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gelfand
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Joined: July 14th, 2002, 3:00 am

Simulate extremes of a Brownian motion

April 27th, 2019, 12:49 pm

To simulate the terminal value of a standard Brownian motion you just need a random generator of normal deviates. What if you want to jointly simulate the high, low, and terminal values of the Brownian motion? It is known that the average maximum value up to time t = 1 is 1/sqrt(pi/2) = 0.7979. I have found empirically that using 100,000 steps to discretize the Brownian motion and using the extremes of this discrete path works pretty well, since the average maximum is 0.7944, but it is computationally intensive. Is there a fast, analytic approach for this simulation?
 
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Alan
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Re: Simulate extremes of a Brownian motion

April 27th, 2019, 9:31 pm

There are many exact formulas found in Borodin & Salminen's 'Handbook' for various related things, such as the joint distribution of the

- max and terminal value
- min, max, and terminal value
- range and terminal value

For example, if [$]M_t[$] is the max of a BM over (0,t), started at [$]x[$], and [$]W_t[$] is the terminal value, then handbook 1.1.1.8 has:

(*) [$] P_x \left\{ M_t \ge y, W_t \in dz \right\} = \frac{1}{\sqrt{2 \pi t}} e^{-(|z - y| +y - x)^2/2 t} \, dz, \quad\quad (y \ge x). [$]

Presumably, you can use this type of thing to get what you want. For example, make a draw of [$]W_t[$], and then use (*) to infer a fast, easy way to make a draw of [$]M_t[$] conditional on knowing [$]W_t[$].Or, forget the Monte Carlos -- just use (*), or related, to compute analytically whatever you are trying to get.
 
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gelfand
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Re: Simulate extremes of a Brownian motion

May 1st, 2019, 8:24 pm

Thanks for your help, Alan.
 
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gelfand
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Re: Simulate extremes of a Brownian motion

May 3rd, 2019, 8:48 pm

I found a paper devoted to my question: The working paper is at http://www.lehrstab-statistik.de/downlo ... rownWP.pdf .

Computational Management Science
January 2010, 7:1
Exact simulation of final, minimal and maximal values of Brownian motion and jump-diffusions with applications to option pricing
  • by Martin Becker
  1. 1.
[font={defaultattr}]
Original Paper
First Online: 28 November 2007[/font]

Abstract
We introduce a method for generating 
[ltr](W(μ,σ)x,T,m(μ,σ)x,T,M(μ,σ)x,T)(Wx,T(μ,σ),mx,T(μ,σ),Mx,T(μ,σ))[/ltr]

 , where 
[ltr]W(μ,σ)x,TWx,T(μ,σ)[/ltr]

 denotes the final value of a Brownian motion starting in x with drift μ and volatility σ at some final time T
[ltr]m(μ,σ)x,T=inf0tTW(μ,σ)x,tmx,T(μ,σ)=inf0≤t≤TWx,t(μ,σ)[/ltr]

 and 
[ltr]M(μ,σ)x,T=sup0tTW(μ,σ)x,tMx,T(μ,σ)=sup0≤t≤TWx,t(μ,σ)[/ltr]

 . By using the trivariate distribution of 
[ltr](W(μ,σ)x,T,m(μ,σ)x,T,M(μ,σ)x,T)(Wx,T(μ,σ),mx,T(μ,σ),Mx,T(μ,σ))[/ltr]

 , we obtain a fast method which is unaffected by the well-known random walk approximation errors. The method is extended to jump-diffusion models. As sample applications we include Monte Carlo pricing methods for European double barrier knock-out calls with continuous reset conditions under both models. The proposed methods feature simple importance sampling techniques for variance reduction.
Keywords
Brownian motion Monte Carlo simulation Jump-diffusions Double barrier options 
 
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Alan
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Re: Simulate extremes of a Brownian motion

May 4th, 2019, 1:56 am

perfect!
 
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DavidJN
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Re: Simulate extremes of a Brownian motion

May 4th, 2019, 6:19 pm

Seeing as both Monte Carlo and binomial trees can be used to price options, can the poster's questions be related to the geometry of the binomial tree? 
 
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bearish
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Re: Simulate extremes of a Brownian motion

May 5th, 2019, 1:07 am

Seeing as both Monte Carlo and binomial trees can be used to price options, can the poster's questions be related to the geometry of the binomial tree? 
It is possible, but not very convenient. There are two general problems: the square root scaling inherent in fully explicit schemes (of which the binomial model is an ugly stepchild) means that you tend to have a relatively coarse spatial resolution, which is not great for catching the realized values of the max and min of the process; and the path dependency of the max/min operators means that you would need to augment the binomial model with two additional spatial dimensions. While locally predictable, they still add complications, mostly of an accounting nature.