Chapter 4 Probability theory for the binomial model

In this chapter, we look at the probability theory behind our pricing of contingent claims in the binomial model. This will allow us to put the important financial concepts on a proper mathematical foundation, and to prepare for Epiphany term. In continuous time, our intuition is less helpful, and we will need to approach the material using a rigorous mathematical theory.

4.1 The probability space

The probability space associated with the binomial model is mathematically equivalent to the coin toss space.

Toss a coin \(T\) times, and let \(\Omega_T\) denote the set of all possible outcomes. Each element \(\omega \in \Omega_T\) can be expressed as \[ \omega = \omega_1 \omega_2 \dots \omega_T, \] where each \(\omega_i\) represents the result of the \(i\)th coin toss, and is either H or T. The set \(\Omega_T\) has \(2^T\) elements in total.

Let \(\mathcal{F}\) be the \(\sigma\)-algebra of all subsets of \(\Omega_T\). We write \(\mathcal{F} = 2^{\Omega_T}\).

We define a probability measure \(\mathbb{P}\) on \(\Omega_T\) as follows. Let \(\#\text{heads}(\omega)\) be the number of heads in \(\omega = \omega_1 \dots \omega_T\), and \(\#\text{tails}(\omega) = T - \#\text{heads}(\omega)\) be the number of tails. Then let \[ \mathbb{P}(\omega) = p^{\#\text{heads}(\omega)} (1-p)^{\#\text{tails}(\omega)}, \] where \(0 < p < 1\) is a fixed real number; and for \(A \in \mathcal{F}\), \[ \mathbb{P}(A) = \sum_{\omega \in A} \mathbb{P}(\omega).\]

For a different \(0 < q < 1\), we can define a different measure \(\mathbb{Q}\) on \(\Omega_T\) with \[ \mathbb{Q}(\omega) = q^{\#\text{heads}(\omega)} (1-q)^{\#\text{tails}(\omega)}, \qquad \mathbb{Q}(A) = \sum_{\omega \in A} \mathbb{Q}(\omega).\]

Recall that two probability measures \(\mathbb{P}\) and \(\mathbb{Q}\) are equivalent to each other if \[ \mathbb{P}(A) > 0 \text{ if and only if } \mathbb{Q}(A) > 0. \] The measures do not need to assign the same probabilities to each event, but they should agree that either \(A\) is a possible event (both \(\mathbb{P}(A)\) and \(\mathbb{Q}(A)\) are strictly positive), or that \(A\) is an impossible event (both are zero). For example, in our case \(\mathbb{P}\) and \(\mathbb{Q}\) are equivalent to each other as long as both \(0 < p < 1\) and \(0 < q < 1\). Typically, we think of \(\mathbb{P}\) as the ‘’objective measure’’, representing our estimate of how likely a head is to occur in reality, and we think of \(\mathbb{Q}\) as the martingale measure that we use for pricing.

Remark. These are the probability measures arising in the recombinant version of the binomial model. In the more general case, \(q_u\) and \(q_d\) can take different values at each node. As a result, for each \(\omega \in \Omega_T\), \(\mathbb{Q}(\omega)\) becomes a product of \(q_u\)s and \(q_d\)s, which can be found by “multiplying along the branches of the tree.”

4.2 Share prices and contingent claims

We define the stock prices \(S_0, S_1, \dots, S_T\) as random variables on \(\Omega_T\) in the following way: \[ S_0(\omega) = s, \quad \text{for all } \omega \in \Omega_T, \] and for each \(t = 1, 2, \dots, T\), \[ S_t(\omega) = Z_t(\omega) S_{t-1}(\omega).\] Here \(Z_1, Z_2, \dots, Z_T\) are independent and identically distributed random variables, with \[ Z_t(\omega) = \begin{cases} u & \text{if } \omega_t = \text{H}, \\ d & \text{if } \omega_t = \text{T}. \end{cases} \]

Remark. The \(T\) random variables \(Z_1, Z_2, \dots, Z_T\) encapsulate all the randomness in the probability space \((\Omega_T, \mathcal{F}, \mathbb{P})\): the \(2^T\) elements of \(\Omega_T\) are in one-to-one correspondence with the \(2^T\) possible real-valued sequences \((z_1, z_2, \dots, z_T)\) describing the possible values of \((Z_1, Z_2, \dots, Z_T)\). In other words, if we know the values of all of \(Z_1, Z_2, \dots, Z_T\), then we know which state \(\omega \in \Omega_T\) we are in, and vice versa.

We say that \(\mathcal{F} = 2^{\Omega_T}\) is equal to the \(\sigma\)-algebra generated by the random variables \(Z_1, Z_2, \dots, Z_T\), and we write \(\mathcal{F} = \sigma(Z_1, Z_2, \dots, Z_T)\).

Example 4.1 In the recombinant market with \(s=4\), \(T=3\), \(u=2\), and \(d=\frac12\), we have \[\begin{align*} S_0(\omega_1 \omega_2 \omega_3) & = 4, \quad \text{ for all } \omega \in \Omega_T; \\ S_1(\omega_1 \omega_2 \omega_3) & = \begin{cases} 8 & \text{if } \omega_1 = \text{H}, \\ 2 & \text{if } \omega_1 = \text{T}; \end{cases} \\ S_2(\omega_1 \omega_2 \omega_3) & = \begin{cases} 16 & \text{if }\omega_1=\omega_2=\text{H},\\ 4& \text{if } \omega_1 \neq \omega_2,\\ 1& \text{if }\omega_1=\omega_2=\text{T}; \end{cases} \end{align*}\] and \[\begin{align*} S_3(\omega_1 \omega_2 \omega_3) & = \begin{cases} 32 & \text{if } \omega_1 = \omega_2 = \omega_3 = \text{H}, \\ 8 & \text{if there are two heads and one tail } \\ 2 & \text{if there are two tails and one head} \\ \frac12 & \text{if } \omega_1 = \omega_2 = \omega_3 = \text{H}. \end{cases} \end{align*}\]

Since the share price \(S_t\) only depends on the first \(t\) coin tosses, we sometimes write \(S_t(\omega_1 \omega_2 \dots \omega_t)\), or even \(S_t (\cdots\omega_{t})\), rather than \(S_t(\omega_1 \omega_2 \dots \omega_T)\).

A contingent claim is a random variable \(X\) defined on this probability space. We write \(X(\omega)\) for the realisation of \(X\) corresponding to the outcome \(\omega\); for example, if \(X\) is a European call option we have \[ X(\omega) = (S_T(\omega) - K)^+. \]

Using this framework, we can find the fair price at any time of general contingent claims. The structure here is largely the same as in Chapter 3, but we can now price claims such as \[ X = (S_1 - K_1)^+ + (S_T - K_2)^+. \]

4.2.1 An algorithm to price any contingent claim

This algorithm allows us to price any contingent claim via its hedging portfolio.

First, we define \(T\) random variables, \(X_{T-1}, \dots, X_1, X_0\), on our probability space, using the recursive formula \[ X_t( \cdots \omega_t) = \frac{1}{1+r} \big( q_u X_{t+1} (\cdots \omega_t \text{H}) + q_d X_{t+1} (\cdots \omega_t \text{T}) \big). \] Here \(q_u\) and \(q_d\) are the martingale probabilities at each node, given by \[ q_u = \frac{1+r-d}{u-d}, \quad q_d = \frac{u-(1+r)}{u-d}.\]

Next, we let \[ y_t (\cdots \omega_{t-1}) = \frac{ X_t(\cdots \omega_{t-1} \text{H}) - X_t(\cdots \omega_{t-1}\text{T}) }{ S_t (\cdots \omega_{t-1} \text{H}) - S_t(\cdots \omega_{t-1} \text{T})}, \quad t = 1 \dots T. \]

Finally, set \(V_0 = X_0\) for all \(\omega \in \Omega_T\). Now, for \(t = 0, \dots, T-1\), let \[ \quad\quad V_{t+1} =y_{t+1}S_{t+1}+(1+r)(V_t-y_{t+1}S_t). \quad\quad\quad(\textbf{Wealth Equation}) \]

Theorem 4.1 The portfolio given by \(y_t(\cdots \omega_{t-1})\) as defined above, with \[ x_t(\cdots \omega_{t-1}) = \frac{ V_{t-1}(\cdots \omega_{t-1}) - y_t(\cdots \omega_{t-1}) S_{t-1}(\cdots \omega_{t-1})}{B_{t-1}} ,\] is self-financing, has value process \(V_t\) as defined above, and replicates the contingent claim \(X\). Moreover, for every \(t\) and \(\omega\), \(V_t(\cdots \omega_t) = X_t(\cdots \omega_t)\).

Proof. First, it is clear from the definition of \(x_t\) that for each \(t\), the relation \[ V_t = x_{t+1} B_t + y_{t+1} S_t \] holds for every \(\omega \in \Omega_T\).

Next, to see that the self-financing condition holds, we check that \[ V_{t+1} = x_{t+1} B_{t+1} + y_{t+1} S_{t+1} \] for \(t = 0, \dots, T\).

We have \[\begin{align} x_{t+1} B_{t+1} + y_{t+1} S_{t+1} & = \frac{V_t - y_{t+1} S_{t}}{B_t} B_{t+1} + y_{t+1} S_{t+1} \\ & = (1+r) (V_t - y_{t+1}S_{t}) + y_{t+1} S_{t+1} \\ & = V_{t+1}, \end{align}\] by definition.

Now, to see that \(V\) and \(X\) always coincide, we use induction. The base case \(V_0 = X_0\) is true by definition. Next, we fix \(\omega_1\dotsm\omega_t\) and use \(V_t(\omega_1\dotsm\omega_t) = X_t(\omega_1\dotsm\omega_t)\) to show that \[\begin{align*} V_{t+1}(\cdots\omega_t\H) &= X_{t+1}(\cdots\omega_t\H), \quad\text{and}\\ V_{t+1}(\cdots\omega_t\T) &= X_{t+1}(\cdots\omega_t\T). \end{align*}\]

First, if \(\omega_{t+1} = \H\), we use the fact that \(S_{t+1}(\cdots \omega_t \H) = u S_t(\cdots \omega_t)\) in the wealth equation to write \[\begin{align} V_{t+1}(\cdots\omega_t\H) &= y_{t+1} u S_t + (1+r) (V_t - y_{t+1}S_t) \\ & = y_{t+1} S_t \big(u - (1+r) \big) + (1+r) V_t. \end{align}\] (Here, we have stopped writing \(\cdots \omega_t\) in the right hand side, to simplify the notation.)

Using the inductive hypothesis and the expression for \(y_{t+1}\), we get \[\begin{align} V_{t+1}(\cdots\omega_t\H) &= \frac{X_{t+1}(\H) - X_{t+1}(\T)}{uS_t - dS_t}S_t \big(u - (1+r)\big) + (1+r)X_t\\ &=( X_{t+1}(\H) - X_{t+1}(\T) ) \frac{u - (1+r)}{u-d} + (1+r) X_t\\ &=( X_{t+1}(\H) - X_{t+1}(\T) ) q_d + [q_u X_{t+1}(\H) + q_d X_{t+1}(\T)]\\ &= X_{t+1}(\H), \end{align}\] as required.

Exercise: Check that, when \(\omega_{t+1} = \T\), we also have \(V_{t+1} = X_{t+1}\).

4.3 Filtrations, conditional expectation and martingales

Definition 4.1 A filtration is a non-descending sequence of \(\sigma\)-algebras \[ \mathcal{F}_0 \subseteq \mathcal{F}_1 \subseteq \dots \subseteq \mathcal{F}_n. \]

For example, in our coin-toss space, we can construct a sequence of \(\sigma\)-algebras in which \(\mathcal{F}_k\) is based on the first \(k\) coin tosses:

Take \(\mathcal{F}_0=\{\emptyset, \Omega\}\). This \(\sigma\)-algebra corresponds to the stock price \(S_0\) at time \(0\).
Let \(A_\H=\{\omega: \omega_1=\H\}, A_\T=\{\omega: \omega_1=\T\}\) and let \[ \mathcal{F}_1=\{\emptyset, \Omega, A_\H, A_\T\}. \]
Let \[ A_{\H\H}=\{\omega: \omega_1=\H, \omega_2=\H\}, A_{\H\T}=\{\omega: \omega_1=\H, \omega_2=\T\}, \] \[ A_{\T\H}=\{\omega: \omega_1=\T, \omega_2=\H\}, A_{\T\T}=\{\omega: \omega_1=\T, \omega_2=\T\} \] and let \[ \mathcal{F}_2=\sigma( A_{\H\H}, A_{\T\H}, A_{\H\T}, A_{\T\T}). \] Recall that we write \(\mathcal{F} = \sigma( A_1, A_2, \dots, A_k)\) when \(\mathcal{F}\) is the smallest \(\sigma\)-algebra containing the sets \(A_1, A_2, \dots, A_k\). In other words, the elements of \(\mathcal{F}_2\) are all the events which can be obtained as intersections, unions, and complements of \(A_{\H\H}\), \(A_{\H\T}\), \(A_{\T\H}\), and \(A_{\T\T}\).
Let \[ \mathcal{F}_3 = \sigma ( A_{\H\H\H}, A_{\H\H\T}, A_{\H\T\T}, A_{\H\T\H}, A_{\T\H\H}, A_{\T\H\T}, A_{\T\T\H}, A_{\T\T\T}), \] where \(A_{\H\H\H} = \{ \omega : \omega_1 = \H, \omega_2 = \H, \omega_3 = \H\}\), and so on.

Continuing in this way, we eventually define \(\mathcal{F}_T = 2^{\Omega_T}\), and we obtain a filtration \(\mathcal{F}_0\subseteq \mathcal{F}_1\subseteq \mathcal{F}_2\subseteq \cdots \subseteq \mathcal{F}_T = 2^{\Omega_T}\).

Remark. We can think of \(\mathcal{F}_k\) as containing all the events whose outcomes are completely determined by knowledge of the first \(k\) coin tosses. We can therefore also write \(\mathcal{F}_k = \sigma(Z_1, \dots, Z_k)\).

Definition 4.2 A random variable \(X : \Omega \to \mathbb{R}\) is measurable with respect to a \(\sigma\)-algebra \(\mathcal{F}\) if for every Borel set \(B \in \mathcal{B}(\mathbb{R})\), \[ \{ \omega \in \Omega: X(\omega) \in B \} \in \mathcal{F}. \]

Example 4.2 We have already met some random variables which are measurable with respect to filtrations:

The holdings in a portfolio at time \(t-1\), \(x_t\) and \(y_t\), are allowed to be random as long as they are measurable with respect to \(\mathcal{F}_{t-1}\).
A random variable \(T\) is a stopping time with respect to the filtration \((\mathcal{F}_t)\) if, for every \(t\), \(\{ T \leq t \} \in \mathcal{F}_{t}\).

Definition 4.3 Given a random variable \(X\) on \(\Omega_T\), the conditional expectation of \(X\) with respect to \(\mathcal{F}_t\) is the unique random variable \(\mathbb{E}_{\mathbb{Q}}[X \vert F_{t}]\) which satisfies:

\(\mathbb{E}_{\mathbb{Q}} [X \vert F_{t}]\) is measurable with respect to \(\mathcal{F}_t\)
\(\mathbb{E}_{\mathbb{Q}} \big[ \mathbb{E}_{\mathbb{Q}}[X \vert F_{t}] \big] = \mathbb{E}_{\mathbb{Q}}[X]\).

If \(X_T = \Phi(S_1, \dots, S_T)\) then the conditional expectation \(\mathbb{E}_{\mathbb{Q}}[X_T | \mathcal{F}_t]\) should be thought of as the expected value of \(X_T\) if we fix the outcomes of \(S_1, \dots, S_t\) and average over the remaining randomness that determines \(S_{t+1}, \dots, S_T\), i.e., over \(Z_{t+1}, \dots, Z_T\).

For \(\omega \in \Omega_T\), we have \[\begin{align} \mathbb{E}_{\mathbb{Q}} [ X_T \vert \mathcal{F}_{t} ] (\cdots \omega_t) = q_u \mathbb{E}_{\mathbb{Q}} [X_T \vert \mathcal{F}_{t+1} ] (\cdots \omega_t \H) + q_d \mathbb{E}_{\mathbb{Q}} [X_T \vert \mathcal{F}_{t+1} ] (\cdots \omega_t \T). \end{align}\]

Example 4.3 We find the conditional expectations \(\mathbb{E}_{\mathbb{Q}} [ S_2 \vert \mathcal{F}_1]\) and \(\mathbb{E}_{\mathbb{Q}}[S_3 \vert \mathcal{F}_1]\) in the model from Example 4.1, when \(r=0\). Firstly, we have \[ q_u = \frac{1-\frac{1}{2}}{2-\frac{1}{2}} = \frac{1}{3}, \quad q_d = \frac{2 - 1}{2 - \frac{1}{2}} = \frac{2}{3}. \]

Now, \[\begin{align} \mathbb{E}_{\mathbb{Q}} [ S_2 \vert \mathcal{F}_1](\H) &= 16 \times \frac{1}{3} + 4 \times \frac{2}{3} = 8, \\ \mathbb{E}_{\mathbb{Q}} [ S_2 \vert \mathcal{F}_1](\T) &= 4 \times \frac{1}{3} + 1 \times \frac{2}{3} = 2. \end{align}\] We see that, in both cases, \(\mathbb{E}_{\mathbb{Q}} [ S_2 \vert \mathcal{F}_1] = S_1\).

Next, \[\begin{align} \mathbb{E}_{\mathbb{Q}} [ S_3 \vert \mathcal{F}_1](\H) &= 32 \times \left(\frac{1}{3}\right)^2 + 8 \times 2 \times \frac{1}{3} \times \frac{2}{3} + 2 \times \left(\frac{2}{3} \right)^2 = 8, \\ \mathbb{E}_{\mathbb{Q}} [ S_3 \vert \mathcal{F}_1](\T) &= 8 \times \left(\frac{1}{3}\right)^2 + 2 \times 2 \times \frac{1}{3} \times \frac{2}{3} + \frac{1}{2} \times \left(\frac{2}{3} \right)^2 = 2. \end{align}\] Once again, we have \(\mathbb{E}_{\mathbb{Q}} [S_3 \vert \mathcal{F}_1] = S_1\).

Important Properties of Conditional Expectations

Linearity. For constants \(a_1,a_2\), we have \[ \mathbb{E}[a_1X+a_2Y\vert\mathcal{F}_t]=a_1\mathbb{E}[X\vert\mathcal{F}_t]+a_2\mathbb{E}[Y\vert\mathcal{F}_t]. \]
Taking out what is known. If \(X\) depends only on the first \(t\) coin flips, then \[ \mathbb{E}[XY\vert\mathcal{F}_t]=X\cdot\mathbb{E}[Y\vert\mathcal{F}_t]. \]
Iterated conditioning. If \(s \leq t\) then \[ \mathbb{E}[\mathbb{E}[X\vert\mathcal{F}_t]\vert\mathcal{F}_s]=\mathbb{E}[X\vert\mathcal{F}_s]. \]
Independence. If \(X\) depends only on coin tosses \(t+1\) to \(T\), then \[ \mathbb{E}[X\vert\mathcal{F}_t]=\mathbb{E}[X]. \]

Definition 4.4 A sequence of random variables \(Y_0, Y_1, \cdots, Y_T\) is called a martingale under the measure \(\mathbb{Q}\) if for each \(t\), the value of \(Y_t\) depends on the outcome of the first \(t\) coin flips (we say the sequence is adapted to the filtration) and \[ \mathbb{E}_{\mathbb{Q}}[Y_{t+1}\vert\F_t]=Y_t, \quad t=0, 1, \dots, T-1. \]

Theorem 4.2 The sequence of discounted stock prices \[ \frac{S_t}{(1+r)^t}, \quad t=0, 1, 2, \dots, T, \] is a martingale under the risk-neutral measure \(\mathbb{Q}\).

Remark. The converse of this statement is also true in the multi-period binomial model. That is to say, the martingale measure \(\mathbb{Q}\) in an arbitrage-free and complete multi-period binomial model is determined by the property that \(\frac{S_t}{(1+r)^t}\) forms a martingale sequence under \(\mathbb{Q}\).

Proof. We have \[ \begin{split} \mathbb{E}_{\mathbb{Q}}\bigg[\frac{S_{t+1}}{(1+r)^{t+1}}\vert\F_t\bigg](\omega_1\cdots \omega_t)&= q_{u }\frac{S_{t+1}(\omega_1\cdots \omega_tH)}{(1+r)^{t+1}}+q_d\frac{S_{t+1}(\omega_1\cdots \omega_tT)}{(1+r)^{t+1}} \\ &=\frac{S_{t}(\omega_1\cdots \omega_t)}{(1+r)^{t+1}}[q_{u }u +q_dd]=\frac{S_{t}(\omega_1\cdots \omega_t)}{(1+r)^t}. \end{split} \]

Theorem 4.3 The discounted value process \[ \frac{V_t}{(1+r)^t},\quad t=0, 1, \dots, T, \] of any self-financing strategy is a martingale under the risk-neutral measure.

Proof. Recall that the self-financing condition implies the wealth equation, \[ V_{t+1}=y_{t+1}S_{t+1}+(1+r)(V_t-y_{t+1}S_t). \] We have \[ \begin{split} \mathbb{E}_{\mathbb{Q}}\bigg[\frac{V_{t+1}}{(1+r)^{t+1}}\vert\F_t\bigg] &=\mathbb{E}_{\mathbb{Q}} \bigg[\frac{y_{t+1}S_{t+1}+(1+r)(V_t-y_{t+1}S_t)}{(1+r)^{t+1}}\vert\F_t\bigg] \\ \text{(linearity)}\to\quad &=\mathbb{E}_{\mathbb{Q}} \bigg[\frac{y_{t+1}S_{t+1}}{(1+r)^{t+1}}\vert\F_t\bigg]+\mathbb{E}_{\mathbb{Q}} \bigg[\frac{(1+r)(V_t-y_{t+1}S_t)}{(1+r)^{t+1}}\vert\F_t\bigg] \\ \text{(taking out what is known)}\to\quad &=y_{t+1}\mathbb{E}_{\mathbb{Q}} \bigg[\frac{S_{t+1}}{(1+r)^{t+1}}\vert\F_t\bigg]+\frac{V_t-y_{t+1}S_t}{(1+r)^t} \\ &=y_{t+1}\frac{S_{t}}{(1+r)^{t}}+\frac{V_t-y_{t+1}S_t}{(1+r)^t}\\ &=\frac{V_t}{(1+r)^t}, \end{split} \] showing that \(V_t/(1+r)^t\) is a martingale.

This proves the correctness of the risk-neutral valuation formula for pricing contingent claims: \[ V_t=\frac{1}{(1+r)^{T-t}}\mathbb{E}_{\mathbb{Q}}[V_T\vert\F_t]. \]

We finish this section with a version of the First Fundamental Theorem for the multi-period binomial model. We require a definition of arbitrage on the multi-period binomial model.

Definition 4.5 A portfolio \(h \equiv \big( h_t = (x_t,y_t), t=0,1,\dots,T+1 \big)\) on the multi-period binomial model \(\mathcal{M} = (B_t, S_t)\) is an arbitrage portfolio if it is self-financing and its value process \(V^h_t = x_{t+1} B_t + y_{t+1} S_t\) satisfies: \[ V_0^h=0, \quad \mathbb{P}(V_T^h\geq 0)=1, \quad \mathbb{P}(V_T^h>0)>0. \]

Theorem 4.4 The following conditions are equivalent for a multi-period binomial model \(\mathcal{M} = (B_t, S_t)\), \(t=0,1,\ldots, T\), with interest rate \(r\).

The model is arbitrage-free according to Definition 4.5.
The condition \(d < 1+r < u\) holds, where \(d < u\) are the two possible values of \(Z_t = S_t/S_{t-1}\) at each time \(t\). (\(Z_t\) equals \(u\) with probability \(p\) and \(d\) with probability \(1-p\) for some \(0<p<1\)).
There is a measure \(\mathbb{Q}\) defined by \[ q_u = \frac{1+r - d}{u - d}, \quad q_d = \frac{u - (1+r)}{u-d} \] at each node of the tree, such that \(\frac{S_t}{(1+r)^t}\) is a martingale under \(\mathbb{Q}\).

Proof. We have done most of the work needed to prove this theorem. Let us show the implications \((1) \Rightarrow (2) \Rightarrow (3) \Rightarrow (1)\).

\((1) \Rightarrow (2)\): Consider the number of periods \(T\) in the model. If \(T=1\) then the implication holds by Theorem 2.1. For \(T > 1\), the 1-period model obtained by observing the market from \(t=0\) to \(t=1\) has no arbitrage, and so \(d < 1+r < u\) by Theorem 2.1.
\((2) \Rightarrow (3)\): This implication is Theorem 4.2 above.
\((3) \Rightarrow (1)\): Suppose \(h_t = (x_t,y_t)\) is a self-financing portfolio that satisfies the conditions \(\mathbb{P}(V^h_0 = 0) =1\) and \(\mathbb{P}(V^h_T \geq 0) = 1\). Since the measure \(\mathbb{Q}\) is equivalent to \(\mathbb{P}\), it follows that \(\mathbb{Q}(V^h_0 = 0) =1\) and \(\mathbb{Q}(V^h_T \geq 0) = 1\). Now, by Theorem 4.3, \(\frac{V_t}{(1+r)^t}\) is a martingale under \(\mathbb{Q}\) and so in particular, \[\mathbb{E}_{\mathbb{Q}} \left [\frac{V_T}{(1+r)^T}\right ] = \mathbb{E}_{\mathbb{Q}} [V_0] = 0.\] This shows that \(\mathbb{E}_{\mathbb{Q}}[V_T] = 0\) and thus \(V_T\) is a non-negative random variable with mean 0, which implies \(V_T\) is identically zero: \(\mathbb{Q}(V_T > 0) = 0\). Consequently, \(\mathbb{P}(V_T > 0) = 0\) as well so \(h\) is not an arbitrage portfolio.