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Posts tagged with convolution

In smile modelling or pictures of the term structure of options or bonds, one speaks of a “volatility landscape” or “risk landscape”.

http://www.science.uva.nl/research/scs/edu/imcs/research/cf-03.png
http://m.rgbimg.com/cache1roJ8k/users/m/mi/micromoth/600/nfUPWLi.jpg
http://www.timera-energy.com/wp-content/uploads/2012/04/Vol-Concepts.png

http://www.math.uni-trier.de/~sachs/scientific_interests/projects/volsurface.png

That is assigning numbers to price-points and time-points; contingencies form a “surface”.

http://upload.wikimedia.org/wikipedia/commons/b/b1/Ivsrf.gif
http://4.bp.blogspot.com/-gwQjSSHR5RY/TxsIHBjijgI/AAAAAAAAAh0/rR0nzyNYT_Y/s1600/farm-flooded%2Brice%2Bfield.jpg
http://www.timera-energy.com/wp-content/uploads/2012/04/Volatility-Surface.jpg

I tend to forget that for farmers, the actual landscape—the actual (sur)face of the Earth is itself the risk landscape.

http://static.auctionservices.com/images/1325775/DCP_9266.JPG
http://www.lavocedelnordest.eu/wp-content/uploads/2014/01/storm-and-rain-over-green-field-jennifer-brindley.jpg
http://images.secondspace.com/p/SUP/5F/C6/93/0F/31/7CXL_14.jpg
http://images.china.cn/attachement/jpg/site1007/20120613/001422373e19114278d84a.jpg

  • Hillocks get more sun (could be good or bad depending on the cooling-degree days
    New Hampshirel CDD 1895-2009
    , the chance of frost, and the abundance of rain)
  • Dells and ravines get more water—which could be good if it’s dry,
    http://www.livemint.com/rf/Image-621x414/LiveMint/Period1/2013/05/23/Photos/drought1--621x414.JPG
    http://news.bbc.co.uk/media/images/38159000/jpg/_38159395_farmer_afp_300.jpg
    http://www.oxfam.org/sites/www.oxfam.org/files/imagecache/space_fullwidth/media/reactions/48865scr-climate-rice-paddy-620.jpg
    http://newsimg.bbc.co.uk/media/images/45437000/jpg/_45437836_farmerindiaafpgetty466.jpg
    or catastrophic in case of flood.
    http://cache.boston.com/universal/site_graphics/blogs/bigpicture/iowa_06_17/iowa9.jpg
    http://assets.inhabitat.com/wp-content/blogs.dir/1/files/2013/01/hay-in-a-flooded-field-537x356.jpg
    http://media.al.com/wire/photo/9657830-large.jpg
    http://4.bp.blogspot.com/_H91cnPsZsBw/TJmZCpXaUiI/AAAAAAAACWk/mOciwQsWgRs/s320/floodedfield-Apaj.JPG
    http://www.johnharveyphoto.com/Life/10_2009/FloodedFieldOfPumpkins_300W.jpg
  • Of course that depends on the crop type. Rice wants to be flooded. Even I know that.
    http://www.vosizneias.com/assets/uploads/news_photos/thumbnails/800_omagpvuyjmnichh6ug5qf8lqjg8vhwkf.jpg
    http://s1.reutersmedia.net/resources/r/?m=02&d=20111107&t=2&i=528335754&w=580&fh=&fw=&ll=&pl=&r=img-2011-11-07T103647Z_01_NOOTR_RTRMDNC_0_India-603600-1
    http://www.vosizneias.com/assets/uploads/news_photos/thumbnails/800_p0gqgzk8813x741owemvapg4pdleoa4g.jpg

  • And just like derivatives, agriculture has its term contingencies. Water in autumn is too late to grow the baby saplings but, too, a flood might not be as bad for the granary as it was for spring's seedlings.
    http://wuppenif.files.wordpress.com/2012/03/eastern-ontario-farmland-soaked-in-spring-runoff.jpg
    http://bloximages.chicago2.vip.townnews.com/santamariatimes.com/content/tncms/assets/v3/editorial/4/8a/48aa336a-551a-11e0-91ef-001cc4c03286/4d8998f8c2a5c.preview-620.jpg
  • Symbiosis between “funded” (planted) neighbours could result in a “value-added merger” if, for example, the bugs which are attracted to one plant fend off another plant’s predators.
  • A monogenetic crop could all be wiped out by the same disease.
  • Diversification, then, would seem to mirror finance as one wants to invest fully in the “cash crop” (let’s say a junk bond), but risks increase as eggs are concentrated in one basket.
  • Or say you wish that lucrative bridge loan’s IRR were applied to your entire portfolio—perhaps this is like a plant with rare seeds, or a plant that only takes in exactly perfect parts of your land.
  • If a farmer could get “negative correlated assets” (half the plants do better in dry; half do better in wet), that would reduce the “portfolio variation”.
  • Is there anything in finance that, like alfalfa, regenerates the “soil” for the next year’s crop?
  • We speak of “exposure” in finance—well, furrows in la terre literally change the exposure to the sun over the course of its chariot ride across the sky!
    http://timjacob.com/wp-content/uploads/2013/01/FEBRUARY-FIELD-36X60.jpg

So you convolve the crop type with the weather it receives localised to its exact spot in the ground. (its place in the "field" — oh! I mean its place in the field!)

Is it possible, then, to apply the lessons of modern portfolio theory to crop selection?

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Given a time-series of one security’s price-train P[t], a low-frequency trader’s job (forgetting trading costs) is to find a step function S[t] to convolve against price changes P[t]

image

with the proviso that the other side to the trade exists.

S[t] represents the bet size long or short the security in question. The trader’s profit at any point in time τ is then given by the above definite integral.

 
  • I haven’t seen anyone talk this way about the problem, perhaps because I don’t read enough or because it’s not a useful idea. But … it was a cool thought, representing a >0 amount of cogitation.
  • This came to mind while reading a discussion of “Monkey Style Trading” on NuclearPhynance. My guess is that monkey style is a Brownian ratchet and as such should do no useful work.
  • If I were doing a paper investigating the public-welfare consequences of trading, this is how I’d think about the problem.

    Each hedge fund / central bank / significant player is reduced to a conditional response strategy, chosen from the set of all step functions uniformly less than a liquidity constraint. This endogenously coughs up the trading volume which really should be fed back into the conditional strategies.
  • Does this viewpoint lead to new risk metrics?
  • Should be mechanical to expand to multiple securities. Would anything interesting come from that?

I wouldn’t usually think that multiplication of functions has anything to do with trading. Maybe some theorems can do a bit of heavy lifting here; maybe not.

It at least feels like an antidote to two wrongful axiomatic habits. For economists who look for real value, logic, and Information Transmission, it says The market does whatever it wants, and the best response is a response to whatever that is. For financial engineering graduates who spent too long chanting the mantraμ dt + σ dBt" this is just another way of emphasising: you can’t control anything except your bet size.

UPDATE: Thanks to an anonymous commenter for a correction.