Oct 30, 2026
The one-period model is specified by the vector of initial prices indexed by market instruments I, x\in\boldsymbol{R}^I, and their final prices X\colon\Omega\to\boldsymbol{R}^I where \Omega is the set of possible outcomes over the period. A fundamental problem in finance is to specify a probability measure on \Omega that can be used to make decisions about an investment \xi\in\boldsymbol{R}^I that results in \xi\cdot X(\omega) when marked-to-market or liquidated at the end of the period.
Models allowing arbitrage are not useful. Every arbitrage-free model has (generally non-unique) risk-neutral “Q” pricing measures but they are blind to the trade-off between risk and return. It is a mistake to assume they are real-world “P” probabilities that traders and portfolio managers care about.
We review well-known results about the one-period Fundamental Theory of Asset Pricing and historical attempts to limit candidate real-world “P” measures that are amenable to estimation by historical and current market information.
Arbitrage exists if there is a position {\xi\in\boldsymbol{R}^I} with {\xi\cdot x < 0} and {\xi\cdot X(\omega)\ge0} for all {\omega\in\Omega}: you make money putting on the position and never lose money unwinding it at the end. Note this definition does not involve probability.
It is not a definition traders and portfolio managers find acceptable. If |\xi\cdot x| is much smaller than |\xi|\cdot|x| they would have to tie up a lot of capital over the period to enter the arbitrage. It makes no sense to use $1MM just to make one buck.
Exercise: Show if \xi is an arbitrage so is t\xi for t > 0.
While this is mathematically true, it eventually fails for large t. At some point you will run out of intruments to purchase.
Exercise: Show if \xi and \nu are arbitrages so is \xi + \nu.
The exercises show the set of arbitrages are a cone: a subset of a vector space that is closed under multiplication by positive scalars and addition.
Exercise: Show cones are convex.
Hint: A subset of a vector space is convex if and only if the points on line between any two points in the set also belong to the set. If x and y are in the set then so is (1 - t)x + ty is for 0 < t < 1.
Exercise: If C is a convex subset then \cup_{t > 0} tC is a cone.
Hint: Note tC = \{tv\mid v\in C\}
The Fundamental Theorem of Asset Pricing for the one-period model is (assuming X is bounded, as it is in the real world) there is no arbitrage if and only if there exists a finitely-additive positive measure D on \Omega with x = \int_\Omega X\,dD. Such measures are called risk-neutral valuation measure. It is not unique except for unrealistic toy models.
The Q-measure D/D(\omega) is called a risk-neutral probability measure but it is a mistake to assume it is the probability of anything. Its existence is equivalent to the geometric fact x belongs to the smallest closed cone containing the range of X.
The realized return on a portfolio \xi\in\boldsymbol{R}^I is R_\xi = \xi\cdot X/\xi\cdot x: investing 1 unit at the beginning of the period results in a payoff of R_\xi.
Exercise: Show for any risk-neutral probability measure E_Q[R_\xi] = 1/D(\Omega).
This shows risk-neutral measures are risk-blind: every portfolio has the same realized return. They cannot be used to evaluate the risk versus return of a portfolio.
\min_{\xi\in(\boldsymbol{R}^I)^*} \frac{1}{2}\operatorname{Var}(\xi^*X) - \lambda(\xi^* x - 1) The minium satsifies 0 = \xi^* V - \lambda x^* where V = E[XX^*] - E[X]E[X]^*.
There is a long history of parametrizing real-world P-measures on \Omega that can be estimated using available information. Historical market prices were originally used for this. As futures and option markets developed they were incorporated to peek into future instrument prices. Recently, more information such as social media and prediction markets are being used.
(Markowitz 1952) limited candidate measures to jointly normal prices. Classical statistics could then be applied to historical data to estimate the mean and covariance matrix assuming stationary, independent, and normally distributed price increments.
Let B(S) be the vector space of all bounded functions on the set S. Its dual can be identified with the vector space of finitely additive measures on S. A bounded linear functional L\colon B(S)\to\boldsymbol{R} gives rise to a measure \lambda\in ba(S) with \lambda(E) = L(1_E).
Exerecise: Use 1_{E\cup F} = 1_E + 1_F - 1_{E\cap F} and 1_\emptyset = \boldsymbol{0} to show lambda is a measure.
Hint: Show \lambda(E\cup F) = \lambda(E) + \lambda(F) - \lambda(E\cap F) and \lambda(\emptyset) = 0.
A simple function is a finite linear combinations of indicator functions: f = \sum_i a_i 1_{A_i} where A_i\subseteq S. Given a measure \lambda\in ba(S) define the integral \int_S f\,d\lambda = \sum_i a_i \lambda(A_i).
Exercise: Show integration is well defined.
Hint: This is clearly true if the sets are pairwise disjoint. Show for any sets E and F we can write E\cup F as the disjoint union (E\setminus F)\cup(E\cap F)\cup F\setminus E.
x = (1, s), X = (R, S).
E[XX^*] - E[X]E[X^*] = \operatorname{Cov}(X_i, X_j)
If S is finite then B(S) can be identified with \boldsymbol{R}^n where n is the cardinality of S.