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How to comprehend this notation?
How to calculate annual returns from daily prices?Understanding the solution of this integralCan someone check this boundary condition for me?How to price 0 floors in csa agreements for negative ois rates?Can someone please verify or disprove this Sharpe Ratio math logic for meHow to Mathematically Prove Markets are Price-Discovering?
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I learned mathematical finance from Bjork's Arbitrage Theory in Continous Time, and never once did I encounter the "quadratic variation"-thingy with the angle brackets.
So now that I am reading Bergomi's book on Stochastic Volatility and I run into this monster on the first chapter, you can understand my confusion: 
Please explain whats' going on here. What is an "average covariation"? I cannot find this on wikipedia. I found what a "quadratic covariation" is, but what does it mean intuitively, especially in this context?
In this context, Bergomi says that he wants to equate implied volatility the future realized volatility. Okay, so I get that the implied volatility is hat-sigma and realized volatility is sigma, and he is weighting them by the "dollar gamma" and then he takes an integral because he wants the average over the period [0, T]. Cool .... but why does he then end by taking those angle-brackets? Why not just equate the two integrals? Why is equating the "covariations" or whatever it is necessary here?
finance-mathematics
asked Jun 14 at 17:02
SarahKuchSarahKuch
161 bronze badge
$endgroup$
add a comment
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$begingroup$
I learned mathematical finance from Bjork's Arbitrage Theory in Continous Time, and never once did I encounter the "quadratic variation"-thingy with the angle brackets.
So now that I am reading Bergomi's book on Stochastic Volatility and I run into this monster on the first chapter, you can understand my confusion:
Please explain whats' going on here. What is an "average covariation"? I cannot find this on wikipedia. I found what a "quadratic covariation" is, but what does it mean intuitively, especially in this context?
In this context, Bergomi says that he wants to equate implied volatility the future realized volatility. Okay, so I get that the implied volatility is hat-sigma and realized volatility is sigma, and he is weighting them by the "dollar gamma" and then he takes an integral because he wants the average over the period [0, T]. Cool .... but why does he then end by taking those angle-brackets? Why not just equate the two integrals? Why is equating the "covariations" or whatever it is necessary here?
finance-mathematics
asked Jun 14 at 17:02
SarahKuchSarahKuch
161 bronze badge
$endgroup$
add a comment
|
$begingroup$
I learned mathematical finance from Bjork's Arbitrage Theory in Continous Time, and never once did I encounter the "quadratic variation"-thingy with the angle brackets.
So now that I am reading Bergomi's book on Stochastic Volatility and I run into this monster on the first chapter, you can understand my confusion:
Please explain whats' going on here. What is an "average covariation"? I cannot find this on wikipedia. I found what a "quadratic covariation" is, but what does it mean intuitively, especially in this context?
In this context, Bergomi says that he wants to equate implied volatility the future realized volatility. Okay, so I get that the implied volatility is hat-sigma and realized volatility is sigma, and he is weighting them by the "dollar gamma" and then he takes an integral because he wants the average over the period [0, T]. Cool .... but why does he then end by taking those angle-brackets? Why not just equate the two integrals? Why is equating the "covariations" or whatever it is necessary here?
finance-mathematics
asked Jun 14 at 17:02
SarahKuchSarahKuch
161 bronze badge
$endgroup$
I learned mathematical finance from Bjork's Arbitrage Theory in Continous Time, and never once did I encounter the "quadratic variation"-thingy with the angle brackets.
So now that I am reading Bergomi's book on Stochastic Volatility and I run into this monster on the first chapter, you can understand my confusion:
Please explain whats' going on here. What is an "average covariation"? I cannot find this on wikipedia. I found what a "quadratic covariation" is, but what does it mean intuitively, especially in this context?
In this context, Bergomi says that he wants to equate implied volatility the future realized volatility. Okay, so I get that the implied volatility is hat-sigma and realized volatility is sigma, and he is weighting them by the "dollar gamma" and then he takes an integral because he wants the average over the period [0, T]. Cool .... but why does he then end by taking those angle-brackets? Why not just equate the two integrals? Why is equating the "covariations" or whatever it is necessary here?
finance-mathematics
finance-mathematics
asked Jun 14 at 17:02
SarahKuchSarahKuch
161 bronze badge
asked Jun 14 at 17:02
SarahKuchSarahKuch
161 bronze badge
asked Jun 14 at 17:02
SarahKuchSarahKuch
161 bronze badge
asked Jun 14 at 17:02
SarahKuchSarahKuch
161 bronze badge
asked Jun 14 at 17:02
SarahKuchSarahKuch
161 bronze badge
161 bronze badge
add a comment
|
add a comment
|
1 Answer
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$begingroup$
In physics (statistical physics), this angle bracket is used to represent average, for example, here is the notation from Van Kampen’s book:
And in stochastic calculus, the quadratic variation is usually represented by the same angle brackets. But like he noted the context should make clear which one is meant.
In the equation you have referenced, an average is meant. So the thing inside the brackets is the P&L of a path, and the angle brackets is then computing the average across the paths.
answered Jun 14 at 17:13
Magic is in the chainMagic is in the chain
3,5441 gold badge4 silver badges10 bronze badges
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$begingroup$
If the "thing" in angle brackets is a random scalar (in this case the P&L), it means an average. If the thing is the trajectory of a stochastic process over time, then it is the quadratic variation.
$endgroup$
– Alex C
Jun 17 at 15:49
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1 Answer
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1 Answer
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$begingroup$
In physics (statistical physics), this angle bracket is used to represent average, for example, here is the notation from Van Kampen’s book:
And in stochastic calculus, the quadratic variation is usually represented by the same angle brackets. But like he noted the context should make clear which one is meant.
In the equation you have referenced, an average is meant. So the thing inside the brackets is the P&L of a path, and the angle brackets is then computing the average across the paths.
answered Jun 14 at 17:13
Magic is in the chainMagic is in the chain
3,5441 gold badge4 silver badges10 bronze badges
$endgroup$
$begingroup$
If the "thing" in angle brackets is a random scalar (in this case the P&L), it means an average. If the thing is the trajectory of a stochastic process over time, then it is the quadratic variation.
$endgroup$
– Alex C
Jun 17 at 15:49
add a comment
|
$begingroup$
In physics (statistical physics), this angle bracket is used to represent average, for example, here is the notation from Van Kampen’s book:
And in stochastic calculus, the quadratic variation is usually represented by the same angle brackets. But like he noted the context should make clear which one is meant.
In the equation you have referenced, an average is meant. So the thing inside the brackets is the P&L of a path, and the angle brackets is then computing the average across the paths.
answered Jun 14 at 17:13
Magic is in the chainMagic is in the chain
3,5441 gold badge4 silver badges10 bronze badges
$endgroup$
$begingroup$
If the "thing" in angle brackets is a random scalar (in this case the P&L), it means an average. If the thing is the trajectory of a stochastic process over time, then it is the quadratic variation.
$endgroup$
– Alex C
Jun 17 at 15:49
add a comment
|
$begingroup$
In physics (statistical physics), this angle bracket is used to represent average, for example, here is the notation from Van Kampen’s book:
And in stochastic calculus, the quadratic variation is usually represented by the same angle brackets. But like he noted the context should make clear which one is meant.
In the equation you have referenced, an average is meant. So the thing inside the brackets is the P&L of a path, and the angle brackets is then computing the average across the paths.
answered Jun 14 at 17:13
Magic is in the chainMagic is in the chain
3,5441 gold badge4 silver badges10 bronze badges
$endgroup$
In physics (statistical physics), this angle bracket is used to represent average, for example, here is the notation from Van Kampen’s book:
And in stochastic calculus, the quadratic variation is usually represented by the same angle brackets. But like he noted the context should make clear which one is meant.
In the equation you have referenced, an average is meant. So the thing inside the brackets is the P&L of a path, and the angle brackets is then computing the average across the paths.
answered Jun 14 at 17:13
Magic is in the chainMagic is in the chain
3,5441 gold badge4 silver badges10 bronze badges
edited Jun 14 at 18:11
answered Jun 14 at 17:13
Magic is in the chainMagic is in the chain
3,5441 gold badge4 silver badges10 bronze badges
answered Jun 14 at 17:13
Magic is in the chainMagic is in the chain
3,5441 gold badge4 silver badges10 bronze badges
answered Jun 14 at 17:13
Magic is in the chainMagic is in the chain
3,5441 gold badge4 silver badges10 bronze badges
3,5441 gold badge4 silver badges10 bronze badges
$begingroup$
If the "thing" in angle brackets is a random scalar (in this case the P&L), it means an average. If the thing is the trajectory of a stochastic process over time, then it is the quadratic variation.
$endgroup$
– Alex C
Jun 17 at 15:49
add a comment
|
$begingroup$
If the "thing" in angle brackets is a random scalar (in this case the P&L), it means an average. If the thing is the trajectory of a stochastic process over time, then it is the quadratic variation.
$endgroup$
– Alex C
Jun 17 at 15:49
$begingroup$
If the "thing" in angle brackets is a random scalar (in this case the P&L), it means an average. If the thing is the trajectory of a stochastic process over time, then it is the quadratic variation.
$endgroup$
– Alex C
Jun 17 at 15:49
$begingroup$
If the "thing" in angle brackets is a random scalar (in this case the P&L), it means an average. If the thing is the trajectory of a stochastic process over time, then it is the quadratic variation.
$endgroup$
– Alex C
Jun 17 at 15:49
add a comment
|
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