SERVING THE QUANTITATIVE FINANCE COMMUNITY

probably
Posts: 175
Joined: May 18th, 2004, 10:46 pm

### Option on variance swap

Hi EiriamjhThere are some which model directly implied vol:Surface Models I know are Brace et al, Cont et al, Haffner's book.Term-Structure models Schoenbucher, and there is a recent paper by Schweizer/Wissel, I heard of.A problem with the former is that it seems very difficult to ensure that the evolution is arb-free, ie that the call prices are free of arb (butterflies etc) at any future point in time.The term-structure models don't suffer from that drawback, but then you don't know what to do if spot moves far away from the initial strike.Hope that helps. Citi made a presentation on a surface model which they use in risk management.Maybe one of them knows how severe the arb-problem actually is?ByePSOption on fwd variance swap is much easier than an option on implied in some way...it's "just" an option on the difference of two spot started var swaps.(btw this gives you a back-test for prices of fwd start options, since they should give (at least roughly) the fwd var swap)
Last edited by probably on June 6th, 2005, 10:00 pm, edited 1 time in total.

eiriamjh
Posts: 462
Joined: October 22nd, 2002, 8:30 pm

### Option on variance swap

thanksindeed a forward-starting VS is iust a calendar spread of VS with appropriate notionals, but I'm not sure what you mean by saying this makes the problem of option valuation 'much easier'perhaps you are saying that you don't need to model skew but only term structure of implied vol?e.

probably
Posts: 175
Joined: May 18th, 2004, 10:46 pm

### Option on variance swap

Well I mean you just have to model term-structure of variance swaps. That is much easier than doing an implied vol surfacein terms of no-arb conditions.So once you have a nice model in place, you can price options on fwd variance etc.

MForde
Posts: 86
Joined: March 30th, 2005, 9:03 am

### Option on variance swap

Hi Hans, no you would have to prove or assume that the sequence of stopping times gives rise to a U.I. Stock price process

probably
Posts: 175
Joined: May 18th, 2004, 10:46 pm

### Option on variance swap

That's what I meant ... if it works, all is fine (I actually like the idea ... but I had exactly that problem).

MForde
Posts: 86
Joined: March 30th, 2005, 9:03 am

### Option on variance swap

HB, another thing u might wanna investigate Re: bounds on the variance of realized variancefrom Ito's lemmaInS_T- lnS_0 = int_{t=0^T} 1/S dS - 1/2 int_{t=0^T} \sigma_t^2 dtbut the variance of the left hand side can be worked out from the implied vol surface, as can the variance of the 1st term on the rhs(see section 3.4 of thesis on http://www.stats.bris.ac.uk/~mamsf), so upper +lower bound on int_{t=0^T} \sigma_t^2 dtcould be obtained by assuming these 2 known quantities have correlation 1 or -1it'd be interesting to compare this against Dupire Skohorod bounds - his is perfectly consistent with a smile at one maturity, this idea is incorporatingsmile info at all maturities, but isn't including ALL the smile info at each maturity

exotiq
Posts: 1888
Joined: October 13th, 2003, 3:45 pm

### Option on variance swap

QuoteOriginally posted by: probablyWell I mean you just have to model term-structure of variance swaps. That is much easier than doing an implied vol surfacein terms of no-arb conditions.So once you have a nice model in place, you can price options on fwd variance etc.I have certainly been one to advocate modeling the term structure of variance swaps, but ask you, how do you calibrate or reasonably estimate what the vol of a forward variance should be?

probably
Posts: 175
Joined: May 18th, 2004, 10:46 pm

### Option on variance swap

Hi ExotiqQuote have certainly been one to advocate modeling the term structure of variance swaps, but ask you, how do you calibrate or reasonably estimate what the vol of a forward variance should be? Well is it a bad idea to use a functional to interpolate the varswap term structure and then to try to find a reasonable vol paramtetrization by using histoical data?Once a reasonable parametrization for the vol is found, you can calibrate it using European prices.

MForde
Posts: 86
Joined: March 30th, 2005, 9:03 am

### Option on variance swap

Exotiq, u probably know this, but if u can observe a forward starting smile, u have the distribution of S_T'/S_T for T'>T, so u could make the independent stoc vol assumption, + use Carr-Lee idea to extract the distN of forward realized variance from T to T'

erstwhile
Posts: 904
Joined: March 3rd, 2003, 3:18 pm

### Option on variance swap

Mforde: your idea for finding the upper and lower bounds of realised variance is interesting. How does it come out? It would be interesting to see some real numbers, i.e., upper and lower bounds on a 3 month call or put on the var of SPX or Eurostoxx50. There are a small number of dealers making prices on var options, and a relatively small number of people who trade them (I am one of the latter), but a fairly large number of people waiting in the wings to see if this market developes. A simple and clear paper showing that these options have well defined upper and lower bounds (maybe published in a nonacademic journal like Risk magazine?) would help to get the market off the ground. And I imagine that risk management departments, which are likely to be struggling to come to terms with the properties of options on variance, would be much more comfortable if they could create scenarios with well defined extreme levels.

MForde
Posts: 86
Joined: March 30th, 2005, 9:03 am

### Option on variance swap

see attached for construction of lower bound for the variance call, what I was talking about was tightest bound for the variance of realized variance

erstwhile
Posts: 904
Joined: March 3rd, 2003, 3:18 pm

### Option on variance swap

Thanks for the notes - very interesting. On the variance of realised variance, i was thinking that knowledge of these bounds ought to imply a sort of weak bound on the value of variance options, right? I mean I know that to price options on realised variance you don't use a Black model and put in the spot var swap and a vol of variance, but surely if you knew limiting values for the vol of realised variance, this should tell you something about limting values for options on variance, right?

BLOBY
Posts: 113
Joined: May 17th, 2004, 5:07 am

### Option on variance swap

In Dupire's article, where does this formula come ?
Attachments
Image2.zip

erstwhile
Posts: 904
Joined: March 3rd, 2003, 3:18 pm

### Option on variance swap

I think the meaning of that is that the expectation of the log(forward) is equal to the sum of a "constant dollar" amount of stock minus 1/K^2 weighting in OTM calls and puts. In particular, integral of 1/K dK from zero to S0 doesn't look very convergent! I was surprised when I read that at first too.

PutorCall
Posts: 166
Joined: November 27th, 2002, 1:45 am

### Option on variance swap

There are of course assumptions behind variance swaps replication.Besides the usual frictionless markets and no arbitrage, These are 1) continuous monitoring of the path in computing the variance swap payoff2) no jumps in the price process3) positivity of the price process (as opposed to nonnegativity).4) continuum of European option strikes at the varaince swap maturity5) continuous trading opportunities in the underlyingIf you do a change of variable in Dupire's formulafrom K = strike to k=ln (K/S0)and express OTM option values in percentage of strike K as a function of k, say theta(k), then the two integrals inhis formula sum to this one:\int_{-infty}^{infty} theta(k) dkNow we have familiar criteria for convergence of the integral