can anyone help? How do you use the Black Scholes equation to price a barrier option whose value is zero if s=L or s=U where L<=S<=U and with payoff V(S,T)=max(S-E,0).I need only find the price in the form of a Laplace transform.THanks

QuoteOriginally posted by: finance345I need only find the price in the form of a Laplace transform.Why? Why not in closed form ?

There is no close-form solution in time domain for double-barrier options. For the solution in Laplace transform domain look athttp://www.wilmott.com/310/today_detail.cfm?articleID=112

Mukuzani,of course there is. here are the relevant papers:(1) Kunitomo, Naoto and Masayuki Ikeda, Pricing options with curved boundaries, Mathematical Finance, 2 (1992): 275-298.(2) Pricing Double Barrier Options using Laplace Transforms by Antoon Pelsser, Finance & Stochastics (search google for year and edition etc).

QuoteOriginally posted by: pb273Mukuzani,of course there is. here are the relevant papers:(1) Kunitomo, Naoto and Masayuki Ikeda, Pricing options with curved boundaries, Mathematical Finance, 2 (1992): 275-298.(2) Pricing Double Barrier Options using Laplace Transforms by Antoon Pelsser, Finance & Stochastics (search google for year and edition etc).Thank you for references. I will look at them. BTW, the second one uses Laplace transformation, as one can see from the title. It is possible to solve the problem by a number of methods. If we rank them from the most classic, the 1st would be Fourier method of variable division that gives in this case the series by the same name, the 2-nd would be an image reflection method: start with the Green function solution and then consider the image of it (negative in our case) beyond the mirror (barrier). The superposition of two sources solves the problem if we have 1 barrier. If we have 2 barriers the image in the first mirror would be reflected in the second one and so on. The result is an infinite series with fast convergence for small time or wide channel - opposite behavior to Fourier series. The combination of these two solution can give you a reasonable approach to solve double-barrier problem. Laplace transform method gives very simple solution in Laplace trasform domain. Further attempt to inverse this solution analytically gives the infinite series again - and it seems that we gained nothing, but more formal and correspondently less error prone approach. It is not bad, but the real advantage of Laplace transform is in existance of excelent numerical algorithms that excelerate the convergence dramatically. The Stehfest algorithm (1970) is very simple and can be implemented in Excel or even calculator.

i wud normally consider an infinite series as a closed form. hence my posts.

I think series solutions are as good as closed form solutions... functions are calculated using series expansion anyway.But what if the series solution is an infinite series of functions, that have to be calculated using another series?I think it should be classified differently than closed form in this case, because it usually takes a long time for these types of series to converge (like gibbs phen. or infinite sum of sinusoids)

QuoteOriginally posted by: pb273i wud normally consider an infinite series as a closed form. hence my posts.ppauper is right, that this is a semantics' issue. Traditionally (in analytical times) close form solution was the one that could be presented in terms of finite combination of special functions or as a "good" integral.An important point: if you can figure out the behavior of you solution with pencil and paper or not.General form infinite series solutions with slow and nonuniform convergence are not of this kind.Another example: PIDE is definitely not a close-form solution, but you can easily obtain infinite series combining explicite finite-difference and iterative methods. If you call it close form solution it will not make it more useful.

I consider solutions in Laplace trasform domain as close form because they are very good for analytical analysis without inversion. At the same time all numerical inversion algorithms are particular sums (tricky ones) of infinite series.

Hello,Thank you for replying.I checked the Antoon Pelsser paper on 'pricing double barrier options using Laplace transforms' but this only gives an outline to solving the problem. I was wondering if you know/have the actual steps for the method of images it uses. Just in case, it uses G=sum from -infinity to infinity of [F(x'-x+2nl,t-t')-F(x+x'-2nl,t-t'/0] where F(x,t)=1/sqrt(4tPi)exp(-x^2/4t).Cheers

QuoteOriginally posted by: finance345Hello,Thank you for replying.I checked the Antoon Pelsser paper on 'pricing double barrier options using Laplace transforms' but this only gives an outline to solving the problem. I was wondering if you know/have the actual steps for the method of images it uses. Just in case, it uses G=sum from -infinity to infinity of [F(x'-x+2nl,t-t')-F(x+x'-2nl,t-t'/0] where F(x,t)=1/sqrt(4tPi)exp(-x^2/4t).CheersYou can use this formula directly starting summation with n=0 and stopping when the next term is less than some small value (e.g. 1e-6). It converges for all finite t, but worse and worse when "t" is growing. If you want to learn more about methods of solutions of parabolic equations, the best source is "Conductions of Heat in Solids" by Carslow and Jaeger - a book that was published in 1946 and reprinted many many times. You will find this and other much more complex problems solved long before BS equation appeared. If you want fast and accurate practical implementation, take the solution in a Laplace transform domain (my paper on wilmott.com is one of many sources) and apply numerical inversion. You can derive this solution youself as a simple excercise if you start with BS equation instead of following the usual financial way: "proving theorems" that BS is valid each time when boundary conditions are slightly changed.I can send you C++ code for Stehfest and Abate & Whitt EULER algorithms of numerical inversion if you are intetrested.

And when time gets large enough that the series is not converging in just a few terms, then the eigenfunction expansion will be rapidly convergent

QuoteOriginally posted by: PatAnd when time gets large enough that the series is not converging in just a few terms, then the eigenfunction expansion will be rapidly convergentPat,I already said in a post below, that it is a reasonable approach.Did you try to construct a solution that matches long and short time series minimizing max error? It should be close to your research in perturbation methods.I prefer Laplace transform with numerical inversion because it is more formal, adjustable to wide range changes in boundary conditions and practically precise.

How about window barrier option???Can I apply a similar formula to it???Or is there other better way of pricing a window barrier option???(Window barrier option specify an active barrier period)