2 The one-period binomial model

2.1 Model Description

We denote the time parameter with the letter $t$. There are two time points, $t=0$ (“today”) and $t=T$ (“tomorrow”) and all the trading takes place on these two dates in the financial market.
We have a risk-free asset: a bond with price process $B_t$.

The bond price is deterministic and is given by \[ B_0=1, \quad B_T=1+r. \] The constant $r$ is the simple rate for the time period $[0, T]$, and we can also interpret the existence of the bond as the existence of a bank with $r$ as its rate of interest.

We have a risky asset: a stock with price process $S_t$.

Here $S_t$ is the price of one share of the stock at time $t$. The dynamical behaviour of the stock is given by \[ S_0=s, \quad S_T=\left \{ \begin{array}{ll} s\cdot u &\; \mbox{with probability $p_{u}$}.\\ s\cdot d& \; \mbox{with probability $p_{d}$}. \end{array} \right. \]

For convenience we write $S_0=s, S_T=s\cdot Z$ where $Z$ is a random variable given by \[ Z=\left \{ \begin{array}{ll} u&\; \mbox{with probability $p_{u}$}.\\ d& \; \mbox{with probability $p_{d}$}. \end{array} \right. \] For notational simplicity we sometimes write $S^{u}_T=s\cdot u$ and $S^d_T=s\cdot d$.

Today’s stock price $s>0$ is known, as are the constants $u>d>0$, and $p_{u}, p_d > 0$, $p_{u}+p_{d}=1$.

We call the pair $\cM=(B_t, S_t)$ a financial market with two assets.

2.2 Portfolios and Arbitrage

A portfolio (also called trading strategy) on the financial market $\cM=(B_t, S_t)$ is any vector $h=(x, y)\in \R^2$ in the plane. The interpretation is as follows:

$x$ is the number of bonds we hold in our portfolio at the time $t=0$ and carry until time $t=T$. If $x<0$, that means that we have sold $x$ bonds and if $x>0$ it means that we have bought $x$ bonds. In financial jargon we have a long position in the bond if $x>0$ and a short position in the bond if $x<0$.
$y$ is the number of units of the stock held at time $t=0$ and carried until time $t=T$. If $y=-3$, this means that we have sold $3$ shares of the stock at time $t=0$ and if $y=1$ it means that we have bought $1$ share of the stock at time $t=0$. Similarly, if $y>0$ we have a long position in the stock and if $y<0$ we have a short position in the stock.

Remark. A short position in risk-free securities means issuing and selling bonds or equivalently borrowing cash. Repaying the loan with interest is referred to as closing the short position. A short position in stock can be realized in practice by short selling. This means that the investor borrows the stock from somebody else who owns it, and sells it (and uses the proceeds to make some other investments). However, the original owner is entitled to receive any dividends due and may wish to sell the stock at any time, so the investor must always have sufficient resources to fulfil the resulting obligations and, in particular, to close the short position in risky assets, that is, to repurchase the stock and return it to the owner. Similarly, the investor must always be able to close a short position in risk-free securities, by repaying the cash loan with interest.

Assumptions

We assume the following:

The market is divisible: this means that fractional holdings and short positions are allowed in the financial market. Namely every vector $h\in \R^2$ can be our trading strategy.
There are no transaction costs and no any other frictions in the financial market.
There is no bid-ask spread: this means that the selling price is equal to the buying price of all assets. (The bid price is the buying price, and the ask price is the selling price).
The market is liquid: this means that one can buy and/or sell unlimited quantities on the market. In particular it is possible to borrow unlimited amounts from the bank (by selling bonds short).

Now, consider a fixed portfolio $h=(x, y)$. This portfolio has a deterministic market value at $t=0$ and a stochastic value at $t=T$.

Definition 2.1 The value process of the portfolio $h=(x, y)$ is defined by \[ V_t^h=xB_t+yS_t, \quad\text{for $t=0, T$}, \] or, in more detail, \[ V_0^h=x+ys, \quad V_T^h=x(1+r)+ysZ. \]

Definition 2.2 An arbitrage portfolio is a portfolio $h$ with the properties \[ V_0^h=0, \quad \P(V_T^h\geq 0)=1, \quad \P(V_T^h>0)>0. \]

Remark. An arbitrage strategy therefore requires no initial investment at time $t=0$ and brings risk-free profits at time $t=T$. A possibility of risk-free profits with no initial investment can emerge when market participants make pricing mistakes.

Example 2.1 Consider a one-period financial market with parameters $r=0.1, s=10, u=1.2, d=1.1, p_{u}=0.2, p_d=0.8$. The possible stock prices at time $t=T$ are $S_T^{u}=12, S_T^d=11$. Consider the portfolio $h=(-10, 1)$. Then \[ V_0^h=-10\times 1+1\times 10=0. \] So the portfolio $h=(-10, 1)$ has zero initial wealth. The value of this portfolio at time $t=T$ is \[ V_T^h=-10\times 1.1+1\times S_T = \begin{cases}1,\\0.\end{cases} \] We conclude that $h=(-10, 1)$ is an arbitrage portfolio on this financial market. The financial meaning of this portfolio is that we sell 10 bonds (or borrow 10 pounds) and with this money we buy one share of stock for the price $s=10$.

Example 2.2 Now suppose that the parameters are $r=0.1, s=10, u=1.2, d=0.7, p_{u}=0.2, p_d=0.8$. The value process of the portfolio $h=(-10, 1)$ is \[\begin{align*} V^h_0 &=0,\\ V^h_T &=-10\times 1.1+1\times S_T=\begin{cases}1,\\-4.\end{cases} \end{align*}\] In this financial market the same portfolio is not an arbitrage strategy.

No-Arbitrage Principle: Equilibrium financial markets are arbitrage free.

Arbitrage opportunities rarely exist in practice. If and when they do, the gains are typically extremely small as compared to the volume of transactions, making them beyond the reach of small investors. In addition, they can be more subtle than the examples above. Situations when the No-Arbitrage Principle is violated are typically short-lived and difficult to spot. The activities of investors (called arbitragers) pursuing arbitrage profits effectively make the market free of arbitrage opportunities. The exclusion of arbitrage in the mathematical model is close enough to reality and turns out to be the most important and fruitful assumption. Arguments based on the No-Arbitrage Principle are the main tools of financial mathematics.

We now give a simple condition for absence of arbitrage in the financial market $\cM=(B_t, S_t)$.

Theorem 2.1 The one-period binomial model is arbitrage-free if and only if the following condition holds: \[\begin{equation} d< 1+r <u. \tag{2.1} \end{equation}\]

Proof. We first show that if the model is arbitrage-free then (2.1) holds. Assume for a contradiction that one of the inequalities in (2.1) does not hold, so that we have, say, the inequality $1+r\geq u$. Then we have $s(1+r)\geq su > sd$. An arbitrage strategy can now be formed by the portfolio $h=(s, -1)$, i.e., we sell the stock short and invest all the money in the bond. For this portfolio we obviously have $V_0^h=0$ and for $t=T$ we have \[ V_T^h=s(1+r)-sZ = \begin{cases} s(1+r)-su & \text{if $Z=u$},\\ s(1+r)-sd & \text{if $Z=d$}. \end{cases} \] Therefore $V_T\geq 0$ when $Z=u$ and $V_T>0$ when $Z=d$ which contradicts the assumption that the model is arbitrage-free. A similar contradiction is reached if $1+r \leq d < u$.

Now assume that (2.1) is satisfied. To show that this implies absence of arbitrage let us consider an arbitrage portfolio $h$ such that $V_0^h=0$. We thus have $x+ys=0,$ i.e., $x=-ys$. Using this relation we can write the value of the portfolio at $t=T$ as \[ V_T^h=\begin{cases} ys[u-(1+r)] & \text{if $Z=u$},\\ ys[d-(1+r)] & \text{if $Z=d$}. \end{cases} \] If $y=0$ then both of $ys[u-(1+r)]$ and $ys[d-(1+r)]$ are zero, so $h$ is not an arbitrage portfolio. But if $y\neq 0$ then one of $ys[u-(1+r)]$ or $ys[d-(1+r)]$ is positive, and the other negative, because of (2.1), and again $h$ is not an arbitrage portfolio.

Remark. The simple return on the risky asset is \[ \frac{S_T-S_0}{S_0}=Z-1 \] and the simple return on the risk-free asset is \[ \frac{B_T-B_0}{B_0}=r. \] So, if condition (2.1) is violated, then either (a) the return on the risky asset is no less than the return on the risk-free asset, or (b) the the return on the risky asset is no more than the return on the risk-free asset. In both cases one can generate arbitrage by short selling the asset with smaller return and investing the proceeds on the asset with dominating return.

2.3 Contingent Claims

Assume that the market is arbitrage free. We study pricing problems for contingent claims.

Definition 2.3 A contingent claim (also known as a financial derivative) is any random variable $X$ depending on the randomness of the market. This represents a contract between two parties, buyer and seller, where seller promises the random payoff $X$ to the buyer at time $T$. For that the buyer pays a certain amount of money to the seller at the time when the contract is initiated (at the time $t=0$ for example). European-style claims are of the form $X=\Phi(S_T)$ i.e., depend only on the value of the share price at time $T$. The function $\Phi$ is called the contract function.

Main Problem: determine the fair price of contingent claims.

If we denote the price of $X$ at $t$ by $\Pi(t; X)$, then it can be seen that at time $t=T$ the problem is easy to solve. In order to avoid arbitrage we must have \[ \Pi(T; X)=X, \] and the hard problem is to determine $\Pi(0;X)$.

We first introduce the following definition.

Definition 2.4 A given contingent claim $X$ is said to be reachable if there exists a portfolio $h$ such that \[ V_T^h=X \] with probability one. In that case we say that the portfolio $h$ is a hedging portfolio or a replicating portfolio. If all claims can be replicated we say that the market is complete.

If a certain claim $X$ is reachable with replicating portfolio $h$, then, from the financial point of view, there is no difference between holding the claim and holding the portfolio. No matter what happens on the stock market, the value of the claim at time $t=1$ will be exactly equal to the value of the portfolio at $t=1$. Thus the price of the claim should equal the market value of the portfolio, and we have the following basic principle.

Pricing Principle: If a claim $X$ is reachable with replication portfolio $h$, then the only reasonable price process for $X$ is given by \[ \Pi(t; X)=V_t^h, \; \; t=0, T. \]

Theorem 2.2 Suppose that a claim $X$ is reachable with replicating portfolio $h$. Then any price at $t=0$ of the claim $X$, other than $V_0^h$, will lead to an arbitrage possibility.

Proof. This is an example of The Law of One Price. Let $\Pi_0$ denote the price of the claim. If $\Pi_0>V_0^h$, then short sell the claim for $\Pi_0$. Deposit $\Pi_0-V_0^h$ in the bank account and use the amount $V_0^h$ to buy the portfolio $h$. At time $T$, your portfolio value $V_T^h$ covers exactly your short position in the claim (which is now $X$). But the money $\Pi_0-V_0^h$ grows to $(\Pi_0-V_0^h)(1+r)$ which gives a risk-free profit.

If $\Pi_0<V_0^h$, short sell the portfolio and buy the claim, and again you can generate arbitrage.

Theorem 2.3 Assuming $u>d$, the one-period binomial model is complete.

Proof. We fix an arbitrary claim $X$ with contract function $\Phi$, and we want to show that there exists a portfolio $h=(x, y)$ such that \[ V_T^h=\left \{ \begin{array}{ll} \Phi(su), & \mbox{if $Z=u$},\\ \Phi(sd), & \mbox{if $Z=d$}. \end{array} \right. \] If we write this in detail, we want to find a solution $(x, y)$ to the following system of equations \[ \begin{array}{cr} (1+r)x+su y=\Phi(su),& \\ (1+r)x+sdy=\Phi(sd).& \end{array} \] Since by assumption $d<u$, this linear system has a unique solution, and a simple calculation shows that it is given by \[\begin{equation} \begin{split} x &=\frac{1}{1+r}\cdot \frac{u \Phi(S_T^d)-d\Phi(S_T^{u})}{u-d},\\ y &=\frac{\Phi(S_T^{u})-\Phi(S_T^d)}{S_T^{u}-S_T^d}. \end{split} \tag{2.2} \end{equation}\] (Recall that $S_T^{u}=su$ and $S_T^d=sd$.)

2.4 Examples of Contingent Claims

A European call option is a contingent claim of the form $X=(S_T-K)^+$. Recall that $K$ is the strike price and $T$ is the expiry date. The corresponding contract function is $\Phi_{\text{call}}(x)=(x-K)^+$.
A European put option is a contingent claim of the form $X=(K-S_T)^+$. Again, $K$ is the strike price and $T$ is the expiry date. The corresponding contract function is $\Phi_{\text{put}}(x)=(K-x)^+$.

A forward contract on the underlying asset $S$ is a contract where the seller (short position) promises (has the obligation) to deliver the underlying asset for some given strike price $K$ at the maturity date $T$. The buyer (long position) of the forward contract has the obligation to receive the underlying asset for $K$ regardless of if $S_T$ is greater or smaller than $K$.

A forward contract is a contingent claim of the form $X=S_T-K$. The contract function is $\Phi_{\text{F}}(x)=x-K$. We have the following obvious relation \[ \Phi_{\text{F}}(x)=\Phi_{\text{call}}(x)-\Phi_{\text{put}}(x). \]

Example 2.3 Consider the financial market in Example 2.2. We would like to find the fair price of a European call option with strike price $K=9$ and maturity date $T$. The corresponding contingent claim is $X=(S_T-9)^+$. By following the idea of Theorem 2.2 we search for a portfolio $(x, y)$ that perfectly hedges the payoff of the call option: \[ x(1+r)+ys{u}=(su-9)^+, \;\;\; x(1+r)+ys{d}=(sd-9)^+. \] By plugging in the values for $r,s,u$ and $d$ we obtain \[ 1.1x+12y=3, \;\;\; 1.1x+7y=0. \] The solution is $y=\frac{3}{5}, x=-\frac{21}{5.5}$. The value of this portfolio at time $t=0$ is \[ V_0=x+ys=-\frac{21}{5.5}+\frac{3}{5}\times 10=\frac{12}{5.5}. \] We conclude that the fair price of this call option is $\frac{12}{5.5}$.

2.5 Martingale Measure

Here we investigate the no arbitrage condition (2.1) further. The relation (2.1) suggests that $1+r$ is a convex combination of $u$ and $d$, i.e., there exists two numbers $0 < q_{u} < 1$ and $0 < q_{d} < 1$ with $q_{u}+q_d=1$ such that \[\begin{equation} q_{u}u+q_{d}d=1+r. \tag{2.3} \end{equation}\]

Observe that: the weights $q_{u}$ and $q_d$ and can be interpreted as probabilities of a probability measure $\Q$ with $\Q(Z=u)=q_{u}, \Q(Z=d)=q_d$.

If we denote the expectation with respect to this measure $\Q$ by $\E_\Q$, we obtain \[ \frac{1}{1+r}\E_\Q[S_T]=\frac{1}{1+r}[q_{u}su+q_ds d]=\frac{1}{1+r}s(1+r)=s. \] We thus have the following relation \[\begin{equation} S_0=\frac{1}{1+r}\E_\Q[S_T]. \tag{2.4} \end{equation}\]

Definition 2.5 The probability measure $\Q$ in (2.4) is called a martingale measure or risk-neutral measure.

Now, we can characterize the absence of arbitrage condition with the existence of martingale measures. This result is a “baby” version of the first fundamental theorem of asset pricing.

Theorem 2.4 The financial market $\cM=(B_t, S_t)$ is arbitrage free if and only if there exists a martingale measure $\Q$.

Proof. We have already proved above that no arbitrage implies the existence of a martingale measure.

So, assume there is a martingale measure $\Q=(q, 1-q)$. Then (2.4) implies that \[ s = S_0 = \frac{1}{1+r}\E_\Q[S_T]=\frac{1}{1+r}[qsu+(1-q)sd]. \] This implies $1+r=qu+(1-q)d$. Since $0<q<1$, we conclude $d<1+r<u$ which is the no-arbitrage condition.

For the binomial model it is easy to calculate the martingale probabilities. We can solve (2.3) for $q_{u}, q_d$ and obtain

\[\begin{equation} \begin{split} q_{u}&=\frac{(1+r)-d}{u-d},\\ q_d&=\frac{u-(1+r)}{u-d}. \end{split} \tag{2.5} \end{equation}\]

Example 2.4 Let the financial market $\cM$ be the same as in Example 2.2. Since $d<1+r<u$, from Theorem 2.1 we conclude that this financial market is arbitrage free. We calculate the risk-neutral measure $\Q=(q_{u}, q_d)$ by using (2.5). We obtain \[ q_{u}=\frac{1.1-0.7}{1.2-0.7}=\frac{4}{5}, \;\; q_d=\frac{1.2-1.1}{1.2-0.7}=\frac{1}{5}, \] and \[ \frac{1}{1+r}\E_\Q[S_T] = \frac{1}{1.1}\bigg[\frac{4}{5}\times 12 + \frac{1}{5}\times 7\bigg] = 10. \]

For our simple model, we can also prove a version of the second fundamental theorem of asset pricing.

Theorem 2.5 Suppose the financial market $(B_t, S_t)$ is arbitrage free. Then it is complete if and only if there is a unique martingale measure.

Proof. Theorem 2.3 tells us that the market is complete, and Theorem 2.4 tells us that there exists a martingale measure, so we just need to show that it is unique. So suppose that $\Q_1 = (q_1,1-q_1)$ and $\Q_2=(q_2,1-q_2)$ are different measures with \[ S_0 = \frac{1}{1+r}\E_{\Q_i}[S_T], \] for $i=1,2$. In other words, $q_i u + (1-q_i) d = 1+r$, for $i=1,2$. But this is possible for two different $q_i$ only if $u = d = 1+r$, contradicting the no-arbitrage condition (2.1).

2.6 Risk Neutral Valuation

Since the binomial model is shown to be complete we can now price any contingent claim. According to the pricing principle of the proceeding section the price at $t=0$ is given by \[ \Pi(0; X)=V_0^h, \] and using the explicit formula (2.2) we obtain after some reshuffling of terms, \[ \Pi(0; X)=x+sy =\frac{1}{1+r}\left\{\frac{1+r-d}{u-d}\cdot \Phi(S_T^{u})+\frac{u-(1+r)}{u-d}\cdot \Phi(S_T^d)\right\}. \] Here we recognize the martingale measure $\Q=(q_{u}, q_d)$, and with this we can write the above formula as follows \[ \Pi(0; X)=\frac{1}{1+r}\{\Phi(S_T^{u})\cdot q_{u}+\Phi(S_T^d)\cdot q_d\}. \] The expression inside the braces can now be interpreted as an expected value under the martingale probability measure $\Q$, so we have proved the following risk neutral pricing formula.

Theorem 2.6 If the binomial model is free of arbitrage, then the arbitrage free price of a contingent claim $X$ is given by the risk neutral valuation formula: \[ \Pi(0;X)=\frac{1}{1+r}\E_\Q[X]. \] Here the martingale measure $\Q$ is uniquely determined by the relation \[ S_0=\frac{1}{1+r}\E_\Q[S_T], \] and the explicit expressions for $q_{u}$ and $q_d$ are given in (2.5). Furthermore, the claim can be replicated using (2.2).

Example 2.5 We would like to find the price of the call option in Example 2.3 by using the risk-neutral valuation formula. The probabilities are \[ q_{u}=\frac{1+r-d}{u-d}=\frac{1.1-0.7}{1.2-0.7}=\frac{4}{5}, \;\ q_d=\frac{u-(1+r)}{u-d}=\frac{1.2-1.1}{1.2-0.7}=\frac{1}{5}. \] By the risk-neutral valuation formula the call price is \[ \Pi(0)=\frac{1}{1+r}\E_\Q[(S_T-9)^+]=\frac{1}{1.1}[3q_{u}+0\times q_d]=\frac{1}{1.1}\times 3\times \frac{4}{5}=\frac{12}{5.5}. \]

Remark. Observe that the only role played by the objective probability $\P=(p_{u}, p_d)$ is that it determines which events are possible and which events are impossible:

if $p_{u}>0$, we should have $q_{u}>0$ and if $p_{u}=0$ we should have $q_{u}=0$ too.
similarly if $p_d>0$, we should have $q_d>0$ and if $p_d=0$ we should have $q_d=0$ too.

In probabilistic language, we say $\P$ and $\Q$ are equivalent measures. This means that an event is possible under the objective measure $\P$ if and only if is also possible under the risk-neutral measure $\Q$. Other than this the objective measure does not have any role in pricing.

2.7 Some Generalisations

We have assumed that $u > d$, mainly so that the market $\mathcal{M}=(B_t,S_t)$ has some interesting features (namely that the “risky” asset really does exhibit random fluctuations in price), but the theory is also valid in the case $u=d$. As mentioned, the market is not so interesting practically, but we include a brief description to see the comparison. The details are not so important, but the main message is that the two fundamental theorems still hold in this setting.

Market $\mathcal{M}=(B_t,S_t)$ with $u=d$

Both assets offer guaranteed rates of payoff per unit of investment: $B_T/B_0 = 1+r$ versus $S_T/S_0 = u = d$. It should be clear that arbitrage will be possible if these rates are not equal (buy the asset which offers the better rate of return, and sell the other asset, to make an arbitrage portfolio). The argument in the proof of Theorem 2.1 is easily modified to show that if $u=d=1+r$ then there is no arbitrage portfolio on the market. Hence the no-arbitrage condition becomes $u=d=1+r$.

Notice that, since $V^h_T = xB_T + yS_T$ is fixed (since $S_T=su=sd$ whatever happens) for any portfolio, the only contingent claims $X$ that are reachable are those which are also constant (otherwise, we could not satisfy $X = V^h_T$ with probability one). This means that the market is not complete. (But, observe that any claim of the form $\Phi(S_T)$ is still reachable.)

Moreover, for all probability measures $\mathbb{Q} = (q_u, q_d)$ we have $u q_u + d q_d = u = d$, so there can only exist a martingale measure (satisfying (2.3)) if $u = d = 1+r$. Therefore the First Fundamental Theorem still holds: no-arbitrage if and only if $u = d = 1+r$ if and only if there exists a martingale measure.

Finally, if there exists a martingale measure, then the above comment shows that there are multiple martingale measures. As we just observed, the market is not complete, and therefore the Second Fundamental Theorem also holds in this setting.

Many assets and many outcomes at time $T$

Here we generalise to markets with more than 2 assets, and/or more than 2 possible states of the market at time $T$. We still assume a single time period, i.e., trades at times $t=0$ and $t=T$ only. (Generalising to multiple periods is the focus of Chapter 3.)

Suppose that the market has $M$ assets whose prices at time 0 are given by the (deterministic) vector $S_0 = (S_0^1, S_0^2, \dots, S_0^M)$. At time $T$ the prices of the assets each change to one of $N$ possible values. In other words, the prices at time $T$ are given by the (random) vector $S_T = (S_T^1, S_T^2, \dots, S_T^M)$ of random variables. We can represent the $N$ possible values of the $M$ different assets as an $M\times N$ matrix $A= (a_{ij})$ (with $M$ rows, $N$ columns), so that the row $A_i = (a_{i1}, a_{i2}, \dots, a_{iN})$ of $A$ lists the possible values for the price of the $i$-th asset at time $T$. We suppose that there are positive probabilities $p_1,\dots,p_N$ with $p_1+\dots + p_N= 1$, and \[ \P(S_T = (a_{1j}, a_{2j}, \dots, a_{Mj})) = p_j, \quad j=1,\dots, N. \]

We define arbitrage on this market in the same way: any portfolio $h = (x_1,\dots, x_M) \in \R^M$ (where $x_i$ specifies the number of units of asset $i$ held from time 0 to time $T$) which satisfies (i) $V^h_0 = 0$, (ii) $\P(V^h_T \geq 0) =1$, and (iii) $\P(V^h_T > 0) = 0$. Observe that $V^h_0 = h \cdot S_0$ is deterministic and $V^h_T = h \cdot S_T$ is random and takes one of the $N$ values in the (row) vector $h A$. Hence arbitrage is a vector $h\in \R^M$ that has $h\cdot S_0 = 0$, and the vector $h A$ has all its entries non-negative, and at least one strictly positive.

We can also define a martingale measure on this market as before: a measure $\Q = (q_1,\dots, q_N)$ satisfying $S_0 = \frac{1}{1+r}\E_\Q[S_T]$. Here, this identity should be viewed as component-wise equality, so that $S_0^i = \frac{1}{1+r}\E_\Q[S_T^i]$ for all $i=1,\dots,M$. In terms of the matrix $A$ of prices, if we write $q = (q_1, \dots, q_N) \in \R^N$ for the vector of probabilities, then a martingale measure satisfies $S_0 = \frac{1}{1+r} q A^\mathsf{T}$ with $q_j > 0$ for all $j=1,..,N$.

It is beyond the scope of this course to prove them, but the Fundamental Theorems of Asset Pricing also hold for this market. The First Fundamental Theorem is a consequence of a simple but powerful result concerning the solvability of systems of linear inequalities, known as Farkas’ lemma. Roughly speaking, the lemma implies that exactly one of the two linear systems above, the one defining $h\in \R^M$ (arbitrage) or the one defining $q \in \R^N$ (martingale measure), has a solution. (Farkas’ lemma also underpins the linear programming duality that you may have seen in Operations Research.)

Finally, the Second Fundamental Theorem follows from rank-nullity for the matrix $A$: completeness of the market is equivalent to the (row) rank of $A$ being equal to $N$; whereas there is a unique solution to $S_0 = \frac{1}{1+r} q A^\mathsf{T}$ if and only if the nullity of $A$ equals 0; these are equivalent by the rank-nullity theorem.