Volatility Swap

alandgd · March 3rd, 2006, 2:26 pm

Dear Colleagues I am trying to calculate the fixed leg of volatility swap using Laplace transform of total variance of the Heston Model, see Valuation of Volatility Derivatives, available at Valuation of Volatility Derivatives,.Unfortunately when a use longer maturity times ( more than 1 y) my calculated values does not converge to expected long term volatility value. I do not know what is wrong with my code, attached below, if someway could help me Id appreciate very much.Thanks in advanceAG

Alan · March 3rd, 2006, 3:25 pm

I see you have a long-run variance of 0.04.Do you find numerically that, as T becomes large, If so, what do you find for C?regards,

alandgd · March 3rd, 2006, 5:12 pm

Alan,Sorry, I didnt understand your question, the expression is unformatted and I am not able to read it. Could you type it again?REgards,Alan

alandgd · March 3rd, 2006, 5:51 pm

Alan,Now It´s ok. I got itStraight to your point, I used the a analitical expression to represent the fixed leg of a volatility swap as contained at attached file (sorry I dont Know use Latex editor)RegarsQuoteOriginally posted by: alandgdAlan,Sorry, I didnt understand your question, the expression is unformatted and I am not able to read it. Could you type it again?REgards,Alan

alandgd · March 3rd, 2006, 5:51 pm

Alan,Now It´s ok. I got itStraight to your point, I used the a analitical expression to represent the fixed leg of a volatility swap as contained at attached file (sorry I dont Know use Latex editor)RegarsQuoteOriginally posted by: alandgdAlan,Sorry, I didnt understand your question, the expression is unformatted and I am not able to read it. Could you type it again?REgards,Alan

Alan · March 3rd, 2006, 6:03 pm

Alan,I read your .pdf and understand the main idea.But I didn't see a value for the constant C that I asked about.It seems to me you are saying your value for C is not what you expect.Without knowing the value, it's hard to comment further.regards,

alandgd · March 3rd, 2006, 6:18 pm

Alan,I read your reply again, but I am not able to answer your question. To me, as T became larger the fixed leg should converge to 0.04 (the long run variance).Using Numerical integration to calculate the expression (I) at pdf file I got the exact value to fixed leg and the constant C doesnt make sense to me.Sorry if I wasnt clear in my arguments.All the BestAlan

Alan · March 3rd, 2006, 7:17 pm

Ok, I'll ask it a different way.In your .pdf you have expression (I) which you have calculated numerically, right?What are your calculated values when T = 1, 10, 100, 1000?

alandgd · March 3rd, 2006, 7:40 pm

Exactly I solved numerically expression (I) to get swaps rate belowUsing my matlab code and BCC parameters I got the following values. Just to remember that T is measured in year, so T =1 implies swap maturity time occurs after 1 yearT = 1 Swap Rate = 0.18324473019382T= 10 Swap Rate = 0.67752551308033In my opinion the swap rate should converge to 0.20 (sqrt(0.04)), but it doesnt. It seems the matlab code has some problems . What do you think?

Alan · March 3rd, 2006, 8:18 pm

I just tried this myself in Mathematica: here are my results:T Your expression (I) Your expression/sqrt(T)1 0.187428 0.18742810 0.614996 0. 194479100 1.99301 0.1993011000 6.32229 0.199928See how it works and what I was talking about earlier?If not, imagine that volatility is a constant V0 and look at the -definition- for <X>_T. It's V0 T, right, -not- V0. edit: fixed a bad digit.

alandgd · March 3rd, 2006, 8:54 pm

Let me organize my ideas,If my aim is calculate the fixed leg of volatility swap I have to compute two quantities1)Numerically integrate the expression (I), 2)Divide the above result by for sqrt(T) Am I right?

Alan · March 3rd, 2006, 9:39 pm

If the market convention defines the vol. payoff asbased upon the -annualized- realized volatility (even when the contractlength is greater than one year), then the answer is yes.You could certainly enter a swap contract with somebody andmutually agree to a payoff without the sqrt{T} factor.What is the swap dealer's convention? My reading of the: "2004 AMERICAS MASTER VARIANCE SWAPCONFIRMATION AGREEMENT" says that it's annualized, butperhaps somebody on the board could confirm that. A final comment: even though the annualized realized volatility is converging to the long run mean, it would be wrong to thinkthat the volatility risk is disappearing as T -> infinity. Its analogousto the mistake of thinking the risk of being in the stock marketdisappears even if the long-run growth rate converges tosome mean.

alandgd · March 7th, 2006, 9:55 pm

Alan,Thank you for your valuable suggestions. By the way, some months ago, when I was a graduate student, you kindly sent a Xerox form a book of yours. This one helped me to finish a chapter, once again, many thanks. Besides have been studying swap volatility I wrote a paper based on my master thesis about asset pricing in general equilibrium and if you dont mind Id like to send it to you. Id appreciate very much your commentsThank in advance Alan

Alan · March 8th, 2006, 3:09 am

You're very welcome, Alan. If the xerox was because my book was out of print, it's back in stock now. Certainly email me your paper.regards,

greg2 · March 29th, 2006, 4:15 pm

Did you manage to evaluate the integral numerically?I can't find the same result as Gatheral for the same parameters, my result seems to be very instable.What method do you use ?