Forward Shooting Grids for Floating-Strike Asian Options
Summary
The document discusses discretizing the running average when using a two-state-variable Forward Shooting Grid Method to price an American floating-strike arithmetic-average call. The option has already been active for part of its life, so the calculation must incorporate the observed running average as well as future asset-price paths. The central implementation point is that the set of feasible average states expands as the lattice advances; each time level needs a range broad enough to include the averages reachable from the prior history and current tree values.
The responses distinguish between keeping a consistent spacing rule for average states and keeping the same number or range of states at every time step. One answer says the range must grow with the tree and reports a comparison price from an equivalent PDE method as a convergence check. The document does not provide code, a complete FSGM specification, or detailed boundary and interpolation rules, and one response acknowledges that variants of the method exist.
Key ideas
- A previously issued Asian option requires the observed running average to initialize the pricing state.
- The feasible range of discretized average states expands as additional lattice steps occur.
- The grid should cover the minimum and maximum reachable averages, including the prior running average.
- A separate PDE valuation can provide a reference for checking lattice convergence.
- The discussion does not specify a complete implementation and notes that method variants exist.
Tags
Full text
# FSGM for previously issued asian options
# FSGM for previously issued asian options
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?
## Answer by Yian Pap (score 1)
https://quant.stackexchange.com/a/30701
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.
## Answer by Alex C (score -1)
https://quant.stackexchange.com/a/30704
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)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.