One-Period Model

Let I be the set of instruments available at the beginning and end of a time period and \Omega be the set of possible outcomes over the period. The prices at the beginning of the period are a vector {x\in\boldsymbol{R}^I = \{x\colon I\to\boldsymbol{R}\}}, the set of functions from I to \boldsymbol{R}. The initial price of instrument i\in I is x(i), which we also write as x_i. The prices at the end of the period are a function X\colon\Omega\to \boldsymbol{R}^I where X(\omega)\in\boldsymbol{R}^I are the final instrument prices if \omega\in\Omega occurred.

Define e_i\in\boldsymbol{R}^I by e_i(j) = 1 if i = j and is zero otherwise, i,j\in I. The set \{e_i\}_{i\in I} is the standard basis of the vector space \boldsymbol{R}^I.

Exercise. For x\in\boldsymbol{R}^I show x = \sum_{i\in I}x(i) e_i.

A portfolio is a linear functional \xi^*\colon\boldsymbol{R}^I\to\boldsymbol{R} where \xi^*(x) = \xi^*x is the cost of acquiring the portfolio. We can identify \xi^* with a vector \xi\in\boldsymbol{R}^I where \xi^*x = \xi\cdot x = \sum_{i\in I}\xi_i x_i.

Exercise. Show \xi(i) = \xi^*(e_i).

Arbitrage exists if there is a portfolio \xi^* with \xi^*x < 0 and \xi^*X(\omega)\ge0 for \omega\in\Omega. You make money setting up the porfolio and never lose when it is unwound at the end of the period. Note this definiton of arbitrage does not depend on a measure.

Even though you make a positive amount with certainty this definition is not good enough for real world applicaton. A measure of the amount of capital involved is |\xi|\cdot |x|. If \xi\cdot x is small relative to this then the return on investment is less attractive.

If \omega_j\in\Omega, D_j > 0 and x = \sum_j X(\omega_j) D_j then \xi^*x = \sum_j \xi^* X(\omega_j) D_j \ge 0 if \xi^*X\ge0. This shows arbitrage does not exist if x is in the smallest cone containing the range of X.

The Fundamental Theorem of Asset Pricing states there is no arbitrage if and only if x belongs to the smallest closed cone containing the range of X. This is equivalent to saying x = \int_\Omega X\,dD for some positive measure D. We call such a measure D a numeraire for the model.

If the market is incomplete then the numeraire D is not unique. If x does not belong to the smallest closed cone containing the range of X and y is the closest point in the cone then \xi = y - x is an arbitrage.

If a derivative pays \phi(X) at the end of the period, where \phi\colon\boldsymbol{R}^I\to\boldsymbol{R} is the payof function, then the value v of the derivative is constrained by (x,v) must belong to the smallest closed cone containing the range of (X,\phi(X)) so {v = \int_\Omega \phi(X)\,dD} for some numeraire D.

If \Omega = \{\omega_0, \omega_1\} has two points and I has two (independent) intruments then x = X(\omega_0) D_0 + X(\omega_1) D_1 for some D_0, D_1\in\boldsymbol{R}. This is called the binomial model and the value of any option is {v = \phi(X_{i_0}(\omega_0)) D_0 + \phi(X_{i_1}(\omega_1)) D_1}. Every payoff is equivalent to a linear payoff in a binomial model.

Exercise. If the model is arbitrage free then D_0,D_1 \ge 0.

IfX_{i_0}, X_{i_1}\in\boldsymbol{R}^\Omega are independent for some i_0,i_1\in I then those determine D_0 and D_1