SERVING THE QUANTITATIVE FINANCE COMMUNITY

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

### Timer options

Thanks for the explanations. EndOfTheWorld, Do you have an idea about how conditional varswaps can be hedged / replicated ?

tw
Posts: 1176
Joined: May 10th, 2002, 3:30 pm

### Timer options

I was just thinking by comparison with a fund that was taking a long position using the Kelly/constant proportion of weath investedin the stock method. That pays out something related to the mean return divided by the variance (i.e. short vol).If, instead of doing that, they buy one of these timer calls, then they have a similar situation: they hope for strong mean returns on lowvol - too much vol on the upside and the strategy is over before it's begun . If they buy the put as insurance, and it gaps down then it's pretty useless.QuoteOriginally posted by: daveangelI dont know if its short vol or you just have a fixed amount of variance to play with.

EndOfTheWorld
Posts: 110
Joined: September 30th, 2008, 8:35 am

### Timer options

Theoritical hedging of the conditional varswap:- the "corridor" varswap (when you reduced the range) is hedged like a variance swap - strip of weighted call/put but strike in the range- for the "proba to be in the range", you replicate with digital options strike at the barrier for every dayIn practice this is more tricky, people already hedge their varswaps with 2 or 3 options, so you can imagine for the conditional what's goign to happend...

plaser
Posts: 39
Joined: August 18th, 2008, 3:43 pm

### Timer options

Biggest risk of these product is daily jump risk because var is observed close to close, if a stock price gets half intraday or gets taken out at double the price, that will lead to massive loss to timer seller. Question is how do you hedge and price this risk.

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

### Timer options

EndOfTheWorld, regarding cond varswaps, why do you trade digital options and not vanilla options strike at the barrier ?

tw
Posts: 1176
Joined: May 10th, 2002, 3:30 pm

### Timer options

Why would that risk be greater than with a vanilla option?QuoteOriginally posted by: plaserBiggest risk of these product is daily jump risk because var is observed close to close, if a stock price gets half intraday or gets taken out at double the price, that will lead to massive loss to timer seller. Question is how do you hedge and price this risk.

EndOfTheWorld
Posts: 110
Joined: September 30th, 2008, 8:35 am

### Timer options

You can break-down the payoff expectation of the conditional variance swap into:1- the expectation of the corridor realised variance (which is the strike of the corridor variance swap) 2- the expectation of the strike times the proportion of price in the range: E{ Strike^2 * ( Sum [ Indicator Function ( S(t) in the range ] / Number of returns ) } (note: don't know how to write fancy formulas)You basically want to count the number of prices in the range every day - your payoff is 0 or 1 => Digital Options. However, you can replicate your digital options with a vanilla call spread (epsilon ->0).Finally, you obtain the expected proportion of price in the range (which is between 0 and 1): when you divide the corridor Var strike (in Var space) by this quantity, that give you the strike of the cond. Var, which is then always greater than the corridor var.

rmeenaks
Posts: 186
Joined: May 1st, 2006, 2:31 pm

### Timer options

EndofTheWorld,Do you have:Structured Flow Handbook: A Guide to Volatility Investingthat is mentioned in the Timer Options PDF?Thanks,Ram

EndOfTheWorld
Posts: 110
Joined: September 30th, 2008, 8:35 am

### Timer options

No sorry. But I'd like to have it

plaser
Posts: 39
Joined: August 18th, 2008, 3:43 pm

### Timer options

Not necessarily bigger than plain all I'm saying is gamma risk is the biggest risk of this product and what makes most sell side reluctant to sell this thing at reasonable price. If no jumps, the price is more or less BlackScholes(var target,S).Also the holder of this option is short put option on var since this thing has finite maturity and if realized vol is 1, you don't relaized your var target before fixed maturity. Lastly the risk is interest rate and dividends due to uncertainty maturity.

mixmasterdeik
Posts: 49
Joined: December 17th, 2009, 7:26 pm

### Timer options

Maybe try Carr & Lee paper "Hedging Variance Options on Continuous Semimartingales" in page 6Cheers

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

### Timer options

To answer to the original question how to price & hedge:To clarify the product payswhere T is a stopping defined as followswhere RVar(t) is classic realized daily variance and where \sigma is a target variance. Assumption: dS(t)/S(t) = q(t) dW(t) (ie, no jumps). Let X(t):=\int_0^t q(s) dW(s), define A(t) := \int_0^t q(s)^2 ds, then the inverse time-change S(t) := \inf{ u: A(u) = s }. Then, B(t) = X( S(t) ) is a Brownian motion with the property that X(t) = B( A(t) ) (all this is known as "Dambin-Dubin-Schwarz", cf. Revuz-Yor). We then find that S(\tau) = exp{ B( \sigma^2 ) - 1/2 \sigma^2 } by definition.In other words, the price of this option under the approximation RVar == Quadraric variation and no interest rates is given as the BS price(*) BS( S(t), \sigma^2 - RV(t), K)where BS(s,var,k) denotes the price of a call in BS with variance var.More interestingly, if you use (*) and write down PnL, then you will see that around the break-even vol of \sigma, the option actually has no gamma. In other words, it's very easy to hedge with equity.
Last edited by probably on January 8th, 2010, 11:00 pm, edited 1 time in total.

mixmasterdeik
Posts: 49
Joined: December 17th, 2009, 7:26 pm

### Timer options

Hi probably,totally agree with that!!For that specific type of option payoff it also has no Vega and Theta. So it has only Delta. Should be the less exotic of the exotic trades.However, sometimes it's traded with a time cap T. So the payoff is payoff(tau) = max (S(tau) - K,0) if RVar(tau)>sigma_set^2 for t<tau<Totherwise it payspayoff(T) = max (S(T) - K,0) at maturity date THas someone taken a look at this other payoff possibility? I have only looked at MC simulation to price it...Cheers,

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

### Timer options

It's very pretty, yes. The "zero gamma" relationship may break down though if realized vol is far of implied if I remember correctly.Not sure about the cap, though. We actually did not trade one. I assume you'll get a standard option towards the end.

mixmasterdeik
Posts: 49
Joined: December 17th, 2009, 7:26 pm

### Timer options

Hi probably,another question I have is the following:In Carr/Lee paper's the underlying seems to be a martingale. In case there's a drift, how can we set up BS equation to match MC price?With no driftBS(S(t),K,Q-RVar) = MC price , where Q is the budget varianceWhen we have a drift should it be:BS( F(t,tau),K, Q - RVar) is an approximation for MC price, where F(t,tau) is the initial forward value at t for tau which is the expected stopping time?Cheers,