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DoubleTrouble
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Compare distribution implied by Dupire's Formula with Black-Scholes

April 30th, 2012, 11:42 am

Hello!I have calculated the implied risk-neutral density of by using Dupire's formula (or whatever you want to call it ):I have market data at time T consisting of 9 different call prices corresponding to 9 different strike prices, volatilities and the rate at time T. I estimated the second derivative using these data points. Now I would like to compare it to the distribution of in the Black-Scholes model. Since is log-normal that is:it is easy to derive an explicit expression for the density function of Now here comes my question. I'm unsure about how to pick the parameters in this equation. Since I shall compare it with the implied density it must be reasonable to have r as the market rate at time T, S0 as the stock price at the first day of the option contracts. But how should I pick sigma? I have 9 different market volatilities. Is it a good idea to just take the mean of these?I appreciate any input!Thank you in advance!
Last edited by DoubleTrouble on April 29th, 2012, 10:00 pm, edited 1 time in total.
 
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foxkingdom
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Compare distribution implied by Dupire's Formula with Black-Scholes

April 30th, 2012, 12:07 pm

I would pick the near ATM one. I think ATM options are more liquid ones and contain more correct information.
 
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DoubleTrouble
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Compare distribution implied by Dupire's Formula with Black-Scholes

April 30th, 2012, 12:24 pm

QuoteOriginally posted by: foxkingdomI would pick the near ATM one. I think ATM options are more liquid ones and contain more correct information.Thank you for your input. And yes, that seems quite reasonable!
 
frolloos
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Compare distribution implied by Dupire's Formula with Black-Scholes

April 30th, 2012, 12:31 pm

why the ATM? if you're comparing the fit of 2 distributions to market prices shouldn't you fit both distributions? so i think you should choose the value of sigma that makes the BS distribution as close as possible to the market implied distribution. it's going to be a bad fit, but it's the least bad fit if you do the optimization right. it's quite possible that sigma = ATM vol does give the best result in the end, but at least the optimization 'proves' this.p.s. it's the Breeden-Litzenberger formula, not Dupire.
Last edited by frolloos on April 29th, 2012, 10:00 pm, edited 1 time in total.
 
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BramJ
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Compare distribution implied by Dupire's Formula with Black-Scholes

May 1st, 2012, 12:20 am

What's your application here? The fact that you have different volatilities for is just a way of the market telling you that the market doesn't believe in a BS world with constant volatility/ a lognormal risk-neutral distribution. So the only question you can ask is which log-normal distribution minimizes the distance between a log-normal distribution and the risk-neutral distribution the market is pricing with. So if you define a metric, then you could back-out the implied volatility that does this I guess.I never really thought about this, but maybe this volatility would be an interesting measure to look at, in the same sense as some people look at at the money volatility or variance swap strikes to have a single number summarizing the level of volatility. Too lazy to think about it now/just thinking out loud, but the volatility minimizing the distance between the actual market distribution and a lognormal distribution might be characterized in terms of the fair value strike of a volatility or variance swap?