A noisy signal when the odds are lopsided

Question

An asset is worth \100 with probability 0.3 and \0 with probability 0.7. Its fair price is therefore \30, and you can buy it for exactly that. Before deciding, you may buy a signal that is 80\% accurate: it says "high" with probability 0.8 when the asset is worth \100, and "low" with probability 0.8 when it is worth \0. What is the most you should pay for the signal?

Accepted Answer

Without the signal. Buying at \30 has expected value 0.3(100) - 30 = 0. Fair price, so you break even, expected profit \0. Signal probabilities. Because the prior is lopsided, so are the signal readings: P(	ext{high}) = 0.3(0.8) + 0.7(0.2) = 0.24 + 0.14 = 0.38, \qquad P(	ext{low}) = 0.62. Posterior after "high": P(V = 100 \mid 	ext{high}) = \frac{0.24}{0.38} \approx 0.632, so buying is worth 0.632(100) - 30 \approx +\33, you buy. Posterior after "low": P(V = 100 \mid 	ext{low}) = \frac{0.7(0.2)}{0.62} = \frac{0.06}{0.62} \approx 0.097, so buying is worth 0.097(100) - 30 \approx -\20, you …

A noisy signal when the odds are lopsided

Your answer

More Market Making questions