Bi/Trinomial Trees

[TheBigShort]

Hey everyone,

I am creating a binomial(eventually trinomial) tree for an SPX 1 month option. What I am looking to do is incorporate local vol into the model. So for example if SPX drops 10% in time t, I will have a vol of vt. Like wise projecting non-constant vol into the future as it goes through the nodes. I am hoping to get some insight from you guys who have experience building non-constant vol bino/trinomial trees.

Thanks

 

[sle]

I am creating a binomial(eventually trinomial) tree for an SPX 1 month option. What I am looking to do is incorporate local vol into the model. So for example if SPX drops 10% in time t, I will have a vol of vt. Like wise projecting non-constant vol into the future as it goes through the nodes. I am hoping to get some insight from you guys who have experience building non-constant vol bino/trinomial trees.

I have some R code for local vol somewhere. The real question is what exactly are you trying to achieve?

 

[TheBigShort]

I have some R code for local vol somewhere. The real question is what exactly are you trying to achieve?

That would be great if you could find it! Well I am trying to get better pricing for these less liquid options. Considering how path dependent vol is, I can't be using BS.

I also want to workout some binomial trees using my own t+n inputs incorporating skew.

 

[panzerman]

Look through the code examples here and see if anything might be helpful.

http://finance.bi.no/~bernt/gcc_prog/recipes/index.html

 

[TheBigShort]

Look through the code examples here and see if anything might be helpful.

http://finance.bi.no/~bernt/gcc_prog/recipes/index.html

Thanks for posting, this will definitely come in handy in the future

 

[ironchef]

Look through the code examples here and see if anything might be helpful.

http://finance.bi.no/~bernt/gcc_prog/recipes/index.html

Thank you, I can definitely benefit from this.

You sir are the reason I keep coming back and read through a lot of nonsense.

 

[Kevin Schmit]

I am creating a binomial(eventually trinomial) tree for an SPX 1 month option. What I am looking to do is incorporate local vol into the model.

What is your vol input? Function? Vector? Matrix? What do you want to pass to the tree pricing function in place of the plain-vanilla v or sigma argument? It may be that what you want is equivalent to the implied binomial tree (circa 1994,Derman & Rubenstein IIRC), for which there is plenty of code available (I think Tian 2012 model is numerically stable but my recollection of it is hazy).

If you want the tree to recombine, you'll need (at least I think so) to specify a deterministic relation between local vol, underlying price, and time (i.e. not path-dependent , remember tree generally only recombines if Sud ~ Sdu), which would imply that vol is not an independently priced risk factor.

 

[TheBigShort]

What is your vol input? Function? Vector? Matrix? What do you want to pass to the tree pricing function in place of the plain-vanilla v or sigma argument? It may be that what you want is equivalent to the implied binomial tree (circa 1994,Derman & Rubenstein IIRC), for which there is plenty of code available (I think Tian 2012 model is numerically stable but my recollection of it is hazy).

If you want the tree to recombine, you'll need (at least I think so) to specify a deterministic relation between local vol, underlying price, and time (i.e. not path-dependent , remember tree generally only recombines if Sud ~ Sdu), which would imply that vol is not an independently priced risk factor.

I want to use my forecasted vol at each node as we go through time. Obviously vol is path dependent and the binomial tree would be perfect for what I am looking to do. There is quite some good stuff in your reply here. Something I am looking at is how should I price the Sv^3 node (as an example) with a stock with high/low debt or high/low PB etc.. ? Could you explain why you think I might need a cov-matrix for this? If I am looking at earnings events, I will need to look at non-recombining trees. I am going through Rubenstein right now. Thanks agian Kev.

 

[Kevin Schmit]

I want to use my forecasted vol at each node as we go through time.

This is exactly the implied binomial tree Derman and Kani derive in their 1994 Risk article "Riding on a Smile." I think that they even use nearly the exact same phrase: "specify vol at each node." In a subsequent article (the name of which escapes me) they derive an impled trinomial tree. The original Deman/Kani tree was unstable and tended to blow up when number of steps was too high.

.

Obviously vol is path dependent

Path dependent vol is incompatible (at least in a recombining tree) with "vol at each node." You need a single vol at each node not multiple vols depending on the path taken to the node. Otherwise you won't have a single probability at each node. You're already fudging the probabilities to constrain them to between 0 and 1, reconciling multiple probabilities at each node would be a Sisyphean task.

how should I price the Sv^3 node (as an example) with a stock with high/low debt or high/low PB etc.. ? Could you explain why you think I might need a cov-matrix for this?

Not sure what you are asking here. Are we still on local-vol? Are your local vols derived from current no-arbed option prices, or is there a model-based component taking into account PB or debt ratios of the underlying? Covar matrix of what? Underlying stock returns?

If I am looking at earnings events, I will need to look at non-recombining trees.

You are probably right, and if you go with non-recombining you might be better off with Monte-Carlo or simmulation methods instead of tree models. Would your earnings spike be modeled as a discrete jump at a single step or between two steps? If so then the standard local vol formula (Dupire) tends to blow up (zero denominator) in the presence of instantaneous discontinuities. There may be other local vol equations that handle jumps better, but I am not familiar with any.

 

[Kevin Schmit]

Recent robust implied binomial tree: