Bayesian decision theory

Idea: One forecast, opposite trades

Both desks see a $40\%$ chance of a rise. Desk A makes $30$ or loses $10$; desk B makes $10$ or loses $30$.

• Same belief, opposite actions
• Decision theory links a belief to an action

Formula: The rule

$\rho (a)=E[L(a,\theta )\mid D],\text{choose}\mathrm{argmin}\rho (a)$

Average each action’s loss over the posterior, then take the action with the smallest value.

Example: To hedge or not

Calm $\frac{1}{2}$, rough $\frac{3}{8}$, crash $\frac{1}{8}$. Unhedged loses $0$, $6$, $20$; a hedge costs $4$ always. Expected loss unhedged?

1. The hedge costs $4$ in every state
2. Unhedged: $\frac{3}{8}\times 6+\frac{1}{8}\times 20=4.75$, so hedge

Formula: The break-even chance

$\text{trade}\text{when}p>\frac{l}{w+l}$

A trade wins w with chance p, else loses l. Desk A needs 1/4, desk B needs 3/4.

Quick check: Desk A’s threshold

A trade makes $30$ with chance $p$, else loses $10$. Above what $p$ is it worth taking?

Idea: The loss picks the estimate

Reporting one number is a decision too.

• Squared loss $(a-\theta {)}^2$: the posterior mean
• Absolute loss $|a-\theta |$: the median
• Cost $s$ short and $o$ over: the $\frac{s}{s+o}$ quantile

Example: How many lots to stock

A client wants $5$, $10$ or $15$ lots with chances $\frac{1}{4}$, $\frac{1}{2}$, $\frac{1}{4}$. A lot short costs $4$, a spare lot $1$. Stock how many?

1. Target $\frac{4}{4+1}=\frac{4}{5}$
2. Cumulative: $\frac{1}{4}$, then $\frac{1}{4}+\frac{1}{2}=\frac{3}{4}$, then $1$
3. First level to reach $\frac{4}{5}$: $15$

Idea: What information is worth

Perfect knowledge caps what any signal can be worth.

• Blind: $\frac{2}{5}\times 30-\frac{3}{5}\times 10=6$
• Knowing the outcome: $\frac{2}{5}\times 30=12$
• Cap: $12-6=6$; noisy signals are worth less

Common mistake: A point forecast in the loss

An option pays $\max (\theta -10,0)$, with $\theta$ at $8$ or $14$. The mean $11$ suggests a value of $1$.

Instead Average the payoff instead: $\frac{1}{2}\times 4=2$. They agree only when the loss is linear in $\theta$.