Risk-neutral measures

Idea: A forecast and a price

The strategist says $60\%$ for a rally, the trader prices with $35\%$. Neither is confused.

• The first is a forecast
• The second is a set of pricing weights, the measure $Q$

Example: Read the weight

Stock $100$ ends the year at $130$ or $90$, cash earns $4\%$. What is $q$?

1. $q=\frac{100\times 1.04-90}{130-90}$
2. $\frac{14}{40}=\frac{7}{20}$

Formula: Discounted prices are martingales

$S_0=\frac{E_Q[S_T]}{1+r}$

Under Q every traded asset grows at the cash rate. The risk premium is priced away.

Idea: The real measure fails the test

With a real $60\%$: $0.6\times 130+0.4\times 90=114$, a $14\%$ expected return.

• The stock pays $10\%$ over cash for risk
• Under $Q$: $\frac{0.35\times 130+0.65\times 90}{1.04}=100$
• No one is assumed indifferent to risk

Idea: State prices

What you pay today for $1$ in a single state and nothing otherwise.

• Up: $\frac{0.35}{1.04}=\frac{35}{104}$, down: $\frac{0.65}{1.04}=\frac{5}{8}$
• Sum $\frac{35}{104}+\frac{5}{8}=\frac{25}{26}$, the price of a sure $1$
• The bad state is dearer: money matters most there

Quick check: Price a claim

Same market. A claim pays $20$ in the up state and $0$ otherwise. What does it cost?

Common mistake: Reading q as a forecast

A $35\%$ weight does not mean the market expects a rally $35\%$ of the time.

Instead It blends real chances with what people pay to avoid each state. A forecast needs a model of risk premia.

Idea: Why the measure exists

No arbitrage gives positive state prices, and scaling them gives a $Q$.

• Complete market: every payoff can be copied, so $Q$ is unique
• A third end price with the same $2$ assets: many measures, a range of prices
• Call spreads read $Q$ from quotes, and prices must be convex in strike

Idea: Say it in the room

Say that $Q$ comes from no arbitrage, not from a forecast.

• Show the martingale check on the stock
• Price the claim, and note that uniqueness needs a complete market