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Traden4Alpha
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Calculating the delta of an option using a monte-carlo simulation of the underlying

October 2nd, 2015, 12:54 pm

QuoteOriginally posted by: barnyQuoteOriginally posted by: outrunThat not the forefront really. :DThe most basic approach is to price an option with MC and then price it again with MC but then do it for S+h and S-h to get the delta/gamma. That's what my 5 year old son would do.Doing that you'll see that there is a lot of sample noise, and that it can be reduced by re-using the sample random numbers so that scenarios will be similar. ..That's what I've seen at risk departments at banks. Banks also use the same technique to reduce day-to-day variation in the VAR. With limited number of MC paths the sample noise will be big, and since VAR is just an imprecise measure or risk anyways with little absolute meaning, people are more interest in changes in VAR.But as we've seen the "bump and revalue" is essentially wrong. It depends in a non-trivial way on your delta S, too small causes problems and too big also. And then there are the problems that Cuch mentions written about by Glasserman, Jaeckel etc.Indeed! The deeper problem is the use of a discrete process to estimate a continuum property. The same problem is seen in problems with roundoff errors in which the discrete set of values of computing floating point numbers fail to represent the continuum of the real numbers.
 
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Cuchulainn
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Calculating the delta of an option using a monte-carlo simulation of the underlying

October 2nd, 2015, 1:43 pm

QuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: barnyQuoteOriginally posted by: outrunThat not the forefront really. :DThe most basic approach is to price an option with MC and then price it again with MC but then do it for S+h and S-h to get the delta/gamma. That's what my 5 year old son would do.Doing that you'll see that there is a lot of sample noise, and that it can be reduced by re-using the sample random numbers so that scenarios will be similar. ..That's what I've seen at risk departments at banks. Banks also use the same technique to reduce day-to-day variation in the VAR. With limited number of MC paths the sample noise will be big, and since VAR is just an imprecise measure or risk anyways with little absolute meaning, people are more interest in changes in VAR.But as we've seen the "bump and revalue" is essentially wrong. It depends in a non-trivial way on your delta S, too small causes problems and too big also. And then there are the problems that Cuch mentions written about by Glasserman, Jaeckel etc.Indeed! The deeper problem is the use of a discrete process to estimate a continuum property. The same problem is seen in problems with roundoff errors in which the discrete set of values of computing floating point numbers fail to represent the continuum of the real numbers.You can experience the same 'theoretical' numerical greek problems even in the absence of roundoff errors. Important as they are, they are for downstream implementation. What I am saying is that there is discrete (in numerical analysis) and discrete (in a computer). They are mutually orthogonal issues in the main.
Last edited by Cuchulainn on October 1st, 2015, 10:00 pm, edited 1 time in total.
 
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Traden4Alpha
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Calculating the delta of an option using a monte-carlo simulation of the underlying

October 2nd, 2015, 3:01 pm

QuoteOriginally posted by: CuchulainnQuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: barnyQuoteOriginally posted by: outrunThat not the forefront really. :DThe most basic approach is to price an option with MC and then price it again with MC but then do it for S+h and S-h to get the delta/gamma. That's what my 5 year old son would do.Doing that you'll see that there is a lot of sample noise, and that it can be reduced by re-using the sample random numbers so that scenarios will be similar. ..That's what I've seen at risk departments at banks. Banks also use the same technique to reduce day-to-day variation in the VAR. With limited number of MC paths the sample noise will be big, and since VAR is just an imprecise measure or risk anyways with little absolute meaning, people are more interest in changes in VAR.But as we've seen the "bump and revalue" is essentially wrong. It depends in a non-trivial way on your delta S, too small causes problems and too big also. And then there are the problems that Cuch mentions written about by Glasserman, Jaeckel etc.Indeed! The deeper problem is the use of a discrete process to estimate a continuum property. The same problem is seen in problems with roundoff errors in which the discrete set of values of computing floating point numbers fail to represent the continuum of the real numbers.You can experience the same 'theoretical' numerical greek problems even in the absence of roundoff errors. Important as they are, they are for downstream implementation. What I am saying is that there is discrete (in numerical analysis) and discrete (in a computer). They are mutually orthogonal issues in the main.Yes, they are orthogonal sources of unexpected error with difference error properties. Yet as N->∞, they merge. Do an MC with floats and >2^32 samples (not an unusual scenario of GPUs) and one might be inclined to use a very small h that hits the roundoff resolution limit of float in the domain and range of the function.
 
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Traden4Alpha
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Calculating the delta of an option using a monte-carlo simulation of the underlying

October 2nd, 2015, 4:47 pm

QuoteOriginally posted by: outrunWith 2^32 samples the stdev in the average estimate due to sampling is still going to be 2^-16 of the std or a single random sample, so you'll have maybe 5 significant decimal digit. That's very low! Sampling noise will typically dominate round off errors. With that in mind one shouldn't risk a too low "h". A too big is only problematic when gamma is high, and if that's the case then you'll have a problem anyway (in hedging,.. You have time frames / translation rates, bid ask spreads)Indeed! And given that precision of the stdev and the use of the bump method, one might be inclined to use quite a small h. Yet if one has two floats that differ by O(2^-16), the actual precision of the difference will be only O(2^-8). So the end result may be accurate to only 2-3 significant digits, half of expected number of digits.
 
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Cuchulainn
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Calculating the delta of an option using a monte-carlo simulation of the underlying

October 4th, 2015, 7:38 pm

QuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: outrunWith 2^32 samples the stdev in the average estimate due to sampling is still going to be 2^-16 of the std or a single random sample, so you'll have maybe 5 significant decimal digit. That's very low! Sampling noise will typically dominate round off errors. With that in mind one shouldn't risk a too low "h". A too big is only problematic when gamma is high, and if that's the case then you'll have a problem anyway (in hedging,.. You have time frames / translation rates, bid ask spreads)Indeed! And given that precision of the stdev and the use of the bump method, one might be inclined to use quite a small h. Yet if one has two floats that differ by O(2^-16), the actual precision of the difference will be only O(2^-8). So the end result may be accurate to only 2-3 significant digits, half of expected number of digits.This is fairly standard issue in much of numerical analysis. For which there are standard solutions.BTW what's a 'bump method'.
 
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quartz
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Calculating the delta of an option using a monte-carlo simulation of the underlying

October 5th, 2015, 9:23 am

QuoteAnd the numerical analysis of MC is underdeveloped in general.If you ever come to MCQMC or MCM &c conferences, there'll be some free drinks ;-)Let's say most quants simply dont have the time nor a sound reason to cope with all that.
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spursfan
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Calculating the delta of an option using a monte-carlo simulation of the underlying

October 6th, 2015, 6:50 am

bump is just forward or backward difference as estimate for first derivative
 
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Cuchulainn
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Calculating the delta of an option using a monte-carlo simulation of the underlying

October 7th, 2015, 8:37 am

QuoteOriginally posted by: spursfanbump is just forward or backward difference as estimate for first derivativeThanks.
 
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quartz
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Calculating the delta of an option using a monte-carlo simulation of the underlying

October 7th, 2015, 11:16 am

QuoteOriginally posted by: spursfanbump is just forward or backward difference as estimate for first derivativeShort for the full expression "bump and revalue" in financial/RM jargon. I thought it would disappear with time and align to finite difference but it is persistent.Now I'd like to know if actuaries use a third name :-)
 
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Cuchulainn
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Calculating the delta of an option using a monte-carlo simulation of the underlying

October 7th, 2015, 12:03 pm

QuoteOriginally posted by: quartzQuoteAnd the numerical analysis of MC is underdeveloped in general.If you ever come to MCQMC or MCM &c conferences, there'll be some free drinks ;-)Let's say most quants simply dont have the time nor a sound reason to cope with all that.I suppose someone giving out free drinks would be called a 'rare event'. How often do these events occur and do the number of drinks exceeds a threshold? Don't feel like getting just one glass of prosecco.
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quartz
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Calculating the delta of an option using a monte-carlo simulation of the underlying

October 7th, 2015, 1:29 pm

QuoteOriginally posted by: CuchulainnQuoteOriginally posted by: quartzQuoteAnd the numerical analysis of MC is underdeveloped in general.If you ever come to MCQMC or MCM &c conferences, there'll be some free drinks ;-)Let's say most quants simply dont have the time nor a sound reason to cope with all that.I suppose someone giving out free drinks would be called a 'rare event'. How often do these events occur and do the number of drinks exceeds a threshold? Don't feel like getting just one glass of prosecco.It would probably be a black swan, I'm not really expecting to see you there... but the payoff might compensate rarity, how many bottles can you integrate? And of course i don't offer drinks to mere mortals, it's already risky as is.
 
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Cuchulainn
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Calculating the delta of an option using a monte-carlo simulation of the underlying

October 7th, 2015, 2:38 pm

QuoteOriginally posted by: quartzQuoteOriginally posted by: CuchulainnQuoteOriginally posted by: quartzQuoteAnd the numerical analysis of MC is underdeveloped in general.If you ever come to MCQMC or MCM &c conferences, there'll be some free drinks ;-)Let's say most quants simply dont have the time nor a sound reason to cope with all that.I suppose someone giving out free drinks would be called a 'rare event'. How often do these events occur and do the number of drinks exceeds a threshold? Don't feel like getting just one glass of prosecco.It would probably be a black swan, I'm not really expecting to see you there... but the payoff might compensate rarity, how many bottles can you integrate? And of course i don't offer drinks to mere mortals, it's already risky as is.I'm looking into it but in general the univariate cdf and pdf solve an ODE while the 2-variate case solves a Goursat PDE. And to compute quantiles.
Last edited by Cuchulainn on October 6th, 2015, 10:00 pm, edited 1 time in total.