What are the important observations about hedging error
What are the important observations about hedging error?
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We can make several significant observations about hedging error.
• It is large: it is O(δt), that is the same order of magnitude as all other terms in the Black–Scholes model. This is typically much bigger than interest received upon the hedged option portfolio.
• On average this is zero: hedging errors are balance out.
• This is path dependent: the bigger gamma, the larger the hedging errors.
• The entire hedging error has standard deviation as of √δt: sum of total hedging error is your last error when you get to expiration. When you want to halve the error you will have to hedge four times as frequently.
• Hedging error is drawn by a chi-square distribution: it’s what φ2 is
• When you are long gamma you will lose money around 68% of the time: it is chi-square distribution in action.
But while you make money this will be from the tails, and big sufficient to provide a mean of zero. But short gamma you lose only 32% of the time, if they will be large losses
• In practice φ is not commonly distributed: the fat tails, high peaks we notice in practice, will make the above observation still more extreme, perhaps a long gamma position will lose 80 percent of the time and win only 20percent. But the mean will be zero.
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