January 26, 2025
The one-period model specifies prices x\in\bm{R}^I for instruments I at the beginning of the period and prices X\in B(\Omega,\bm{R}^I) at the end of the period depending on what outcome \omega\in\Omega occured, where B(\Omega,\bm{R}^I) is the set of all bounded \bm{R}^I valued functions on \Omega. Let P\in ba(\Omega) (F in Ross’s notation) be the “real-world” probabilty measure on outcomes in \Omega, where ba(\Omega) is the set of bounded finitely-additive measures on \Omega.
The FTAP states there is no arbitrage if and only if there exists a positive measure {D\in ba(\Omega)} with {x = \int_\Omega X\,dD}. In general D is not unique. Define a “probability measure” Q = D/D(\Omega) that is not the probability of anything. Under this risk neutral probabilty measure we have x = E^*[X]D(\Omega).
If there exists \zeta\in\bm{R}^I with {\zeta^*X(\omega) = 1} for \omega\in\Omega then {\zeta^*x = \int_\Omega dD = D(\Omega)}. The discount D = D(\Omega) is the price of the zero coupon bond \zeta.
Given the “real-world” measure P and a risk-neutral measure Q the pricing kernel is \phi = dQ/dP is
Put Ross’s notation on a rigorous foundation.
I. Basic Framework
\tag{1}p_g = \int g(\theta)\,dP(\theta) \quad\leftrightarrow\quad x = \int_\Omega X(\omega)\,dQ(\omega) where g corresponds to X and Q is Ross’s P.
\tag{2} p_g = e^{-r(\theta^0)T}E^*[g(\theta)] = E[g(\theta)\phi(\theta)] \quad\leftrightarrow\quad x = DE^*[X] = E[X\phi] where \phi = D dQ^*/dP = dQ/dP.
Ross goes from a one-period model to a multi-period model to claim \tag{3} Q(\theta_i, \theta_j, T) = \int_\theta Q(\theta_i, \theta, t) Q(d\theta, \theta_j, T - t) “where Q(\theta_i, \theta_j, T) is the forward martingale probability transition function for going from state \theta_i to state \theta_j. in T periods and where the integration is over the intermediate state \theta at time t. Notice that the transition function depends on the time interval and is independent of calendar time.”
The multi-period model specifies prices X_{t_j}\in B(\mathcal{A}_j,\bm{R}^I), j\in\bm{N}, where \mathcal{A}_j is the information available at time t_j, and a probability measure P on the set of outcomes \Omega. The information available at time t_j is a partition \mathcal{A}_j of \Omega.
The FTAP for multi-period models states there is no arbitrage if and only if there exist positive measures D_j\in ba(\mathcal{A}_j) making X_j D_j a martingale measure X_j D_j = (X_k D_k)|\mathcal{A}_j, j \le k
Exercise. Show Y = E[X\mid\mathcal{A}] if and only if Y(P\mid\mathcal{A}) = (XP)\mid\mathcal{A}.
Hint: Y = E[X\mid\mathcal{A}] if and only if Y is \mathcal{A}-measurable and \int_A Y\,dP = \int_A X\,dP for all A\in\mathcal{A}.
A process (X_j) is a martingale if X_j = E[X_k\mid\mathcal{A}_j], j\le k. A process is Markov is Y_j = E[Y_k\mid \mathcal{A}(X_j)], j\le k where \mathcal{A}(X) is the smallest algebra for which X is measurable.
The binomial model has sample space \Omega = [0, 1). The probability measure is Lesbegue measure. The information available at time t_j = j is the partition \mathcal{A}_j = \{A_{ij}\mid 0\le i < 2^j\} where A_{ij} = [i/2^j, (i + 1)/2^j). Each atom of the partition determines the first j digits of the base 2 expansion of \omega\in\Omega. Define V_j(\omega) = \omega_1 + \cdots + \omega_j where \omega = \sum_{i>0} \omega_i 2^-i and \omega_i\in\{0,1\} is the base 2 expansion of \omega.
Exercise. Show V_jP = (V_{j+1}P)|A_j + 1/2.
Hint: V_{j+1} = V_j + \omega_{j+1} and \omega_{j+1}P(A_{j,j+1}) = 1/2. ## Markov Process
Let S be a finite set of states and non-negative transition probabilties \pi_{st} for s,t\in S with \sum_{t\in S} \pi_{st} = 1, s\in S.
and \Omega = S^\bm{N}. Define X_j\colon\Omega\to\bm{R} by X_0 = x_0 and