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### Finite difference and control variates

Posted: **June 21st, 2015, 7:35 pm**

by **mutley**

We know control variates improve Monte Carlo convergence for derivative with no closed-form solution using the estimated error for one that does possess a closed-form on the same path.Could the same be applied to FDM? i.e. on a given grid, you are pricing something exotic and you also price something similar but with a closed-form. Idea being to try to adjust out the biases imposed by a sparse grid / bad FDM scheme.At each spatial node on each time slice, you adapt the regressed price of the exotic by the observed bias of that of the regressed vanilla?Does this make sense conceptually? I know there are many methods for reducing computational burden from FDM pricing (better schemes, change of variable, etc) but was thinking about this for a structure i am looking at and the analogue between MC and FDM appeared interesting.Any thoughts welcome, I've not yet tried it out myself.

### Finite difference and control variates

Posted: **June 22nd, 2015, 9:43 am**

by **Cuchulainn**

Hi mutley.In general MC and FDM don't mix. The maths is all different. MC lacks the concept of pointwise convergence, so it's like mixing oil with water.In 1,2,3 dimension FDM is fine; after that MC is probably the only way.QuoteAt each spatial node on each time slice, you adapt the regressed price of the exotic by the observed bias of that of the regressed vanilla?Do you have an example to make it clear?

### Finite difference and control variates

Posted: **June 24th, 2015, 9:11 am**

by **emac**

QuoteOriginally posted by: outrunA simple example would be an American option? That's a classical control variates method when using trees/FDM grids.* use FDM to price both European and American options on the grid* use the difference between the closed form and FDM for the European option as measure of numerical error/bias* correct the American price on the grid with the numerical error(edit: a different view would be that you price just the difference between the European and American)The problem I see is that the bias you compute for the European is a global bias in the value instead of a local bias in the computing of the value as a function of surrounding grid values. As the value of the European and American option diverge across the the grid, then so will the discretisation biases of the two. We had good results (20 years ago) with American options though.. Effectively you're computing the early exercise premium part of the option with FDM which is a smaller value than the full option. If FDM gives a percentage error then it makes sense to decompose the computation in a large closed form part and a small exotic part that gets approximated.If you use a discretisation + MC method e.g. Euler-Maruyama, the error splits into variance + bias. Control variates are for reducing the variance not the bias.Since FDM method has no error due to variance, the question is ...... misguided.

### Finite difference and control variates

Posted: **June 24th, 2015, 9:21 am**

by **emac**

QuoteOriginally posted by: Cuchulainn MC lacks the concept of pointwise convergence, so it's like mixing oil with water.Ummmm ... a standard Monte Carlo approximation converges almost surely. In what sense does it lack a concept of pointwise convergence.

### Finite difference and control variates

Posted: **June 24th, 2015, 7:27 pm**

by **Cuchulainn**

QuoteOriginally posted by: emacQuoteOriginally posted by: Cuchulainn MC lacks the concept of pointwise convergence, so it's like mixing oil with water.Ummmm ... a standard Monte Carlo approximation converges almost surely. In what sense does it lack a concept of pointwise convergence.We've been through this before in an earlier thread. How do you get (deterministic) penny accuracy with MC? You can't IMO.

### Finite difference and control variates

Posted: **June 25th, 2015, 1:01 pm**

by **mutley**

hello sorry didnt mean to neglect this thread after asking it. had zero time to put something together to test it / determine whether it was a load of horse. will try this weekend. gah, work.

### Finite difference and control variates

Posted: **June 25th, 2015, 3:56 pm**

by **Cuchulainn**

QuoteOriginally posted by: mutleyhello sorry didnt mean to neglect this thread after asking it. had zero time to put something together to test it / determine whether it was a load of horse. will try this weekend. gah, work.No prob, take your time.

### Finite difference and control variates

Posted: **June 25th, 2015, 4:02 pm**

by **Cuchulainn**

This thread is ambiguous. Are we talking about SDE or PDE? In the latter case the words variance and bias are meaningless.

### Finite difference and control variates

Posted: **June 26th, 2015, 2:12 pm**

by **mutley**

It is the latter, PDE, and it wasn't variance of solution i was looking to reduce / mitigate but rather implementation bias."Idea being to try to adjust out the biases imposed by a sparse grid / bad FDM scheme."So say you use Euler cause it is quick & easiest to code - but you also know it is first-order accurate and suffers from conditional stability. An example of what i was thinking is what Outrun mentions - American exercise.you regress a European and an American on two grids, otherwise identical but for the exercise condition on the latter.As you roll back the European grid, you observe the pricing errors on each grid point with what your BS calculator would give for the result.You use the observed "implementation bias" of your Euler method for the European to adapt the price of the Euler method for the American.Every time the Euler PDE steps out of line with the BlackScholes price, you bring it back into line - using these tiny adjustments to improve the accuracy of the American PDEas i said, this idea could be complete rubbish - but thought worth finding out if it is rotten or not (which i will do this weekend)

### Finite difference and control variates

Posted: **June 26th, 2015, 2:28 pm**

by **Cuchulainn**

QuoteYou use the observed "implementation bias" of your Euler method for the European to adapt the price of the Euler method for the American.Sorry, I don't get this.Besides no one uses explicit Euler for PDE. "implementation bias" -> numerical error is a bit better. No one in PDE uses 'bias'. mutley,here's a good discussion + numbers on PDE for American The good old days.

### Finite difference and control variates

Posted: **June 28th, 2015, 9:50 am**

by **Cuchulainn**

QuoteJoshi and Tian did stuff like that. Possibly, but not for PDE. (AFAIK) QuoteIt might also be a good thing to tilt the grid so that at one end you'll have S0 and at the other end K in the middle of the grid -or al least exactly on a node, or between two nodes-. That makes the results you're investigating less noisy/oscillating due to arbitrary node placements and gives you a clearer view of the benefits of your method.The issue is the lack of smoothness of the payoff function. There are several ways of doing this, again whether you are talking about lattices (the present case?) or PDE/FDM. I would say this thesis is a very good source -Stefanie Mueller QuoteI haven't seen this variant where you estimate correction terms on each grid point using the European grid, and then correct the American price with that difference in each step. The way I've seen it done is to only do a single correction term at the end of the computation instead of on each grid point. ..so I'd be interested to see if improves things -or maybe worsen things-. By grid point, I assume the corrections at each S_j point for each time level. Then OK. Other alternative probably are less optimal. in any case, this correction results in 1st order convergence in general.So, for American options see (2.74) and (2.75) in the thesis. The PDE can use a similar well-known approach.All this has been around for a while..

### Finite difference and control variates

Posted: **June 28th, 2015, 10:38 am**

by **Cuchulainn**

Regarding payoff smoothing, I use the averaging of the payoff (originally due to H.O. Kreiss) produced QuoteWe have tested the Tian method (with and without averaging) on the above option problem. In general ? and as expected ? payoff averaging produces more accurate results than the Tian method applied to the unprocessed payoff function. For example, the exact solution is 2.6710 and we got the following results on the step sizes {100, 200, 500, 700, 1000, 500}: . Tian (unprocessed payoff): {2.48041,2.60347,2.70589,2.73202,2.65752}. . Tian and Simpson: {2.59016,2.68143,2.65892,2.69196,2.69195,2.67826}. . Tian and Simpson 3/8 {2.56272,2.66194,2.67105,2.70198,2.68334,2.68204}.In general, Tian is not always robust. At the end of the day, FDM (or FEM) is probably a better choice. Because of unconditional stability and 2nd order accuracy.