Forums  > Pricing & Modelling  > a Bachelier model vs shifted Black model  
     
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pj


Total Posts: 3522
Joined: Jun 2004
 
Posted: 2020-06-11 15:18
Ladies and gentelmen,

All the developers are adamant that the negative prices
in commodities do not exist.

So the question arises,
how could one approximate the Bachelier model with shifted Black model (i.e. add a shift to the modeled commodity)?

Any ideas, thoughts, references?

< EDIT > to the attention of them, nitpickers. I mean modeled price.

The older I grow, the more I distrust the familiar doctrine that age brings wisdom Henry L. Mencken

deeds


Total Posts: 478
Joined: Dec 2008
 
Posted: 2020-06-11 15:22

A significant series in literature from a couple of angles around implying three moment shifted lognormal (in the manner of Turnbull Wakeman for Asians) for baskets, spreads and for negative rates.

I know you have great resources, lmk if you don't see

pj


Total Posts: 3522
Joined: Jun 2004
 
Posted: 2020-06-11 16:22
This one seems to be promising.
Walter Schachermayer And Josef Teichmann how close are the option pricing formulas of Bachelier
and Black–Merton–Scholes?

But maybe something more down to Earth?

@ deeds, thank you, will take a look.
(Although I have been burned on
the normal approximations of lognormals)

The older I grow, the more I distrust the familiar doctrine that age brings wisdom Henry L. Mencken

deeds


Total Posts: 478
Joined: Dec 2008
 
Posted: 2020-06-11 16:55

Thanks, pj, possible to share details?

bad fit in a range or more global problem?

pj


Total Posts: 3522
Joined: Jun 2004
 
Posted: 2020-06-11 17:12
If I recall well, bad fit.
Finally we yielded and integrated everything numerically.

The older I grow, the more I distrust the familiar doctrine that age brings wisdom Henry L. Mencken

nikol


Total Posts: 1124
Joined: Jun 2005
 
Posted: 2020-06-11 22:01
Price of commodity itself is positive (thx God). The negative is price of obligation to deliver, where storage_price exceeds sell_price, so you may think of something like:

delivery_price = sell_price - storage_price < 0

My point is that the price of commodity at delivery point (DDP) can be broken down into chain of deliveries in accordance to Incoterms (https://en.wikipedia.org/wiki/Incoterms)

Every step like load, offload, transshipping, transporting, storage etc. adds a bit to the final price. If you agreed to deliver at 100 and locked that price with futures and all your contractual obligations suddenly popped up above 100, then you have to pay to get rid of the product, hence negative price.

Bachelier and Black are saved!

deeds


Total Posts: 478
Joined: Dec 2008
 
Posted: 2020-06-12 10:20

Thanks, have looked at incoterms

...confirming understanding...commodity price is modeled as a stochastic process separately from costs of carry. In the equation above is storage_price modeled as a fixed amount for each contract?

nikol


Total Posts: 1124
Joined: Jun 2005
 
Posted: 2020-06-12 13:52
I never modelled these costs, so cannot say.

However. IMHO. it should be dynamic. Look as an example storage costs around Oklahoma when prices became negative. To make it properly designed you have to account for demand and available capacity. This is what happened - oil was supplied at usual volume speed, while its consumption completely ceased, so the storage capacity depleted and was in high demand. Perhaps, it should be some model of "message delivery in the pipeline system with random intensities and limited capacity"

pj


Total Posts: 3522
Joined: Jun 2004
 
Posted: 2020-06-12 14:18
So you propose to model the storage costs as well.
Should I use another Black Scholes for that? Tongue out

The older I grow, the more I distrust the familiar doctrine that age brings wisdom Henry L. Mencken

nikol


Total Posts: 1124
Joined: Jun 2005
 
Posted: 2020-06-12 14:54
Whatever you please. :)
You should model reality. Otherwise you will end up in the imaginary world with negative commodity prices.

frolloos


Total Posts: 121
Joined: Dec 2007
 
Posted: 2020-06-12 16:02
I don't know anything about commodities but just saw this paper which may be of interest to you:

https://arxiv.org/pdf/2006.06076.pdf

No vanna, no cry

nikol


Total Posts: 1124
Joined: Jun 2005
 
Posted: 2020-06-12 16:39
@froloos

Interesting approach.

I disagree with footnote about Russian hotel. That example does not mention repo contract hence the whole story is unrealistic. The recent oil story is better one. Or even epic long lasting story with negative rates :)

frolloos


Total Posts: 121
Joined: Dec 2007
 
Posted: 2020-06-12 17:08
I don't know much about Russian hotels either :) The only time I was at a Russian hotel was before I met my wife, on a typical cold Moscow winter night (pre global warming). There was a good party, and the hotel seemed far from dilapidated, although I suspect there were some repo transactions going on, but now I'm really going off topic.

No vanna, no cry

vertigo


Total Posts: 8
Joined: Dec 2015
 
Posted: 2020-06-17 19:11
let z by the price of the underlying asset ... then we have the classical lognormal model:

dz=v_ln z dw

where v_ln is the lognormal implied vol

we also have the classic normal model:

dz = v_n dw

where v_n is the normal implied vol

now consider a new model, where we have

dz = d(z-a) = (z-a) v dw

where v is a (new) implied vol and a is some positive number.

this is the shifted lognormal model. z has the solution

z_t - a = (z_0-a) exp [ v*w_t - 0.5*v^2*t ]

the price of the european call E[(z_t-K)^+] can then be computed in the standard manner

... maybe one day ...

ronin


Total Posts: 585
Joined: May 2006
 
Posted: 2020-06-17 20:28
> https://arxiv.org/pdf/2006.06076.pdf

Richard Martin is a smart guy. I knew him back in the day.

But this is still a one factor model, or two factor if you incorporate the convenience yield. The oil curve has more than two factors. Also, how do you get the vol and correls for the convenience yield?

Last time I was pricing oil options, we were actually doing it as a multifactor curve. Each future was lognormal with a flat forward, and the rolling contract was piecewise constant interpolation between roll dates. Vols came from options on futures, and correls were bootstrapped from swaptions. Basically, LMM for commodities.

@pj, I guess something like that would still work, if you just make the futures normal or shifted lognormal instead of lognormal.

"There is a SIX am?" -- Arthur
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