Sunday, July 3, 2016

implied volatility - Calendar Arbitrage in a Vol Surface


I am trying to determine the condition such that my implied vol surface doesn't have calendar arbitrage. I have done research and found that one such condition is that total variance should increase along the time axis. However, I want to find a different condition using the call option price or forwards, or something to that extent.


Furthermore, I do not want to assume proportional dividends, same forward moneyness, etc. The information I do know is option price and forward prices.



My approach is something as follows. Let X and Y be unknowns. at t=0, I would need to pay (or receive) XC(t1)+YC(t2) where C(T)=exp(rT)BS(FT,K,T,r,σ). Note that I am assuming that we are working with the same strike K. Let X=1 for simplicty. At t=1, if $S_{t_1}K,thenmypayoffwouldbeS_{t_1}-K-Y\exp(rT_1)C(t_2)$.


I'm not sure how I would continue my argument from here, though perhaps I want to use the fact that C(t)exp(rt)(FtK). I know I would first need to find out the quantity of Y first.


Any help would be greatly appreciated. Jim




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