should model a smile each for call and put, or just one

[tesla1060]

Hi all,



I have a question regarding modeling volatility smile.



If I use BSM to compute volatility for the same strike call and put with a pre-defined r, and using the underlying price, the iv will be different thus I will get two smile.



If I use put-call parity to compute the implied forward, therefore the implied r, I will be getting the same iv for the same strike call and put. but for different strike, implied forward might be slightly different.



My question is, if I want to do market making, should I model two volatility curves, or should I just use put-call parity to imply forward, and form one curve, in the later case, should I use near atm implied forward for all strikes in calculating the iv?



I would like to know how market pratitioners approach this problem, and the reasoning behind.



Thanks.

[nikol]

Underlying is one, its forward p.d.f. is one hence smile is one too.

You can confirm it by thinking about arbitrage between calls, puts and futures at each strike using call-put parity. All three instruments are free to bounce within bid-ask spread.



However, every option at every strike is an individual market where matching happens separately. Quote dynamics in these silos share same underlying and are constrained by non-arbitrage requirements (add: strike-strike arb, time arb). In the rest these markets evolve independently.

[TakeItAndRun]

I think your question and beyond has already been answered [url=/Show%20Post.aspx?PostIDKey=147191]here[/url].

[Strange]

The general idea is that you are making markets in volatility so there is a single volatility curve subject to the arbitrage relationships. On top of that, an MM applies an appropriate bid/offer and the rest is either hedge out or marked conservatively.



However, the "true vol market" has an implicit assumption of symmetric long/short treatment - i.e that borrow, financing treatment and corporate rights are simply the opposite (with some bid-offer) for both long and short positions. Sometimes these things are not true - e.g. a stock is hard to borrow or have shorting restrictions, an underlying can have asymmetrical financing for regulatory reasons, an underlying can have implicit long but not short optionality. In that case, as an MM you have to correct for that effect somehow; either build a fancy model that does it for you (the "right" way) or you can maintain a separate volatility curve for calls and puts.

[tesla1060]

what may go wrong, if I use implied forward from near atm options to get the same iv for same strike call and put, thus model only one volatility curve, and price both my call and put from this curve?

[nikol]

Yes, but that curve gives you only mid-vol (mid-price) level. You have to add your margin to account for risks, ie. to stay at best bid-ask or even deeper in the book.



What can go wrong with option trading?

Sell ATM put just before the next "Lehman Brothers" event. It will go through your margin.



Deep OTM might be illiquid, so your bid-ask ivol spread will generate prices with

bid=0 (hence no bids),

ask=1 tick (very few limit orders hang).



Deep ITM is similar.

[Strange]

"what may go wrong, if I use implied forward from near atm options to get the same iv for same strike call and put, thus model only one volatility curve, and price both my call and put from this curve?"



I assume you are asking what can go wrong from the perspective of "oops, this does not look right", not the risk aspect (cause a lot of things can go wrong there). It depends on many things, mostly relating to the nature of the underlying - usually that is the type of stuff where put/call parity does not hold for structural reasons.



Is it an equity that you are trying to make markets on? Also, if you are a market maker, why do you need to imply stuff from the market - you should have a forward model as well as an idea where vols should be bought and sold?

[tesla1060]

Hi, it is option on equity etf.



Yes, what I mean is there any fundamentally misconception in using implied forward to build one smile curve to price options.



@Strange, what do you mean by forward model?



Here is my understanding. What I need to do is to price option correctly based on a model that takes in input parameter such as underlying movement(I am probably assuming underlying leads option a bit)



dC/dS = Delta_bsm + Vega * dSigma/dS



So the total delta will be BSM Delta + Delta caused by iv change. What I should do is make a model that can best find the second term of the above equation, as dS is either known, or if I can predict that will be another topic.



My understanding is that I first build a smile based on market prices, there is a general assumption that the shape of smile will not change much in short period of time. As underlying moves, if under sticky delta assumption, the iv of a particular strike will move along the smile a bit due to undelrying move, thus I can get dSigma/dS from the smile.



I would like to know if there any misconception in the above?

[nikol]

looks like you are on the right path



> thus I can get dSigma/dS from the smile



remember that dC2/d2K ~ p.d.f. (K is strike), so Sigma comes out easily.