interval. The left hand [$]t_1[$] of the interval is excluded as BS hedged portfolio at [$]t_1[$] should be adjusted by adding or withdrawing a portion of stocks keeping the portfolio risk neutral. Such adjustments over [ t , T ] should adjust the BS price similar to as a coupon payments adjusts the price of zero coupon bond. Using this analogy we can compare BS price to the zero coupon bond price up to the first coupon payment. If we consider a discrete time stochastic cash flow [$]C ( t_j , S ( t_j ))[$] , j = 1, 2, ... n it looks like one should first define a criterion optimality for exercising such random chain. 
 
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Alan
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Re: Exercise boundaries for american call and put

July 16th, 2016, 2:14 pm

In principle, if you are only told the fair value of your put, if it's at parity, you know you should exercise.
the value is  parity on the exercise boundary
In the area where the put should already have been exercised, would the scheme report a value below parity?
No, the fair value is parity if [$]S(t) < S_c(t)[$], so a decent numerical scheme should report parity.
 
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list1
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Re: Exercise boundaries for american call and put

July 16th, 2016, 2:36 pm

In principle, if you are only told the fair value of your put, if it's at parity, you know you should exercise.
the value is  parity on the exercise boundary
In the area where the put should already have been exercised, would the scheme report a value below parity?
No, the fair value is parity if [$]S(t) < S_c(t)[$], so a decent numerical scheme should report parity.
I have thought that optimal exercise of an asset on an interval is the random moment when rate of return on the asset is maximum. It is possible it should be look like 
[$] \max_{ t \le u\le T}\,  \frac{ B ( t , u ) \,C ( u . S ( u ) ) }{ C ( t . S ( t ) ) } [$]
Thus the problem of early exercise of AO is reduced to finding statistical characteristics of such random moment.  
Last edited by list1 on July 16th, 2016, 3:40 pm, edited 1 time in total.
 
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list1
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Re: Exercise boundaries for american call and put

July 16th, 2016, 3:38 pm

the value is  parity on the exercise boundary
In the area where the put should already have been exercised, would the scheme report a value below parity?
No, the fair value is parity if [$]S(t) < S_c(t)[$], so a decent numerical scheme should report parity.
I have thought that optimal exercise of an asset on an interval is the random moment when rate of return on the asset is maximum. It is possible it should be look like 
[$] \max_{ t \le u\le T}\,  \frac{ B ( t , u ) \,C ( u . S ( u ) ) }{ C ( t . S ( t ) ) } [$]

and thus the problem of early exercise is reduced to finding statistical characteristics of such random moment.
 
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Alan
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Re: Exercise boundaries for american call and put

July 16th, 2016, 3:49 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 [$]u = t[$].  
 
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list1
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Re: Exercise boundaries for american call and put

July 16th, 2016, 7:46 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 [$]u = t[$].  
Alan, It makes sense but idea was not fully realised. Indeed the deterministic function 
[$] E[e^{-r (u - t)} (K - S_u)^+ | S_t][$]
if we assume that S ( t ) = x of the variable u represents the value of the option at a future moment u given information about underlying stock at a current moment t. Taking maximum in u represents maximum BS value of the option during lifetime of the option. Nevertheless if  buyer purchases option at t for C ( t , S ( t ) ) then the value of the option at a future moment u will be C ( u , S ( u )). Here S ( u ) is the real world not the risk neutral value of stock which is involved in definition of the C ( u , S ).. Hence we should look at the maximum in of the return at unknown u which presents maximum of  C ( u , S ( u ))/ C ( t , S ( t )) . C ( t , S ( t ) ) is a constant and we actually need to know max C ( u , S ( u )) and here we have random  function. I do not think that maximum u is reached within set of markov moments as far as for a more simpler problem to find max S ( u ) on [ 0 , T ] we should know S for each [$]\omega[$] on all [ o , T ]. Of course if we simplify the problem and decide to use a reduced optimal principal we can follow reduced optimal function given in your message. One can see that using the reduced goal function we lose something in accuracy pricing rule. Lost is the market risk that  can be expressed as a difference between two goal functions. In general the difference could be illustrated in similar situation
[$]\max_{ t < u < T }\, S ( u )[$] and [$]\max_{ t < u < T }\, E S ( u )[$]

 
 
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Kamil90
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Re: Exercise boundaries for american call and put

August 2nd, 2016, 7:36 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?
 
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bearish
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Re: Exercise boundaries for american call and put

August 2nd, 2016, 9:32 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. 
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