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Hi All, hope that you can help me with the following confusion:
Assume zero discount rate and suppose a modeller prices a 1-year call option on a stock using the BS model with vol v1. It turns out that the true dynamics of the stock is indeed a geometric brownian motion, but with vol v2 != v1, and with drift zero under the real world measure (the drift is not really relevant for this question).
Also assume that continous time heding is possible. Now the "assumed" model and the true model, though having different vols, should still have the same path space, namely the set of continous positive functions over the time horizon, if the initial stock price is positive.
The theory says, if one follows the BS hedge in continous time, one would be almost surely replicate the call option payoff perfectly if the stock does follows a GBM with vol v1. On the other hand, although in our case the true vol is v2, almost surely each possible sample path under the true model is also a possible path under the assumed model. If that's the case, wont we able to attain perfect hedge even the vol used is wrong (because when I hedge, all that matters is the realized sample path I am looking at)?
Thanks,
TW
Assume zero discount rate and suppose a modeller prices a 1-year call option on a stock using the BS model with vol v1. It turns out that the true dynamics of the stock is indeed a geometric brownian motion, but with vol v2 != v1, and with drift zero under the real world measure (the drift is not really relevant for this question).
Also assume that continous time heding is possible. Now the "assumed" model and the true model, though having different vols, should still have the same path space, namely the set of continous positive functions over the time horizon, if the initial stock price is positive.
The theory says, if one follows the BS hedge in continous time, one would be almost surely replicate the call option payoff perfectly if the stock does follows a GBM with vol v1. On the other hand, although in our case the true vol is v2, almost surely each possible sample path under the true model is also a possible path under the assumed model. If that's the case, wont we able to attain perfect hedge even the vol used is wrong (because when I hedge, all that matters is the realized sample path I am looking at)?
Thanks,
TW