Binomial Model
1 The Binomial Model
The binomial model is a discrete-time model of a financial market in which the price of a risky asset moves, in each period, to one of only two possible values. Despite its simplicity it contains the main ideas of Option Pricing: replicating a payoff with a self-financing portfolio, pricing by no arbitrage, and the risk-neutral measure. It is also easy to compute with, handles early exercise (American options) naturally, and converges to the Black-Scholes-Merton Model as the time step goes to zero.
This note builds the model, shows how to price a derivative in it, proves put-call parity, works through an example and extends the results to a stock paying continuous dividends.
2 Constructing the Model
2.1 One Period
Consider one period of length \(h\), with times \(t\in\{0,h\}\), and two assets.
The market consists of
a risk-free bond with price \(B_0=1\) and \(B_h=e^{rh}\), where \(r\) is the continuously compounded risk-free rate (see Interest), and
a stock with price \(S_0=S\) and
\[ S_h = \begin{cases} uS & \text{with probability } p, \\ dS & \text{with probability } 1-p, \end{cases} \qquad 0<d<u, \]
where \(u\) and \(d\) are the up and down factors.
A portfolio holding \(x\) bonds and \(y\) shares has value \(V_t = xB_t + yS_t\).
The real-world probability \(p\) will play no role in pricing. The model has two possible outcomes and two assets, which is what makes it possible to replicate any payoff.
2.2 No Arbitrage
The one-period model is free of arbitrage if and only if
\[ d < e^{rh} < u. \]
Proof. If \(e^{rh}\le d\), borrow \(S\) and buy one share. The cost at time \(0\) is zero and the value at time \(h\) is \(S_h - Se^{rh}\), which is at least \(0\) in both states and strictly positive in the up state, an arbitrage. If \(e^{rh}\ge u\), short one share and lend the proceeds, which is an arbitrage in the other direction.
Conversely, suppose \(d<e^{rh}<u\) and define
\[ q = \frac{e^{rh}-d}{u-d} \in (0,1). \]
Then \(q\,uS + (1-q)\,dS = e^{rh}S\), so the discounted stock price is a martingale when the up probability is \(q\). For any portfolio with zero initial value, the discounted terminal value \(e^{-rh}V_h\) therefore has expectation \(0\) under \(q\). Since \(q\) and \(1-q\) are both positive, \(V_h\ge 0\) in both states forces \(V_h=0\) in both states, so there is no arbitrage.
We assume this condition from now on.
2.3 Pricing by Replication
A derivative pays \(V_u\) if the stock moves up and \(V_d\) if it moves down. For a call option with strike \(K\) these are \(V_u=(uS-K)^+\) and \(V_d=(dS-K)^+\).
To replicate, hold \(\Delta\) shares and \(B\) units of cash (lent if \(B>0\), borrowed if \(B<0\)) so that the portfolio pays exactly the derivative in both states,
\[ \begin{aligned} \Delta\,uS + Be^{rh} &= V_u, \\ \Delta\,dS + Be^{rh} &= V_d. \end{aligned} \]
Subtracting the equations and solving gives
\[ \Delta = \frac{V_u - V_d}{S(u-d)}, \qquad B = e^{-rh}\,\frac{uV_d - dV_u}{u-d}. \]
By the no-arbitrage principle the derivative has the same price as the replicating portfolio,
\[ V_0 = \Delta S + B. \]
The number \(\Delta\) is the hedge ratio: it measures how sensitive the derivative is to the stock. A call is replicated by buying shares and borrowing (\(\Delta>0\), \(B<0\)), and a put by shorting shares and lending (\(\Delta<0\), \(B>0\)). These formulas hold for any payoff \((V_u,V_d)\).
2.4 Risk-Neutral Pricing
Substituting \(\Delta\) and \(B\) into \(V_0=\Delta S+B\) and collecting the \(V_u\) and \(V_d\) terms gives
\[ V_0 = e^{-rh}\left[\,q\,V_u + (1-q)\,V_d\,\right], \qquad q = \frac{e^{rh}-d}{u-d}. \]
The no-arbitrage condition \(d<e^{rh}<u\) is exactly the condition \(0<q<1\). The number \(q\) is the risk-neutral probability of an up move. It is not the real-world probability \(p\). It is the probability under which the stock earns the risk-free rate on average,
\[ e^{-rh}\,\mathbb{E}_{\mathbb{Q}}[S_h] = S, \]
so the price of the derivative is its discounted expected payoff under the risk-neutral measure \(\mathbb{Q}\), as in Risk-Neutral Pricing. Replication and risk-neutral expectation give the same price because the second is derived from the first.
2.5 Multiple Periods
Divide \([0,T]\) into \(n\) periods of length \(h=T/n\). In each period the stock moves up by \(u\) or down by \(d\), independently of the past, so after \(j\) up moves and \(k-j\) down moves at time \(kh\) the price is \(S u^j d^{k-j}\). The tree recombines: an up move followed by a down move ends at the same price as a down move followed by an up move, so there are only \(k+1\) nodes at time \(kh\).
Prices are found by backward induction:
At maturity set the node value equal to the payoff \(h(S_T)\).
At each earlier node apply the one-period formulas to the two values that follow it,
\[ V = e^{-rh}\left[\,q\,V_{\text{up}} + (1-q)\,V_{\text{down}}\,\right]. \]
Repeat until time \(0\).
Since \(u\), \(d\) and \(r\) do not change, \(q\) is the same at every node. The hedge \(\Delta\) and the cash position \(B\) do change from node to node, so the replicating portfolio is rebalanced at each node. The rebalancing is self-financing: the portfolio held coming into a node is worth exactly the cost of the new portfolio.
Unrolling the recursion gives a closed form.
A European derivative with payoff \(h(S_T)\) has time-\(0\) price
\[ V_0 = e^{-rT}\sum_{j=0}^{n}\binom{n}{j}\,q^{\,j}(1-q)^{n-j}\, h\!\left(S u^{j}d^{\,n-j}\right). \]
This is the discounted expectation of \(h(S_T)\) when the number of up moves is \(\text{Binomial}(n,q)\). At every node the discounted stock price \(e^{-rkh}S_{kh}\) is a \(\mathbb{Q}\)-martingale (see Martingale Theory).
2.6 Choosing \(u\) and \(d\)
The factors \(u\) and \(d\) are model parameters. To match a stock with volatility \(\sigma\) the Cox-Ross-Rubinstein (CRR) choice is
\[ u = e^{\sigma\sqrt{h}}, \qquad d = \frac{1}{u} = e^{-\sigma\sqrt{h}}. \]
The no-arbitrage condition \(d<e^{rh}<u\) holds for small enough \(h\). As \(n\to\infty\) the binomial price of a European option converges to its Black-Scholes price (see European Options), because the log-return over \([0,T]\) is a sum of \(n\) independent steps and tends to a normal distribution by the central limit theorem.
3 Put-Call Parity
In the binomial model, a European call and put with the same strike \(K\) and maturity \(T\) on a stock paying no dividends have prices satisfying
\[ C_0 - P_0 = S_0 - Ke^{-rT}. \]
Proof. The payoffs satisfy, in every state,
\[ (S_T-K)^+ - (K-S_T)^+ = S_T - K. \]
By the pricing formula above, which is linear in the payoff,
\[ C_0 - P_0 = e^{-rT}\,\mathbb{E}_{\mathbb{Q}}\left[(S_T-K)^+ - (K-S_T)^+\right] = e^{-rT}\,\mathbb{E}_{\mathbb{Q}}[S_T - K]. \]
Since \(e^{-rt}S_t\) is a \(\mathbb{Q}\)-martingale, \(\mathbb{E}_{\mathbb{Q}}[S_T] = e^{rT}S_0\), hence
\[ C_0 - P_0 = e^{-rT}\left(e^{rT}S_0 - K\right) = S_0 - Ke^{-rT}. \qquad\blacksquare \]
The parity can also be seen from replication in one period. The call and put have values \((uS-K)^+\), \((dS-K)^+\) and \((K-uS)^+\), \((K-dS)^+\), and their difference \(C-P\) has value \(uS-K\) in the up state and \(dS-K\) in the down state. Applying the replication formulas with \(V_u=uS-K\) and \(V_d=dS-K\),
\[ \Delta_C-\Delta_P = \frac{(uS-K)-(dS-K)}{S(u-d)} = 1, \qquad B_C-B_P = e^{-rh}\,\frac{u(dS-K)-d(uS-K)}{u-d} = -Ke^{-rh}, \]
so \(C-P=S-Ke^{-rh}\). A long call and short put therefore replicate a long share financed by borrowing \(Ke^{-rh}\), which is a forward contract. Note that the proof uses only the no-arbitrage condition, not the values of \(u\) and \(d\) or the real-world probability \(p\). The parity holds at every node of the tree, with \(S\) and the remaining time to maturity taken at that node.
4 Example
Price a European call and put on a stock with \(S_0=100\), strike \(K=100\), maturity \(T=1\) and rate \(r=5\%\), using \(n=2\) periods (\(h=0.5\)) with \(u=1.2\) and \(d=0.85\).
Parameters. \(e^{rh}=e^{0.025}=1.025315\), so \(d<e^{rh}<u\) and there is no arbitrage. The risk-neutral probability and discount factor are
\[ q = \frac{1.025315-0.85}{1.2-0.85} = 0.500900, \qquad e^{-rh} = 0.975310. \]
Stock tree. \(S_u=120\), \(S_d=85\), \(S_{uu}=144\), \(S_{ud}=102\) and \(S_{dd}=72.25\).
Terminal payoffs.
| Node | \(S_T\) | Call \((S_T-100)^+\) | Put \((100-S_T)^+\) |
|---|---|---|---|
| \(uu\) | 144 | 44 | 0 |
| \(ud\) | 102 | 2 | 0 |
| \(dd\) | 72.25 | 0 | 27.75 |
Backward induction. Applying \(V=e^{-rh}[qV_{\text{up}}+(1-q)V_{\text{down}}]\):
| Node | \(S\) | Call \(C\) | Put \(P\) |
|---|---|---|---|
| \(u\) | 120 | 22.4690 | 0.0000 |
| \(d\) | 85 | 0.9771 | 13.5081 |
| root | 100 | 11.4525 | 6.5754 |
For example, the call at the root is \(C_0 = 0.975310\,[\,0.5009(22.4690)+0.4991(0.9771)\,] = 11.4525\).
Hedge at the root (call). With \(V_u=22.4690\) and \(V_d=0.9771\),
\[ \Delta = \frac{22.4690-0.9771}{100(0.35)} = 0.6141, \qquad B = 0.975310\cdot\frac{1.2(0.9771)-0.85(22.4690)}{0.35} = -49.9531, \]
and indeed \(\Delta S+B = 61.41-49.95 = 11.4525\). To replicate the call, buy \(0.6141\) shares and borrow \(\$49.95\).
Put-call parity check.
\[ C_0-P_0 = 11.4525-6.5754 = 4.8771, \qquad S_0-Ke^{-rT} = 100-100e^{-0.05} = 4.8771. \;\checkmark \]
5 Continuous Dividends
Suppose the stock pays dividends continuously at a constant yield \(\delta\), which are reinvested in the stock. Then \(\Delta\) shares held at time \(0\) grow to \(\Delta e^{\delta h}\) shares at time \(h\). The model is otherwise the same, and every result above carries over with \(r\) replaced by \(r-\delta\) in the stock’s drift.
No arbitrage. The model is free of arbitrage if and only if
\[ d < e^{(r-\delta)h} < u. \]
The proof is the same as before, comparing the stock with the bond after reinvesting dividends.
Replication. The replicating portfolio must satisfy
\[ \begin{aligned} \Delta\,uS\,e^{\delta h} + Be^{rh} &= V_u, \\ \Delta\,dS\,e^{\delta h} + Be^{rh} &= V_d, \end{aligned} \]
which gives
\[ \Delta = e^{-\delta h}\,\frac{V_u - V_d}{S(u-d)}, \qquad B = e^{-rh}\,\frac{uV_d - dV_u}{u-d}, \qquad V_0 = \Delta S + B. \]
Risk-neutral pricing. Substituting gives
\[ V_0 = e^{-rh}\left[\,q\,V_u + (1-q)\,V_d\,\right], \qquad q = \frac{e^{(r-\delta)h}-d}{u-d}. \]
The no-arbitrage condition is again \(0<q<1\). Under \(q\) the stock has expected growth \(e^{(r-\delta)h}\) per period, so the stock with its dividends reinvested earns the risk-free rate, \(e^{-rt}e^{\delta t}S_t\) is a \(\mathbb{Q}\)-martingale, and \(\mathbb{E}_{\mathbb{Q}}[S_T] = e^{(r-\delta)T}S_0\).
Multiple periods. Backward induction uses the new \(q\) and the new \(\Delta\) at each node, and the closed form is unchanged,
\[ V_0 = e^{-rT}\sum_{j=0}^{n}\binom{n}{j}\,q^{\,j}(1-q)^{n-j}\, h\!\left(S u^{j}d^{\,n-j}\right). \]
For the CRR tree \(u=e^{\sigma\sqrt{h}}\), \(d=1/u\), the condition \(d<e^{(r-\delta)h}<u\) again holds for small \(h\).
For a stock paying a continuous dividend yield \(\delta\),
\[ C_0 - P_0 = S_0e^{-\delta T} - Ke^{-rT}. \]
Proof. As before, \((S_T-K)^+ - (K-S_T)^+ = S_T-K\), so by linearity \(C_0-P_0 = e^{-rT}\,\mathbb{E}_{\mathbb{Q}}[S_T-K]\). Using \(\mathbb{E}_{\mathbb{Q}}[S_T]=e^{(r-\delta)T}S_0\),
\[ C_0 - P_0 = e^{-rT}\left(e^{(r-\delta)T}S_0 - K\right) = S_0e^{-\delta T} - Ke^{-rT}. \qquad\blacksquare \]
In one period the replication formulas with \(V_u=uS-K\) and \(V_d=dS-K\) give \(\Delta_C-\Delta_P=e^{-\delta h}\) and \(B_C-B_P=-Ke^{-rh}\), so a long call and short put replicate a forward contract on the stock with reinvested dividends, financed by borrowing \(Ke^{-rh}\).
An option on an exchange rate fits this framework, with \(S\) the exchange rate in domestic currency per unit of foreign currency and \(\delta\) the foreign interest rate, since a unit of foreign currency held in a foreign bank account grows at that rate.
6 Extensions
American options. The holder may exercise at any node, so backward induction takes the larger of the exercise value and the continuation value,
\[ V = \max\left(\text{payoff},\; e^{-rh}\left[\,q\,V_{\text{up}} + (1-q)\,V_{\text{down}}\,\right]\right), \]
where \(V_{\text{up}}\) and \(V_{\text{down}}\) are themselves American values. An American call on a stock with no dividends is never exercised early, but an American put can be, as can a call when \(\delta>0\) is large. Parity then holds only as an inequality.
Other payoffs. The same tree prices any path-independent payoff, and with extra bookkeeping path-dependent options such as Barrier Options.
8 References
- Cox, J. C., Ross, S. A. and Rubinstein, M. (1979). Option pricing: a simplified approach. Journal of Financial Economics, 7(3), 229-263.
- McDonald, R. L. Derivatives Markets, 3rd ed. Pearson. Chapters 10 and 11.
- Shreve, S. E. Stochastic Calculus for Finance I: The Binomial Asset Pricing Model. Springer.