Matt Wolf
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 Jun 15 comment Physical or Real-world Probability Measure ...in the same probability space you can have random variables that have two expectations, "one under the original probability measure $P$..., and the other under the new probability measure $\tilde P$.,... (Shreve, Stochastic Calculus for Finance II, 2004 Edition, P.210). Both probability measures live in the same probability space. Jun 15 comment Physical or Real-world Probability Measure ...but admittedly the dice example is not a good one, I apologize for that. But it does not change my stance that in the same probability space you can have random variables that have two expectations, "one under the original probability measure $P$..., and the other under the new probability measure $\tilde P$.,... (Shreve, Stochastic Calculus for Finance II, 2004 Edition, P.210). Both probability measures live in the same probability space. Jun 15 comment Physical or Real-world Probability Measure that is not what I said. But to clear up the terminology confusion I seemingly have, you are basically saying both dice experiments are underlying one and the same identical real-valued probability function (because that is how your cited Wiki article defines a probability measure. Jun 15 comment Physical or Real-world Probability Measure I am happy to stand corrected, could you please help to clear up my confusion in case I am missing something or are we just rubbing elbows in terminology space? Jun 15 comment Physical or Real-world Probability Measure what about the example of a 6-sided vs 10-sided dice? Both reside in the real world, both in the same probability space, yet they both represent 2 very different probability measures, meaning, both represent their own real valued functions, defined on events in a shared probability space, and both satisfy measure properties. Jun 15 comment Physical or Real-world Probability Measure @AFK, I guess I am confused what you are trying to say here. Are you saying $P$ is the probability measure belonging to probability space ($\Omega, F, P$) and $\tilde P$ belongs to probability space ($\Omega, F, \tilde P$)? I am not a mathematician but this seems to disagree with how Shreve in his book Stochastic Calculus for Finance II defines probability spaces and measures. Jun 14 comment Physical or Real-world Probability Measure @Ulysses, one probability space ($\Omega, F, P$) already has two probability measures, $P$ and $\tilde P$ Jun 12 comment Physical or Real-world Probability Measure Fully concur that terminology is key here. It is easy to get confused about probability space, probability measures, driving brownian motions/ random variables. And I agree that for the purposes OP described all 3 assets (domestic money market account, stock, and foreign money market account) do not have to be martingales. Jun 12 comment Physical or Real-world Probability Measure My point was that the precise reason risk-neutral probability measures are used is that there are uncountably many probability measures due to different risk preferences. As long as a probability measure lies between [0;1], the entire probability space sums to 1 and the empty set is 0 and the countable additivity property is satisfied you have a probability measure. There is one probability space but there can and there are many probability measures. Jun 12 comment Physical or Real-world Probability Measure ...finding such is what drives market practitioners to taking the detour via risk-neutral pricing. Jun 12 comment Physical or Real-world Probability Measure every single random variable already has two expectations, one under the "real world" probability measure, one under a "new" probability measure. Radon Nikodym comes to mind and the same Radon Nikodym derivative can be applied to a whole derivative process. So maybe I misunderstood and we are talking on one hand about many different expectations vs a single probability measure. Certainly, it might be possible to not just derive a single additional "risk-free" probability measure but several such, but yes, only one "real world" probability measure exists... Jun 12 comment Parametrizing the Radon Nikodym I think what might be referred to here is the "change of measure" process. You can move a real random variable from a "real" probability space to a new probability space by defining a new random variable within the new probability space. Jun 12 comment Physical or Real-world Probability Measure so then let me ask you, how do you discount your terminal payoff when you price a derivative under the real-world probability measure ? Jun 12 comment Physical or Real-world Probability Measure Can you elaborate on your statement because in its current form I would tend to disagree. There are many real world probability measures, in fact there are countless real world probability measures even for a single asset hence the interest of market participants to price in risk neutral terms. Real world probability measures are a function of utility. Jun 12 comment Calculating the rate of return over a year then the data for a year before does not exist Very simple, you calculate the absolute return over the two data points and then convert to an annualized rate...done. Jun 11 comment Black-Scholes under stochastic interest rates is this homework or an assignment? Jun 11 comment Distribution of Black Scholes call option price at time 0