Risk neutral probabilities and physical probabilities

Definition Consider a finite model with states $\Omega = \{\omega_1, \dots, \omega_K\}$ in which every subset of $\Omega$ is an event. The physical measure $\mathbb{P}$ is the probability measure the model uses to describe how likely the states are, with $p_j = \mathbb{P}(\{\omega_j\})$. It is the measure behind forecasts, expected returns, risk measures and statistical estimation. In this unit every listed state has $p_j > 0$, so the only event of physical probability zero is the empty set, and a state the model regards as impossible is left out of $\Omega$. Physical probabilities are model inputs, often estimated from data, and two analysts can hold different ones for the same market.

Definition Let the market trade a cash account with $B_0 = 1$ and $B_1 = R > 0$ in every state. A risk-neutral measure, or equivalent martingale measure for the cash numeraire, is a probability measure $\mathbb{Q}$ on $\Omega$ that is equivalent to $\mathbb{P}$ and prices every traded asset by its discounted expected payoff,

where $c_n$ is the date-$0$ price and $A_n$ the date-$1$ payoff of asset $n$. Its state probabilities are $q_j = R\pi_j$, where $\pi$ is a state-price vector. A positive state-price vector gives such a measure, because the cash account forces $\sum_j q_j = R\sum_j \pi_j = 1$, and conversely $\pi_j = q_j/R$ turns such a measure into positive state prices. So $\mathbb{Q}$ exists exactly when positive state prices exist, which by The fundamental theorem of asset pricing means exactly when there is no arbitrage. When it exists, it is unique exactly when the market is complete, that is, when the payoff matrix has rank $K$. In a two-state market with a stock and the cash account,

which lies strictly between $0$ and $1$ exactly when $S_d < R S_0 < S_u$. The name does not mean that investors ignore risk or that prices actually move according to $\mathbb{Q}$.

Definition Two probability measures $\mathbb{P}$ and $\mathbb{Q}$ on the same events are equivalent if they have the same null events,

for every event $A$. On a finite $\Omega$ where every state has $p_j > 0$, this means $q_j > 0$ for every $j$. A pricing measure must be equivalent to $\mathbb{P}$, and matching the traded prices is not enough. If a measure that prices every traded asset gave a possible state the weight $0$, every traded portfolio with a nonnegative payoff that is positive only in that state would cost $0$, which makes it an arbitrage. Pricing with such a measure would hide the arbitrage instead of ruling it out.

Definition For equivalent measures $\mathbb{P}$ and $\mathbb{Q}$ on a finite $\Omega$ with every $p_j > 0$, the likelihood ratio of $\mathbb{Q}$ with respect to $\mathbb{P}$ is the random variable $Z$ with $Z(\omega_j) = q_j / p_j$. It is positive in every state, and

for every random variable $X$. Conversely, any $Z > 0$ with $\mathbb{E}^{\mathbb{P}}[Z] = 1$ defines an equivalent measure by $q_j = p_j Z(\omega_j)$. With the cash numeraire, the pricing kernel of One period replication and state prices is $m = Z/R$, so every attainable claim has the price

The ratio $Z$ changes probabilities and has mean $1$, while division by $R$ converts date-$1$ currency into date-$0$ currency. They are separate operations that the kernel combines. In continuous models $Z$ becomes a Radon-Nikodym derivative, which later lessons treat.

Definition A numeraire is a traded asset whose price is strictly positive in every state at every date, used as the unit of account. The cash numeraire is the money-market account $B$ with $B_0 = 1$. A price expressed in units of $B$ is a discounted price $S_t/B_t$, and the risk-neutral measure for the cash numeraire makes every discounted traded price a martingale. In one period this reads

When $B_1 = R$ is known at date $0$, the factor $1/R$ moves outside the expectation. When the growth of the account is random, as over several periods with changing rates, the ratio must stay inside, and replacing the account by its mean is wrong in general, because the expectation of a ratio is not the ratio of the expectations. Another numeraire $N$ gives another pricing measure $\mathbb{Q}^N$, with $V_0/N_0 = \mathbb{E}^{\mathbb{Q}^N}[X/N_1]$ and the same currency price for every attainable claim. A risk-neutral probability is therefore incomplete without its numeraire, and expectations for different numeraires, such as the $T$-forward measures of later units, cannot be mixed without a change of measure.

Definition Forecasting probabilities, given by $\mathbb{P}$, describe how often outcomes occur and give expected payoffs and expected returns. Pricing probabilities, given by $\mathbb{Q}$ for a stated numeraire, are fixed by current prices and give prices through expectations in numeraire units. Under the cash-numeraire $\mathbb{Q}$ every traded asset has expected gross return $R$, while under $\mathbb{P}$ expected returns can differ from $R$, and in general the price of a claim is not $\mathbb{E}^{\mathbb{P}}[X]/R$. In a complete finite market without arbitrage, $\mathbb{Q}$ and the price of every claim are determined by the prices and payoffs of the traded assets and depend on $\mathbb{P}$ only through which states have positive probability. Changing $\mathbb{P}$ while the prices and the possible states stay fixed changes $Z$ and the kernel $m$ but leaves $\mathbb{Q}$ and every replication price unchanged. In an incomplete market several equivalent measures fit the prices, and valuing a claim that cannot be replicated needs a further rule or the price intervals of Market completeness and nonunique prices.