Chapter 2 The one-period binomial model
2.1 The model
In this model, we have two time points: \(t=0\) (today) and \(t=T\) (or \(t=1\), if you prefer; tomorrow). All trading happens at one of these two time points.
The market, \(\mathcal{M}\), consists of two assets: a bond, and a stock.
The bond is our risk-free asset. We write \(B_t\) for its price at time \(t\) (\(t=0,T\)), and we have \[ B_0 = 1, \qquad B_T = 1 + r .\] The price of the bond is deterministic (because it is risk-free), and \(r\) is the simple interest rate for the time period \([0,T]\).
The stock is our risky asset. We write \(S_t\) for the price of one share of the stock at time \(t\); we know that \(S_0 = s\), and that \[ S_T = \begin{cases} s \cdot u & \mbox{ with probability } p_u \\ s \cdot d & \mbox{ with probability } p_d \end{cases}. \] We will sometimes write \(S_T = s \cdot Z\), where \[ Z = \begin{cases} u & \mbox{ with probability } p_u \\ d & \mbox{ with probability } p_d \end{cases}. \] The constants \(s > 0\), \(u > d > 0\), and \(p_u, p_d \in (0,1), p_u + p_d = 1\) are known.
Assumptions
We make the following assumptions about the market:
It is divisible: we are allowed to hold negative, and non-integer, quantities of both the bond and the stock.
It is liquid: there are no caps to the quantities we can buy and/or sell. In particular, we can short sell as many bonds as we like with no limit from the bank.
There is no bid-ask spread: for both the bond and the stock, the bid (buying) price and the ask (selling) price are the same.
There are no transaction costs and no other frictions in the market.
2.2 Portfolios and arbitrage
In this market, a portfolio (or a trading strategy) is any vector \(h = (x,y) \in \mathbb{R}^2\). We interpret the portfolio in the following way:
buy \(x\) bonds in the risk-free asset, and \(y\) shares of the stock, at time 0. (If \(x\) and/or \(y\) is negative, this represents our short selling bonds and/or shares at time 0.)
sell \(x\) bonds and \(y\) shares at time \(T\). We will assume that we always “close out” our position, that is, we are not holding onto shares for a later date; and if we have short sold any bonds or shares, we must buy them back at time \(T\).
Our divisibility assumption means that any \(h \in \mathbb{R}^2\) is a valid trading strategy.
Value and arbitrage
The value process of a portfolio \(h\) is the process \(V_t^h\), given by \[ V_t^h = x B_t + y S_t, \qquad t \in \{0,T\};\] in other words, \[ V_0^h = x + ys, \qquad V_T^h = x(1+r) + y s Z. \]
An arbitrage portfolio is any portfolio \(h\) for which \[ V_0^h = 0, \qquad \mathbb{P}(V_T^h \geq 0) = 1, \qquad \mathbb{P}(V_T^h > 0) > 0.\] Arbitrage portfolios are opportunities for risk-free profits: they have no initial cost, and at worst zero return.
Example 2.1 Consider a one-period binomial market with \(r = 0.1\), \(s=10\), \(u=1.2\), \(d=1.1\), \(p_u=0.2\), \(p_d=0.8\).
We will look at the portfolio \(h = (-10, 1)\). We have \[ V_0^h = -10 \times 1 + 1 \times 10 = 0; \] and \[ V_T^h = -10 \times 1.1 + 1 \times S_T = \begin{cases} 1 & \mbox{wp } 0.2 \\ 0 & \mbox{wp } 0.8 \end{cases}. \]
We have found an arbitrage portfolio on this market: if we sell 10 bonds (or borrow 10 pounds, if you prefer) and buy one share of the stock at time 0, at worst we regain our money and we have a chance to make a profit.
Example 2.2 Let’s change the parameters from the previous example, so that \(d=0.7\) instead of \(d=1.1.\) Now \(S_T\) is either 1.2 or 0.7, so that the value process of the portfolio \(h = (-10, 1)\) becomes \[\begin{align*} V^h_0 &=0,\\ V^h_T &=-10\times 1.1+1\times S_T=\begin{cases}1 & \mbox{wp } 0.2\\-4 & \mbox{wp } 0.8\end{cases}. \end{align*}\]
This is no longer an arbitrage portfolio, as we can now lose money through it.
No-arbitrage principle: In practice, arbitrage opportunities rarely exist and, when they do, they are short-lived. This gives rise to one of the central assumptions of financial mathematics: that all derivative assets are priced such that arbitrage opportunities do not exist.
Theorem 2.1 The one-period binomial model is arbitrage-free if and only if we have \[\begin{equation} d< 1+r <u. \tag{2.1} \end{equation}\]
Proof. First, we suppose that Equation (2.1) is true, and look for an arbitrage portfolio \(h = (x,y)\). Since \[ V_0^h = x + sy = 0,\] we must have \(x = -sy\).
Now, \[ V_T^h = - s y (1+r) + s yZ = \begin{cases} s y(u - (1 + r)) & \mbox{if } Z=u \\ s y( d - (1 + r)) & \mbox{if } Z = d \end{cases}. \]
By Equation (2.1), \(u - (1+r)\) must be positive and \(d - (1+r)\) must be negative, so we can only have \(\mathbb{P}(V_t^h \geq 0) = 1\) if \(y=0\). But then, \(\mathbb{P} (V_t^h > 0) = 0\) and so no arbitrage portfolio exists.
Now, we suppose that one of the inequalities in Equation (2.1) does not hold; for instance, we have \((1+r) \geq u\). Then we have \(s(1+r) \geq su > sd\) (Remember that \(u>d\) is always true.)
Consider the portfolio \(h = (s, -1)\): we sell one share short and invest all the money in the bond. We have \(V_0^h = 0\), and \[ V_T^h s(1+r) - sZ = \begin{cases} s(1+r - u) & \mbox{if } Z=u \\ s(1 + r - d) & \mbox{if } Z = d \end{cases} . \] So \(V_T^h \geq 0\) when \(Z=u\), and \(V_T^h > 0\) when \(Z = d\); we have \(\mathbb{P}(V_T^h \geq 0) = 1\) and \(\mathbb{P}(V_T^h > 0) > 0\), so we have found an arbitrage portfolio.
We can do a similar calculation if \(1 + r \leq d\) (try it!) so that
2.3 Contingent claims
A contingent claim or financial derivative is any product based on the market, whose value can be described in terms of the (random) values of the assets in the market. From an economics perspective, it is a contract between the buyer and the seller, in which the seller promises the random payoff \(X\) to the buyer at time \(T\).
Mathematically, a random variable \(X\) is a contingent claim if we can find a contract function \(\Phi\) such that \(X = \Phi(\{ S_t: t \in [0,T]\})\).
Example 2.3 Here are some examples of the contract functions for contingent claims we’ve already seen.
In a European call option, we have \(X = \Phi_{\text{call}}(S_T) = (S_T - K)^+\)
If the contingent claim is “one unit of every asset on the market,” then \(X=\Sigma_{j=1}^m S_T^j\).
A forward contract on \(S\) is a contract in which the the asset is to be sold at a strike price \(K\) at expiry time \(T\), and both buyer and seller are obliged to complete the transaction. In this case, the contract function is \(\Phi_F(x) = x - K\), and \(X = \Phi(S_T) = S_T - K\).
In general, in European-style claims \(\Phi\) only depends on the price at time \(T\), \(S_T\), but in general it can depend on the values of \(S_t\) at any (or all) times \(t \in [0,T]\).
We say that a contingent claim is reachable if there exists a portfolio \(h\) such that \[ \mathbb{P} (V_T^h = X) = 1.\] In this case, \(h\) is a hedging portfolio or replicating portfolio for \(X\).
If every contingent claim \(X\) is reachable, we say that the market is complete.
Theorem 2.2 If \(u > d\), the one-period binomial model is complete.
Proof. For any claim \(X\) with contract function \(\Phi\), we need to show that there exists a portfolio \(h=(x,y)\) with \[ V_T^h = \begin{cases} \Phi(su) & \mbox{if } Z = u \\ \Phi(sd) & \mbox{if } Z = d \end{cases}. \]
In other words, we want to find a solution to the system \[ (1+r) x + su y = \Phi(su) \\ (1+r) x + sd y = \Phi(sd). \]
If \(u > d\), this system has a unique solution, namely, \[\begin{align} x & = \frac{1}{1+r} \frac{ u \Phi(sd) - d \Phi(su)}{u-d} \\ y & = \frac{\Phi(su) - \Phi(sd)}{s(u-d)}. \end{align}\]
Question: what is a fair price for a contingent claim \(X\)?
At \(t=T\), this is an easy problem to solve: we know how to calculate its value, by finding \(X\) using \(\Phi\). Writing \(\Pi(X,t)\) for the price of \(X\) at time \(t\), we must have \(\Pi(X,T) = X\).
To calculate the fair price at time 0, we use the following pricing principle:
If a claim \(X\) is reachable with replication portfolio \(h\), then the only reasonable price process for \(X\) is \[ \Pi(X,t) = V_t^h, \quad t = 0, T .\]
Theorem 2.3 Suppose that a claim \(X\) is reachable with replication portfolio \(h\), and that at time \(t=0\) the price of \(X\) is different from \(V_0^h\). Then there is an opportunity for arbitrage.
Proof. (This is an example of the Law of One Price, Theorem 1.1.)
Let \(\Pi(X,0)\) be the price of the claim, and consider the case \(\Pi(X,0) > V_0^h\). At time 0, we can short sell the claim, buy the portfolio \(h\), and be left with \(\Pi(X,0) - V_0^h\), which we deposit in bonds.
Now at time \(T\), the portfolio value \(V_T^h\) will exactly cover the claim we sold short, and we have a risk-free profit coming from our bonds, which are now worth \((\Pi(X,0) - V_0^h)(1+r)\).
In the case \(\Pi(X,0) < V_0^h\), we can follow a similar argument, this time by short selling the portfolio and buying the claim.
Example 2.4 In the market from Example 2.2, let’s find the fair price for a European call option with strike price \(K=9\) and maturity date \(T\). The corresponding contingent claim is \(X = (S_T - 9)^+\).
A hedging portfolio for \(X\) is a portfolio \(h = (x,y)\) such that \[ (1+r) x + su y = (su - 9)^+, \qquad \qquad (1+r) x + sd y = (sd - 9)^+; \] in other words, \[ 1.1 x + 12 y = 3, \qquad \qquad 1.1 x + 7y = 0. \] The solution here is \(y = \frac{3}{5}\), \(x = - \frac{42}{11}\), so the value of this portfolio at time \(t\) is \[ V_0 = x + ys = \frac{-42}{11} + \frac{3}{5} \times 10 = \frac{24}{11}. \] We conclude that the fair price for this call option is \(\frac{24}{11}\).
2.4 The martingale measure
We say that a process \((X_0, X_T)\) is a martingale under a measure \(\mathbb{Q}\) if \[ \mathbb{E}_{\mathbb{Q}} [ X_T] = X_0.\] (We will see a more precise definition of martingales in more general contexts, later in the term.)
In this section, we’ll look at the conditions we need to place on the model to guarantee the existence of such a measure for the process \((S_0, \alpha S_T)\) and determine what that measure must be.
Since \(\alpha = \frac{1}{1+r}\) and \(S_0 = s\), we are really looking for a measure under which \[\begin{align*} s &= \frac{1}{1+r} \mathbb{E}_{\mathbb{Q}} [S_T]. \end{align*}\] If we write \[\begin{align*} \mathbb{Q}(Z=u) &= q_u \\ \mathbb{Q}(Z=d) &= q_d, \end{align*}\] our martingale condition becomes \[ \frac{1}{1+r} (s u q_u + s d q_d) = s, \] and our question is really: under which conditions do there exist \(q_u\) and \(q_d\) which solve \[\begin{align} u q_u + d q_d &= 1 + r \\ q_u + q_d & = 1 \tag{2.2}\\ 0 < q_u , q_d &< 1? \end{align}\]
Surprisingly, we have already met such conditions in Equation (2.1).
Theorem 2.4 The financial market \(\mathcal{M} = (B_t, S_t)\) is arbitrage free if and only if there exists a martingale measure \(\mathbb{Q}\).
Proof. From Theorem 2.1, we know that the market is arbitrage free if and only if \[ d < 1+r < u.\]
To see that Equations (2.2) have a solution if and only if \(d < 1+r < u\), we could use some convex analysis: this is exactly the condition under which \(1+r\) can be written as a convex combination of \(u\) and \(d\).
Otherwise, we can look for the solutions directly. Solving the system of linear equations, we get \[\begin{align} q_u & = \frac{(1+r) - d}{u-d} \\ q_d & = \frac{u - (1+r)}{u-d}. \tag{2.3} \end{align}\]
Both \(q_u\) and \(q_d\) are positive if and only if the no-arbitrage condition holds.
Theorem 2.4 is a version of the first fundamental theorem of asset pricing, which we will see in full detail later; we can also prove a simple version of the second fundamental theorem of asset pricing.
Theorem 2.5 Suppose the financial market is arbitrage free. Then it is complete if and only if there is a unique martingale measure.
Proof. If the market is arbitrage free, we must have \(d < 1+r < u\), and in particular \(d < u\). By Theorem 2.2, the market must be complete, and by Theorem 2.4 we know that a martingale measure exists.
To see that it is unique, suppose we have found two martingale measures \(\mathbb{Q}_1\) and \(\mathbb{Q}_2\), such that \[ \frac{1}{1+r} \mathbb{E}_{\mathbb{Q}_i} [S_T] = S_0 \] holds for each of \(i=1,2\). In other words, we have found \(q_1, q_2\) which both satisfy \[ q_i u + (1-q_i) d = 1+r. \]
The only way \(q_1\) and \(q_2\) can be different is if \(u = 1+r = d\), which contradicts our no-arbitrage assumption.
Example 2.5 Using the market \(\mathcal{M}\) from Example 2.2, with parameters \(r=0.1\), \(s=10\), \(u=1.2\), and \(d=0.7\), we can check that \(d < 1+r < u\), so the market is arbitrage free; it is complete, so there exists a unique martingale measure. We have \[ q_u = \frac{1.1 - 0.7}{1.2 - 0.7} = \frac{4}{5}, \qquad q_d = \frac{1.2-1.1}{1.2-0.7} = \frac{1}{5}, \] and we can check that \[ \frac{1}{1+r} \mathbb{E}_{\mathbb{Q}} \left[ S_T \right] = \frac{1}{1.1} \left[ \frac{4}{5} \times 12 + \frac{1}{5} \times 7 \right] = 10 = S_0. \]
2.5 Risk neutral valuation
Now that we know whether binomial market is complete, we can price any contingent claim. As we saw in Theorem 2.2, if \(u > d\) any contingent claim \(X\) with contract function \(\Phi\) is reachable, with hedging portfolio \(h= (x,y)\) given by \[\begin{align} x & = \frac{1}{1+r} \frac{ u \Phi(sd) - d \Phi(su)}{u-d} \\ y & = \frac{\Phi(su) - \Phi(sd)}{s(u-d)}. \tag{2.4} \end{align}\]
The price at time \(t=0\) of this portfolio is given by \[\begin{align*} \Pi(X,0) & = V_0^h \\ & = x + sy \\ & = \frac{1}{1+r} \left( \frac{ u \Phi(sd) - d \Phi(su)}{u-d} + (1+r) \frac{\Phi(su) - \Phi(sd)}{u-d} \right) \\ & = \frac{1}{1+r} \left( \Phi(su) q_u + \Phi(sd) q_d \right), \end{align*}\] where \(q_u\) and \(q_d\) are exactly the probabilities coming from the martingale measure \(\mathbb{Q}\)! We can interpret this as an expectation under \(\mathbb{Q}\), to get the following pricing formula.
Theorem 2.6 If the binomial model is free of arbitrage, then the arbitrage-free price of a contingent claim \(X\) at time \(t=0\) is given by the risk neutral valuation formula \[ \Pi(X,0) = \frac{1}{1+r} \mathbb{E}_{\mathbb{Q}} [X], \] where \(\mathbb{Q}\) is the martingale measure (or risk-neutral measure) uniquely determined by the relation \[ \frac{1}{1+r} \mathbb{E}_{\mathbb{Q}}[ S_T] = S_0 \] or, equivalently, given explicitly in Equations (2.3). Furthermore, the claim can be replicated using the portfolio set out in Equations (2.4).
Example 2.6 Using our same market from the other examples in this chapter, let’s price the call option with strike price \(K=9\). We have \(q_u = \frac{4}{5}\) and \(q_d = \frac{1}{5}\), so by the risk-neutral valuation formula, \[\begin{align*} \Pi(X, 0) & = \frac{1}{1+r} \mathbb{E}_{\mathbb{Q}} [ (S_T - 9)^+] \\ & = \frac{1}{1.1} \big( 3 \times q_u + 0 \times q_d \big) \\ & = \frac{1}{1.1} \times 3 \times \frac{4}{5} = \frac{24}{11}. \end{align*}\]
2.6 Some generalisations
We can extend the one-period binomial model slightly, and the first and second fundamental theorems of asset pricing will still hold. Here are two (non-examinable) examples.
What happens when \(u=d\)
If \(u=d\), both assets offer guaranteed rates of payoff per unit of investment: we have \[ \frac{B_T}{B_0} = 1 + r \qquad \mbox{and} \qquad \frac{S_T}{S_0} = u = d.\]
If these rates are not the same, there will be arbitrage in the market, with an arbitrage portfolio formed by buying the asset with the higher rate of payoff, and selling the asset with the lower rate. Meanwhile, we can slightly modify the argument from the proof of Theorem 2.1 to show that if \(u = d = 1+r\), there is no arbitrage portfolio on the market. Hence the no-arbitrage condition becomes \(u = d = 1+r\).
For any portfolio \(h\), \(V_T^h = x B_T + y S_T\) has a fixed value; so the only contingent claims \(X\) which are reachable are those which are also constant. Any contingent claim whose final value is genuinely random cannot be reached in this market: the market is not complete. On the other hand, any claim of the form \(X= \Phi(S_T)\) is still reachable.
For any probability measure \(\mathbb{Q} = (q_u, q_d)\), we have \[ u q_u + d q_d = u = d, \] so martingale measures exist precisely when \(u = d = 1+r\). This means that the first fundamental theorem still holds: we have no arbitrage, if and only if \(u = d = 1+r\), if and only if there exists a martingale measure.
Finally, if any martingale measure exists, we must have multiple martingale measures. However, as we have observed, the market is not complete, so the second fundamental theorem also holds in this setting.
Many assets, and/or many outcomes, at time \(T\)
Finally, while keeping a single time period (that is, trades at time \(t=0\) and \(t=T\) only), we can consider markets with more than 2 assets and/or more than 2 possible states of the market at time \(T\).
Suppose the market has \(m\) assets, whose prices at time \(0\) are given by the deterministic vector \(S_0 = (S_0^1, \dots, S_0^m)\), and that at time \(T\) the price vector \(S_T\) takes one of \(n\) possible values. In other words, there are \(n\) vectors \(\mathbf{a}_1, \dots, \mathbf{a}_n \in \mathbb{R}^m\) and \(n\) positive probabilities \(p_1, \dots, p_n\) such that \[ \mathbb{P}(S_T = \mathbf{a}_j) = p_j , \qquad j = 1, \dots, n.\]
These vectors also define an \(m \times n\) matrix \(A\), in which the \(i\)th row lists the possible values for the price of the \(i\)th asset at time \(T\), and the \(j\)th column is the vector \(\mathbf{a}_j\).
Arbitrage on this market is defined in the same way as on the simple market: any portfolio \(h = (x_1, \dots, x_m) \in \mathbb{R}^m\) is an arbitrage portfolio if it satisfies (i) \(V_0^h = 0\); (ii) \(\mathbb{P}(V_T^h \geq 0) = 1\); and (iii) \(\mathbb{P}(V_T^h > 0) > 0\). The value of \(h\) at time \(T\), \(V_T^h\), is random, and takes one of the \(n\) values in the row vector \(h A\). So finding an arbitrage portfolio is equivalent to finding a vector \(h \in \mathbb{R}^m\) such that \(h \cdot S_0 = 0\), and \(hA\) has all non-negative entries, with at least one strictly positive.
To define a martingale measure on this market, we look for a measure \(\mathbb{Q} = (q_1, \dots, q_n)\) such that, for each \(i = 1, \dots, m\), \(S_0^i = \frac{1}{1+r} \mathbb{E}_{\mathbb{Q}} [S_T^i]\). In terms of the matrix \(A\), a martingale measure is equivalent to a vector \(q = (q_1, \dots, q_n) \in \mathbb{R}^n\) such that \(S_0 = \frac{1}{1+r} q A^T\), and such that each \(q_j \in (0,1)\).
The first and second fundamental theorems of asset pricing both hold here. The first is a consequence of a result known as Farkas’ Lemma, concerning the solvability of systems of linear equalities: it tell us, roughly speaking, that given the system of equations defining \(h\), and the system of equations defining \(q\), exactly one of these has a solution. For the second, we can relate the completeness of the market to the (row) rank of \(A\) being equal to \(n\), while the uniqueness of a solution to \(S_0 = \frac{1}{1+r} q A^T\) corresponds to the nullity of \(A\) being 0. By the rank-nullity theorem, the first situation holds if and only if the second does too.