SERVING THE QUANTITATIVE FINANCE COMMUNITY

frolloos
Topic Author
Posts: 1566
Joined: September 27th, 2007, 5:29 pm
Location: Netherlands

### what does put-call symmetry imply

AFAIK there is no generalized put-call symmetry. The results I know are restricted to symmetric smiles. But supposing there is a generalized put-call symmetry, then would such a relationship imply that given the put smile the call smile would be fully determined (and vice versa)? That doesn't sound plausible to me, in other words I doubt that generalized put-call symmetry relationship can exist unless there are some (stringent) model assumptions?

outrun
Posts: 4573
Joined: April 29th, 2016, 1:40 pm

### Re: what does put-call symmetry imply

European put call symmetry is derivable from the contract terms and rational decision making, there is not need for a model.
If you look at a synthetic underlying (long call, short put) then the right to buy and the obligation end up in a guaranteed fixed cashflow at expiration no matter what the underlying does. There is interest rate risk because it's a future cashflow, but that can be hedged with a bond today. There is also pegging risk when the underlying lands exactly at the strike on expiration but that's technical details (although I ran into that once big time, had luck and made money, but realized I could just as easily have lost as much)

frolloos
Topic Author
Posts: 1566
Joined: September 27th, 2007, 5:29 pm
Location: Netherlands

### Re: what does put-call symmetry imply

I think you mean put-call parity whereas I meant put-call symmetry:

http://risklatte.com/Articles/Quantitat ... e/QF67.php

outrun
Posts: 4573
Joined: April 29th, 2016, 1:40 pm

### Re: what does put-call symmetry imply

frolloos wrote:
I think you mean put-call parity whereas I meant put-call symmetry:

http://risklatte.com/Articles/Quantitat ... e/QF67.php

Ah, I misunderstood indeed!

That rewrite you linked to looks pretty much model free?
Except that it isn't static?

frolloos
Topic Author
Posts: 1566
Joined: September 27th, 2007, 5:29 pm
Location: Netherlands

### Re: what does put-call symmetry imply

outrun wrote:
frolloos wrote:
I think you mean put-call parity whereas I meant put-call symmetry:

http://risklatte.com/Articles/Quantitat ... e/QF67.php

Ah, I misunderstood indeed!

That rewrite you linked to looks pretty much model free?
Except that it isn't static?

Well according to the papers I have read, PCS is not model free. However its not hard to use Black Scholes PCS, using only high school maths, to say something about the smile in a model free manner.

If you are interested I can send it to you.

outrun
Posts: 4573
Joined: April 29th, 2016, 1:40 pm

### Re: what does put-call symmetry imply

frolloos wrote:
outrun wrote:
frolloos wrote:
I think you mean put-call parity whereas I meant put-call symmetry:

http://risklatte.com/Articles/Quantitat ... e/QF67.php

Ah, I misunderstood indeed!

That rewrite you linked to looks pretty much model free?
Except that it isn't static?

Well according to the papers I have read, PCS is not model free. However its not hard to use Black Scholes PCS, using only high school maths, to say something about the smile in a model free manner.

If you are interested I can send it to you.

Yes always interested!

The problem is that the "weights" aren't static, it's not a static replication, ..so then afiak it's no longer model free.

frolloos
Topic Author
Posts: 1566
Joined: September 27th, 2007, 5:29 pm
Location: Netherlands

### Re: what does put-call symmetry imply

Outrun, sent you a 2-pager. Happy to hear your questions/thoughts. What I find interesting is that it appears if a market smile is observed at time t (today), then the formula can be used in theory to check whether the call prices (vols) for strikes K > S are consistent with the put prices (vols) for strikes K < S. On the other hand the formula could also be a tautology in which case there really is nothing to check? Don't know yet.

Alan
Posts: 9399
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: what does put-call symmetry imply

frolloos wrote:
AFAIK there is no generalized put-call symmetry. The results I know are restricted to symmetric smiles.

I don't think there's a model-independent relation, but consider the class of risk-neutral stoch. vol. models where

(*) $dS_t = (r - \delta) S_t dt + \sqrt{V_t} S_t dB_t$,

$dV_t = b(V_t) dt + a(V_t) dW_t, \quad dB_t dW_t = \rho dt$.

Then you can show there is a "put-call duality" (I like "duality" better than "symmetry"), where

$C(S,K,V,t) = S K \hat{P}(\frac{1}{S},\frac{1}{K},V,t) = \hat{P}(K,S,V,t)$.

Here $K$ is the option strike,  $C$ is the call value under (*), and $\hat{P}$ is the put value under the dual model:

(**) $dS_t = (\delta - r) S_t dt + \sqrt{V_t} S_t dB_t$,

$dV_t = \hat{b}(V_t) dt + a(V_t) dW_t, \quad dB_t dW_t = -\rho dt$,

where $\hat{b}(V) = b(V) + \rho a(V) \sqrt{V}$.

So, basically, in the dual model:
$r \leftrightarrow \delta$, $\rho \rightarrow -\rho$, and the vol drift gets adjusted when there is a non-zero correlation.

If you specialize further to a parameterized vol. model, sometimes the vol. drift adjustment translates to simply a parameter change under the same model -- e.g., try the Heston or 3/2 model. But this is not a general property.

If you take the second dual (the dual of the dual model), you get back to where you started.

There is discussion in "Option Valuation under Stochastic Volatility" Chapt 8 (my year 2000 book).

bearish
Posts: 3260
Joined: February 3rd, 2011, 2:19 pm

### Re: what does put-call symmetry imply

At the risk of stating the obvious (which I don't think has been stated yet in this thread) a put is a call and a call is a put. What we have is an option to exchange one asset for another, so it is merely a matter of perspective whether to label a given variation a put or a call. This is obvious in an FX setting, but somehow tends to get obscured in an equity context. But it helps keep things straight when e.g. thinking about things like early exercise of an American call on a non-dividend paying stock which, notwithstanding Merton's result, can be optimal in a negative interest rate world (even ignoring extra complications like borrowing cost).

Alan
Posts: 9399
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: what does put-call symmetry imply

Good point. So the vol drift adjustment I posted can be looked upon as simply the stochastic vol. change needed when the asset perspectives are swapped -- with the other parameter changes being 'obvious' from the FX point of view.

agnoatto
Posts: 24
Joined: November 23rd, 2011, 8:12 am

### Re: what does put-call symmetry imply

Put call duality is model independent. See page 19 of

On the duality principle in option pricing: semimartingale setting by Ernst Eberlein Antonis Papapantoleon and Albert N. Shiryaev.

You can find it here: http://page.math.tu-berlin.de/~papapan/ ... uality.pdf

Hope this helps.
Alessandro
Prof Alessandro Gnoatto, PhD
Department of Economics
University of Verona
Via Cantarane 24 - 37129 Verona - Italy
37129, Verona, Italy

Alan
Posts: 9399
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: what does put-call symmetry imply

Good one -- it seems we've arrived at the most general statement.

frolloos
Topic Author
Posts: 1566
Joined: September 27th, 2007, 5:29 pm
Location: Netherlands

### Re: what does put-call symmetry imply

I was travelling with very limited Wilmott access - thank you all for the useful replies.

I glanced at, not read, the paper suggested by agnoatto, looks quite technical for those with limited background in stochastics (like myself), but will give it a go.

There is a paper by Bates where he gives PCS for the CEV model. That paper and Alan's remarks should be good for my purposes. Thanks again.

katastrofa
Posts: 5808
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: what does put-call symmetry imply

Topical paper (generalising PCS): http://www.math.uchicago.edu/~rl/PCSR22.pdf