Discrete Martingale Theory

In the previous chapter, we introduced binomial model for option pricing where the price is determined by constructing a replicating portfolio consisting of the underlying asset and the risk-free asset. As a consequence of this construction, the option price is expressed as the expected value of the discounted payoff under a risk-neutral probability measure. The risk-neutral probability is uniquely determined by the no-arbitrage condition, and the existence of a replicating portfolio ensures that the derivative can be perfectly hedged.

In this chapter, we place these ideas in a more general mathematical framework for discrete-time financial markets. In particular, we develop the martingale theory and introduce the concept of an equivalent martingale measure, which is exactly the risk-neutral probability measure in the binomial model. We further show that under such a measure, the discounted price processes of traded assets become martingales and the option prices can be expressed as discounted expectation of their future payoffs.

The martingale framework provides a general approach to pricing and hedging options in discrete time setup, and it also provides a natural extension of the pricing model to continuous-time framework.

Basic Concepts

In this section, we introduce two concepts, namely, the normalized market and conditional expectation.

Normalized Market and Numéraire

Recall that binomial models express option prices in terms of discounted payoffs. This approach extends to any option pricing models, in general. Thus, it is convenient to work within a normalized market where price processes are discounted.

Generally, it is assumed that the value of the risk-free asset in a portfolio at any time is strictly positive, i.e., \(B_t>0\). Often one considers the normalized price of the risky assets given by

\[ \widetilde{\boldsymbol{S}}_k = \frac{\boldsymbol{S}_k}{B_k}, k=0,1,2,\ldots, n. \]

In the normalized form, the finance market is called the normalized finance market or discounted market, denoted by \((1,\widetilde{\boldsymbol{S}})\). The process \(\{B_k\}\) is called a numéraire.

In a discounted market, the value of a portfolio \(\Pi_k\) at time \(t=t_k\), \(k=1,2,\ldots,n\), is given by

\[ \widetilde{V}_k = \phi_k + \boldsymbol{\theta}_k \cdot \widetilde{\boldsymbol{S}}_k \]

and is called the discounted value. The corresponding value process \(\{\widetilde{V}_k\}\) is called the discounted value process.

Note:

Note that at every time level \(t_k\), since the value of the risk-free asset is normalized, we have \(\widetilde{B}_k = 1\). This makes the value of this asset at time \(t_{k+1}\) as \(B_{k+1} = \widetilde{B}_k I(\tilde{r})=I(\tilde{r})\). Further, the price in the normalized market becomes \(\widetilde{B}_{k+1}=1\).

Characterization of Self-financing condition

Our next task is to find a necessary and sufficient condition for a strategy to be self-financing. Let us first rewrite the self-financing condition in a normalized market.

Remark:

The self-financing condition in Definition «Click Here» is equivalent to the condition

\[ \Delta \phi_k B_k + \Delta \boldsymbol{\theta}_k \cdot \boldsymbol{S}_k =0, \]

where

\[ \Delta \phi_k = (\phi_{k+1}-\phi_k)~\text{and}~ \Delta \boldsymbol{\theta}_k= (\boldsymbol{\theta}_{k+1}-\boldsymbol{\theta}_k), \]

for \(k=1,2,\ldots, n-1.\)

With respect to the discounted finance market, the self-finance condition can be re-written as

\[ \Delta \phi_k + \Delta \boldsymbol{\theta}_k \cdot \widetilde{\boldsymbol{S}}_k =0, \]

for \(k=1,2,\ldots, n-1.\)

Note:

With a slight abuse of notation, we use \(V_0\) to denote the initial value of a strategy, i.e., \(V_0 = V_0(\Pi_1)\). Accordingly, the discounted initial value is denoted by \(\widetilde{V}_0\).

Problem:

Show that a strategy \(\{(\phi_k,\boldsymbol{\theta_k})~|~k=1,2,\ldots, n\}\) is self-financing if and only if

\[ \phi_{k+1} = \widetilde{V}_0 - \sum_{j=0}^k \Delta \boldsymbol{\theta}_j \cdot \widetilde{\boldsymbol{S}}_j, ~~k=0,1,\ldots,n-1, \]

where \(\boldsymbol{\theta}_0=\boldsymbol{0}\).

Lemma:

Show that a strategy \(\{(\phi_k,\boldsymbol{\theta_k})~|~k=1,2,\ldots, n\}\) is self-financing if and only if, for every \(k=1,2,\ldots, n\),

\[ \widetilde{V}_k = \widetilde{V}_0 + \widetilde{G}_k, \]

where

\[ \widetilde{G}_k = \sum_{j=1}^{k} \boldsymbol{\theta}_j \cdot \Delta \widetilde{\boldsymbol{S}}_{j-1} \]

is called the discounted gain for the predictable process \(\{\boldsymbol{\theta}_k\}\).

Proof:

For \(k=1,\ldots,n,\) the normalized value process is given by

\[ \widetilde{V}_k = \phi_k + \boldsymbol{\theta}_k \cdot \widetilde{\boldsymbol{S}}_k. \]

Using Problem «Click Here» in the above expression, we see that the strategy is self-financing if and only if

\[ \widetilde{V}_k = \widetilde{V}_0 - \sum_{j=0}^{k-1} \Delta \boldsymbol{\theta}_j \cdot \widetilde{\boldsymbol{S}}_j + \boldsymbol{\theta}_k \cdot \widetilde{\boldsymbol{S}}_k. \]

Expanding the RHS sum, we see that the strategy is self-financing if and only if

\begin{eqnarray} \widetilde{V}_k &=& \widetilde{V}_0 + \Big(\boldsymbol{\theta}_1 \cdot (\widetilde{\boldsymbol{S}}_1 - \widetilde{\boldsymbol{S}}_0) +\boldsymbol{\theta}_2\cdot (\widetilde{\boldsymbol{S}}_2 - \widetilde{\boldsymbol{S}}_1) +\boldsymbol{\theta}_3\cdot (\widetilde{\boldsymbol{S}}_3 - \widetilde{\boldsymbol{S}}_2)\\ &&~~~~~~~~~~~~+ \ldots +\boldsymbol{\theta}_{k-1}\cdot (\widetilde{\boldsymbol{S}}_{k-1}-\widetilde{\boldsymbol{S}}_{k-2}) +\boldsymbol{\theta}_k\cdot (\widetilde{\boldsymbol{S}}_k - \widetilde{\boldsymbol{S}}_{k-1})\Big). \end{eqnarray}

(6.1)

Thus, we proved that the strategy is self-financing if and only if

\begin{eqnarray} \widetilde{V}_k &=& \widetilde{V}_0 + \sum_{j=1}^k \boldsymbol{\theta}_j \cdot (\widetilde{\boldsymbol{S}}_j - \widetilde{\boldsymbol{S}}_{j-1})\\ &=&\widetilde{V}_0 + \sum_{j=1}^k \boldsymbol{\theta}_j\cdot \Delta \widetilde{\boldsymbol{S}}_{j-1} =\widetilde{V}_0 + \widetilde{G}_k. \end{eqnarray}

(6.2)

Note:

We can choose \(B_0=1\) so that the discounted initial value \(\widetilde{V}_0\) coincides with the initial value \(V_0\). This choice leads to \(B_1 = I(\tilde{r})\) at time \(t_1\), and consequently the numéraire value \(\widetilde{{V}}_1\) equals the discounted value of \(V_1\) justifying the name mentioned in Remark «Click Here» .

Conditional Expectation

In many financial applications, including option pricing models, it is essential to estimate the price of an asset based on the available information at a given time. As we have seen in our studies so far, the asset price at a fixed time \(t\) is modeled by a random variable \(X_t\) and the available information is captured by a \(\sigma\)-field \(\mathcal{F}_t\). The estimated price is then expressed as the expectation of \(X_t\) given \(\mathcal{F}_t\), which is denoted by \(\mathbb{E}(X_t ~|~ \mathcal{F}_t)\).

While deriving a model to obtain a fair price of an option using binomial models (in Remark «Click Here» ), we arrived at an explicit formula. We noted that this formula is precisely the conditional expectation of a random variable given a \(\sigma\)-field. We now give a rigorous definition of this concept, as it also plays a central role in martingale theory.

Definition:

[Conditional Expectation]

Let \(X\) be a random variable defined on a probability space \((\Omega,\mathcal{F},\mathbb{P})\) with \(\mathbb{E}(|X|)<\infty\) and let \(\mathcal{F}_1\subseteq \mathcal{F}\) be a \(\sigma\)-field. A \(\mathcal{F}_1\)-measurable random variable \(Y\) defined on \((\Omega,\mathcal{F}, \mathbb{P})\) with \(\mathbb{E}(|Y|)<\infty\) is said to be a conditional expectation of \(X\) given \(\mathcal{F}_1\), denoted by \(Y=\mathbb{E}(X~|~\mathcal{F}_1)\), if

\[ \mathbb{E}(Y{1\hspace{-0.09in}1}_A) = \mathbb{E}(X{1\hspace{-0.09in}1}_A), \text{ for all }A\in \mathcal{F}_1, \]

where \({1\hspace{-0.09in}1}_A\) denotes the indicator function of the set \(A\).

Remark:

A random variable \(Y'\) defined on \((\Omega,\mathcal{F}_1,\mathbb{P})\) that differs from \(Y\) only on a set of probability zero is also a conditional expectation of \(X\) given \(\mathcal{F}_1\). Hence, conditional expectations are not unique. As a consequence, properties regarding conditional expectations generally hold only almost surely. However, we omit to mention a.s. as it holds in all the results concerning conditional expectation. This is an important technical detail that one has to keep in mind always.

In particular, if \(\mathcal{F}_1 = \sigma(Z)\), for some random variable \(Z\) on \((\Omega,\mathcal{F}_1,\mathbb{P})\), then we can also use the notation \(\mathbb{E}(X~|~Z)\).

Note:

By the notation \(\mathbb{E}(X~|~X_1,X_2,\ldots, X_n)\), we mean \(\mathbb{E}(X~|~\mathcal{F}^{X}_n)\)

Example:

Consider the filtration probability space \((\Omega,\mathcal{F}^*,\mathbb{P},\{\mathcal{F}_k\})\) defined on \(\mathbb{T}_n=\{t_0,t_1,\ldots,t_n\}\). Here, we take the probability measure as

\[ \mathbb{P}(\boldsymbol{\omega}) = {p^*}^{U(\boldsymbol{\omega})}(1-p^*)^{D(\boldsymbol{\omega})}, ~ \boldsymbol{\omega}=(\omega_1,\omega_2,\ldots,\omega_n)\in \Omega = \Omega_1\times \Omega_2\times \ldots \times \Omega_n, \]

where for each \(k=1,2,\ldots n,\) \(\Omega_k = \{\text{up},\text{down}\}\), \(U(\boldsymbol{\omega})\) is the number of upward movements and \(D(\boldsymbol{\omega})\) is the number of downward movements of the stock. Also, recall that each \(\mathcal{F}_k = \sigma(\texttt{P}_k)\), where \(\texttt{P}_k\) is the collection of sets \(B_{\boldsymbol{\omega}_k}\) given by

\[ B_{\boldsymbol{\omega}_k} = \{\boldsymbol{\omega}\in \Omega~|~ \text{the first } k \text{ components of }\boldsymbol{\omega} \text{ equal }\boldsymbol{\omega}_k\}. \]

The \(\sigma\)-field \(\mathcal{F}_k\) represents the information generated by the movement of the stock at the first \(k\) time levels.

For any given \(k\in \{1,2,\ldots,n\}\) we use the notation

\[ \Omega_k' = \Omega_1\times\ldots\times \Omega_k ~\text{and}~ \Omega_k'' = \Omega_{k+1}\times\ldots\times \Omega_n. \]

With these notations, we can write \(\Omega = \Omega_k'\times\Omega_k''\), for each \(k=1,2,\ldots,n\), and therefore any \(\boldsymbol{\omega}\in \Omega\) can be written \(\boldsymbol{\omega} = (\boldsymbol{\omega}_k',\boldsymbol{\omega}_k'')\) with \(\boldsymbol{\omega}_k'\in \Omega_k'\) and \(\boldsymbol{\omega}_k''\in \Omega_k''\). Thus we can write, for any given \(\boldsymbol{\omega}_k'\in \Omega_k'\),

\[ B_{\boldsymbol{\omega}_k'} = \{(\boldsymbol{\omega}_k',\boldsymbol{\omega}_k'')\in \Omega~|~ \boldsymbol{\omega}_k''\in \Omega_k''\}. \]

For a given \(k\in \{1,2,\ldots,n\}\) and for any random variable \(X\) defined on the filtered probability space \((\Omega,\mathcal{F}^*,\mathbb{P},\{\mathcal{F}_k\})\), we claim that the conditional expectation is given by

\begin{eqnarray} \mathbb{E}(X~|~\mathcal{F}_k)(\boldsymbol{\omega}) = \sum_{\boldsymbol{\alpha}\in \Omega_k''} {p^*}^{U(\boldsymbol{\alpha})}(1-p^*)^{D(\boldsymbol{\alpha})}X(\boldsymbol{\omega}_k',\boldsymbol{\alpha}), \end{eqnarray}

(6.3)

for every \(\boldsymbol{\omega} = (\boldsymbol{\omega}_k',\boldsymbol{\omega}_k'')\in \Omega\). The realistic interpretation of the above expression is that the best prediction of \(X\) given the information provided by the first \(k\) movements of the stock (``known") is the average of \(X\) over the remaining outcomes \(\alpha\) (``unknown").

Let us denote the right-hand side sum by \(Y(\boldsymbol{\omega})\). Note that \(Y\) is actually a function of \(\boldsymbol{\omega}_k'\) only, i.e. \(Y(\boldsymbol{\omega})=Y(\boldsymbol{\omega}_k')\). Hence, \(Y\) is \(\mathcal{F}_k\)-measurable. Thus, as per Definition «Click Here» , we have to show that

\[ \mathbb{E}(Y{1\hspace{-0.09in}1}_A) = \mathbb{E}(X{1\hspace{-0.09in}1}_A),~\text{for all}~A\in \mathcal{F}_k. \]

In fact, it is enough to show the above equation for \(A=B_{\boldsymbol{\omega}_k'}\), for every \(\boldsymbol{\omega}_k'\in \Omega_k'.\)

We have

\begin{eqnarray} \mathbb{E}\left(Y{1\hspace{-0.09in}1}_{B_{\boldsymbol{\omega}_k'}}\right) &=&\sum_{\boldsymbol{\omega}\in B_{\boldsymbol{\omega}_k'}} Y(\boldsymbol{\omega})\mathbb{P}(\boldsymbol{\omega})\\ \end{eqnarray}

(6.4)

Using the definition of \(\mathbb{P}\) and substituting the right hand side of (6.3) in the place of \(Y\), we get \begin{multline*} \mathbb{E}\left(Y{1\hspace{-0.09in}1}_{B_{\boldsymbol{\omega}_k'}}\right) =\left({p^*}^{U(\boldsymbol{\omega}_k')}(1-p^*)^{D(\boldsymbol{\omega}_k')}\right)\\ \left(\sum_{\boldsymbol{\alpha}\in \Omega_k''}{p^*}^{U(\boldsymbol{\alpha})}(1-p^*)^{D(\boldsymbol{\alpha})}X(\boldsymbol{\omega}_k',\boldsymbol{\alpha})\right)\\ \left(\sum_{\boldsymbol{\alpha}\in \Omega_k''} {p^*}^{U(\boldsymbol{\alpha})}(1-p^*)^{D(\boldsymbol{\alpha})}\right). \end{multline*} Since

\[ \sum_{\boldsymbol{\alpha}\in \Omega_k''} {p^*}^{U(\boldsymbol{\alpha})}(1-p^*)^{D(\boldsymbol{\alpha})}=1, \]

we have the desired result.

The following problems provide some important properties of condition expectation.

Problem:

[Linearity property]

Let \(X_1\) and \(X_2\) be random variables on \((\Omega,\mathcal{F},\mathbb{P})\). For any \(a,b\in \mathbb{R}\), show that

\[ \mathbb{E}(aX_1+bX_2~|~\mathcal{F_1}) = a\mathbb{E}(X_1~|~\mathcal{F}_1) + b\mathbb{E}(X_2~|~\mathcal{F}_1), \]

where \(\mathcal{F}_1\subseteq \mathcal{F}\) is a \(\sigma\)-field on \(\Omega\).

Problem:

[Tower property]

Let \(X\) be a random variable on \((\Omega,\mathcal{F},\mathbb{P})\) and let \(\mathcal{F}_1\subseteq \mathcal{F}_2 \subseteq\mathcal{F}\). Show that

\[ \mathbb{E}\big(\mathbb{E}(X~|~\mathcal{F}_2)~|~\mathcal{F}_1\big) = \mathbb{E}(X~|~\mathcal{F}_1). \]

Lemma:

[Factor Property]

Let \(X\) be a random variable on \((\Omega,\mathcal{G},\mathbb{P})\) and \(Y\) be a random variable on \((\Omega,\mathcal{F},\mathbb{P})\). Then show that

\[ \mathbb{E}(XY~|~\mathcal{G}) = X\mathbb{E}(Y~|~\mathcal{G}). \]

Proof:

Since \(\mathbb{E}(Y~|~\mathcal{G})\) is a \(\mathcal{G}\)-measurable random variable, \(X\mathbb{E}(Y~|~\mathcal{G})\) is a \(\mathcal{G}\)-measurable random variable. So, it is enough to show that

\[ \mathbb{E}(X\mathbb{E}(Y~|~\mathcal{G}){1\hspace{-0.09in}1}_A) = \mathbb{E}(XY{1\hspace{-0.09in}1}_A),~~A\in \mathcal{G}. \]

We prove this for the special case that the range of \(X\) is a finite set \(\{x_1,\ldots,x_n\}\).

Set \(A_j=\{X=x_j\}\). Then \(A_j\in \mathcal{G}\) and

\[ {1\hspace{-0.09in}1}_AX = \sum_{j=1}^n x_j {1\hspace{-0.09in}1}_{AA_j}. \]

Multiplying this equation by \(\mathbb{E}(Y~|~\mathcal{G})\) and using linearity of conditional expectation, we get the required result.

Martingale Process and Measure

The aim of this section is to generalize the binomial model pricing idea to general discrete-time markets and connect it to the Fundamental Theorem of Asset Pricing (FTAP). To formalize this idea in general market models we first need to introduce martingales and equivalent martingale measures.

Discrete Martingale Process

We now define the notion of discrete martingale.

Definition:

[Discrete Martingale]

A stochastic process \(\{M_k\}\) defined on a probability space \((\Omega, \mathcal{F}^*,\mathbb{P})\) is a martingale with respect to a filtration \(\{\mathcal{F}_k\}\) if

the process \(\{M_k\}\) is adapted to \(\{\mathcal{F}_k\}\),
for each \(k=0,1,2,\ldots,n-1\), \(\mathbb{E}(|M_k|)<\infty\), and
\[ \mathbb{E}(M_{k+1}~|~\mathcal{F}_{k}) = M_k, ~~~\text{a.s.} \]

We say that \(\{M_n\}\) is a \(\mathcal{F}_k\)-martingale. Further,

If
\[ \mathbb{E}(M_{k+1}~|~\mathcal{F}_{k}) \le M_k, ~~~\text{a.s.},~\text{for each } k=0,1,2,\ldots,n-1, \]
then the process \(\{M_k\}\) is a supermartingale.
If
\[ \mathbb{E}(M_{k+1}~|~\mathcal{F}_{k}) \ge M_k, ~~~\text{a.s.},~\text{for each } k=0,1,2,\ldots,n-1, \]
then the process \(\{M_k\}\) is a submartingale.

Note:

The above definition can be extended to a multidimensional stochastic process, where each component process should be martingale. The above definition also holds if \(\{M_k\}\) is an infinite sequence.

Problem:

Let \(X\) be a random variable on (\(\Omega, \mathcal{F}^*,\mathbb{P})\) and let \(\{\mathcal{F}_k\}\) be a filtration. Define \(M_k=\mathbb{E}(X~|~\mathcal{F}_k)\), for \(k=0,1,2,\ldots\). Show that \(\{M_n\}\) is a martingale with respect to \(\{\mathcal{F}_k\}\).

Remark:

[Fair Game Interpretation]

The martingale conditions can be interpreted as gamblers' fair game condition. We can see this through the following steps:

Let us first interpret the process \(\{M_k\}\) as the winning amount of a gambler at different time levels, hence \(M_{k+1}-M_k\) denotes the gain accumulated during the period \((t_{k},t_{k+1}]\);
Since the process is adapted, \(M_k\) is \(\mathcal{F}_k\)-measurable. This may be interpreted as the information about all possibilities of \(M_k\) (and also \(M_j\), \(j=0,1,\ldots,k\)) are included in \(\mathcal{F}_k\).
Assume that we are at time \(t=t_k\) and a gambler would like to know the gain at time \(t=t_{k+1}\). Intuitively, we may agree that a 'fair game' should not give any information (to the gambler) about the future gain at the present time. Martingale condition represents this intuitive meaning of 'fair game' mathematically.
Indeed, since \(M_k\) is \(\mathcal{F}_k\)-measurable, we always have
\[ \mathbb{E}(M_k~|~\mathcal{F}_{k}) = M_k. \]
In fact, this can be seen using factor property. Indeed,
\[ \mathbb{E}(M_k~|~\mathcal{F}_{k}) = M_k\mathbb{E}(1~|~\mathcal{F}_{k}) =M_k. \]
By linearity of conditional expectation, we have
\[ \mathbb{E}(M_{k+1} - M_k ~|~\mathcal{F}_{k}) = \mathbb{E}(M_{k+1}~|~\mathcal{F}_{k}) - \mathbb{E}(M_k~|~\mathcal{F}_{k}). \]
Using martingale condition (2) in the first term and the predicted process condition on the second term, we get
\[ \mathbb{E}(M_{k+1} - M_k ~|~\mathcal{F}_{k}) = M_k - M_k = 0. \]
Thus, the expectation of the gain given all the relevant information ( i.e. given \(\mathcal{F}_k\) at time \(t=t_k\)) is zero. That means, the gambler has no significant information (based on the past \(k\) games) about the gain of the next game that needs to be initiated at time \(t=t_k\) and goes till \(t=t_{k+1}\).

Problem:

[Linearity]

Let \(\{M_k\}\) and \(\{M_k'\}\) be \(\mathcal{F}_k\)-martingales. Show that \(\{aM_k + bM_k'\}\) is a \(\mathcal{F}_k\)-martingale, for any \(a,b\in \mathbb{R}\).

Problem:

Let \(\{M_k\}\) be a \(\mathcal{F}_k\)-martingale. For any \(j < k\), show that

\[ \mathbb{E}(M_k~|~\mathcal{F}_j) = M_j. \]

Problem:

Show that any predictable martingale is almost surely constant.

Note:

The above result says that if we know the future value, then the best prediction of it is the known value itself.

Problem:

Consider the discounted stock price process \(\{\widetilde{S}_k\}\) obtained from a multi-step binomial model with the risk-neutral probability. Let \(\{\widetilde{S}_k\}\) be adapted to a filtration \(\{\mathcal{F}_k\}\). Show that in an arbitrage free market, \(\{\widetilde{S}_k\}\) is an \(\mathcal{F}_k\)-martingale.

If we interpret a stock investment as a game between buyer and seller, then the above result shows that the game is a fair game under the risk-neutral probability measure.

Problem:

Let \(\{\Pi_k\}\) be a self-financing strategy predictable with respect to a filtration \(\{\mathcal{F}_k\}\), and let \(\{\widetilde{S}_k\}\) be a discounted price process obtained from a multi-step binomial model with the risk-neutral probability. If \(\{\widetilde{V}_k\}\) is the corresponding discounted value process, then show that \(\{\widetilde{V}_k\}\) is a \(\mathcal{F}_k\)-martingale.

Fundamental Theorem of Asset Pricing

The mathematical concepts we developed in this section so far justify our earlier heuristic arguments and they also unlock two important theorems of quantitative finance, the Fundamental Theorems of Asset Pricing (FTAP), which connects no-arbitrage principles to the existence of risk-neutral measures.

Our aim in this subsection is to discuss the two fundamental theorems of asset pricing which ensures the existence of the risk-neutral probability in a market where there is more than one risky asset, but only finitely many (called finite market model).

Let a finite market \((B,\boldsymbol{S})\), where \(\boldsymbol{S}=(S^1,S^2,\ldots,S^m)\), be defined on a filtered probability space \((\Omega,\mathcal{F}^*,\mathbb{P}, \{\mathcal{F}_k\})\).

Definition:

Two probability measures \(\mathbb{P}_1\) and \(\mathbb{P}_2\) on \((\Omega,\mathcal{F})\) are equivalent (or mutually absolutely continuous) if for each \(A\in \mathcal{F}\),

\[ \mathbb{P}_1(A) = 0 ~\text{ if and only if }~ \mathbb{P}_2(A) = 0. \]

Definition:

[Equivalent Martingale Measure]

Given a market \((B,\boldsymbol{S})\) and a market model \((\Omega, \mathcal{F},\mathbb{P},\{\mathcal{F}_k\})\).

An equivalent martingale measure (EMM) is a probability measure \(\mathbb{P}^*\) defined on \((\Omega,\mathcal{F})\) such that

\(\mathbb{P}^*\) is equivalent to \(\mathbb{P}\) and
\(\{\widetilde{\boldsymbol{S}}_k\}\) is \(\mathcal{F}_k\)-martingale with respect to \(\mathbb{P}^*\), i.e., for each \(k=0,1,\ldots\), we have \(\mathbb{E}^*(\widetilde{\boldsymbol{S}}_{k+1}~|~\mathcal{F}_k) = \widetilde{\boldsymbol{S}}_k,\) where \(\mathbb{E}^*\) denotes the expectation under the probability measure \(\mathbb{P}^*\).

Remark:

Recall that in the finite market model, \(\mathbb{P}\) gives positive probability to every \(\boldsymbol{\omega}\in \Omega\). Thus, for the probability measure \(\mathbb{P}^*\) to be equivalent to \(\mathbb{P}\), we need \(\mathbb{P}^*(\{\boldsymbol{\omega}\})>0\), for all \(\boldsymbol{\omega}\in \Omega\).

Remark:

An EMM is often called a risk-neutral probability.

The following result is a general form of Problem «Click Here» .

Problem:

Let \(\mathbb{P}^*\) be an EMM and \(\{\Pi_k\}\) be a self-financing strategy in the finite market model. Then show that the discounted value process \(\{\widetilde{V}_k\}\) is a \(\mathcal{F}_k\)-martingale with respect to \(\mathbb{P}^*\).

The following theorem is very important in option pricing theory. We only prove the 'if ' part of the theorem and omit the 'only if' part for the course.

Theorem:

[First Fundamental Theorem of Asset Pricing]

A finite market is viable if and only if there exists an equivalent martingale measure \(\mathbb{P}^*\).

Proof:

Assume that there exists an EMM \(\mathbb{P}^*\). We have to show that the market \((B,\boldsymbol{S})\) is arbitrage free. We assume the contrary that the market allow arbitrage. That is, there exists an arbitrage portfolio. Hence, we can construct a self-finance strategy \(\{\Pi_k\}\) such that

\[ V_0 = 0,~~V_n \ge 0, \mathbb{P}\text{-a.s.} ~\text{and}~\mathbb{P}(\{V_n>0\})>0. \]

We can see that the corresponding discounted values also satisfy the same conditions. Further, since \(\mathbb{P}^*\) is equivalent to \(\mathbb{P}\), we get

\[ \mathbb{P}^*(\{\widetilde{V}_{n}>0\})>0. \]

However, since \(\{\widetilde{V}_{k}\}\) is \(\mathcal{F}_k\)-martingale with respect to \(\mathbb{P}^*\), by Problem «Click Here» we have

\[ \mathbb{E}^*(\widetilde{V}_{n}) = 0, \]

which is a contradiction.

The next theorem provides an equivalent condition for market completeness. Before stating the theorem, we first give the definition of a complete market.

Definition:

[Complete Market]

A market is said to be complete if every contingent claim is replicable.

Problem:

Show that the binomial model for European options is a complete market model.

Theorem:

[Second Fundamental Theorem of Asset Pricing]

A viable market \((B, \boldsymbol{S})\) is complete if and only if there exists a unique EMM with numéraire \(B\).

Proof:

Assume that the market is viable and complete.

Since the market is viable, by the first fundamental theorem, there is an EMM \(\mathbb{P}^*\). Assume the contrary that there exists another EMM \(\mathbb{P}^{**}\). Let \(H\) be the payoff of a European option and let \(\{\Pi_k\}\) be a self-financing and replicative strategy with discounted value process \(\{\widetilde{V}_k\}\). By Problem «Click Here» , we have

\[ \widetilde{H}_n = \widetilde{V}_n = \widetilde{V}_0 + \sum_{j=1}^{n} \boldsymbol{\theta}_j \cdot \Delta \widetilde{\boldsymbol{S}}_{j-1}, \]

Since the discounted stock price process is a \(\mathcal{F}_k\)-martingale with respect to \(\mathbb{P}^*\) and \(\{\boldsymbol{\theta}_k\}\) is predictable, we have (using factor property)

\[ \mathbb{E}^*(\boldsymbol{\theta}_j\cdot\Delta\widetilde{\boldsymbol{S}}_{j-1}~|~\mathcal{F}_{j-1}) = \boldsymbol{\theta}_j\cdot \mathbb{E}^*(\Delta\widetilde{\boldsymbol{S}}_{j-1}~|~\mathcal{F}_{j-1}) =0. \]

Taking expectation on both sides, we get

\[ \mathbb{E}^*(\boldsymbol{\theta}_j\cdot\Delta\widetilde{\boldsymbol{S}}_{j-1})=0,~j=1,2,\ldots,n. \]

Therefore, we have

\[ \mathbb{E}^*(\widetilde{H}_n) = \widetilde{V}_0. \]

We now have

\[ \mathbb{E}^*(\widetilde{H}_n) = E^{**}(\widetilde{H}). \]

The above equality holds for any \(H\). In particular, this holds for claims of the form \(H_n={1\hspace{-0.09in}1}_A\), \(A\in \mathcal{F}^*\). Therefore, \(\mathbb{P}^*(A) = \mathbb{P}^{**}(A),\) for all \(A\in \mathcal{F}^*\). That is \(\mathbb{P}^* = \mathbb{P}^{**}\). This proves the necessary part.

We omit the proof of the converse.

General Pricing Model

Having proved the fundamental theorems of asset pricing, we can now generalize the binomial model’s logic to arbitrary discrete-time markets, proving that arbitrage-free prices are expectations of discounted payoffs under an EMM.

Theorem:

Suppose the finite market model is viable and complete, and \(\mathbb{P}^*\) is the unique EMM. Then for any European contingent claim with payoff \(H\), the process

\[ \{B_kE^*(H^*~|~\mathcal{F}_k) ~|~ k=0,1,\ldots,n\} \]

is the unique arbitrage free price process for the European contingent claim, where \(H^*=H/B_n\) is the discounted payoff.

Any discussion on this theorem (including proof) is omitted for this course.