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aligama
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vega of Asian Option

July 4th, 2002, 1:05 pm

there is any reasonable explanation for a negative vega of an Asian option, coming out from a Monte Carlo simulation pricing function?More generally, I' m wondering if there is a specific literature on the vega for Asian Option
 
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Collector
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vega of Asian Option

July 5th, 2002, 2:24 am

What type of Asian option? A standard Asian should have positive vega I believe. Let's keep it simple and assume geometric average Asian. Such an option can be valued using black-scholes replacing the vol of the continuous average with SpotVol/sqrt(3).The vega function will still be the same, that is positive. Are you sure you not simply are getting wrong vega values because inaccurate MC simulation. ?? or are you looking at some strange type of Asian option, barrier Asian for example?
Last edited by Collector on July 4th, 2002, 10:00 pm, edited 1 time in total.
 
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Chukchi
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vega of Asian Option

July 5th, 2002, 5:11 am

Most probably Vega sometimes could be negative for Asian options.Indeed for Geometrical Asians not only volatility 'Sigma' should be substituted by effective Vol=Sigma/Sqrt[3] but also the divident 'q' should become effective divident Q=(r+q+(Sigma^2)/6)/2.When you start to differentiate the usual Black-Scholes formula to get Vega all the terms containing effective divident Q should produce additional terms. After standard magical cancellations you'll be left with just two terms that have different signs.The negative term is -S*Exp[-Q*t]*N[d1]*Sigma*t/6.It could be large enough to overpower the positive term that is proportional to n[d1]*Sqrt[t/3]*S*Exp[-Q*t].
 
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mj
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vega of Asian Option

July 5th, 2002, 8:25 am

how are you computing the vega? it sounds like a bug to me.if you are computing it bumping the inputs make sure you are using the same random numbers for eachsim. for greater accuracy use the Broadie-Glasserman likelihood ratio method see P. Jaeckel's book for exampleif you post the precise parameters and outputs you have i can easily check if you have a bugmj
 
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aligama
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vega of Asian Option

July 5th, 2002, 11:32 am

To be more specific, I found this result for Asian basket option dicrete arithmetic average with a floor and making the valuation of the vega close to the end of inquiry dates and having a very large Historical increment.In the following attach you can find some thought about the problem, trying to give an explanation to the result.
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negative vega.zip
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Collector
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vega of Asian Option

July 5th, 2002, 12:08 pm

yeah Chukchi is natural right, stupid me forgot the vol in the dividend adjustment.here I have a simple Asian option Java calculator, including vega:Asian Java
 
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Alan
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vega of Asian Option

July 6th, 2002, 5:18 am

there is any reasonable explanation for a negative vega of an Asian option, coming out from a Monte Carlo simulation pricing function?More generally, I' m wondering if there is a specific literature on the vega for Asian Option >>I vote for the vega of the (continuous) arithmetic average option, with a put or call payout, alwaysbeing positive (well non-negative to be really picky).I haven't done this carefully, but I believe here's how you prove it. You start withthe PDE developed by Roger's and Shi. For example, for the Asian call option, they show C = S f(x,t),where 0 = df/dt + (1/2)sig^2 x^2 (d^2 f/dx^2) - (r x + 1/T) df/dx with terminal conditionf(x,T) = -min[x,0]. Also x= (K-Sbar)/S, where K=Strike and Sbar is the average-to-date for a seasoned optionand 0 for a fresh option.Anyway, what you want to prove is that g(x,t) = df/dsig >= 0. To do that, you basically follow thetype of argument used in, for example, Romano & Touzi (1997, Math. Finance, 7, 399-412).Briefly, the method is to differentiate the PDE with respect to sig. This gives you a new PDEfor g. You write a Feynman-Kac style solution for that PDE, which shows that g(x,t) has the same sign as d^2f/dx^2.Then you go through the same exercise for h(x,t) = df/dx and k(x,t) = d^2f/dx^2, getting PDEs for them. It finallyboils down to d^2f/dx^2 being non-negative for all (x,t) if the payoff is convex (which it is for puts/calls).Hence, vega must be non-negative. Regards,Alan
 
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Collector
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vega of Asian Option

July 6th, 2002, 12:38 pm

Never say never, there is always an exception , for a Geometric average I beleive option supersymmetry is holding up, that is you can price a call as a put with negative vol and multiply result with minus one, and opposite. In this case an increase in vol will be a reduction in vol, for example from -60% to -59% vol, and the option value will naturally go down in value. So it all depends if we are in the matter or antimatter world!! It is nothing mathematical wrong with this, well except we probably also should change the definition of a vol increase to be a negative one, in that case I assume we all agree with Alan the mastermind of stocahstic vol
Last edited by Collector on July 5th, 2002, 10:00 pm, edited 1 time in total.
 
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Chukchi
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vega of Asian Option

July 7th, 2002, 1:39 am

Ordinary seasoned arithmetic Asian options are known to be exposed to the knock-out possibilities.The knock-out barrier condition looks like t*Sbar > K*T. It is an Asian Put that is worhless at this barrier.The intuition for a possiblity of negative Vega is similar to that of the ordinary knock-out barrier options.When 'Sbar' is close to K*T/t the aged Asians behave more or less as ordinary barrier options.It could mean that it is possible to expect a negative Vega and/or Gamma.
 
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Alan
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vega of Asian Option

July 7th, 2002, 2:38 pm

Thanks for the book plug, Collector. I have attached a .gif fileshowing Vega vs. volatility for two cases: the geometric Asiancall and the Arithmetic Asian call. The parameters are S = K = 100,T = 1 year, r = 15%. The call is freshly written. The x-axisis various volatilities sigma running from 1% to 10%. The lowergrahp is the geometric case, confirming, as Chukchi has arguedthat vega can be negative for that. The upper is the arithmetic case,which is positive throughout. I still believe this one is alwayspositive, but the graph is not an example of close-to-the barrier. (The arithmetic values were computed using Giles Thompson's excellentlower bound approximation).Regards,Alan
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TwoVegas.zip
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Collector
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vega of Asian Option

July 8th, 2002, 1:49 am

Since we are speaking about Asian options, there are rumors of P. W. visiting Asia, is it reflected in the market? In other words have implied Asian vols gone up? sorry if I am off topic again
Last edited by Collector on July 7th, 2002, 10:00 pm, edited 1 time in total.
 
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Energetic
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vega of Asian Option

July 8th, 2002, 3:25 pm

I agree with Chukchi. Increasing volatility makes the KO event more probable. Therefore, negative vega is possible.
 
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gyuan1
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vega of Asian Option

October 27th, 2002, 8:43 pm

Thanks Aligama for your message. Do you have a few lines VBA or C++ codes to price American type Asian options and their greeks?Thanks from George. Here is my e-mail address: [email protected]