A confusion needed to be resolved

[forwardlooking]

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

 

[koupparis]

The distribution of final stock prices will look very different for different vols. Think of N(0,1) vs N(0,10). So yes you may hit the same final stock price, but the frequency of the hit (or the chance you'll hit it) is very different and hence determines different prices too.

 

[HyperVolatility]

you could potentially get the same price but in terms of returns distributions you are talking about 2 different worlds.

Look at it in terms of REAL PORTFOLIO RETURNS

accademically speaking you could get the same price even using different vols but in real terms a wrong volatility estimation would cost you a fortune

 

[financeguy]

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

ok just came across this post.. i think the following is what you're looking for, the answers you've been given seem to not quite hit the spot do they

so i think the answer is that gamma and implied vol level have an inverse relationship (so the higher the vol, the less gamma you have - think about why this is the case!).. basically what that means is that the level of implied vol you model with directly affects the delta you see and trade off of and will therefore affect your pnl

so lets go through an example.. say you've bought this option at v1, and trade it as if the true vol is v1, which is less than the true vol v2.. so the spot is really more volatile than you thought it would be.. spot goes up and you've got some spot to sell naturally as you're long gamma from buying the option, but how much spot do you have to sell to be delta neutral again? so you have more gamma than you should have (since v1<v2), so you've got more spot to sell with v1 than with v2.. what does this mean? this means that as spot runs higher, as it will since the real vol is v2, you are less long spot than you should be which means you're making less money as spot runs further! essentially you're taking profit too quickly and not letting your deltas run long enough.. if you had traded off of vol v2, you wouldnt have sold as much spot lower down and would be able to sell more higher as you'd have had less gamma.. so if you buy this option at v1 with the plan to delta hedge it and make money as spot realizes a higher vol, you've got to trade the gamma as if it's worth v2 otherwise you wont realize all that spot volatility in your delta hedging

there's also the subtlety of having non-zero vanna (d delta / d vol) outside of the ATM point, but this is a much smaller effect than gamma of course

does this make sense?