Stochastic Processes and Financial Mathematics
(part one)

$\newcommand{\footnotename}{footnote}$ $\def \LWRfootnote {1}$ $\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}$ $\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}$ $\let \LWRorighspace \hspace $ $\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }$ $\newcommand {\mathnormal }[1]{{#1}}$ $\newcommand \ensuremath [1]{#1}$ $\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } $ $\newcommand {\setlength }[2]{}$ $\newcommand {\addtolength }[2]{}$ $\newcommand {\setcounter }[2]{}$ $\newcommand {\addtocounter }[2]{}$ $\newcommand {\arabic }[1]{}$ $\newcommand {\number }[1]{}$ $\newcommand {\noalign }[1]{\text {#1}\notag \\}$ $\newcommand {\cline }[1]{}$ $\newcommand {\directlua }[1]{\text {(directlua)}}$ $\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}$ $\newcommand {\protect }{}$ $\def \LWRabsorbnumber #1 {}$ $\def \LWRabsorbquotenumber "#1 {}$ $\newcommand {\LWRabsorboption }[1][]{}$ $\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }$ $\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }$ $\def \mathcode #1={\mathchar }$ $\let \delcode \mathcode $ $\let \delimiter \mathchar $ $\def \oe {\unicode {x0153}}$ $\def \OE {\unicode {x0152}}$ $\def \ae {\unicode {x00E6}}$ $\def \AE {\unicode {x00C6}}$ $\def \aa {\unicode {x00E5}}$ $\def \AA {\unicode {x00C5}}$ $\def \o {\unicode {x00F8}}$ $\def \O {\unicode {x00D8}}$ $\def \l {\unicode {x0142}}$ $\def \L {\unicode {x0141}}$ $\def \ss {\unicode {x00DF}}$ $\def \SS {\unicode {x1E9E}}$ $\def \dag {\unicode {x2020}}$ $\def \ddag {\unicode {x2021}}$ $\def \P {\unicode {x00B6}}$ $\def \copyright {\unicode {x00A9}}$ $\def \pounds {\unicode {x00A3}}$ $\let \LWRref \ref $ $\renewcommand {\ref }{\ifstar \LWRref \LWRref }$ $ \newcommand {\multicolumn }[3]{#3}$ $\require {textcomp}$ $\newcommand {\intertext }[1]{\text {#1}\notag \\}$ $\let \Hat \hat $ $\let \Check \check $ $\let \Tilde \tilde $ $\let \Acute \acute $ $\let \Grave \grave $ $\let \Dot \dot $ $\let \Ddot \ddot $ $\let \Breve \breve $ $\let \Bar \bar $ $\let \Vec \vec $ $\DeclareMathOperator {\var }{var}$ $\DeclareMathOperator {\cov }{cov}$ $\def \ra {\Rightarrow }$ $\def \to {\rightarrow }$ $\def \iff {\Leftrightarrow }$ $\def \sw {\subseteq }$ $\def \wt {\widetilde }$ $\def \mc {\mathcal }$ $\def \mb {\mathbb }$ $\def \sc {\setminus }$ $\def \v {\textbf }$ $\def \p {\partial }$ $\def \E {\mb {E}}$ $\def \P {\mb {P}}$ $\def \R {\mb {R}}$ $\def \C {\mb {C}}$ $\def \N {\mb {N}}$ $\def \Q {\mb {Q}}$ $\def \Z {\mb {Z}}$ $\def \B {\mb {B}}$ $\def \~{\sim }$ $\def \-{\,;\,}$ $\def \|{\,|\,}$ $\def \qed {$\blacksquare $}$ $\def \1{\unicode {x1D7D9}}$ $\def \cadlag {c\`{a}dl\`{a}g}$ $\def \p {\partial }$ $\def \l {\left }$ $\def \r {\right }$ $\def \F {\mc {F}}$ $\def \G {\mc {G}}$ $\def \H {\mc {H}}$ $\def \Om {\Omega }$ $\def \om {\omega }$

Chapter 5 The binomial model

We now return to financial mathematics. We will extend the one-period model from Chapter 1 and discover a surprising connection between arbitrage and martingales.

5.1 Arbitrage in the one-period model

Let us recall the one-period market from Section 1.2. We have two commodities, cash and stock. Cash earns interest at rate $r$, so:

• If we hold $x$ units of cash at time $0$, they become worth $x(1+r)$ at time $1$.

At time $t=0$, a single unit of stock is worth $s$ units of cash. At time $1$, the value of a unit of stock changes to

\[S_1= \begin {cases} sd & \text { with probability }p_d,\\ su & \text { with probability }p_u, \end {cases} \]

where $p_u+p_d=1$.

Note that roles of $u$ and $d$ are interchangeable – we would get the same model if we swapped the values of $u$ and $d$ (and $p_u$ and $p_d$ to match). So, we lose nothing by assuming that $d<u$. We also assume that all of $r, p_d, p_d, d$ and $u$ are strictly positive.

The price of our stock changes as follows:

• If we hold $y$ units of stock, worth $ys$, at time $0$, they become worth $yS_1$ at time $1$.

Recall that we can borrow cash from the bank (provided we pay it back with interest at rate $r$, at some later time) and that we can borrow stock from the stockbroker (provided we give the same number of units of stock back, at some later time). Thus, $x$ and $y$ are allowed to be negative, with the meaning that we have borrowed.

Recall also that we use the term portfolio for the amount of cash/stock that we hold at some time. We can formalize this: A portfolio is a pair $h=(x,y)\in \R ^2$, where $x$ is the amount of cash and $y$ is the number of (units of) of stock.

Definition 5.1.1 The value process or price process of the portfolio $h=(x,y)$ is the process $V^h$ given by
$\seteqnumber{0}{5.}{0}$
\begin{align*} V^h_0&=x+ys\\ V^h_1&=x(1+r)+yS_1. \end{align*}

We can also formalize the idea of arbitrage. A portfolio is an arbitrage if it makes money for free:

Definition 5.1.2 A portfolio $h=(x,y)$ is said to be an arbitrage possibility if:
$\seteqnumber{0}{5.}{0}$
\begin{align*} V^h_0&=0\\ \P [V^h_1\geq 0]&=1\\ \P [V^h_1>0]&>0. \end{align*} We say that a market is arbitrage free if there do not exist any arbitrage possibilities.

It is possible to characterize exactly when the one-period market is arbitrage free. In fact, we have already done most of the work in exercise 1.3.

Proposition 5.1.3 The one-period market is arbitrage free if and only if $d<1+r<u$.

Proof: $(\Rightarrow ):$ Recall that we assume $d<u$. Hence, if $d<1+r<u$ fails then either $1+r\leq d<u$ or $d<u\leq 1+r$. In both cases, we will construct an arbitrage possibility.

In the case $1+r\leq d<u$ we use the portfolio $h=(-s,1)$ which has $V^h_0=0$ and

\[V^h_1=-s(1+r)+S_1\geq s(-(1+r)+d)\geq 0,\]

hence $\P [V^h_1\geq 0]=1$. Further, with probability $p_u>0$ we have $S_1=su$, which means $V^h_1>s(-(1+r)+d)\geq 0$. Hence $\P [V^h_1>0]>0$. Thus, $h$ is an arbitrage possibility.

If $0<d<u\leq 1+r$ then we use the portfolio $h'=(s,-1)$, which has $V^{h'}_0=0$ and

\[V^{h'}_1=s(1+r)-S_1\geq s(1+r-u)\geq 0,\]

hence $\P [V^{h'}_1\geq 0]=1$. Further, with probability $p_d>0$ we have $S_1=sd$, which means $V^{h'}_1>s(-(1+r)+u)\geq 0$. Hence $\P [V^{h'}_1>0]>0$. Thus, $h'$ is also an arbitrage possibility.

Remark 5.1.4 In both cases, at time $0$ we borrow whichever commodity (cash or stock) will grow slowest in value, immediately sell it and use the proceeds to buy the other, which we know will grow faster in value. Then we wait; at time $1$ we own the commodity has grown fastest in value, so we sell it, repay our debt and have some profit left over.

$(\Leftarrow ):$ Now, assume that $d<1+r<u$. We need to show that no arbitrage is possible. To do so, we will show that if a portfolio has $V^h_0=0$ and $V^h_1\geq 0$ then it also has $V^h_1=0$.

So, let $h=(x,y)$ be a portfolio such that $V^h_0=0$ and $V^h_1\geq 0$. We have

\[V^h_0=x+ys=0.\]

The value of $h$ at time $1$ is

\[V^h_1=x(1+r)+ysZ.\]

Using that $x=-ys$, we have

\begin{equation} \label {eq:value_cc} V^h_1= \begin{cases} ys(u-(1+r)) & \text { if }Z=u,\\ ys(d-(1+r)) & \text { if }Z=d.\\ \end {cases} \end{equation}

Since $\P [V^h_1\geq 0]=1$ this means that both (a) $ys(u-(1+r))\geq 0$ and (b) $ys(d-(1+r))\geq 0$. If $y<0$ then we contradict (a) because $1+r<u$. If $y>0$ then we contradict (b) because $d<1+r$. So the only option left is that $y=0$, in which case $V^h_0=V^h_1=0$. ∎

Expectation regained

In Proposition 5.1.3 we showed that our one period model was free of arbitrage if and only if

\[d<1+r<u.\]

This condition is very natural: it means that sometimes the stock will outperform cash and sometimes cash will outperform the stock. Without this condition it is intuitively clear that our market would be a bad model. From that point of view, Proposition 5.1.3 is encouraging since it confirms the importance of (no) arbitrage.

However, it turns out that there is more to the condition $d<1+r<u$, which we now explore. It is equivalent to asking that there exists $q_u,q_d\in (0,1)$ such that both

\begin{equation} \label {eq:interest_convex} q_u+q_d=1\hspace {1pc}\text { and }\hspace {1pc}1+r=uq_u+dq_d. \end{equation}

In words, (5.2) says that $1+r$ is a weighted average of $d$ and $u$, where $d$ has weight $q_d$ and $u$ has weight $q_u$. Solving these two equations gives

\begin{equation} \label {eq:q_probs} q_u=\frac {(1+r)-d}{u-d},\quad \quad q_d=\frac {u-(1+r)}{u-d}. \end{equation}

Now, here is the key: we can think of the weights $q_u$ and $q_d$ as probabilities. Let’s pretend that we live in a different world, where a single unit of stock, worth $S_0=s$ at time $0$, changes value to become worth

\[ S_1= \begin {cases} sd & \text { with probability } q_d,\\ su & \text { with probability } q_u.\\ \end {cases} \]

We have altered – the technical term is tilted – the probabilities from their old values $p_d,p_u$ to new values $q_d,q_u$. Let’s call this new world $\Q $, by which we mean that $\Q $ is our new probability measure: $\Q [S_1=sd]=q_d$ and $\Q [S_1=su]=q_u$. This is often called the risk-neutral world, and $q_u,q_d$ are known as the risk-neutral probabilities¹.

Since $\Q $ is a probability measure, we can use it to take expectations. We use $\E ^\P $ and $\E ^\Q $ to make it clear if we are taking expectations using $\P $ or $\Q $.

We have

\begin{align*} \frac {1}{1+r}\E ^\Q [S_1] &=\frac {1}{1+r}\Big (su\Q [S_1=su]+sd\Q [S_1=sd]\Big )\\ &=\frac {1}{1+r}(s)(uq_u+dq_d)\\ &=s. \end{align*} The price of the stock at time $0$ is $S_0=s$. To sum up, we have shown that the price $S_1$ of a unit of stock at time $1$ satisfies

\begin{equation} \label {eq:risk_free_1} S_0=\frac {1}{1+r}\E ^\Q [S_1]. \end{equation}

This is a formula that is very well known to economists. It gives the stock price today ($t=0$) as the expectation under $\Q $ of the stock price tomorrow ($t=1$), discounted by the rate $1+r$ at which it would earn interest.

Equation (5.4) is our first example of a ‘risk-neutral valuation’ formula. Recall that we pointed out in Chapter 1 that we should not use $\E ^\P $ and ‘expected value’ prices. A possible cause of confusion is that (5.4) does correctly calculate the value (i.e. price) of a single unit of stock by taking an expectation. The point is that we (1) use $\E ^\Q $ rather than $\E ^\P $ and (2) then discount according to the interest rate. We will see, in the next section, that these two steps are the correct way to go about arbitrage free pricing in general.

Moreover, in Section 5.4 we will extend our model to have multiple time steps. Then the expectation in (5.4) will lead us to martingales.

¹ We will discuss the reason for the name ‘risk-neutral’ later.

Stochastic Processes and Financial Mathematics (part one)

Chapter 5 The binomial model

5.1 Arbitrage in the one-period model

Expectation regained

Stochastic Processes and Financial Mathematics
(part one)