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Consider an American floating-strike arithmetic-average call option which was initiated 0.125 year ago, and still has 0.25 year to its expiry. The underlier has a current price of \$1.05\, volatility of 0.40, dividend yield of 0.02 and a running average of $0.95 (taken over the earlier period of 0.125 year). The risk free rate is 0.05. Write a Matlab function to implement the two-state-variable FSGM (Forward Shooting Grid Method) for pricing the above option.

With $\rho$ = $\frac{1}{2}$, use your function to generate results for the number of time periods in the lattice being 40, 80 and 160 respectively.

I'm having some issues when taking account with the running average. Are the discretized average states the same for each time level?

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The question should be clearer. At first it wasn't even obvious what method you were referring to, hence Alex clarified that. But there are many variations of this method too. Then for example, what is $\rho$ ? Just because you may know what it refers to, it doesn't mean others do immediately. Because looking at the HW paper Alex provided for example, there's no mention of $\rho$. Looking further I guess I found what you mean (the ratio of the average spacing to the asset spacing at each time-step), but you should've stated it.

Then to the question itself, the 2 or 3 papers I've now seen on this all state that no, they are not the same, with each progressing time step you have to use more discretized average states/values. Which is logical because as the tree's asset range expands, so should the possible average range. You should just make sure that at every time step this range covers the maximum and minimum possible average, based on the asset values the tree has taken thus far (plus the running average in your case).

Funnily enough the first result that comes up searching for forward shooting grid method is a matlab code that implements it for arithmetic Asians, albeit with no running average. This page though seems to have been taken down recently, maybe your supervisor owns it? :) Of course there is still the cached version...

Finally, it may help you indirectly if I told you the right price for this option is 0.11274 (calculated with an equivalent PDE method). As you increase the number of time periods in the lattice you should be converging to that, otherwise something's wrong.

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I don't know the method by Barraquand and Pudet. But usually in this kind of method all the discretized average values are of the same form for each time step. For example in Hull and White they are all of the form $S_0 e^{mh}$ where h is a constant and $m=\cdots-2,-1,0,1,2,\cdots$. However the range of values which can occur (i.e. the range of $m$) increases at each time step (lower values as well as higher values than were possible in the previous time step can occur in the next time step. In the first step only $m=0$ can occur).

An example is given in this paper by Hull and White (1993)

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  • $\begingroup$ So you never looked at Barraquand and Pudet: Pricing of American Path dependent Claims, Mathematical Finance, Volume 6, Issue 1, January 1996, Pages 17–51 but you are guessing what it says based on a paper published by someone else 3 years earlier? $\endgroup$ – noob2 Oct 24 '16 at 15:16

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