Mathematical Finance In Discrete Time

Charles Edouard Boutibonne (1816-1897) was a French painter.

The topics are

Finite Spot Markets
Finite Futures Market
Futures Prices Versus Forward Prices
The Cox-Ross-Rubinstein Model

We will study so-called finite markets. We work with a finite probability space $(\Omega ,{\cal F},\mathbb{P})$ with a finite number $|\Omega |$ of points $\omega$ and positive probability $\mathbb{P}(\{\omega\})>0$. We use a filtration consisting of $\sigma$-fields ${\cal F}_{0}\subset {\cal F}_{1}\subset\cdots\subset {\cal F}_{T}$. We take ${\cal F}_{0}=\{\emptyset ,\Omega\}$ and ${\cal F}_{T}= {\cal F}={\cal P}(\Omega )$, where ${\cal P}(\Omega )$ is the power set of $\Omega$.

The financial market contains $d+1$ financial assets. The usual interpretation is to assume one risk-free asset (bond or bank account) labelled $0$, and $d$ risky assets (stocks) labelled $1$ to $d$. The prices of the assets at time $t$ are random variables $S_{t}^{(0)},S_{t}^{(1)},\cdots ,S_{t}^{(d)}$ which are nonnegative and ${\cal F}_{t}$-measurable. We write ${\bf S}_{t}= (S_{t}^{(0)},S_{t}^{(1)},\cdots ,S_{t}^{(d)})$ for the vector of prices at time $t$. Since the underlying probability space and the set of dates are both finite sets, all random variables and all stochastic processes considered here are necessarily bounded. We may also assume ${\cal F}_{t}={\cal F}_{t}^{{\bf S}}\equiv\sigma ({\bf S}_{0},\cdots,{\bf S}_{t})$, which means that the underlying filtration $\{{\cal F}_{t}\}_{t\leq T}$ is generated by the observations of the price process ${\bf S}$. We assume that all assets are perfectly divisible and the market is frictionless, i.e., no restrictions on the short-selling of assets, nor transaction costs or taxes, are present.

\begin{equation}{\label{a}}\tag{A}\mbox{}\end{equation}

Finite Spot Markets.

A trading strategy $\boldsymbol{\phi}$ is a $\mathbb{R}^{d+1}$ vector stochastic process
\[\boldsymbol{\phi}=\left\{\boldsymbol{\phi}_{t}\right\}_{t=1}^{T}
=\left\{\left (\phi_{t}^{(0)},\phi_{t}^{(1)},\cdots ,\phi_{t}^{(d)}\right )\right\}_{t=1}^{T}\]
which is predictable in the sense that each $\phi_{t}^{(i)}$ is ${\cal F}_{t-1}$-measurable for $t\geq 1$. Here $\phi_{t}^{(i)}$ denotes the number of shares of asset $i$ held in the portfolio at time $t-$, which is determined on the basis of available information before time $t$. In other words, the investor selects his/her time $t$ portfolio after observing the prices ${\bf S}_{t-1}$. However, the portfolio $\boldsymbol{\phi}_{t}$ must be established before announcement of the prices ${\bf S}_{t}$. The components $\phi_{t}^{(i)}$ may assume negative as well as positive numbers, which reflects the fact that we allow short sales and assume that the assets are perfectly divisible.

Definition. The value of the portfolio at time $t$ is the scalar product
\[V_{t}(\boldsymbol{\phi})=\boldsymbol{\phi}_{t}\cdot {\bf S}_{t}=\sum_{i=0}^{d}\phi_{t}^{(i)}\cdot S_{t}^{(i)}\mbox{ for }
t=1,2,\cdots ,T\mbox{ and }V_{0}(\boldsymbol{\phi})=\boldsymbol{\phi}_{1}\cdot {\bf S}_{0}.\]
The process $V_{t}(\boldsymbol{\phi})$ is called the wealth or value process of the trading strategy. $\sharp$

The initial wealth $V_{0}(\boldsymbol{\phi})$ is called the initial investment or endowment of the investor. Now $\boldsymbol{\phi}_{t}\cdot {\bf S}_{t-1}$ reflects the market value of the portfolio just after it has been established at time $t-1$, whereas $\boldsymbol{\phi}_{t}\cdot {\bf S}_{t}$ is the value just after time $t$ prices are observed, but before changes are made in the portfolio. Therefore, we have
\[\boldsymbol{\phi}_{t}\cdot ({\bf S}_{t}-{\bf S}_{t-1})=\boldsymbol{\phi}_{t}\cdot\Delta {\bf S}_{t}\]
is the change in the market value due to changes in security prices which occur between time $t-1$ and $t$.

Definition. The gains process $G_{\boldsymbol{\phi}}$ of a trading strategy $\boldsymbol{\phi}$ is given by
\[G_{t}(\boldsymbol{\phi})=\sum_{\tau =1}^{t}\boldsymbol{\phi}_{\tau}\cdot ({\bf S}_{\tau}-{\bf S}_{\tau -1})=
\sum_{\tau =1}^{t}\boldsymbol{\phi}_{\tau}\cdot\Delta {\bf S}_{\tau}\mbox{ for }t=1,2,\cdots ,T.\]

Definition. The strategy $\boldsymbol{\phi}$ is self-financing when
\begin{equation}{\label{bineq23}}\tag{1}
\boldsymbol{\phi}_{t}\cdot {\bf S}_{t}=\boldsymbol{\phi}_{t+1}\cdot {\bf S}_{t}
\end{equation}
for $t=1,2,\cdots ,T-1$. $\sharp$

The meaning is interpreted as follows. When new prices ${\bf S}_{t}$ are quoted at time $t$, the investor adjusts his portfolio from $\boldsymbol{\phi}_{t}$ to $\boldsymbol{\phi}_{t+1}$, without bringing in or consuming any wealth. We denote by $\boldsymbol{\Phi}$ the set of all self-financing strategies. It is clear that the class $\boldsymbol{\Phi}$ is a vector space; for every $\boldsymbol{\phi},\boldsymbol{\psi}\in\boldsymbol{\Phi}$ and arbitrary real numbers $c,d$, the linear combination $c\boldsymbol{\phi}+d\boldsymbol{\psi}$ also represents a self-financing strategy.

Definition. A numeraire is a price process $\{X_{t}\}_{t=0}^{T}$, which is strictly positive for all $t\in\{0,1,\cdots ,T\}$. $\sharp$

We use $S^{(0)}$ without further specification as a numeraire. We furthermore take $S_{0}^{(0)}=1$, and define $\beta_{t}=1/S_{t}^{(0)}$ as a discount factor.

Proposition. (Numeraire Invariance). Let $X_{t}$ be a numeraire. A trading strategy $\boldsymbol{\phi}$ is self-financing with respect to ${\bf S}_{t}$ if and only if $\boldsymbol{\phi}$ is self-financing with respect to ${\bf S}_{t}/X_{t}$.

Proof. Since $X_{t}$ is strictly positive for all $t=0,1,\cdots ,T$, we have the following equivalence
\[\boldsymbol{\phi}_{t}\cdot {\bf S}_{t}=\boldsymbol{\phi}_{t+1}\cdot {\bf S}_{t}\mbox{ if and only if }
\boldsymbol{\phi}_{t}\cdot X_{t}^{-1}{\bf S}_{t}=\boldsymbol{\phi}_{t+1}\cdot X_{t}^{-1}{\bf S}_{t}.\]
This completes the proof. $\blacksquare$

Define $\bar{{\bf S}}_{t}=(1,\beta_{t}S_{t}^{(1)},\cdots ,\beta_{t}S_{t}^{(d)})$, the vector of discounted prices, and consider the discounted value process
\[\bar{V}_{t}(\boldsymbol{\phi})=\beta_{t}(\boldsymbol{\phi}_{t}\cdot {\bf S}_{t})=\boldsymbol{\phi}_{t}\cdot\bar{{\bf S}}_{t}\mbox{ for }t=1,2,\cdots ,T\]
and the discounted gains process
\[\bar{G}_{t}(\boldsymbol{\phi})=\sum_{\tau =1}^{t}\boldsymbol{\phi}_{\tau}\cdot (\bar{{\bf S}}_{\tau}-
\bar{{\bf S}}_{\tau -1})=\sum_{\tau =1}^{t}\boldsymbol{\phi}_{\tau}\cdot\Delta\bar{{\bf S}}_{\tau}\mbox{ for }t=1,2,\cdots ,T.\]
Observe that the discounted gains process reflects the gains from the trading with assets $1$ to $d$ only. We also have
\begin{equation}{\label{bineq21}}\tag{2}
\bar{{\bf S}}_{0}=(1,S_{0}^{(1)},\cdots ,S_{0}^{(d)})={\bf S}_{0}
\end{equation}
since $S_{0}^{(0)}=1$ and $\beta_{0}=1/S_{0}^{(0)}=1$.

\begin{equation}{\label{binc1}}\tag{3}\mbox{}\end{equation}

Corollary \ref{binc1}. A trading strategy $\boldsymbol{\phi}$ is self-financing with respect to ${\bf S}_{t}$ if and only if $\boldsymbol{\phi}$ is self-financing with respect to $\bar{{\bf S}}_{t}$. That is to say, $(\boldsymbol{\phi}_{t}-\boldsymbol{\phi}_{t+1}) \cdot\bar{{\bf S}}_{t}=0$ for $t=1,2,\cdots ,T-1$.

\begin{equation}{\label{binp17}}\tag{4}\mbox{}\end{equation}

Proposition \ref{binp17}. A trading strategy $\boldsymbol{\phi}$ is self-financing if and only if
\[V_{t}(\boldsymbol{\phi})=V_{0}(\boldsymbol{\phi})+G_{t}(\boldsymbol{\phi})\]
and
\begin{equation}{\label{bineq22}}\tag{5}
\bar{V}_{t}(\boldsymbol{\phi})=V_{0}(\boldsymbol{\phi})+\bar{G}_{t}(\boldsymbol{\phi})
\end{equation}
for $t=1,2,\cdots ,T$. On the other hand, if we do not assume $S_{0}^{(0)}$ to be $S_{0}^{(0)}=1$, then we obtain
\[\bar{V}_{t}(\boldsymbol{\phi})=\bar{V}_{0}(\boldsymbol{\phi})+\bar{G}_{t}(\boldsymbol{\phi}).\]

Proof. Assume $\boldsymbol{\phi}$ is self-financing strategy. Then, we have
\begin{align*}
V_{0}(\boldsymbol{\phi})+\bar{G}_{t}(\boldsymbol{\phi}) & =\boldsymbol{\phi}_{1}\cdot {\bf S}_{0}+\sum_{\tau=1}^{t}
\boldsymbol{\phi}_{t}\cdot (\bar{{\bf S}}_{\tau}-\bar{{\bf S}}_{\tau -1})\\
& =\boldsymbol{\phi}_{1}\cdot\bar{{\bf S}}_{0}+\left [\boldsymbol{\phi}_{t}\cdot\bar{{\bf S}}_{t}-
\boldsymbol{\phi}_{1}\cdot\bar{{\bf S}}_{0}-\sum_{\tau =1}^{t-1}(\boldsymbol{\phi}_{\tau}-\boldsymbol{\phi}_{\tau +1})
\cdot\bar{{\bf S}}_{\tau}\right ]\mbox{ (by (\ref{bineq21}))}\\
& =\boldsymbol{\phi}_{t}\cdot\bar{{\bf S}}_{t}\mbox{ (by Corollary \ref{binc1})}\\
& =\bar{V}_{t}(\boldsymbol{\phi})
\end{align*}
Assume now that (\ref{bineq22}) holds true. Summing up to $t=2$, (\ref{bineq22}) is
\[\boldsymbol{\phi}_{2}\cdot\bar{{\bf S}}_{2}=\boldsymbol{\phi}_{1}\cdot\bar{{\bf S}}_{0}+
\boldsymbol{\phi}_{1}\cdot (\bar{{\bf S}}_{1}-\bar{{\bf S}}_{0})+\boldsymbol{\phi}_{2}\cdot (\bar{{\bf S}}_{2}-\bar{{\bf S}}_{1}).\]
Substracting $\boldsymbol{\phi}_{2}\cdot\bar{{\bf S}}_{2}$ on both sides gives $\boldsymbol{\phi}_{2}\cdot\bar{{\bf S}}_{1}=\boldsymbol{\phi}_{1}\cdot\bar{{\bf S}}_{1}$. Proceeding similarly, or by induction, we can show that $\boldsymbol{\phi}_{t}\cdot\bar{{\bf S}}_{t}=\boldsymbol{\phi}_{t+1}\cdot\bar{{\bf S}}_{t}$. From Corollary \ref{binc1}, the proof is complete. $\blacksquare$

We are allowed to borrow (so $\phi_{t}^{(0)}$ may be negative) and sell short (so $\phi_{t}^{(i)}$ may be negative for $i=1,\cdots ,d$).

Proposition. If $(\phi_{t}^{(1)},\cdots ,\phi_{t}^{(d)})$ is predictable and $V_{0}$ is ${\cal F}_{0}$-measurable, there is a unique predictable process $\{\phi_{t}^{(0)}\}_{t=1}^{T}$ such that $\boldsymbol{\phi}=(\phi^{(0)}, \phi^{(1)},\cdots ,\phi^{(d)})$ is self-financing with initial value of the corresponding portfolio $V_{0}(\boldsymbol{\phi})=V_{0}$.

Proof. If $\boldsymbol{\phi}$ is self-financing, then by Proposition \ref{binp17},
\[\bar{V}_{t}(\boldsymbol{\phi})=V_{0}+\bar{G}_{t}(\boldsymbol{\phi})=V_{0}+\sum_{\tau =1}^{t}
(\phi_{\tau}^{(1)}\cdot\Delta \bar{S}_{\tau}^{(1)}+\cdots +\phi_{\tau}^{(d)}\cdot\Delta \bar{S}_{\tau}^{(d)}).\]
On the other hand,
\[\bar{V}_{t}(\boldsymbol{\phi})=\boldsymbol{\phi}_{t}\cdot
\bar{{\bf S}}_{t}=\phi_{t}^{(0)}+\phi_{t}^{(1)}\cdot \bar{S}_{t}^{(1)}+\cdots +\phi_{t}^{(d)}\cdot \bar{S}_{t}^{(d)}.\]
Equate these, we have
\[\phi_{t}^{(0)}=V_{0}+\sum_{\tau =1}^{t} (\phi_{\tau}^{(1)}\cdot\Delta
\bar{S}_{\tau}^{(1)}+\cdots +\phi_{\tau}^{(d)}\cdot\Delta \bar{S}_{\tau}^{(d)})
-(\phi_{t}^{(1)}\cdot \bar{S}_{t}^{(1)}+\cdots +\phi_{t}^{(d)}\cdot \bar{S}_{t}^{(d)}).\]
which defines $\phi_{t}^{(0)}$ uniquely. The terms in $\bar{S}_{t}^{(i)}$ are
\[\phi_{t}^{(i)}\Delta \bar{S}_{t}^{(i)}-\phi_{t}^{(i)}\bar{S}_{t}^{(i)}=-\phi_{t}^{(i)}\bar{S}_{t-1}^{(i)},\]
which is ${\cal F}_{t-1}$-measurable. So
\[\phi_{t}^{(0)}=V_{0}+\sum_{\tau =1}^{t-1} (\phi_{\tau}^{(1)}\cdot\Delta
\bar{S}_{\tau}^{(1)}+\cdots +\phi_{\tau}^{(d)}\cdot\Delta \bar{S}_{\tau}^{(d)})
-(\phi_{t}^{(1)}\cdot \bar{S}_{t-1}^{(1)}+\cdots +\phi_{t}^{(d)}\cdot \bar{S}^{(d)}_{t-1}),\]
where as $\phi^{(1)},\cdots ,\phi^{(d)}$ are predictable, all terms on the right-hand side are ${\cal F}_{t-1}$-measurable, so $\phi_{0}$ is predictable. $\blacksquare$

Arbitrage Price.

We pursue an analysis of the spot market model ${\cal M}=({\bf S}, \boldsymbol{\Phi})$, where ${\bf S}$ is an adapted stochastic process, and $\boldsymbol{\Phi}$ stands for the class of all self-financing (spot) trading strategies.

Definition. Let $\tilde{\boldsymbol{\Phi}}$ be a subset of self-financing strategies $\boldsymbol{\Phi}$. A strategy $\boldsymbol{\phi}\in \tilde{\boldsymbol{\Phi}}$ is called an arbitrage opportunity or arbitrage strategy with respect to $\tilde{\boldsymbol{\Phi}}$ if $\mathbb{P}\{V_{0}(\boldsymbol{\phi})=0\}=1$, and the terminal wealth of $\boldsymbol{\phi}$ satisfies $\mathbb{P}\{V_{T}(\boldsymbol{\phi}) \geq 0\}=1$ and $\mathbb{P}\{V_{T}(\boldsymbol{\phi})>0\}>0$. $\sharp$

So, an arbitrage opportunity is a self-financing strategy with zero initial value, which produces a nonnegative final value with probability one and has a positive probability of a positive final value.

Definition. We say that the market ${\cal M}$ is arbitrage-free when there are no arbitrage opportunities in the class $\boldsymbol{\Phi}$ of trading strategies (i.e., $\tilde{\boldsymbol{\Phi}}=\emptyset$). $\sharp$

Definition. A contingent claim $X$ with maturity date $T$ is an arbitrarynonnegative ${\cal F}_{T}$-measurable random variable. We denote the class of all contingent claims by ${\cal X}^{+}$. $\sharp$

A typical example of a contingent claim $X$ is an option on some underlying asset ${\bf S}$ (e.g. for the case of a European call option with maturity date $T$ and strike $K$), then we have the functional relation $X=f({\bf S})$ with some function $f$ (e.g. $X=(S_{T}-K)^{+}$).

A {\bf replicating strategy} for the contingent claim $X$ with maturity date $T$ is a self-financing strategy $\boldsymbol{\phi}\in\boldsymbol{\Phi}$ such that $V_{T}(\boldsymbol{\phi})=X$. Given a contingent claim $X$, we denote by $\boldsymbol{\Phi}_{X}$ the class of all self-financing strategies which replicate $X$. The wealth process $V_{t}(\boldsymbol{\phi})$ for $t\leq T$ of an arbitrary $\boldsymbol{\phi}$ from $\boldsymbol{\Phi}_{X}$ is called a {\bf replicating process} of $X$ in ${\cal M}$. We say that the claim $X$ is {\bf attainable} if it admits at least one replicating strategy $\boldsymbol{\phi}\in\boldsymbol{\Phi}$. So the replicating strategy generates the same time $T$ cash-flow as $X$ does.

Definition. We say that the contingent claim $X$ is uniquely replicated in ${\cal M}$ when it admits a unique replicating process in ${\cal M}$; that is, the equality $V_{t}(\boldsymbol{\phi})=V_{t}(\boldsymbol{\psi})$ for all $t\leq T$ holds for arbitrary trading strategies $\boldsymbol{\phi},\boldsymbol{\psi}$ belonging to $\boldsymbol{\Phi}_{X}$. $\sharp$

\begin{equation}{\label{musp311}}\tag{6}\mbox{}\end{equation}

Proposition \ref{musp311}. Suppose that the market ${\cal M}$ is arbitrage-free. Then, any attainable contingent claim $X$ is uniquely replicated in the market ${\cal M}$.

Proof. Suppose there is an attainable contingent claim $X$ and strategies $\boldsymbol{\phi}$ and $\boldsymbol{\psi}$ such that
\begin{equation}{\label{bineq25}}\tag{7}
V_{T}(\boldsymbol{\phi})=V_{T}(\boldsymbol{\psi})=X,
\end{equation}
but there exists a $\tau <T$ such that $V_{u}(\boldsymbol{\phi})=V_{u}(\boldsymbol{\psi})$ for every $u<\tau$ and $V_{\tau}(\boldsymbol{\phi})\neq V_{\tau}(\boldsymbol{\phi})$. Define $A\equiv\{\omega\in\Omega:V_{\tau}(\boldsymbol{\phi}(\omega )> V_{\tau}(\boldsymbol{\phi}(\omega )\}$, then $A\in {\cal F}_{\tau}$ and $\mathbb{P}(A)>0$ (otherwise just rename the strategies). Define the ${\cal F}_{\tau}$-measurable random variable $Y\equiv V_{\tau}(\boldsymbol{\phi})-V_{\tau}(\boldsymbol{\psi})$ and consider the trading strategy $\boldsymbol{xi}$ defined by
\[\boldsymbol{\xi}_{u}=\left\{\begin{array}{ll}\boldsymbol{\phi}_{u}-\boldsymbol{\psi}_{u}, & u\leq\tau\\
1_{A^{c}}\cdot (\boldsymbol{\phi}_{u}-\boldsymbol{\psi}_{u})+1_{A}\cdot (Y\cdot\beta_{\tau},0,\cdots ,0), & \tau <u\leq T.
\end{array}\right .\]
Then $\boldsymbol{\xi}$ is predictable and the self-financing condition (\ref{bineq23}) is clearly true for $t\neq\tau$, and for $t=\tau$ we have using that $\boldsymbol{\phi},\boldsymbol{\psi}\in\boldsymbol{\Phi}$
\begin{equation}{\label{bineq24}}\tag{8}
\boldsymbol{\xi}_{\tau}\cdot {\bf S}_{\tau}=(\boldsymbol{\phi}_{\tau}-\boldsymbol{\psi}_{\tau})
\cdot {\bf S}_{\tau}=V_{\tau}(\boldsymbol{\phi})-V_{\tau}(\boldsymbol{\psi}),
\end{equation}
\begin{align*}
\boldsymbol{\xi}_{\tau +1}\cdot {\bf S}_{\tau} & =
1_{A^{c}}\cdot (\boldsymbol{\phi}(\tau +1)-\boldsymbol{\psi}
(\tau +1))\cdot {\bf S}_{\tau}+1_{A}\cdot Y\cdot\beta_{\tau}\cdot S_{0}(\tau )\\
& =1_{A^{c}}\cdot (\boldsymbol{\phi}_{\tau}-\boldsymbol{\psi}
(\tau ))\cdot {\bf S}_{\tau}+1_{A}\cdot (V_{\tau}(\boldsymbol{\phi})-
V_{\tau}(\boldsymbol{\psi}))\cdot\beta_{\tau}\cdot\beta^{-1}_{\tau}
\mbox{ (since $\boldsymbol{\phi},\boldsymbol{\psi}\in\boldsymbol{\Phi}$)}\\
& =\left\{\begin{array}{ll}
V_{\tau}(\boldsymbol{\phi})-V_{\tau}(\boldsymbol{\psi}) & \mbox{if $\omega\in A$}\\
(\boldsymbol{\phi}_{\tau}-\boldsymbol{\psi}_{\tau})\cdot
{\bf S}_{\tau} & \mbox{if $\omega\in A^{c}$}
\end{array}\right .\\
& =V_{\tau}(\boldsymbol{\phi})-V_{\tau}(\boldsymbol{\psi})
\mbox{ (by (\ref{bineq24}))}.
\end{align*}

Hence $\boldsymbol{\xi}$ is a self-financing strategy with initial value equal to zero. Furthermore we have

\begin{align*}V_{T}(\boldsymbol{\xi}) & =1_{A^{c}}\cdot (\boldsymbol{\phi}_{t}-\boldsymbol{\psi}_{T}\cdot {\bf S}_{t}+1_{A}\cdot (Y\cdot\beta_{\tau},0,\cdots ,0)\cdot {\bf S}_{t}\\ & =\left\{\begin{array}{ll}Y\cdot\beta_{\tau}\cdot S_{t}^{0}\mbox{if $\omega\in A$}\\ (\boldsymbol{\phi}_{t}-\boldsymbol{\psi}-{T})\cdot {\bf S}_{t}=0 & \mbox{if $\omega\in A^{c}$}\end{array}\right .\mbox{ (by (\ref{bineq25}))}\\ & =1_{A}\cdot Y\cdot\beta_{\tau}\cdot S_{t}^{0}\geq 0\end{align*}
and
\[\mathbb{P}\left\{V_{T}(\boldsymbol{\xi})>0\right\}=\mathbb{P}(A)>0.\]
Hence the market contains an arbitrage opportunity with respect to the class $\boldsymbol{\Phi}$ of self-financing strategies. But this contradicts the assumption that the market is arbitrage-free. $\blacksquare$

This uniqueness property allows us to define the concept of an arbitrage price process.

\begin{equation}{\label{bind1}}\tag{9}\mbox{}\end{equation}

Definition \ref{bind1}. Suppose the market ${\cal M}$ is arbitrage-free. Let $X$ be any attainable contingent claim with time $T$ maturity. The arbitrage price process $\Pi_{t}(X)$, $0\leq t\leq T$ or simply arbitrage price of $X$ is given by the wealth process $V_{t}(\boldsymbol{\phi})$ of any replicating strategy $\boldsymbol{\phi}$ for $X$. $\sharp$

The converse implication in Proposition \ref{musp311} is not valid. In other words, the uniqueness of the wealth process of any attainable contingent claim does not imply the arbitrage-free property of a market in general. Indeed, it is trivial to construct a finite market in which all claims are uniquely replicated, but there exists a strictly positive claim, say $Y$, which admits a replicating strategy with negative initial investment.

Risk-Neutral Valuation Formula.

Definition. A probability measure $\bar{\mathbb{P}}$ on $(\Omega ,{\cal F}_{T})$ equivalent to $\mathbb{P}$ (resp. absolutely continuous with respect to $\mathbb{P}$) is called a martingale measure for $\bar{{\bf S}}$ (resp. a generalized martingale measure for $\bar{{\bf S}}$) when the relative process (discounted asset price process) $\bar{{\bf S}}$ follows a $\bar{\mathbb{P}}$-martingale with respect to the filtration $\{{\cal F}_{t}\}_{t=0}^{T}$. $\sharp$

We denote by ${\cal P}(\bar{{\bf S}})$ and $\bar{{\cal P}}(\bar{{\bf S}})$ the class of all martingale measures for $\bar{{\bf S}}$, and the class of all generalized martingale measure for $\bar{{\bf S}}$, respectively. Then, we have ${\cal P}(\bar{{\bf S}})\subset\bar{{\cal P}}(\bar{{\bf S}})$.

Definition. A probability measure $\bar{\mathbb{P}}$ on $(\Omega ,{\cal F}_{T})$ that is equivalent to $\mathbb{P}$ (resp. absolutely continuous with respect to $\mathbb{P}$) is called a martingale measure for ${\cal M}=({\bf S},\boldsymbol{\phi})$ (resp. a generalized martingale measure for ${\cal M}=({\bf S},\boldsymbol{\phi}))$ when, for every trading strategy $\boldsymbol{\phi}\in\boldsymbol{\Phi}$, the relative (discounted) wealth process $\bar{V}_{t}(\boldsymbol{\phi})$ follows a $\bar{\mathbb{P}}$-martingale with respect to the filtration $\{{\cal F}_{t}\}_{t=0}^{T}$. $\sharp$

We denote by ${\cal P}({\cal M})$ and $\bar{{\cal P}}({\cal M})$ the class of all martingale measures for ${\cal M}$, and the class of all generalized martingale measure for ${\cal M}$, respectively. The goal is to show that the equalities ${\cal P}(\bar{{\bf S}})={\cal P}({\cal M})$ and $\bar{{\cal P}}(\bar{{\bf S}})=\bar{{\cal P}}({\cal M})$ are satisfied.

\begin{equation}{\label{binp19}}\tag{10}\mbox{}\end{equation}

Proposition \ref{binp19}. Let $\bar{\mathbb{P}}$ be a (generalized) martingale measure and $\boldsymbol{\phi}\in\boldsymbol{\Phi}$ be any self-financing strategy. Then, the wealth process $\bar{V}_{t}(\boldsymbol{\phi})$ is a $\bar{\mathbb{P}}$-martingale with respect to the filtration $\{{\cal F}_{t}\}_{t=0}^{T}$. The result is still valid when $S_{0}^{(0)}$ is not assumed to be $S_{0}^{(0)}=1$.

Proof. From Proposition \ref{binp17}, we have $\bar{V}_{t}(\boldsymbol{\phi})=V_{0}(\boldsymbol{\phi})+\bar{G}_{t}(\boldsymbol{\phi})$ for $t=1,\cdots ,T$. Therefore, we obtain
\begin{equation}{\label{museq35}}\tag{11}
\bar{V}_{t+1}(\boldsymbol{\phi})-\bar{V}_{t}(\boldsymbol{\phi})=\bar{G}_{t+1}(\boldsymbol{\phi})-
\bar{G}_{t}(\boldsymbol{\phi})=\boldsymbol{\phi}_{t+1}\cdot(\bar{{\bf S}}_{t+1}-\bar{{\bf S}}_{t}).
\end{equation}
For $\boldsymbol{\phi}\in\boldsymbol{\Phi}$, we see that $\bar{V}_{t}(\boldsymbol{\phi})$ is the martingale transform of the $\bar{\mathbb{P}}$-martingale $\bar{{\bf S}}$ by $\boldsymbol{\phi}$, which also says a $\bar{\mathbb{P}}$-martingale itself by Proposition \ref{binp18}. More explicitly, we obtain
\begin{align*}\mathbb{E}_{\bar{\mathbb{P}}}[\bar{V}_{t+1}(\boldsymbol{\phi})-\bar{V}_{t}(\boldsymbol{\phi})|{\cal F}_{t}] & =\mathbb{E}_{\bar{\mathbb{P}}}[\boldsymbol{\phi}_{t+1}\cdot (\bar{{\bf S}}_{t+1}-\bar{{\bf S}}_{t})|{\cal F}_{t}]\mbox{ (by (\ref{museq35}))}\\ & =\boldsymbol{\phi}_{t+1}\cdot \mathbb{E}_{\bar{\mathbb{P}}}[\bar{{\bf S}}_{t+1}-\bar{{\bf S}}_{t})|{\cal F}_{t}]\\ & \quad\mbox{($\boldsymbol{\phi}$ is predictable, i.e., $\boldsymbol{\phi}_{t+1}$ is ${\cal F}_{t}$-measurable)}\\ & =0\mbox{ ($\bar{{\bf S}}$ is a $\bar{\mathbb{P}}$-martingale)}.\end{align*}
This completes the proof. $\blacksquare$

In view of Proposition \ref{binp19}, we have the following result

Proposition. A probability measure $\bar{\mathbb{P}}$ on $(\Omega ,{\cal F}_{T})$ is a martingale measure for the spot market model ${\cal M}$ if and only if it is a (generalized) martingale measure for the relative process $\bar{{\bf S}}$, i.e., ${\cal P}(\bar{{\bf S}})={\cal P}({\cal M})$ and $\bar{{\cal P}}(\bar{{\bf S}})=\bar{{\cal P}}({\cal M})$.

\begin{equation}{\label{binp20}}\tag{12}\mbox{}\end{equation}

Proposition \ref{binp20}. If an equivalent martingale measure exists, that is, if ${\cal P}(\bar{{\bf S}})\neq\emptyset$, then the market ${\cal M}$ is arbitrage-free. If the market ${\cal M}$ is arbitrage-free, then the class ${\cal P}(\bar{{\bf S}})$ of equivalent martingale measures is nonempty.

Proof. Assume such a $\bar{\mathbb{P}}$ exists. By Proposition \ref{binp19}, $\bar{V}_{t}(\boldsymbol{\phi})$ is a $\bar{\mathbb{P}}$-martingale. Therefore, we have
\[\mathbb{E}_{\bar{\mathbb{P}}}[\bar{V}_{t}(\boldsymbol{\phi})]=\mathbb{E}_{\bar{\mathbb{P}}}[\bar{V}_{0}(\boldsymbol{\phi})]=\mathbb{E}_{\bar{\mathbb{P}}}[V_{0}(\boldsymbol{\phi})].\]

If the strategy is an arbitrage opportunity, its initial value, the right-hand side above, is zero. Therefore, the left-hand side $\mathbb{E}_{\bar{\mathbb{P}}}[\bar{V}_{t}(\boldsymbol{\phi})]$ is zero, but $\bar{V}_{t}(\boldsymbol{\phi})\geq 0$ by definition. Also each $\bar{\mathbb{P}}(\{\omega \})>0$ (by assumption, each $\mathbb{P}(\{\omega\})>0$, so by equivalence $\bar{\mathbb{P}}(\{\omega\})>0$). This and $\bar{V}_{t}(\boldsymbol{\phi})\geq 0$ force $\bar{V}_{t}(\boldsymbol{\phi})=0$. So no arbitrage is possible. (The converse is difficult) $\blacksquare$

From Proposition \ref{binp20}, we have the following central theorem.

Theorem (No-Arbitrage Theorem). The market ${\cal M}$ is arbitrage-free if and only if there exists a probability measure $\bar{\mathbb{P}}$ equivalent to $\mathbb{P}$ under which the discounted asset price process $\bar{{\bf S}}$ is a $\bar{\mathbb{P}}$-martingale.

\begin{equation}{\label{binp41}}\tag{13}\mbox{}\end{equation}

Proposition \ref{binp41}. The arbitrage price process of any attainable contingent claim $X$ is given by the risk-neutral valuation formula
\begin{equation}{\label{museq36}}\tag{14}
\Pi_{t}(X)=\frac{1}{\beta_{t}}\cdot E_{\bar{\mathbb{P}}}[X\beta_{T}|{\cal F}_{t}]=
S_{t}^{(0)}\cdot E_{\bar{\mathbb{P}}}\left [\left .\frac{X}{S_{T}^{(0)}}\right |{\cal F}_{t}\right ]\mbox{ for all }t=0,1,2,\cdots ,T,
\end{equation}
where $E_{\bar{\mathbb{P}}}$ is the expectation operator with respect to an equivalent martingale measure $\bar{\mathbb{P}}$. The formula is still valid when $S_{0}^{(0)}$ is not assumed to be $S_{0}^{(0)}=1$.

Proof. Since we assume that the market is arbitrage-free there exists an equivalent martingale measure $\bar{\mathbb{P}}$. For any contingent claim $X$ with maturity $T$ and any replicating trading strategy $\boldsymbol{\phi} \in\boldsymbol{\Phi}$ we have, for
each $t=0,1,2,\cdots ,T$,
\begin{align*}\Pi_{t}(X) & =V_{t}(\boldsymbol{\phi})\mbox{ (by Definition \ref{bind1})}\\ & =(\beta_{t})^{-1}\bar{V}_{t}(\boldsymbol{\phi})=(\beta_{t})^{-1} \cdot E_{\bar{\mathbb{P}}}[\bar{V}_{T}(\boldsymbol{\phi})|{\cal F}_{t}] \mbox{ (as $\bar{V}_{t}(\boldsymbol{\phi})$ is $\bar{\mathbb{P}}$-martingale by Proposition \ref{binp19})}\\ & =(\beta_{t})^{-1}\cdot E_{\bar{\mathbb{P}}}[\beta_{T}\cdot V_{T}(\boldsymbol{\phi})|{\cal F}_{t}]\mbox{ (undoing the discounting)}\\ & =(\beta_{t})^{-1}\cdot E_{\bar{\mathbb{P}}}[\beta_{T}X|{\cal F}_{t}] \mbox{ ($\boldsymbol{\phi}$ is a replicating strategy for $X$)}.\end{align*}
This completes the proof. $\blacksquare$

In a more general setting (e.g., in a continuous-time framework), a generalized martingale measure no longer play the role of a pricing measure; that is, equality (\ref{museq36}) may fail to hold if a martingale measure $\bar{\mathbb{P}}$ is merely absolutely continuous with respect to an underlying probability measure $\mathbb{P}$. The reason is that the Ito stochastic integral (as opposed to a finite sum) is not invariant with respect to an absolutely continuous change of a probability measure.

Price Systems.

Recall that, since $\Omega =\{\omega_{1},\cdots .\omega_{n}\}$, the space ${\cal X}_{+}$ of all contingent claims which settle at time $T$ may be identified with the finite-dimensional linear space $\mathbb{R}^{n}$. For any $X\in {\cal X}_{+}$, we write

\[X=(X(\omega_{1}),\cdots ,X(\omega_{n}))=(x_{1},\cdots ,x_{n})\in \mathbb{R}^{n}.\]
We say that a linear functional $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ is strictly positive if $f({\bf x})>0$ for every ${\bf x}=(x_{1},\cdots ,x_{n})$ with $x_{i}\geq 0$ for all $i$ and $x_{j}>0$ for some $j$.

By a price system, we mean an arbitrary strictly positive linear functional $\widehat{\Pi}:{\cal X}_{+}\rightarrow \mathbb{R}$. For any price system $\widehat{\Pi}$, there exists a (unique) vector ${\bf y}=(y_{1},\cdots ,y_{n})$ such that $y_{i}>0$ for all $i$ and $\widehat{\Pi}(X)=\mathbb{E}_{\mathbb{P}}[X\cdot {\bf y}]$ for every $X\in {\cal X}_{+}$ (The Riesz representation). This vector is commonly referred to as the state-price vector.

Definition. A price system $\widehat{\Pi}$ is compatiable with arbitrage pricing in the market model ${\cal M}$ whenever $\widehat{\Pi}(X)=\Pi_{0}(X)$ for any contingent claim $X$ attainable in ${\cal M}$. $\sharp$

Definition. If ${\cal M}=({\bf S},\boldsymbol{\Phi})$ is a finite model of security market (not necessarily arbitrage-free), then a price system $\widehat{\Pi}$ is said to be consistent with ${\cal M}$ whenever $\widehat{\Pi}(V_{T}(\boldsymbol{\phi}))=V_{0}(\boldsymbol{\phi})$ for every $\boldsymbol{\phi}\in\boldsymbol{\Phi}$. $\sharp$

Proposition. There is a one-to-one correspondence between the class ${\cal P}({\cal M})$ of all martingale measures for ${\cal M}$ and the set of all price systems consistent with ${\cal M}$. It is given by the following formulas
\[\widehat{\Pi}(X)=S_{0}^{(0)}\cdot E_{\bar{\mathbb{P}}}\left [\left .\frac{X}
{S_{t}^{(0)}}\right |{\cal F}_{t}\right ]\mbox{ for all }X\in {\cal X}_{+}\]
and
\[\bar{\mathbb{P}}(A)=\frac{\widehat{\Pi}(S_{t}^{(0)}\cdot 1_{A})}{S_{0}^{(0)}}\mbox{ for all }A\in {\cal F}_{T}.\]
The result is still valid if $S_{0}^{(0)}$ is not assumed to be $S_{0}^{(0)}=1$. $\sharp$

Complete Markets.

In a highly efficient security market, we expect that the law of one price holds true, that is for a specified cash-flow there exists only one price at any instant. Otherwise arbitrageurs would use the opportunity to cash in a riskless profit. So the no-arbitrage condition implies that for an attainable contingent claim its time $t$ price must be given by the value of any replicating strategy.

We now return to the main question: given an contingent claim $X$, i.e., a cash-flow at time $T$, how can we determine its value (price) at time $t<T$? For an attainable contingent claim this value should be given by the value of any replicating strategy at time $t$, i.e., there should be a unique value process, say $V_{t}(X)$, representing the time $t$ value of the simple contingent claim $X$.

Definition. A market ${\cal M}$ is complete when every contingent claim is attainable. In other words, for every nonnegative ${\cal F}_{T}$-measurable random variable $X\in{\cal X}^{+}$, there exists a replicating self-financing strategy $\boldsymbol{\phi}\in\boldsymbol{\Phi}$ satisfying $V_{T}(\boldsymbol{\phi})=X$. $\sharp$

Using a self-financing trading strategy the investor’s wealth may go negative at time $t<T$, but he must be able to cover his debt at the final date. To avoid negative wealth the concept of admissible strategies is introduced. A self-financing trading strategy $\boldsymbol{\phi}\in\boldsymbol{\Phi}$ is called admissible when $V_{t}(\boldsymbol{\phi})\geq 0$ for each $t=1,2,\cdots ,T$. We write $\boldsymbol{\Phi}_{a}$ for the class of admissible trading strategies. The modeling assumption of admissible strategies reflects the economic fact that the broker should be protected from the unbounded short sales. From the mathematical point of view it is not really needed and we use self-financing strategies $\boldsymbol{\phi}\in\boldsymbol{\Phi}$ when addressing the mathematical aspects of the theory. In fact, one can show that a security market which is arbitrage-free with respect to $\boldsymbol{\Phi}_{a}$ is also arbitrage-free with respect to $\boldsymbol{\Phi}$.

Proposition. In the case of an arbitrage-free market, one can even insist on replicating contingent claims by an admissible strategy $\boldsymbol{\phi}\in\boldsymbol{\Phi}_{a}$.

Proof. Since the market is arbitrage-free, the equivalent martingale measure $\bar{\mathbb{P}}$ exists. From Proposition \ref{binp19}, $\bar{V}_{t}(\boldsymbol{\phi})$ is a $\bar{\mathbb{P}}$-martingale. So

\[\bar{V}_{t}(\boldsymbol{\phi})=\mathbb{E}_{\bar{\mathbb{P}}} [\bar{V}_{t}(\boldsymbol{\phi})|{\cal F}_{t}]\]

for $t=0,1,\cdots ,T$. If $\boldsymbol{\phi}$ replicates $X$, $V_{t}(\boldsymbol{\phi})=X\geq 0$, so discounting, $\bar{V}_{t}(\boldsymbol{\phi})\geq 0$, so the above equation gives $\bar{V}_{t}(\boldsymbol{\phi})\geq 0$ for each $t$. Thus all the values at each time $t$ are nonnegative, not just the final value at time $T$, so $\boldsymbol{\phi}$ is admissible. $\blacksquare$

\begin{equation}{\label{binp43}}\tag{15}\mbox{}\end{equation}

Theorem \ref{binp43} (Completeness Theorem). An arbitrage-free market ${\cal M}$ is complete if and only if there exists a unique probability measure $\bar{\mathbb{P}}$ equivalent to $\mathbb{P}$ under which discounted asset prices are martingales; that is, there exists a unique martingale measure $\bar{\mathbb{P}}$ for ${\cal M}$. $\sharp$

To say that every contingent claim can be replicated means that every $\bar{\mathbb{P}}$-martingale can be written, or represented, as a martingale transform (of the discounted prices) by a replicating (perfect-hedge) trading strategy $\boldsymbol{\phi}$. This says that all $\bar{\mathbb{P}}$-martingales can be represented as martingale transforms of discounted prices.

Assume now that we are in an arbitrage-free complete market and let $X\in {\cal X}^{+}$ be any contingent claim, $\boldsymbol{\phi}$ an admissible strategy replicating it, then $V_{T}(\boldsymbol{\phi}) =X$. As $\bar{V}_{t}(\boldsymbol{\phi})$ is a martingale transform of the $\bar{\mathbb{P}}$-martingale $\bar{{\bf S}}_{t}$ by $\boldsymbol{\phi}_{t}$, $\bar{V}_{t}(\boldsymbol{\phi})$ is a $\bar{\mathbb{P}}$-martingale. Therefore, we have
\begin{align*}V_{0}(\boldsymbol{\phi}) & =\bar{V}_{0}(\boldsymbol{\phi})=\mathbb{E}_{\bar{\mathbb{P}}}[\bar{V}_{T}(\boldsymbol{\phi})|{\cal F}_{0}]\mbox{ (since $\bar{V}_{t}(\boldsymbol{\phi})$ is a$\bar{\mathbb{P}}$-martingale)}\\ & =\mathbb{E}_{\bar{\mathbb{P}}}[\bar{V}_{T}(\boldsymbol{\phi})]\mbox{ (since ${\cal F}_{0}=\{\emptyset ,\Omega\}$)}=\mathbb{E}_{\bar{\mathbb{P}}}[\beta_{T}X]\end{align*}
More generally, since $\bar{V}_{t}(\boldsymbol{\phi})$ is a $\bar{\mathbb{P}}$-martingale, the same arguments gives
\begin{align*}\bar{V}_{t}(\boldsymbol{\phi}) & =\mathbb{E}_{\bar{\mathbb{P}}}[\bar{V}_{T}(\boldsymbol{\phi})|{\cal F}_{t}]\mbox{ ($\bar{V}_{t}(\boldsymbol{\phi})$ is the $\bar{\mathbb{P}}$-martingale)}\\ & =\mathbb{E}_{\bar{\mathbb{P}}}[\beta_{T}V_{T}(\boldsymbol{\phi})|{\cal F}_{t}]=\mathbb{E}_{\bar{\mathbb{P}}}[\beta_{T}X|{\cal F}_{t}].\end{align*}
Since $\bar{V}_{t}(\boldsymbol{\phi})=\beta_{t}V_{t}(\boldsymbol{\phi})$, we have
\[V_{t}(\boldsymbol{\phi})=\frac{1}{\beta_{t}}\mathbb{E}_{\bar{\mathbb{P}}}[\beta_{T}X|{\cal F}_{t}]\mbox{ for }t=0,1,2,\cdots ,T.\]
We can also refer to Proposition \ref{binp41} about the above formula. It is natural to call $V_{0}(\boldsymbol{\phi})=\Pi_{0}(X)$ above the arbitrage price (or more exactly, arbitrage-free price) of the contingent claim at time $0$, and $V_{t}(\boldsymbol{\phi})=\Pi_{t}(X)$ above the arbitrage price (or more exactly, arbitrage-free price) of the contingent claim $X$ at time $t$. For, if an investor sells the calim $X$ at time $t$ for $V_{t}(\boldsymbol{\phi})$, he can follow strategy $\boldsymbol{\phi}$ to replicate $X$ at time $T$ and clear the claim; an investor selling for this value is perfectly hedged. To sell the claim for any other amount would provide an arbitrage opportunity. To summarize, we have the following theorem.

Theorem (Risk-Neutral Pricing Formula). In an arbitrage-free complete market, arbitrage prices of contingent claims are their discounted expected values under the risk-neutral (equivalent martingale) measure $\bar{\mathbb{P}}$. $\sharp$

Let us return to the Taqq and Willinger \cite{taq} approach. They assume a weaker notion of attainability of a contingent claim; namely, they say a contigent claim $X$ is attainable under $Q$ when there exists a self-financing trading strategy $\boldsymbol{\phi}$ such that equality $X=V_{T}(\boldsymbol{\phi})$ holds $Q$-a.s. Similarly, a market is called complete under $Q$ if any contingent claim is attainable under $Q$. This coincides with the notion of attainability of contingent claims and completeness of the market if and onyl if probability measure $Q$ is equivalent to the original probability measure $\mathbb{P}$, i.e., when $Q\{\omega_{i}\}>0$ for all $i$. The main result of Taqq and Willinger \cite{taq} in this regard states that the market is complete under $Q$ if and only if $Q$ is an extremal point of the convex set $\bar{{\cal P}}({\cal M})$ of generalized martingale measures.

\begin{equation}{\label{b}}\tag{B}\mbox{}\end{equation}

Finite Futures Market.

Let ${\bf Z}=(Z^{(1)},\cdots ,Z^{(k)})$ be a price process. It will be essential to assume that the price process of at least one asset follows a strictly positive process. Therefore, we assume, without loss of generality, that the inequality $Z_{t}^{(k)}>0$ is satisfied for every $t\leq T$. We now assume that $Z^{(1)},\cdots ,Z^{(k-1)}$ are interpreted as futures prices of some financial assets $S^{(1)},\cdots ,S^{(k-1)}$. More precisely, $Z_{t}^{(i)}=f_{S^{i}}(t,T_{i})$ represents the value of the futures price at time $t$ of the $i$th spot security corresponding to the delivery date $T_{i}$. We assume here that $T_{i}\geq T$ for every $i=1,\cdots ,k-1$. For ease of notation, we shall write briefly $f_{t}^{(i)}=f_{S_{i}}(t,T_{i})$. We also write ${\bf f}_{t}=(f_{t}^{(1)},\cdots f_{t}^{(k-1)})$. The last coordinate, $Z^{(k)}=S^{(k)}$, is assumed to model the spot price of a certain security. The process $S^{(k)}$ may be taken to model the price of an arbitrary risky asset. We shall typically assume that $S^{(k)}$ represents the price process of a riskless asset, i.e., of a savings account. The rationale for this assumption lies in the fact that we prefer to express spot prices of derivative securities, such as futures options, in term of a spot security. In other words, the price process $S^{(k)}$ will play the role of a benchmark for the prices of derivative securities. It is convenient to assume that the price process $S^{(k)}$ is strictly positive.

Self-financing Futures Strategies.

By a futures trading strategy, we mean an arbitrary $\mathbb{R}^{k}$-valued adapted process $\boldsymbol{\phi}_{t}$ for $t\leq T$. The coordinates $\phi_{t}^{(i)}$, $i=1,\cdots ,k-1$, represent the number of long or short positions in a given futures contract assumed by an individual at time $t$. On the other hand, $\phi_{t}^{(k)}\cdot S_{t}^{(k)}$ stands for the cash investment in the spot security $S^{(k)}$ at time $t$. As in the case of sopt markets, we write $B=S^{(k)}$. In view of specific features of futures contracts, namely the fact that it costs nothing to enter such a contract, we alter slightly the definition of the wealth process.

Definition. The wealth process $V^{({\bf f})}({\boldsymbol{\phi}})$ of a futures trading strategy $\boldsymbol{\phi}$ is an adapted stochastic process given by the equality
\begin{equation}{\label{museq311}}\tag{16}
V_{t}^{({\bf f})}(\boldsymbol{\phi})=\phi_{t}^{(k)}\cdot B_{t}
\end{equation}
for all $t\leq T$. $\sharp$

In particular, the initial investment $V_{0}^{({\bf f})}(\boldsymbol{\phi})$ of any futures portfolio $\boldsymbol{\phi}$ equals $V_{0}^{({\bf f})}(\boldsymbol{\phi})=\phi_{0}^{(k)}\cdot B_{0}$. Let us denote by $\boldsymbol{\phi}^{({\bf f})}= (\phi^{(1)},\cdots ,\phi^{(k-1)})$ the futures position. Though $\boldsymbol{\phi}^{({\bf f})}$ does not present in formula (\ref{museq311}), which defines the wealth process, it appears explicitly in the self-financing condition (\ref{museq312}) as well as (\ref{museq313}) below, which describes the gains process (this in turn reflects the marking to market feature of a futures contract).

Definition. A futures trading strategy $\boldsymbol{\phi}$ is said to be self-financing if and only if the condition
\begin{equation}{\label{museq312}}\tag{17}
\boldsymbol{\phi}^{({\bf f})}_{t-1}\cdot ({\bf f}_{t}-{\bf f}_{t-1})+\boldsymbol{\phi}^{(k)}_{t-1}\cdot B_{t}=\phi_{t}^{(k)}\cdot B_{t}
\end{equation}
is satisfied for any $t=1,\cdots ,T$. The gains process $G_{t}^{({\bf f})}(\boldsymbol{\phi})$ of any futures trading strategy $\boldsymbol{\phi}$ is given by the equality
\begin{equation}{\label{museq313}}\tag{18}
G_{t}^{({\bf f})}(\boldsymbol{\phi})=\sum_{s=0}^{t-1}\boldsymbol{\phi}^{({\bf f})}_{s}\cdot ({\bf f}_{s+1}-{\bf f}_{s})+
\sum_{s=0}^{t-1}\phi_{s}^{(k)}\cdot (B_{s+1}-B_{s})
\end{equation}
for all $t\leq T$. $\sharp$

Let us denote by $\boldsymbol{\Phi}^{({\bf f})}$ the vector space of all self-financing futures trading strategies.

Proposition. A futures trading strategy $\boldsymbol{\phi}$ is self-financing if and only if we have
\[V_{t}^{({\bf f})}(\boldsymbol{\phi})=V_{0}^{({\bf f})}(\boldsymbol{\phi})+G_{t}^{({\bf f})}(\boldsymbol{\phi})\]
for all $t\leq T$.

Proof. Taking into account (\ref{museq312}) and (\ref{museq313}), for any self-financing futures strategy, we get
\begin{align*}
V_{t}^{({\bf f})}(\boldsymbol{\phi}) & =V_{0}^{({\bf f})}(\boldsymbol{\phi})+
\sum_{s=0}^{t-1}\left (V_{s+1}^{({\bf f})}(\boldsymbol{\phi})-V_{s}^{({\bf f})}(\boldsymbol{\phi})\right )\\
& =V_{0}^{({\bf f})}(\boldsymbol{\phi})+\sum_{s=0}^{t-1}\left (\phi^{(k)}_{s+1}\cdot B_{s+1}-\phi_{s}^{(k)}\cdot B_{s}\right )\\
& =V_{0}^{({\bf f})}(\boldsymbol{\phi})+\sum_{s=0}^{t-1}\left (\boldsymbol{\phi}^{({\bf f})}_{s}\cdot ({\bf f}_{s+1}-{\bf f}_{s})+
\phi_{s}^{(k)}\cdot B_{s+1}-\phi_{s}^{(k)}\cdot B_{s}\right )\\
& =V_{0}^{({\bf f})}(\boldsymbol{\phi})+\sum_{s=0}^{t-1}\boldsymbol{\phi}^{({\bf f})}_{s}\cdot ({\bf f}_{s+1}-{\bf f}_{s})+
\sum_{s=0}^{t-1}\boldsymbol{\phi}^{({\bf f})}_{s}\cdot (B_{s+1}-B_{s})\\
& =V_{0}^{({\bf f})}(\boldsymbol{\phi})+G_{t}^{({\bf f})}(\boldsymbol{\phi})
\end{align*}
for every $t=0,1,\cdots ,T$. This proves the “only if” part. $\blacksquare$

We say that a futures trading strategy $\boldsymbol{\phi}\in\boldsymbol{\Phi}^{({\bf f})}$ is an arbitrage opportunity when $\mathbb{P}\{V_{0}^{({\bf f})}(\boldsymbol{\phi})=0\}=1$, and the terminal wealth of $\boldsymbol{\phi}$ satisfies $V_{T}^{({\bf f})}(\boldsymbol{\phi}) \geq 0$ and $\mathbb{P}\{V_{T}^{({\bf f})}(\boldsymbol{\phi})>0\}>0$. We say that a futures market ${\cal M}^{({\bf f})}=({\bf f},B, \boldsymbol{\Phi}^{({\bf f})})$ is arbitrage-free when there are no arbitrage opportunities in the class of $\boldsymbol{\phi}^{({\bf f})}$ of all futures trading strategies. The notions of a contingent claim, replication and completeness, as well as of a wealth process of an attainable contingent claim, remain the same, with obvious terminological modifications. For instance, we say that a claim $X$ which settles at time $T$ is attainable in ${\cal M}^{({\bf f})}$ if there exists a self-financing futures trading strategy $\boldsymbol{\phi}$ such that $V_{T}^{({\bf f})}(\boldsymbol{\phi})=X$. The following result can be proved along the same lines as Proposition \ref{musp311}.

Proposition. Suppose the market ${\cal M}^{({\bf f})}$ is arbitrage-free. Then, any attainable contingent claim $X$ is uniquely replicated in the market ${\cal M}^{({\bf f})}$. $\sharp$

Suppose that the futures market ${\cal M}^{({\bf f})}$ is arbitrage-free. Then, the wealth process of an attainable contingent claim $X$ which settles at time $T$ is called the arbitrage price process of $X$ in the market model ${\cal M}^{({\bf f})}$. We denote it by $\Pi_{t}^{({\bf f})}(X)$ for $t\leq T$.

Martingale Measures for a Futures Market.

Recall that a probability measure $\bar{\mathbb{P}}$ on $(\Omega,{\cal F}_{T})$, equivalent to $\mathbb{P}$, is called a martingale measure for ${\bf f}$ if the process ${\bf f}$ follows a $\bar{\mathbb{P}}$-martingale with respect to the filtration $\{{\cal F}_{t}\}_{t=0}^{T}$. Note that we take here simply the futures prices, as opposed to the case of a spot market in which we dealt with relative prices. We denote by ${\cal P}({\bf f})$ the class of all martingale measures for ${\bf f}$. A probability measure $\bar{\mathbb{P}}$ on $(\Omega ,{\cal F}_{T})$ equivlent to $\mathbb{P}$ is called a martingale measure for ${\cal M}^{({\bf f})}({\bf f},B,\boldsymbol{\Phi}^{({\bf f})})$ if the relative wealth process $\bar{V}^{({\bf f})}(\boldsymbol{\phi})= V^{({\bf f})}(\boldsymbol{\phi})/B$ of any self-financing futures trading strategy $\boldsymbol{\phi}$ follows a $\bar{\mathbb{P}}$-martingale with respect to the filtration $\{{\cal F}_{t}\}_{t=0}^{T}$. The class of all martingale measures for ${\cal M}^{({\bf f})}$ is denoted by ${\cal P}({\cal M}^{({\bf f})})$.

In the next result, it is essential to assume that the discrete-time process $B$ is predictable. Intuitively, the future value of $B_{t+1}$ is known already at time $t$. Such a specific property of a savings account $B$ may arise naturally in a discrete-time model with an uncertain rate of interest. Indeed, it is common to assume that at any date $t$, the rate of interest $r_{t}$ that prevails over the next time period $[t,t+1]$ is known already at the beginning of this period. For instance, if the $\sigma$-field ${\cal F}_{0}$ is trivial, the rate $r_{0}$ is a real number, so that the value $B_{1}$ of a savings account at time $1$ is also deterministic. Then, at time $1$, at any rate $r_{1}$ is known, so that $B_{2}$ is a ${\cal F}_{1}$-measurable random variable and so forth.

\begin{equation}{\label{musl322}}\tag{19}\mbox{}\end{equation}

Proposition \ref{musl322}. For any self-financing futures trading strategy $\boldsymbol{\phi}$ up to time $T$, the relative wealth process $\bar{V}^{({\bf f})}(\boldsymbol{\phi})$ admits the following representation
\begin{equation}{\label{museq2024}}\tag{20}
\bar{V}^{({\bf f})}_{t}(\boldsymbol{\phi})=\bar{V}^{({\bf f})}_{0}(\boldsymbol{\phi})+\sum_{s=0}^{t-1}
\frac{\boldsymbol{\phi}^{({\bf f})}_{s}\cdot ({\bf f}_{s+1}-{\bf f}_{s})}{B_{s+1}}\mbox{ for all }t\leq T.
\end{equation}
Consequently, for any martingale measure $\bar{\mathbb{P}}\in {\cal P}({\bf f})$, the relative wealth process $\bar{V}^{({\bf f})}(\boldsymbol{\phi})$ of a self-financing futures trading stratefy $\boldsymbol{\phi}$ follows a martingale under $\bar{\mathbb{P}}$.

Proof. Let us denote $V^{({\bf f})}=V^{({\bf f})}(\boldsymbol{\phi})$ and $\bar{V}^{({\bf f})}=\bar{V}^{({\bf f})}(\boldsymbol{\phi})$. Then, we get
\begin{align*}
\bar{V}_{t+1}^{({\bf f})}-\bar{V}_{t}^{({\bf f})} & =\frac{V^{({\bf f})}(t+1)}{B_{t+1}}-\frac{V^{({\bf f})}(t)}{B_{t}}=
\frac{(\phi_{t+1}^{(k)}\cdot B_{t+1})}{B_{t+1}}-\phi_{t}^{(k)}\mbox{(by (\ref{museq311}))}\\
& =\frac{\left (\boldsymbol{\phi}^{({\bf f})}_{t}\cdot ({\bf f}_{t+1}-{\bf f}_{t})+\phi_{t}^{(k)}\cdot B_{t+1}\right )}
{B_{t+1}}-\phi_{t}^{(k)}\mbox{(by (\ref{museq312}))}\\
& =\frac{\boldsymbol{\phi}^{({\bf f})}_{t}\cdot ({\bf f}_{t+1}-{\bf f}_{t})}{B_{t+1}}.
\end{align*}
Then (\ref{museq2024}) follows immediately from the above expression. Now, we also get
\[\mathbb{E}_{\bar{\mathbb{P}}}\left [\left .\bar{V}_{t+1}^{({\bf f})}-\bar{V}_{t}^{({\bf f})}\right |{\cal F}_{t}\right
]=\mathbb{E}_{\bar{\mathbb{P}}} \left [\left .\frac{\boldsymbol{\phi}^{({\bf f})}_{t}\cdot ({\bf f}_{t+1}-{\bf
f}_{t})}{B_{t+1}}\right |{\cal F}_{t}\right ]= \frac{\boldsymbol{\phi}^{({\bf f})}_{t}}{B_{t+1}} \cdot \mathbb{E}_{\bar{\mathbb{P}}}[{\bf f}_{t+1}-{\bf f}_{t}|{\cal F}_{t}]=0\] as the random variable $\boldsymbol{\phi}^{({\bf f})}_{t}/B_{t+1}$ is ${\cal F}_{t}$-measurable and the process ${\bf f}$ follows a martingale under $\bar{\mathbb{P}}$. This shows that $\bar{V}^{({\bf f})}(\boldsymbol{\phi})$ follows a martingale under $\bar{\mathbb{P}}$. $\blacksquare$

The following result can be easily established from Proposition \ref{musl322}.

Proposition. A probability measure $\bar{\mathbb{P}}$ on $(\Omega ,{\cal F}_{T})$ is a martingale measure for the futures market model ${\cal M}^{({\bf f})}$ if and only if $\bar{\mathbb{P}}$ represents a martingale measure for the futures price process ${\bf f}$; that is, ${\cal P}({\bf f})= {\cal P}({\cal M}^{({\bf f})})$. $\sharp$

Proposition. Assume that the class ${\cal P}({\cal M}^{({\bf f})})$ of futures martingale measures is nonempty. Then the futures market model ${\cal M}^{({\bf f})}$ is arbitrage-free. Moreover, the arbitrage price in ${\cal M}^{({\bf f})}$ of any attainable contingent claim $X$ which settles at time $T$ is given by the risk-neutral valuation formula
\begin{equation}{\label{museq134}}\tag{21}
\Pi_{t}^{({\bf f})}(X)=B_{t}\cdot E_{\bar{\mathbb{P}}}\left [\left .\frac{X}{B_{T}}\right |{\cal F}_{t}\right ]\mbox{ for all }t\leq T,
\end{equation}
where $\bar{\mathbb{P}}$ is any martingale measure from the class ${\cal P} ({\cal M}^{({\bf f})})$.

Proof. It is easy to check the absence of arbitrage opportunities in ${\cal M}^{({\bf f})}$. Also, equality (\ref{museq134}) is a straightforward consequence of Proposition \ref{musl322}. Indeed, it follows immediately from the martingale property of the discounted wealth of a strategy that replicates $X$. $\blacksquare$

Proposition. We have the following properties.

(i) A finite futures market ${\cal M}^{({\bf f})}$ is arbitrage-free if and only if the class ${\cal P}({\cal M}^{({\bf f})})$ of martingale measures is nonempty.

(ii) An arbitrage-free futures market ${\cal M}^{({\bf f})}$ is complete if and only if the uniqueness of a martingale measure $\bar{\mathbb{P}}$ for ${\cal M}^{({\bf f})}$ holds.

Proof. Both statements can be proved by means of the same arguments as in Propositions \ref{binp20} and \ref{binp43}.

\begin{equation}{\label{c}}\tag{C}\mbox{}\end{equation}

Futures Prices Versus Forward Prices.

We write $S^{(i)}$, for $i=1,\cdots ,k$, to denote spot prices of primary assets. Also we assume the strictly positive process $B=S^{(k)}$ plays the role of numeraire asset. It is convenient to introduce an auxiliary family of derivative spot securities, referred to as zero-coupon bonds. For any $t\leq T$, we denote by $p(t,T)$ the time $t$ value of a security which pays to its holder one unit of cash at time $T$ (no intermediate cash flows are paid before this date). We refer to $p(t,T)$ as the spot price at time $t$ of a (default-free) zero-coupon bond of maturity $T$, or briefly, the price of a $T$-maturity bond. Assume that the spot market model is arbitrage-free so that the class ${\cal P}({\cal M})$ of spot martingale measures is nonempty. Taking for granted the attainability of the European claim $X=1$ which settles at time $T$ (and which represents the bond’s payoff), we obtain
\begin{equation}{\label{museq2025}}\tag{22}
p(t,T)=B_{t}\cdot E_{\bar{\mathbb{P}}}\left [\left .\frac{1}{B_{T}}\right |{\cal F}_{t}\right ]\mbox{ for all }t\leq T.
\end{equation}
Let us fix $i$ and denote $S^{(i)}=S$ (as usual, we shall refer to $S$ as the stock price process). We know that a forward contract with the settlement date $T$ is represented by the contingent claim $X=S_{T}-K$ that settles at time $T$. Recall that the forward price $F_{S}(t,T)$ at time $t\leq T$ is defined as that level of the delivery price $K$ (determined at time $t$, i.e., ${\cal F}_{t}$-measurable; that is to say, $X=S_{T}-F_{S}(t,T)$) which makes the forward contract worthless at time $t$ equal to zero.

\begin{equation}{\label{musp3331}}\tag{23}\mbox{}\end{equation}

Proposition \ref{musp3331}. The forward price at time $t\leq T$ of a stock $S$ for the settlement date $T$ equals
\begin{equation}{\label{museq3315}}\tag{24}F_{S}(t,T)=\frac{S_{T}}{p(t,T)}\end{equation}
for all $t\leq T$.

Proof. In view of (\ref{museq36}), we obtain
\[0=X=\Pi_{t}(X)=B_{t}\cdot E_{\bar{\mathbb{P}}}\left [\left .\frac{S_{T}-F_{S}(t,T)}{B_{T}}\right |{\cal F}_{t}\right ].\]
On the other hand, since by assumption $F_{S}(t,T)$ is ${\cal F}_{t}$-measurable and $E_{\bar{\mathbb{P}}}[\bar{S}_{T}|{\cal F}_{t}]= \bar{S}_{t}$, we get
\[\Pi_{t}(X)=B_{t}\cdot E_{\bar{\mathbb{P}}}[\bar{S}_{T}|{\cal F}_{t}]-B_{t}\cdot
F_{S}(t,T)\cdot E_{\bar{\mathbb{P}}}[1/B_{T}|{\cal F}_{t}]= S_{t}-p(t,T)\cdot F_{S}(t,T)\]
by (\ref{museq2025}). This completes the proof. $\blacksquare$

Though we have derived (\ref{museq3315}) using risk-neutral valuation approach, it is clear that equality (\ref{museq3315}) can also be easily established using standard no-arbitrage arguments.

Let us denote by $f_{S}(t,T)$ the futures price of the stock $S$. In particular, $f_{S}(T,T)=S_{T}$. We change slightly our setting, namely, instead of focusing on a savings account, we assume that we are given the price process of the $T$-maturity bond. We make a rather strong assumption that $p(t,T)$ follows a predictable process, i.e., for any $t\leq T-1$, the random variable $p(t+1,T)$ is ${\cal F}_{t}$-measurable. Intuitively, this means that on each date we know the bond price which will prevail on the next date (though hardly a realistic assumption, it is nevertheless trivially satisfied in any security market model which assumes a deterministic savings account; note that if all bond prices with different maturities are predictable, the bond prices (and thus the savings account) are deterministic).

\begin{equation}{\label{musp332}}\tag{25}\mbox{}\end{equation}

Proposition \ref{musp332}. Let the bond price $p(t,T)$ follow a predictable process. The combined spot-futures market is arbitrage-free if and only if the futures and forward prices agree; that is, $f_{S}(t,T)=F_{S}(t,T)$ for all $t\leq T$.

\begin{equation}{\label{d}}\tag{D}\mbox{}\end{equation}

The Cox-Ross-Rubinstein Model.

We take $d=1$, that is, the model consists of two basic securities. The time horizon is $T$ and the set of dates in the financial market model is $t=0,1,\cdots ,T$. Assume that the first of the given basic securities is a riskless bond or bank account $B$ with price process
\[B_{t}=(1+r)^{t}\mbox{ for }t=0,1,\cdots ,T,\]
implying that the bond yields a riskless rate of return $r$ in each time interval $[t,t+1]$. Furthermore we have a risky asset (stock) with price process
\[S_{t+1}=\left\{\begin{array}{ll}u\cdot S_{t} & \mbox{with probability $\mathbb{P}$},\\ d\cdot S_{t} & \mbox{with probability $1-p$}\end{array}\right .\mbox{ for }t=0,1,\cdots ,T-1\]
with $0<d<u$, $S_{0}\in \mathbb{R}_{+}$. Alternatively, we write this as
\[Z_{t+1}\equiv\frac{S_{t+1}}{S_{t}}\mbox{ for }t=0,1,\cdots ,T-1.\]
We set up a probabilistic model by considering the $Z_{t}$ for $t=0,1,\cdots ,T$ as random variables defined on probability space $(\tilde{\Omega},\tilde{{\cal F}}_{t},\bar{\mathbb{P}}_{t})$ with
\begin{align*} & \tilde{\Omega}_{t}=\tilde{\Omega}=\{d,u\},\\ & \tilde{{\cal F}}_{t}=\tilde{{\cal F}}={\cal P}(\Omega )=\{\emptyset ,u,d,\tilde{\Omega}\},\\ & \bar{\mathbb{P}}_{t}=\bar{\mathbb{P}}\mbox{ with }\bar{\mathbb{P}}(\{u\})=p\mbox{ and }\bar{\mathbb{P}}(\{d\})=1-p\mbox{ for }p\in (0,1).\end{align*}
On these probability spaces we define
\[Z_{t}(u)=u\mbox{ and }Z_{t}(d)=d\mbox{ for }t=1,2,\cdots ,T.\]
The aim is to define a probability space on which we can model the basic securities $(B,S)$. Since we can write the stock price as
\[S_{t}=S_{0}\prod_{\tau =1}^{t}Z_{\tau}\mbox{ for }t=1,2,\cdots ,T,\]
the above definitions suggest using as the underlying probabilistic model of the financial market the product space $(\Omega ,{\cal F},P)$, i.e.,
\[\Omega =\tilde{\Omega}_{1}\times\cdots\times\tilde{\Omega}_{T}=\tilde{\Omega}^{T}=\{d,u\}^{T}\]
with each $\boldsymbol{\omega}\in\Omega$ representing the successive values of $Z_{t}$ for $t=1,2,\cdots ,T$. Hence, each $\omega\in\Omega$ is a $T$-tuple $\boldsymbol{\omega}=(\tilde{\omega}_{1},\cdots ,\tilde{\omega}_{T})$ and $\tilde{\omega}_{t}\in\tilde{\Omega}=\{d,u\}$. For the $\sigma$-field we use ${\cal F}={\cal P}(\Omega )$ (the power set of $\Omega$) and the probability measure is given by
\[\mathbb{P}(\{\boldsymbol{\omega}\})=\bar{\mathbb{P}}_{1}(\{\tilde{\omega}_{1}\})\times\cdots\bar{\mathbb{P}}_{T}(\{\tilde{\omega}_{T}\})=\bar{\mathbb{P}}(\{\tilde{\omega}_{1}\})\times\cdots\bar{\mathbb{P}}(\{\tilde{\omega}_{T}\}).\]
The role of a product space is to model independent replication of a random experiment. The $Z_{t}$ above are two-valued random variables, so can be thought of as tosses of a biased coin; we need to build a probability space on which we can model a succession of such independent tosses.

Now, we redefine (with a slight abuse of notation) the $Z_{t}$ for $t=1,\cdots ,T$ as random variables on $(\Omega ,{\cal F},P)$ as (the $t$-th projection) $Z_{t}(\boldsymbol{\omega})=Z_{t}(\tilde{w}_{t})$. Observe that by this definition and the above construction, $Z_{1},\cdots ,Z_{T}$ are independent and identically distributed with
\[\mathbb{P}\{Z_{t}=u\}=p=1-\mathbb{P}\{Z_{t}=d\}.\]
To model the flow of information in the market we use the obvious filtration
\begin{align*} & {\cal F}_{0}=\{\emptyset ,\Omega\}\\ & {\cal F}_{t}=\sigma(Z_{1},\cdots ,Z_{t})=\sigma (S_{1},\cdots ,S_{t})\\ & {\cal F}_{T}={\cal F}={\cal P}(\Omega )\mbox{ (the class of all subsets of $\Omega$)} \end{align*}
This construction emphasises again that a multi-period model can be viewed as a sequence of single-period models. Indeed, in the Cox-Ross-Rubinstein case we use identical and independent single-period models. We now turn to the pricing of derivative assets in the Cox-Ross-Rubinstein market model. To do so we first have to discuss whether the Cox-Ross-Rubinstein model is arbitrage-free and complete. We use the bond price process $B_{t}$ as numeraire. The first task is to find an equivalent martingale measure of the same class as $\mathbb{P}$, i.e., a probability measure $Q$ defined as a product measure via a measure $\tilde{Q}$ on $(\tilde{\Omega},\tilde{{\cal F}})$ satisfying $\tilde{Q}=(\{u\})=q$ and $\tilde{Q}(\{d\})=1-q$. We call the set of all measures of that type ${\cal P}$.

\begin{equation}{\label{binp42}}\tag{26}\mbox{}\end{equation}

Proposition \ref{binp42}. We have the following properties.

(i) A martingale measure $Q\in {\cal P}$ for the discounted stock price $S^{*}$ exists if and only if
\begin{equation}{\label{bineq41}}\tag{27} d<1+r<u.\end{equation}

(ii) If (\ref{bineq41}) holds true, then there is a unique such measure in ${\cal P}$ characterized by
\[q=\frac{1+r-d}{u-d}.\]

Proof. Since $S_{t}=S^{*}_{t}B_{t}=S^{*}_{t}\cdot (1+r)^{t}$, we have $Z_{t+1}=S_{t+1}/S_{t}=(S^{*}_{t+1}/S^{*}_{t})\cdot(1+r)$. Therefore, the discounted price $S^{*}_{t}$ is a $Q$-martingale if and only if $\mathbb{E}_{Q}[S^{*}_{t+1}|{\cal F}_{t}]=S^{*}_{t}$ for $t=0,1,\cdots ,T-1$, i.e., $\mathbb{E}_{Q} [(S^{*}_{t+1}/S^{*}_{t})|{\cal F}_{t}]=1$. Then we have $\mathbb{E}_{Q}[Z_{t+1}|{\cal F}_{t}]=1+r$. But $Z_{1},\cdots ,Z_{T}$ are mutually independent and hence $Z_{t+1}$ is independent of ${\cal F}_{t}=\sigma (Z_{1},\cdots ,Z_{t})$. So
\[1+r=\mathbb{E}_{Q}[Z_{t+1}|{\cal F}_{t}]=\mathbb{E}_{Q}[Z_{t+1}]=u\cdot q+d\cdot (1-q).\]
Since $q\in [0,1]$, we see that $1+r\in [d,u]$ (since we assume $d<u$). Now if $1+r=d$ or $u$ then $q=0$ or $1$, i.e., $\tilde{Q}(\{u\})=0$ or $\tilde{Q}(\{d\})=0$. We know that $\mathbb{P}$ has no nonempty null sets. Therefore if $Q$ is to be equivalent to $\mathbb{P}$, then $1+r=d$ and $u$ should be excluded and (\ref{bineq41}) is proved. To prove the uniqueness and to find the value of $q$ we simply observe that under (\ref{bineq41}) $u\cdot q+d\cdot (1-q)=1+r$ has a unique solution. Solving it for $q$ leads to the above formula. This completes the proof. $\blacksquare$

Corollary. The Cox-Ross-Rubinstein model is arbitrage-free and complete.

Proof. The result follows from Proposition \ref{binp42} and Theorem \ref{binp43}. $\blacksquare$

Since every probability measure is a product measure, uniqueness of the equivalent martingale measure on the product space is equivalent to uniqueness of the individual martingale measures on the component space. (Observe that this is true for general product space situations, it merely says that for independent random variables, the individual distributions determine the joint distribution.) We thus have the following result.

Corollary. The multi-period model is complete if and only if every underlying single-period model is complete.

\begin{equation}{\label{binp44}}\tag{28}\mbox{}\end{equation}

Proposition \ref{binp44}. The arbitrage price process of a contingent claim $X$ in the Cox-Ross-Rubinstein model is given by
\[\Pi_{t}(X)=B_{t}\mathbb{E}_{\bar{\mathbb{P}}}[X/B_{T}|{\cal F}_{t}]\mbox{ for }t=0,1,\cdots ,T,\]
where $\mathbb{E}_{\bar{\mathbb{P}}}$ is the expectation operator with respect to the unique equivalent martingale measure $\bar{\mathbb{P}}$ characterized by $p=(1+r-d)/(u-d)$.

Proof. This follows from Proposition\ref{binp41} since the Cox-Ross-Rubinstein model is arbitrage-free and complete. $\blacksquare$

Corollary. Consider a European call option with expiry $T$ and strike price $K$ written on (one share of) the stock $S$. The arbitrage price process $C_{t}$ for $t=0,1,\cdots ,T$ of the option is given by
\[C_{t}=(1+r)^{-(T-t)}\sum_{i=1}^{T-t}\left (\begin{array}{c}
T-t\\ i\end{array}\right )p^{i}(1-p)^{T-t-i}\cdot (S_{t}\cdot u^{i}\cdot d^{T-t-i}-K)^{+}.\]

Proof. Recall that
\[S_{t}=S_{0}\prod_{i=1}^{t}Z_{i}\mbox{ for }t=1,2,\cdots T.\]
By Proposition \ref{binp44}, the price $C_{t}$ of a call option with strike price $K$ at time $t$ is
\begin{align*} C_{t} & =(1+r)^{-(T-t)}\cdot\mathbb{E}_{\bar{\mathbb{P}}}[(S_{T}-K)^{+}|{\cal F}_{t}]\mbox{ (since $B_{t}=(1+r)^{t}$ and $X=(S_{T}-K)^{+}$)}\\ & (1+r)^{(-T-t)}\cdot \mathbb{E}_{\bar{\mathbb{P}}}\left [\left .\left (S_{0}\prod_{i=1}^{T}Z_{i}-K\right )^{+}\right |{\cal F}_{t}\right ]\\ & =(1+r)^{(-T-t)}\cdot \mathbb{E}_{\bar{\mathbb{P}}}\left [\left .\left (S_{0}\prod_{i=1}^{t} Z_{i}\cdot\prod_{i=t+1}^{T}Z_{i}-K\right )^{+}\right |{\cal F}_{t}\right ]\\ & =(1+r)^{(-T-t)}\cdot \mathbb{E}_{\bar{\mathbb{P}}}\left [\left .\left (S_{t}\prod_{i=t+1}^{T} Z_{i}-K\right )^{+}\right |{\cal F}_{t}\right ]\\ & =(1+r)^{(-T-t)}\cdot \mathbb{E}_{\bar{\mathbb{P}}}\left [\left (S_{t}\prod_{i=t+1}^{T}Z_{i}-K\right )^{+}\right ]\\ & =(1+r)^{-(T-t)}\sum_{i=1}^{T-t}\left (\begin{array}{c} T-t\\ i\end{array}\right )p^{i}(1-p)^{T-t-i}\cdot (S_{t}\cdot u^{i}\cdot d^{T-t-i}-K)^{+}, \end{align*}
where the fifth equality follows from the fact that $S_{t}$ is ${\cal F}_{t}$-measurable, $Z_{t+1},\cdots ,Z_{T}$ are independent of ${\cal F}_{t}$ and $(x-K)^{+}$ is a nonnegative function. The last equality follows from
\[\mathbb{E}[f(Z_{t+1},\cdots ,Z_{T})]=\sum f(z_{t+1},\cdots ,z_{T})\cdot\mathbb{P}\{Z_{t+1}=z_{t+1},\cdots ,Z_{T}=z_{T}\}.\]
This completes the proof. $\blacksquare$

Since the Cox-Ross-Rubinstein model is complete we can find unique hedging strategies for replicating contingent claims. Recall that this
means we can find a portfolio $\boldsymbol{\phi}=(\phi_{t}^{(0)},\phi_{t}^{(1)})$, $\boldsymbol{\phi}$ predictable, such that the value process $V_{t}(\boldsymbol{\phi})=\phi_{t}^{(0)}B_{t}+\phi_{t}^{(1)}S_{t}$ satisfies $\Pi_{t}(X)=V_{t}(\boldsymbol{\phi})$ for all $t=0,1,\cdots ,T$. Using the bond as numeraire we get the discounted equation
\begin{equation}{\label{bineq47}}\tag{29}
\tilde{\Pi}_{X}(t)=\bar{V}_{t}(\boldsymbol{\phi})=\phi_{t}^{(0)}+\phi_{t}^{(1)}S_{t}\mbox{ for all }t=0,1,\cdots ,T.
\end{equation}
The equation (\ref{bineq47}) has to be true for each $\boldsymbol{\omega}=(\tilde{\omega}_{1},\cdots ,\tilde{\omega}_{T}) \in\Omega$ and each $t=1,\cdots ,T$. (observe that $\Pi_{X}(0)=V_{0}(\boldsymbol{\phi})=\phi_{0}(1)+\phi_{1}(1)S_{0}$). Given such a $t$ we only can use information up to (and including) time $t-1$ to ensure that $\boldsymbol{\phi}$ is predictable. Therefore we know the coordinates $\tilde{\omega}_{1},\cdots ,\tilde{\omega}_{t-1}$, but $\tilde{\omega}_{t}$ can only have one of the values $d$ or $u$. This leads to the following system of equations, which can be solved for $\phi_{t}^{(0)}$ and $\phi_{t}^{(1)}$ uniquely. With a slight abuse of notation we write $\tilde{\boldsymbol{\omega}}=(\tilde{\omega}_{1},\cdots ,\tilde{\omega}_{t-1})$, $\tilde{\boldsymbol{\omega}}_{u}=(\tilde{\omega}_{1},\cdots ,\tilde{\omega}_{t-1},u)$ and $\tilde{\boldsymbol{\omega}}_{d}=(\tilde{\omega}_{1},\cdots ,\tilde{\omega}_{t-1},d)$. Then
\[\tilde{\Pi}_{X}(t,\tilde{\boldsymbol{\omega}}_{u})=\phi_{0}(t,\tilde{\boldsymbol{\omega}})+
\phi_{1}(t,\tilde{\boldsymbol{\omega}})S^{*}_{t}(\tilde{\boldsymbol{\omega}}_{u})\] and
\[\tilde{\Pi}_{X}(t,\tilde{\boldsymbol{\omega}}_{d})=\phi_{0}(t,\tilde{\boldsymbol{\omega}})+
\phi_{1}(t,\tilde{\boldsymbol{\omega}})S^{*}_{t}(\tilde{\boldsymbol{\omega}}_{d}).\]
The solution is given by
\[\phi_{0}(t,\tilde{\boldsymbol{\omega}})=\frac{S^{*}_{t}(\tilde{\boldsymbol{\omega}}_{u})
\tilde{\Pi}_{X}(t,\tilde{\boldsymbol{\omega}}_{d})-S^{*}_{t}(\tilde{\boldsymbol{\omega}}_{d})
\tilde{\Pi}_{X}(t,\tilde{\boldsymbol{\omega}}_{u})}{S^{*}_{t}(\tilde{\boldsymbol{\omega}}_{u})-
S^{*}_{t}(\tilde{\boldsymbol{\omega}}_{d})}=\frac{u\cdot\tilde{\Pi}_{X}(t,\tilde{\boldsymbol{\omega}}_{d})-
d\cdot\tilde{\Pi}_{X}(t,\tilde{\boldsymbol{\omega}}_{u})}{u-d}\]
and
\[\phi_{1}(t,\tilde{\boldsymbol{\omega}})=\frac{\tilde{\Pi}_{X}(t,\tilde{\boldsymbol{\omega}}_{u})-
\tilde{\Pi}_{X}(t,\tilde{\boldsymbol{\omega}}_{d})}{S^{*}_{t}(\tilde{\boldsymbol{\omega}}_{u})-
S^{*}_{t}(\tilde{\boldsymbol{\omega}}_{d})}=\frac{\tilde{\Pi}_{X}(t,\tilde{\boldsymbol{\omega}}_{u})-
\tilde{\Pi}_{X}(t,\tilde{\boldsymbol{\omega}}_{d})} {S^{*}_{t-1}(\tilde{\boldsymbol{\omega}})(u-d)}.\]
Observe that we only need to know $\tilde{\boldsymbol{\omega}}$ to compute $\boldsymbol{\phi}_{t}$, hence $\boldsymbol{\phi}$ is predictable.

We make this rather abstract construction more transparent by constructing the hedge portfolio for the European call. We write
\[C_{t}(x)=(1+r)^{-(T-t)}\sum_{i=1}^{T-t}\left (\begin{array}{c}
T-t\\ i\end{array}\right )p^{i}(1-p)^{T-t-i}\cdot (x\cdot u^{i}\cdot d^{T-t-i}-K)^{+},\]
Then $C_{t}(x)$ is value of the call at time $t$ given that $S_{t}=x$.

Proposition. The perfect hedging strategy $\boldsymbol{\phi}=(\phi_{0},\phi_{1})$ replicating the European call option with time of expiry $T$ and strike price $K$ is given by
\[\phi_{t}^{(0)}=\frac{u\cdot C_{t}(d\cdot S_{t-1})-d\cdot C_{t}(u\cdot S_{t-1})}{(1+r)^{t}\cdot (u-d)}\]

and
\[\phi_{t}^{(1)}=\frac{C_{t}(u\cdot S_{t-1})-C_{t}(d\cdot S_{t-1})}{S_{t-1}\cdot (u-d)}.\]

Proof. $C_{t}(x)$ must be the value of the portfolio at time $t$ if the strategy $\boldsymbol{\phi}$ replicates the claim
\[\phi_{t}^{(0)}B_{t}+\phi_{t}^{(1)}S_{t}=\phi_{t}^{(0)}(1+r)^{t}+\phi_{t}^{(1)}S_{t}=C_{t}(S_{t}).\]
Now $S_{t}=S_{t-1}\cdot Z_{t}=S_{t-1}\cdot u$ or $S_{t-1}\cdot d$, so
\[\phi_{t}^{(0)}(1+r)^{t}+\phi_{t}^{(1)}S_{t-1}\cdot u=C_{t}(u\cdot S_{t-1})\]

and
\[\phi_{t}^{(0)}(1+r)^{t}+\phi_{t}^{(1)}S_{t-1}\cdot d=C_{t}(d\cdot S_{t-1}).\]
Then we get
\[\phi_{t}^{(1)}S_{t-1}\cdot (u-d)=C_{t}(u\cdot S_{t-1})-C_{t}(d\cdot S_{t-1}).\]
So $\phi_{t}^{(1)}$ in fact dependens only on $S_{t-1}$, thus yielding the predictability of $\boldsymbol{\phi}$ and
\[\phi_{t}^{(1)}=\frac{C_{t}(u\cdot S_{t-1})-C_{t}(d\cdot S_{t-1})}{S_{t-1}\cdot (u-d)}.\]
Using any of the equations in the above system and solving for $\phi_{t}^{(0)}$ completes the proof. $\blacksquare$

Corollary. When the payoff function is a nondecreasing function of the asset price $S_{t}$, the perfect-hedging strategy replicating the claim does not involve short-selling of the risky asset.