In an upcoming paper, we present a deterministic method to deal with evolution of stochastic processes which are a function of Brownian motion. We start with presenting a deterministic analogue of Ito formula for variables which are functions of arithmetic Brownian motion with piecewise constant vol. We find densities of conditional and unconditional values of functions of arithmetic Brownian motion. We also find conditional and unconditional densities of dt-integrals and dz-integrals of functions of arithmetic Brownian motion. We use these dt- and dz-integrals with Girsanov formula to find functions of arithmetic Brownian motion with possibly non-linear drift. We further extend our work, using Girsanov formula, to find evolution of functions of SDEs with possibly non-linear noise and non-linear drift. We give both conditional density(expected values conditional on the value of independent variable) and unconditional density of the stochastic process in evolution. There is no simulation or finite-differences. The basic procedure exploits some nice properties of normal random variable. We hope to release the paper on SSRN in 7-10 days. I will put some code for the method on this thread.

In my previous paper on the analytic solution of PDES by Brownian bridging and Girsanov, I did a naiive change of coordinates which went along the mean of Brownian bridge, it was admittedly not the best approach to the problem but was an attempt at solution of the problem that showed that the problem in fact can be solved. This time, I have done a better change of coordinates with careful interpolation of vol and quadratic variations and dt-integrals can be solved with very good precision even at extreme volatilities for even higher moments of diffusion. We can even give the densities of these dt-integrals very easily. I do want to take some time to make sure that this time my paper is free of initial errors.

I will try to put my paper on SSRN within a few days. But here are the main ideas. I use the idea of probability preserving tubes forward in time. Each tube is dealt with independent of other tubes and functions of random variable(arithmetic Brownian motion) are calculated in each tube forward in time. Since the probability densities that are functions of normal random variables are obtained by rescaling, possibly non-linearly, the base of the probability density, functions of evolving normal random variable are calculated by assigning them the same probability as the one for each tube but rescaled by one dimensional change of variable (analog of m-dimensional jacobian)so that we have perfect normalization. I do not numerically normalize like I did in previous papers. Amazingly the densities obtained are very precise using this technique. There is no monte carlo or finite differences.

Here are Two graphs that visually show the fit between my method using simple analytics and monte carlo density. We can see that analytic density is quite exact for dt-integral but has some mismatch for dz-integral. I am trying to improve this mismatch. It is just a matter of better interpolation. dz-integral is defined as integral of Z(t)^p dz(t), Z(0)=4.0, Vol=.35, T=1.0;In the above graph p=2.0dt-integral is defined as integral of Z(t)^p dt, Z(0)=4.0, Vol=.35, T=1.0;;In the above graph p=2.0I will post more graphs tomorrow for p=.5, p=-.5, p=4.0;

Last edited by Amin on July 11th, 2013, 10:00 pm, edited 1 time in total.

These graphs show the fit between my method using simple analytics and monte carlo density. We can see that analytic density is quite exact for dt-integral but has some mismatch for dz-integral. I am trying to improve this mismatch. It is a matter of better interpolation. dz-integral is defined as integral of Z(t)^p dz(t), Z(0)=1.0, Vol=.35, T=1.0;In the above graph p=0.5; where dZ(t)=Vol*dz(t)dt-integral is defined as integral of Z(t)^p dt, Z(0)=1.0, Vol=.35, T=1.0;;In the above graph p=0.5; where dZ(t)=Vol*dz(t)I will post more graphs tomorrow for p=-.5, p=4.0; The red analytic graph is cut close to zero to avoid instabilities. The problems with zero will be refined later.Please notice that title of first graph is wrong. It is integral of sqrt(Z(t))dt instead.

Last edited by Amin on July 11th, 2013, 10:00 pm, edited 1 time in total.

- Cuchulainn
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QuoteI am trying to improve this mismatch. It is a matter of better interpolation. The approximate curve looks a bit rough, indeed. Which interpolation method are you using? Is there noise in the data?

Rough? Are you confusing green curve as analytic which is a bit rough due to variance. I could have increased the number of paths but monte carlo is accurate enough to tell us where the analytic curve should fall. The red curve(analytic) should fall exactly in the centre of green curve.My framework is different from monte carlo and I have to work out formulas which are analogue of monte carlo in my framework. There is no randomness in real sense of the word in my framework. Everything is deterministic.

- Cuchulainn
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QuoteOriginally posted by: AminRough? Are you confusing green curve as analytic which is a bit rough due to variance. I could have increased the number of paths but monte carlo is accurate enough to tell us where the analytic curve should fall. The red curve(analytic) should fall exactly in the centre of green curve.My framework is different from monte carlo and I have to work out formulas which are analogue of monte carlo in my framework. There is no randomness in real sense of the word in my framework. Everything is deterministic.Maybe I am looking at the curve in the wrong way.

Yes, if you meant that fit is a bit rough, you are perfectly right in that. For derivatives pricing, we need precision.

Here I again show a graph comparing the density of the above dz-integral with monte carlo with refined algorithm. It is the same dz-integral as shown in the previous graph. Now the match is much better though we can still see very little deviation between the two graphs at the right side of the density.The parameters are given again as dz-integral is defined as integral of Z(t)^p dz(t), Z(0)=1.0, Vol=.35, T=1.0;In the above graph p=0.5; where dZ(t)=Vol*dz(t)I am writing a small optimization program to find the best fit universal interpolation constants. I hope that will remove even the slight mismatch between the densities.

Last edited by Amin on July 13th, 2013, 10:00 pm, edited 1 time in total.

- Cuchulainn
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Not sure if it's the same problem but there is a discussion of interpolation here

Daniel, I have the exact values of functions of normal variable in each probability tube or standard deviation fraction at different points in time but the evolution of stochastic variables and more complex functions in between follows Ito like dynamics with non-standard but universal weights on different Ito terms. These weights were figured out easily for dt-integrals but I am trying optimization to find the weights for dz-integrals. Another interesting point is that since we do not have random numbers, I am using a proxy for vol which is of order dt in my deterministic framework as opposed to order sqrt(dt) in monte carlo. So it adds up directly to quadratic variations or other terms. It is these weights that I was referring to when I mentioned interpolation.

I wrote a very brief description of my work and method for Wilmott friends. Since my work is not totally complete, I am not putting it on SSRN yet. It may take a few days before I can put together last part on stochastic volatility option pricing by Girsanov. I will continue to update as my work completes.

Last edited by Amin on July 16th, 2013, 10:00 pm, edited 1 time in total.

I got the idea of moving along standard deviations from Brownian bridging. When I asked myself the question what is special about Brownian bridge, I realized that it would be interesting to go a step further and try standard deviations because they preserve probability mass. It is a wonderful way to integrate normal along time. It is surprising nobody worked on this before as far as I know. I think it would be very interesting for younger people doing research in the universities to see how stochastic calculus can be adapted to this powerful setting i.e stochastic calculus along standard deviations. I believe, everything you can do with monte carlo, you can do with this method only without excruciating pain. Really, I can see Bermudans being priced using stochastic calculus along standard deviations with only some minor numerical work.

For those who would like to experiment, you can get around instabilities around zero by using an LCEV like treatment. This is explained in detail in Andersen and Andreassen's paper Volatility Skews and Extensions of LIBOR Market Model where When LIBOR rates with very small power get closer to zero, they becomes like lognormal with very high vol. We can get good stability close to zero in our deterministic setup with a similar treatment.

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