A fill just changed what the stock is worth

Question

You quote around a fair value of 50.00. The stock is really worth either 50.10 (good news pending) or 49.90 (bad news pending), each equally likely. Half the traders you face are informed (\alpha = 0.5) and trade on the true value; the rest are noise, buying or selling at random. Someone just lifted your offer (bought from you). What is your updated fair value, and where should you re-center your quotes?

Accepted Answer

Why a buy is informative. A buy is more likely to come when the value is high (informed buyers appear only then). So "they bought" is evidence the value is 50.10, and your fair value must tick up. Compute how much. Probability of a buy under each value. A buy happens if it's an informed trader and value is high, or a noise trader choosing to buy (50\%): P(	ext{buy} \mid 50.10) = \alpha + (1-\alpha)	frac12 = 0.5 + 0.25 = 0.75, P(	ext{buy} \mid 49.90) = 0 + (1-\alpha)	frac12 = 0.25. Posterior (Bayes, equal priors). P(50.10 \mid 	ext{buy}) = \frac{0.75}{0.75 + 0.25} = 0.75. Updated fair val…

A fill just changed what the stock is worth

Your answer

More Market Making questions