[R-SIG-Finance] Implied Probability Distribution

[Krishna Kumar]

> Using non overlapping data of one month call option I have estimated the
> implied volatility using the GBSVolatility of fOptions. Using these
> implied volatilities I have obtained delta using GBSGreeks of the same
> package. 
>
> Using the natural splines I have plotted the implied volatility and
> delta.
>
>  
>
> I have two questions (this might be trivial but would be glad for
> insight):
>
> 1) How do I extrapolate to obtain the tail?
>   
I am afraid there is no easy answer to this either you could flat 
forward extrapolate or use your fitted functional form/spline to 
extrapolate and obtain the vols for those strikes in the wings. 
Stability is the key and using something that is too flexible could 
sometimes hurt in this case.

> 2) How do I back out the option price? (We can use the Black- Scholes
> model (GBSOption of fOptions), however my understanding is it requires
> implied volatility and strike, but we have implied volatility and delta)
>
>
>   
I couldn't find a version of the Breeden & Litzenberger paper but here 
is an easy read on this use eqn 3 on Page-4. 
http://www.bcb.gov.br/ingles/estabilidade/2002_nov/ref200201c62i.pdf

In essence you use a continuum of Call prices and take the 2nd 
derivative with respect to Strike(K) and for this take the centered 
difference
for starters there are some higher order differences that you can mess 
with later.

Also if you have the delta you can always map that to a strike K using 
the formula in http://www.mathfinance.org/formulas_u/Vanilla/node20.html
If you need further help it would help if you provide further 
information and possibly the data that you are using for this list to be 
of further help.

Best
Krishna





>  
>
> Thank you in advance for the help.
>
> Regards,
>
> Ravi
>
>
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