Chapter 3 - State Prices

Two Probability Measures covered so far:

1. Actual Probability Measure: found by empirical estimation
2. Risk Neutral Probability: discounted prices of assets are martingales Disagree on correct positive probabilities, agree on which paths are possible.

Radon Nikodym Derivative Consider $\Omega$ in which we have $\mathbb{P}$ and $\tilde{\mathbb{P}}$. Let’s assume that the two measures both given positive probability to every element of $\Omega$ forming the quotient:

$Z$ is a random variable since it depends on the outcome $\omega$ of a random experiment.

Properties:

1. $\mathbb{P}(Z > 0) = 1$
2. $\mathbb{E}Z = 1$
3. For any random variable $Y$ we have $\tilde{\mathbb{E}}Y = \mathbb{E}[ZY]$

Proofs left as exercise, pretty easy proofs that follow from assumption of $\tilde{\mathbb{P}}(\omega) > 0$ for $\Omega$.

e.g., Consider the probability space:

We take $p = \frac{2}{3}$ and $q = \frac{1}{3}$ then the actual probability measures are:

Then, the risk-neutral probabilities are computed as:

Using the Radon-Nikodym derivative, we have:

Given $V_3$’s we can take the expected value of the discounted prices at time $3$ to get $0$:

Definition In the $N$-period binomial model with actual probability measure $\mathbb{P}$ and risk-neutral probability measure $\tilde{\mathbb{P}}$, let $Z$ denote the Radon-Nikodým derivative of $\tilde{\mathbb{P}}$ with respect to $\mathbb{P}$;

where $#H(\omega_1 \dots \omega_N)$ denotes the number of heads appearing in the sequence $\omega_1 \dots \omega_N$ and $#T(\omega_1 \dots \omega_N)$ denotes the number of tails appearing in this sequence. The state price density random variable is given by: \(\zeta(\omega) = \frac{Z(\omega)}{(1 + r)^N},\) and $\zeta(\omega) \mathbb{P}(\omega)$ is called the state price corresponding to $\omega$. Let $\bar{\omega} = \bar{\omega}_1 \dots \bar{\omega}_N$ be a particular coin toss sequence in the $N$-period model, and consider a derivative security that pays off 1 if $\bar{\omega}$ occurs and otherwise pays off 0; i.e.,

According to the risk-neutral pricing formula, value of this derivative security at time zero is:

We see that the state price $\zeta(\bar{\omega}) \mathbb{P}(\bar{\omega})$ tells the price at time zero of a contract that pays 1 at time $N$ if and only if $\bar{\omega}$ occurs. This price should include a risk-adjusted discounting factor.

Theorem: $Z$ is an r.v. (Radon Nikodym Derivative), in an N-period binomial model,

$Z_n$ for $n = 0, 1, \dots, N$ is a martingale under actual probability measure $\mathbb{P}$.

e.g., $n = 0, 1, 2, 3$ define $Z_n = \mathbb{E}_n [Z]$, specifically:

$Z_1 = \mathbb{E}_1 [Z]$ but we compute it using the martingale $Z_1 = \mathbb{E}[Z_2]$, giving:

Then we further “discount” by martingales to achieve the following:

Definition: The Radon Nikodym derivative process is given by:

Specifically, $Z_N = Z$ and $Z_0 = 1$.

Lemma: Assume conditions for the derivative process above, let $Y$ be an r.v. depending only on the first $n$ coin tosses:

Use theorem to prove, and definition $Z_n$ to define:

$Y$ takes value 1 if and only if the first $n$ coin tosses result in the particular sequence $\overline{\omega_1}, \dots, \overline{\omega_n}$ we have fixed in advance. The coin tosses $\omega_{n + 1}, \dots, \omega_N$ are irrelevant. For each $n$, $Z_n(\omega_1, \dots, \omega_n$) is the ratio of the risk-neutral probability and the actual probability of obtaining the sequence of coin tosses $\omega_1, \dots, \omega_n$. Lemma, asserts that, if $Y$ depends only on the first $n$ coin tosses, then we do not need to consider the coin tosses after time $n$.

Note: Might use $Z_n$ as a surrogate for the Radon Nikodym derivative $Z$ in the formula:

Lemma: Assume conditions of our definition, and $Y$ is a random variable depending only on the first $m$ coin tosses:

Theorem: Price of Derivative Security at $n$ by Radon-Nikodym Process Consider an $N$-period binomial model with $0 < d < 1 + r < u$. Assume that the actual probability for head, $p$, and the actual probability for tail, $q$, are positive. The risk-neutral probabilities for head and tail are given, as usual, by:

and these are also both positive. Let $\mathbb{P}$ and $\tilde{\mathbb{P}}$ denote the corresponding actual and risk-neutral probability measures, respectively. Let $Z$ be the Radon-Nikodým derivative of $\tilde{\mathbb{P}}$ with respect to $\mathbb{P}$, and let $Z_n$, for $n = 0, 1, \dots, N$, be the Radon-Nikodým derivative process. Consider a derivative security whose payoff $V_N$ may depend on all $N$ coin tosses. For $n = 0, 1, \dots, N$, the price at time $n$ of this derivative security is:

where the state price density process $\zeta_n$ is defined by:

Problem: Optimal Investment: Given $X_0$, find an adapted portfolio process $\Delta_0, \Delta_1, \dots, \Delta_{N - 1}$ that maximizes:

subject to the wealth equation:

Risk averse agents use utility functions $U$ to capture trade-off between actual risk and actual return. It does not make sense to do this under a risk-neutral measure because under the risk-neutral measure, both the stock and money-market account have the same rate of return.

e.g., Consider a two-period model with $S_0 = 4$ and $u = 2, d = \frac{1}{2}$ and assume that $r = \frac{1}{4}$

Where we are aiming to maximize $\mathbb{E} \ln X_2$.

This gives us three linear equations with 4 unknowns, where we can get:

Problem: Given $X_0$ we want to find a random variable $X_N$ without regard to the portfolio process, that maximizes:

subject to:

Lemma: Suppose we have $\Delta_0^, \Delta_1^, \dots, \Delta_{N - 1}^$ is an optimal portfolio process for the optimal investment problem and $X_N^$ is the corresponding optimal wealth random variable at time $N$. Then $X_N^$ is optimal for the Maximizing Utility Function problem. Conversely, suppose $X_N^$ is optimal for the utility function problem. Then there is a portfolio process that starts with initial wealth $X_0$ and has value $X_N^*$ at time $N$, and this portfolio process is optimal for the optimal investment problem.

e.g., Let’s take the same Radon-Nikodym derivative from the problem we previously went though with the derivative of $\tilde{\mathbb{P}}$ wrt $\mathbb{P}$.

We simplify by using subscripts to denote different values of the state price density $\zeta$.

We use notation below to denote the respective probabilities of the possible $\Omega$ sequences:

We denote the wealth, given a sequence, by:

Thus, we have the problem:

which we can take the log of and expand:

This is clearly a lagrangian, where we can use the lagrangian multiplier and get partials as:

Thus, we get:

We can solve for $1 / \lambda$ which can be substituted into our formula:

Our optimal wealth at time 2 is given as,

Now, we can find $\Delta_0, \Delta_1(H), \Delta_1(T)$ as we did following the previous discounting method. We re-write the Radon-Nikodym derivative of $Z$ without reference to the risk-neutral measure,

In an N-period model, we have $2^N$ possible coin toss sequences $\omega$ which are listed as

where $M = 2^N$. Thus we formalize this as, given $X_0$ we wish to find $(x_1, x_2, \dots, x_M)$ that maximizes the following summation:

The lagrangian, as defined specifically, previously, is now generally written as:

With multiplier equations:

We can reduce to the above, recalling how $x_m$ and $\zeta_m$ were defined: