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For starters, one can argue they provide a better fit to the distribution of asset returns than a Normal distribution simply because stable distributions allow for more degrees of freedom. I had a discussion with a very well-known financial mathematician on the subject of using stable distributions as the return process for derivatives pricing, and his ...

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finally: in $t=0$: sell 19 Bonds, sell 2 calls for $V_0=0.5$, buy 2 shares for 10 => $$P_0=19+2*0.5-20=0$$ in $t=1,S_1=10$ $$P_1 = 2*10-19.9481-2*0=0.069>0$$ in $t=1,S_1=11$ $$P_1=2*11-19.9481-2*1=0.069>0$$ gosh, that took ages.. Thanks anyway!

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it's a complete market so the price of the call option is determinable. There are multiple ways to do it. If the strike is 10 it is particularly easy since the pay-off is $S_1 - 10$ so the value is $S_0 - 10/\beta$ as we can replicate precisely with one unit of $S$ and cash that will be worth $-10$ at time $1$ which $-10/\beta$ units of the cash account ...

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