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laoliyan2
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Posts: 15
Joined: January 2nd, 2018, 1:33 pm

hedge with implied volatility

January 21st, 2018, 2:52 pm

why we can  write down the following equation?
Please give me some advice.
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QQ图片20180121225513.png
 
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bearish
Posts: 5188
Joined: February 3rd, 2011, 2:19 pm

Re: hedge with implied volatility

January 21st, 2018, 8:10 pm

So, the interpretation of P(S(t),I(t),t) is as the conditional expectation of I(T), as of time t (<T) given the current values S(t) and I(t). Since it is a conditional expectation process, it's a martingale and thus has expected increments equal to zero, which explains the right hand side, of the underlines equation. The left hand side follows from applying Ito's lemma to P and substituting in the drift and volatility terms of S(t) and I(t). I(t), in fact, has zero instantaneous volatility (quadratic variation) and thus no second order Ito term. 
 
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laoliyan2
Topic Author
Posts: 15
Joined: January 2nd, 2018, 1:33 pm

Re: hedge with implied volatility

January 22nd, 2018, 6:59 am

So, the interpretation of P(S(t),I(t),t) is as the conditional expectation of I(T), as of time t (<T) given the current values S(t) and I(t). Since it is a conditional expectation process, it's a martingale and thus has expected increments equal to zero, which explains the right hand side, of the underlines equation. The left hand side follows from applying Ito's lemma to P and substituting in the drift and volatility terms of S(t) and I(t). I(t), in fact, has zero instantaneous volatility (quadratic variation) and thus no second order Ito term. 
dt is in every part,  where is dX?
 
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User avatar
bearish
Posts: 5188
Joined: February 3rd, 2011, 2:19 pm

Re: hedge with implied volatility

January 22nd, 2018, 11:58 am

So, the interpretation of P(S(t),I(t),t) is as the conditional expectation of I(T), as of time t (<T) given the current values S(t) and I(t). Since it is a conditional expectation process, it's a martingale and thus has expected increments equal to zero, which explains the right hand side, of the underlines equation. The left hand side follows from applying Ito's lemma to P and substituting in the drift and volatility terms of S(t) and I(t). I(t), in fact, has zero instantaneous volatility (quadratic variation) and thus no second order Ito term. 
dt is in every part,  where is dX?
 
The left hand side is the expected value of dP.