Implied Probability Distribution
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 [[alternative HTML version deleted]]
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