Reading 64 - Section 6.1.3 How can there be an arbitrage opportunity if the calculated price of the option is based on the assumptions of u and d? Couldn’t you very easily change the u and d assumptions to equate the model price with the market price? Thanks.
Up move (u) and down move (d) are not arbit values but are based on volatility assumptions so if there is a change in the volatility estimate then only u and d can be changed. I think u = exp(volatility*sqrt(T)). So I think your question should be whether we can change the volatility assumption. I think we can but not sure. Its the same question we face in BS model, that is what volatility estimate to use in the formula.
Kabhii, where did you get this formula for u? Just curious to know the source.
I think its there in John C. Hull options futures book you can also go to the following URL: URL: http://en.wikipedia.org/wiki/Binomial_options_pricing_model
bigshibu Wrote: ------------------------------------------------------- > Reading 64 - Section 6.1.3 > > How can there be an arbitrage opportunity if the > calculated price of the option is based on the > assumptions of u and d? Couldn’t you very easily > change the u and d assumptions to equate the model > price with the market price? Thanks. I disagree here that there is an arbitrage in the sense that capturing the difference in value is not risk-free. The value in the traded option price would in fact be using a different value for the volatility of the underlying security, that is, it has an implied volatility that differs from that of the option being priced. Nonetheless, I believe the text is referring to a relative value arbitrage. What the text conveniently leaves out is the formula for u and d, which imo would have helped in getting a better understanding in this section as to how one arrives at the future value of S+ and S-. As one can plainly see, S+ and S- are a direct function of volatility and time. This example assumes a volatility that is lower than the implied volatility being priced in to the option.
I don’t know about that…those formulas for S+ and S- seem pretty arbitrary because the exact formula doesn’t matter. You can come up with an infinite number of S+ and S- for any vol. Explaining why is beyond the curriculum.
Joey As per the formula if we use any volatility estiamte then we can surely get different S+ and S- values but does it make sense to use any volatility estimate. As per my info there are some models which are used for predicting future volatility estimates: 1. Historical estimation method 2. GARCH 3. EWMA There might be other models, in fact there is a whole chapter on the volatility estimation model in Hull. Now which estimate to use for a volatility is dependent on our view of future volatility. As far as implied volatility is concerned its not a estimate derived from historical prices but obtained from current market price of options. But if a trader thinks that the implied volatility is not the correct estimate of vol to use then he might make some profit by doing an arbitrage. Darkhelmet, I think I have answered your question also.