Update: Now I think physical probability is not observable and utterly unusable in the math including the numerical methods. In contrast, RN probabilities can be derived from observed prices.

Therefore, now I feel physical measure is completely irrelevant to option math.

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RN measure is the "first" practical measure for derivative pricing. Most theories/models are formulated in RN measure. T-Forward measure and stock numeraire are convenient when using these models...

Therefore, now I feel physical measure is completely irrelevant to option math.

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RN measure is the "first" practical measure for derivative pricing. Most theories/models are formulated in RN measure. T-Forward measure and stock numeraire are convenient when using these models...

Physical measure is an impractical measure for pricing. Physical measure is personal feeling, not related to any market prices. Physical measure is mentioned only for teaching purpose. There's no "market data" on physical measure.

Market prices reflect RN (not physical) probabilities.

Consider cash-or-nothing bet that pays $100 iff team A wins a playoff. The bet is selling for $30, so the RN Pr(win) = 30%. I am an insider and I rig the game so physical Pr() = 80% and Meimei (my daughter) may feel it's 50-50 but these personal opinions are irrelevant for pricing any derivative.

Instead, we use the option price $30 to back out the RN probabilities. Namely, Calibrate the pricing curves using liquid options, then use the RN probabilities to price less liquid derivatives.

Professor Yuri is the first to point out (during my oral exam!) that option prices are the input, not the output to such pricing systems.

Market prices reflect RN (not physical) probabilities.

Consider cash-or-nothing bet that pays $100 iff team A wins a playoff. The bet is selling for $30, so the RN Pr(win) = 30%. I am an insider and I rig the game so physical Pr() = 80% and Meimei (my daughter) may feel it's 50-50 but these personal opinions are irrelevant for pricing any derivative.

Instead, we use the option price $30 to back out the RN probabilities. Namely, Calibrate the pricing curves using liquid options, then use the RN probabilities to price less liquid derivatives.

Professor Yuri is the first to point out (during my oral exam!) that option prices are the input, not the output to such pricing systems.

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