Aha! (1+V)/(1-V) looks like it’s decaying to −1 at negative V, and blows up at V=1 - meaning it’s risk loving. So we went from risk neutral to risk loving, and to compensate we made bad outcomes more likely.
That is, the analogy to the risk neutral measure seems to check out: a JB rotation increases your risk tolerance, and compensates by increasing the probability of the bad outcomes.
Specifically, if you take the derivatives you get 2/(1-V)^2 V’ for the first and 2/(1-V)^2 (V″ + 2V’^2) for the second. The (negative) ratio is -V″/V’ − 2 V’, and -V″/V’ is often used as a measure of risk aversion. If instead we divide by V’^2 we’d get -V″/V’^2 − 2. A log utility a + bln(x) has a constant value of -V″/V’^2 equal to 1/b, so this is saying that we reduce this measurement of risk aversion by a constant.
I think you should be able to write the risk neutral measure from finance this way, but it’s not obvious to me how to do so, and I don’t know enough mathematical finance to easily deconfuse myself here.
Aha! (1+V)/(1-V) looks like it’s decaying to −1 at negative V, and blows up at V=1 - meaning it’s risk loving. So we went from risk neutral to risk loving, and to compensate we made bad outcomes more likely.
That is, the analogy to the risk neutral measure seems to check out: a JB rotation increases your risk tolerance, and compensates by increasing the probability of the bad outcomes.
Specifically, if you take the derivatives you get 2/(1-V)^2 V’ for the first and 2/(1-V)^2 (V″ + 2V’^2) for the second. The (negative) ratio is -V″/V’ − 2 V’, and -V″/V’ is often used as a measure of risk aversion. If instead we divide by V’^2 we’d get -V″/V’^2 − 2. A log utility a + bln(x) has a constant value of -V″/V’^2 equal to 1/b, so this is saying that we reduce this measurement of risk aversion by a constant.
I think you should be able to write the risk neutral measure from finance this way, but it’s not obvious to me how to do so, and I don’t know enough mathematical finance to easily deconfuse myself here.