Stochastic Processes and Financial Mathematics
(part one)

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5.5 Portfolios, arbitrage and martingales

Since we now have multiple time steps, we can exchange cash for stock (and vice versa) at all times $t=0,1,\ldots ,T-1$. We need to expand our idea of a portfolio to allow for this.

The filtration corresponding to the information available to a buyer/seller in the binomial model is

\[\mc {F}_t=\sigma (Z_1,Z_2,\ldots , Z_{t}).\]

In words, the information in $\mc {F}_t$ contains changes in the stock price up to and including at time $t$. This means that, $S_0,S_1,\ldots ,S_t$ are all $\mc {F}_t$ measurable, but $S_{t+1}$ is not $\mc {F}_t$ measurable.

When we choose how much stock/cash to buy/sell at time $t-1$, we do so without knowing how the stock price will change during $t-1\mapsto t$. So we must do so using information only from $\mc {F}_{t-1}$.

We now have enough terminology to define the strategies that are available to participants in the binomial market.

Definition 5.5.1 A portfolio strategy is a stochastic process

\[h_t=(x_t,y_t)\]

for $t=1,2,\ldots ,T$, such that $h_t$ is $\mc {F}_{t-1}$ measurable.

The interpretation is that $x_t$ is the amount of cash, and $y_t$ the amount of stock, that we hold during the time step $t-1\mapsto t$. We make our choice of how much cash and stock to hold during $t-1\mapsto t$ based on knowing the value of $S_0,S_1,\ldots ,S_{t-1}$, but without knowing $S_{t}$. This is realistic.

Definition 5.5.2 The value process of the portfolio strategy $h=(h_t)_{t=1}^T$ is the stochastic process $(V^h_t)_{t=0}^T$ given by
$\seteqnumber{0}{5.}{9}$
\begin{align*} V^h_0&=x_1+y_1S_0,\\ V^h_t&=x_t(1+r)+y_tS_t, \end{align*} for $t=1,2,\ldots ,T$.

At $t=0$, $V^h_0$ is the value of the portfolio $h_1$. For $t\geq 1$, $V^h_t$ is the value of the portfolio $h_t$ at time $t$, after the change in value of cash/stock that occurs during $t-1\mapsto t$. The value process $V^h_t$ is $\mc {F}_{t}$ measurable but it is not $\mc {F}_{t-1}$ measurable.

We will be especially interested in portfolio strategies that require an initial investment at time $0$ but, at later times $t\geq 1,2,\ldots ,T-1$, any changes in the amount of stock/cash held will pay for itself. We capture such portfolio strategies in the following definition.

Definition 5.5.3 A portfolio strategy $h_t=(x_t,y_t)$ is said to be self-financing if

\[V^h_{t}=x_{t+1}+y_{t+1}S_{t}.\]

for $t=0,1,2,\ldots ,T$.

This means that the value of the portfolio at time $t$ is equal to the value (at time $t$) of the stock/cash that is held in between times $t\mapsto t+1$. In other words, in a self-financing portfolio at the times $t=1,2,\ldots $ we can swap our stocks for cash (and vice versa) according to whatever the stock price turns out to be, but that is all we can do.

Lastly, our idea of arbitrage must also be upgraded to handle multiple time steps.

Definition 5.5.4 We say that a portfolio strategy $(h_t)$ is an arbitrage possibility if it is self-financing and satisfies
$\seteqnumber{0}{5.}{9}$
\begin{align*} V^h_0&=0\\ \P [V^h_T\geq 0]&=1.\\ \P [V^h_T>0]&>0. \end{align*}

In words, an arbitrage possibility requires that we invest nothing at times $t=0,1,\ldots ,T-1$, but which gives us a positive probability of earning something at time $T$, with no risk at all of actually losing money.

It’s natural to ask when the binomial model is arbitrage free. Happily, the condition turns out to be the same as for the one-period model.

Proposition 5.5.5 The binomial model is arbitrage free if and only if $d<1+r<u$.

The proof is quite similar to the argument for the one-period model, but involves more technical calculations and (for this reason) we don’t include it as part of the course.

Recall the risk-neutral probabilities from (5.3). In the one-period model, we use them to define the risk-neutral world $\Q $, in which on each time step the stock price moves up (by $u$) with probability $q_u$, or down (by $d$) with probability $q_d$. This provides a connection to martingales:

Proposition 5.5.6 If $d<1+r<u$, then under the probability measure $\Q $, the process

\[M_t=\frac {1}{(1+r)^t}S_t\]

is a martingale, with respect to the filtration $(\mc {F}_t)$.

Proof: We have commented above that $S_t\in m\mc {F}_t$, and we also have $d^tS_0\leq S_t \leq u^tS_0$, so $S_t$ is bounded and hence $S_t\in L^1$. Hence also $M_t\in m\mc {F}_t$ and $M_t\in L^1$. It remains to show that

\begin{align*} \E ^\Q [M_{t+1}\|\F _t] &=\E ^\Q \l [M_{t+1}\1_{\{Z_{t+1}=u\}}+M_{t+1}\1_{\{Z_{t+1}=d\}}\|\mc {F}_t\r ]\\ &=\E ^\Q \l [\frac {uS_t}{(1+r)^{t+1}}\1_{\{Z_{t+1}=u\}}+\frac {dS_t}{(1+r)^{t+1}}\1_{\{Z_{t+1}=d\}}\|\mc {F}_t\r ]\\ &=\frac {S_t}{(1+r)^{t+1}}\l (u\E ^\Q \l [\1_{\{Z_{t+1}=u\}}\|\mc {F}_t\r ]+d\E ^\Q \l [\1_{\{Z_{t+1}=d\}}\|\mc {F}_t\r ]\r )\\ &=\frac {S_t}{(1+r)^{t+1}}\l (u\E ^\Q \l [\1_{\{Z_{t+1}=u\}}\r ]+d\E ^\Q \l [\1_{\{Z_{t+1}=d\}}\r ]\r )\\ &=\frac {S_t}{(1+r)^{t+1}}\l (u\Q [Z_{t+1}=u]+d\Q \l [Z_{t+1}=d\r ]\r )\\ &=\frac {S_t}{(1+r)^{t+1}}(uq_u+dq_d)\\ &=\frac {S_t}{(1+r)^{t+1}}(1+r)\\ &=M_t. \end{align*} Here, from the second to third line we take out what is known, using that $S_t\in m\mc {F}_t$. To deduce the third line we use linearity, and to deduce the fourth line we use that $Z_{t+1}$ is independent of $\mc {F}_t$. Lastly, we recall from (5.2) that $uq_u+dq_d=1+r$. Hence, $(M_t)$ is a martingale with respect to the filtration $\mc {F}_t$, in the risk-neutral world $\Q $. ∎

Remark 5.5.7 Using Lemma 3.3.6 we have $\E ^\Q [M_0]=\E ^\Q [M_1]$, which states that $S_0=\frac {1}{1+r}\E ^\Q [S_1]$. This is precisely (5.4).

Stochastic Processes and Financial Mathematics (part one)

5.5 Portfolios, arbitrage and martingales

Stochastic Processes and Financial Mathematics
(part one)