Hi, is there a detailed document analysing the pnl of a delta hedged option. I am trying to understand the variation in the pnl with a variation in hedged volatility. Went through the paper from wilmott and it talks about implied and realised case with a cursory mathematical explanation. Is there a paper or a book that gives a rigorous/thorough treatment to this subject.
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PnL of a delta hedged option
- Thread starter gver
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There is no more rigorous treatment then realising that pricing is done under the RN measure, while the hedge portfolio evolves under the historical measure.
In the simplest case of the BS, this implies that implied vol should always equal the historical and the realized vol. Then if you compute your delta further assuming that the stock's return is risk free rate, then you are delta hedged irrespectively of the actual return, while your theta should, on average, compensate for gamma.
In reality, option explicitly trades vol with its movement driven by supply demand, so delta hedged position will have residual vol risk. You are, essentially, a non-linear portfolio manager.
In the simplest case of the BS, this implies that implied vol should always equal the historical and the realized vol. Then if you compute your delta further assuming that the stock's return is risk free rate, then you are delta hedged irrespectively of the actual return, while your theta should, on average, compensate for gamma.
In reality, option explicitly trades vol with its movement driven by supply demand, so delta hedged position will have residual vol risk. You are, essentially, a non-linear portfolio manager.
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