Value at risk
- Thread starter Anas
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i think you are very confused. your question does not make any sense from a technical stand point or from a risk management stand point...
first, it is NOT enough to say "given a stochastic process X" as your process to be used to model some underlying. is X measurable? what space have you defined it on? etc... it is like saying "given an arbitrary method to model the future, assume that the method of this type "... nonsense.
a stochastic process is not defined by a symbol X, but by the underlying information - its law, the probability space its defined on, its dominating measure, its characteristics (measurable? continuous? finite mean? etc)...
second, a diffusion is a strong markov process with continuous paths... a process with jumps, by definition, does not have continuous paths... you are talking about a levy process, which does not have continuous paths unless its jump component is identical to zero, i.e. the diffusion with jump process you consider is a shifted brownian motion.
third, why would you be interested in the VaR of a levy process? keep in mind that the VaR will, on a day T, tell you how much to hold on T for the day T+1. have you considered the potential issues with jump discontinuities? suppose that the VaR you hold without a jump discontinuity is $5 million and with a jump discontinuity is $20 million.. which is the correct VaR? you have not specified that...
fourth, why do you need to use extreme value theory? essentially you are trying to find the number x such that P(X > x ) = 1 - alpha. a simple but commonly used levy process, such as X = linear term + brownian motion + compound poisson process, has analytical results for its distribution (i.e. lognormal), see Glasserman - Monte Carlo Methods in Financial Engineering, page 134. there is no need too use extreme value theory in your question unless you specify some more information why you want to do so.
first, it is NOT enough to say "given a stochastic process X" as your process to be used to model some underlying. is X measurable? what space have you defined it on? etc... it is like saying "given an arbitrary method to model the future, assume that the method of this type "... nonsense.
a stochastic process is not defined by a symbol X, but by the underlying information - its law, the probability space its defined on, its dominating measure, its characteristics (measurable? continuous? finite mean? etc)...
second, a diffusion is a strong markov process with continuous paths... a process with jumps, by definition, does not have continuous paths... you are talking about a levy process, which does not have continuous paths unless its jump component is identical to zero, i.e. the diffusion with jump process you consider is a shifted brownian motion.
third, why would you be interested in the VaR of a levy process? keep in mind that the VaR will, on a day T, tell you how much to hold on T for the day T+1. have you considered the potential issues with jump discontinuities? suppose that the VaR you hold without a jump discontinuity is $5 million and with a jump discontinuity is $20 million.. which is the correct VaR? you have not specified that...
fourth, why do you need to use extreme value theory? essentially you are trying to find the number x such that P(X > x ) = 1 - alpha. a simple but commonly used levy process, such as X = linear term + brownian motion + compound poisson process, has analytical results for its distribution (i.e. lognormal), see Glasserman - Monte Carlo Methods in Financial Engineering, page 134. there is no need too use extreme value theory in your question unless you specify some more information why you want to do so.
Hello , here are the details : Fix a time horizon T
where b are \gamma are an integrable processs , for all T>t>0
Nt is a poisson process , the jumps are nonnegative ie : Y_i are iid and Y_1 \geq 0
the process :
is integrable.
Now we define the value at risk by :
where :
Now in order to test some theoritical bounds for small \alpha , I need to plot the curve for the VAR , I have read that Monte carlo doesn't work for small \alpha and EVT is a better approach ,
where b are \gamma are an integrable processs , for all T>t>0
Nt is a poisson process , the jumps are nonnegative ie : Y_i are iid and Y_1 \geq 0
the process :
Now we define the value at risk by :
where :
Now in order to test some theoritical bounds for small \alpha , I need to plot the curve for the VAR , I have read that Monte carlo doesn't work for small \alpha and EVT is a better approach ,
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Hello , here are the details : Fix a time horizon T
where b are \gamma are an integrable processs , for all T>t>0
Nt is a poisson process , the jumps are nonnegative ie : Y_i are iid and Y_1 \geq 0
the process :is integrable.
Now we define the value at risk by :
where :
Now in order to test some theoritical bounds for small \alpha , I need to plot the curve for the VAR , I have read that Monte carlo doesn't work for small \alpha and EVT is a better approach ,
is this from a book? pdf? post the source and ill take a look at it. where did you hear that monte carlo doesn't work for small alpha and e v t is better? source?
Here is the article ESTIMATION OF VALUE AT RISK AND RUIN PROBABILITY FOR DIFFUSION PROCESSES WITH JUMPS - Denis - 2009 - Mathematical Finance - Wiley Online Library
I don't remember where I read about EVT , but the article says that there are some problems with MONTE CARLO ( see the introduction ) , can you help me with implementing the EVT ?
diegosanaz
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The problems I guess is that you would need a huge amount of draws for very small alphas. It is not easy to get precise VaR values. Specially since there is no close solution for what the process will look like and therefore would need some sort of Euler scheme.
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