Pricing options when strike is zero or negative

[Kevin Schmit]

I am having trouble pricing (IV, greeks) options where strike or underlying can go to zero or negative. For example, Eurodollar futures options, priced/settled on rate (100 minus price), with a strike of 100 (effectively a zero strike, these have been listed with non-zero volume and OI in last decade).

Obviously ln(F/K) returns NaN. Alternatives I have got so far are a) replacing lognormal with normal -- (F - K) instead of ln(F/K) but then IV loses scale and normal doesn't come close to empirical distribution; or, b) shifted lognormal -- where, in ED case, ln(F/K) is replaced by ln((F+2)/(K+2)), which has its own set of problems.

Anyone have any alternative analytical pricing formulas?

 

[destriero]

Kev, have you read this (I have not): https://www2.deloitte.com/content/d...x-be-aers-fsi-pricing-with-negative-rates.pdf

 

[Kevin Schmit]

Kev, have you read this (I have not): https://www2.deloitte.com/content/d...x-be-aers-fsi-pricing-with-negative-rates.pdf

No, haven't seen this one. Thanks for the link!

I have been googling on this, downloading papers, all day. Nothing really satisfactory so far, but then I haven't read all the papers yet. Definitely no code posted anywhere, as far as I can tell.

 

[Martinghoul]

The common approach used nowadays is called "shifted lognormal", which you've mentioned. If you google it, you'll find lots of literature.

There are lots of other techniques out there, mostly variants of SABR of one type or another.

 

[Kevin Schmit]

The common approach used nowadays is called "shifted lognormal",

Yeah, that's what I think I've settled on. Derive IV's on positive part of per-expiry curve without and with (minimal that covers the negative portion) shift (two curves). Derive interpolation formula from shifted to not shifted. Then derive shifted on negative portion and adjust those vols (negative portion of per-expiry curve) using the local interpolation forumula. That way curves are consistent across periods.

I just got the first cut at this working a few minutes ago. Whether these vol curves will match anyone else's is another question entirely.

 

[Martinghoul]

Yeah, that's what I think I've settled on. Derive IV's on positive part of per-expiry curve without and with (minimal that covers the negative portion) shift (two curves). Derive interpolation formula from shifted to not shifted. Then derive shifted on negative portion and adjust those vols (negative portion of per-expiry curve) using the local interpolation forumula. That way curves are consistent across periods.

I just got the first cut at this working a few minutes ago. Whether these vol curves will match anyone else's is another question entirely.

I can help you validate, if you need...