8.1 Cox-Ross-Rubinstein Approach to Option Pricing

As the simplest example to price an option we consider the approach of Cox, Ross and Rubinstein (CRR) which is based on the assumption of a binomial model, and which can be interpreted as a numerical method to solve the Black-Scholes equation. We treat exclusively European options and assume for the time being that the underlying pays no dividends within the time to maturity. Again, we discretize time and consider solely the points in time $ t_0=0,
t_1=\Delta t, t_2=2\Delta t,... , t_n=n\Delta t=T$ with $ \Delta t
= \frac{T}{n}.$ The binomial model proceeds from the assumption that the discrete time stock price process $ S_j$ follows a geometric random walk (see Chapter 4), which is the discrete analogue of the geometric Brownian motion. The binomial model has the feature that at any point of time the stock price has only two possibilities to move: Using the notation introduced above, if the stock price in time $ t_j$ is equal to $ S_j^k$ then in time $ t_{j+1}$ it can take only the values $ u \cdot S_j^k$ and $ d \cdot S_j^k.$ The probabilities $ p$ and $ q$ are independent of $ j.$ All other probabilities $ p_{kl}^j$ associated to $ S_{j+1}^l \neq u \cdot S_j^k$ and $ \neq d \cdot S_j^k$ are $ 0.$

In order to approximate the Black-Scholes differential equation by means of the Cox-Ross-Rubinstein approach, the probabilities $ p,q$ as well as the rates $ u,d$ have to be chosen such that in the limit $ \Delta t \rightarrow 0$ the binomial model converges to a geometric Brownian motion. That is, arguing as in (6.23) the conditional distribution of $ \ln S_{j+1}$ given $ S_j$ must be asympotically a normal distribution with expectation parameter $ \ln S_j + (b -\frac{1}{2} \sigma^2) \Delta t$ and variance parameter $ \sigma^2\Delta t.$ However, the conditional distribution of $ \ln S_{j+1}$ given $ S_j$ implied by the binomial model is determined by $ \ln (u \cdot S_j), \ln (d \cdot S_j)$ and their associated probabilities $ p$ and $ q.$ We set the parameters of the geometric random walk such that the conditional expectations and variances implied by the binomial model are equal to their asymptotic values for $ \Delta t \rightarrow 0.$ Taking into account that $ p+q=1$ we obtain three equations for the four unknown variables $ p,q,u$ and $ d:$

$\displaystyle p + q$ $\displaystyle =$ $\displaystyle 1,$  
$\displaystyle E \stackrel{\mathrm{def}}{=}p\ln (u \cdot S_j)+q\ln (d \cdot S_j)$ $\displaystyle =$ $\displaystyle \ln (S_j)+(b - \frac{1}{2}\sigma^2)\Delta t,$  
$\displaystyle p\{ \ln (u \cdot S_j)-{E} \}^2+q\{ \ln (d \cdot
S_j)-{E}\}^2$ $\displaystyle =$ $\displaystyle \sigma^2\Delta t.$  

Due to the first equation, the current stock price $ S_j$ disappears from the remaining equations. By substituting $ q=1-p$ into the latter two equations, we obtain, after some rearrangements, two equations and three unknown variables:
$\displaystyle p\ln (\frac {u}{d})+\ln d$ $\displaystyle =$ $\displaystyle (b - \frac{1}{2}\sigma^2 )\Delta t,$  
$\displaystyle (1-p)p \{ \ln \left(\frac {u}{d} \right) \}^2$ $\displaystyle =$ $\displaystyle \sigma^2\Delta t.$  

To solve this nonlinear system of equations we introduce a further condition

$\displaystyle u \cdot d=1,$

i.e. if the stock price moves up and subsequently down, or down and subsequently up, then it takes its initial value two steps later. This recombining feature is more than only intuitively appealing. It simplifies the price tree significantly. At time $ t_j$ there are only $ m_j = j+1$ possible values the stock price $ S_j$ can take. More precisely, given the starting value $ S_0$ at time $ t_0$ the set of possible prices at time $ t_j$ is

$\displaystyle S_j^k = S_0 u^k d^{j-k} , \; k=0,...,j,$

because it holds $ S_{j+1}^{k+1} = u \cdot S_j^k$ and $ S_{j+1}^k = S_j^k/u.$ In the general case there would be $ m_j = 2j$ possible states since the not only the number of up and down movements would determine the final state but also the order of up and down movements.

Solving the system of three equations for $ p,u,d$ and neglecting terms being small compared to $ \Delta t$ it holds approximatively:

$\displaystyle p = \frac 12+\frac 12 (b -\frac{1}{2}\sigma^2 )\frac {\sqrt {\Delta t}}{\sigma}, \quad u = e^{\sigma\sqrt {\Delta t}}, \quad d=\frac 1{u}.$ (8.2)

For the option price at time $ t_j$ and a stock price $ S_j = S_j^k$ we use the shortcut $ V_j^k = V(S_j^k,t_j).$ As in equation (7.1) we obtain the option price at time $ t_j$ by discounting the conditional expectation of the option price at time $ t_{j+1}:$

$\displaystyle V_j^k = e^{-r\Delta t}\{ pV_{j+1}^{k+1}+(1-p)V_{j+1}^k \} .$ (8.3)

At maturity $ T=t_n$ the option price is known. In case of a European option we have

$\displaystyle V_n^k = \max \{0,S_n^k-K\} , \; k=0,...,n.$ (8.4)

Beginning with equation (7.1) and applying equation (7.3) recursively all option values $ V_j^k, \, k=0,...,j, \, j=n-1,n-2,...,0$ can be determined.

Example 8.1  
An example of a call option is given in Table 7.1. First the tree of stock prices is computed. Since $ \Delta t =
\tau/n = 0.1$ it follows from equation (7.2) that $ u=1.0823.$ Given the current stock price $ S_0 = 230$ the stock can either increase to $ S_1^1 = uS_0 = 248.92$ or decrease to $ S_1^0 = S_0/u = 212.52$ after the first time step. After the second time step, proceeding from state $ S_1 = S^1_1$ the stock price can take the values $ S_2^2 = uS_1 = 269.40$ or $ S_2^1 =
S_1/u = 230$, proceeding from $ S_1 = S^0_1$ it can move to $ S_2^1
= 230$ or $ S_2^0 = 196.36$ and so on. At maturity, after 5 time steps, the stock price $ S_5$ can take the following
six values $ S_5^5 = u^5S_0 = 341.51, S_5^4 = u^3S_0 = 291.56, ...,
S_5^0= S_0/u^5 = 154.90.$

Following, given the tree of stock prices, we compute the option price at maturity applying equation (7.4), for example $ V_5^4 = V(S_5^4,t_5) = S_5^4 -K =
81.561$ or $ V_5^1 = 0$, since $ S_5^1 = 181.44 < K.$ Equation (7.2) implies $ p=0.50898$, since the cost of carry $ b$ are equal to the risk free interest rate $ r$ when no dividends are paid. Proceeding from the options' intrinsic values at maturity we compute recursively the option values at preceding points of time by means of equation (7.3). With $ V_5^4 = 81.561, V_5^3 = 38.921$ we obtain the option value $ V_4^3 = 60.349$ at time $ t_4 = 0.4$ corresponding to a stock price $ S_4 = S_4^3 =
269.40$ by substituting the known values of $ p, r, \Delta t.$ Analogously we obtain the option value $ V_0^0 = 30.378$ at time $ t_0=0$ and current stock price $ S_0 = 230$ by means of equation (7.3) and the time $ t_1=0.1$ option values $ V_1^1 =
44.328, V_1^0 = 16.200.$

Using only 5 time steps $ 30.378$ is just a rough approximation to the theoretical call value. However, comparing prices implied by the Black-Scholes formula (6.24) to prices implied by the Cox-Ross-Rubinstein approach for different time steps $ n$ the convergence of the numerical binomial model solution to the Black-Scholes solution for increasing $ n$ is evident (see Table 7.2).


Table 7.1: Evolution of option prices (no dividend paying underlying)
Current stock price $ S_t$ 230.00
Exercise price $ K$ 210.00
Time to maturity $ \tau$ 0.50
Volatility $ \sigma$ 0.25
Risk free rate $ r$ 0.04545
Dividend none
Time steps 5
Option type European call
Stock prices Option prices
341.50558           131.506
315.54682         106.497  
291.56126       83.457   81.561
269.39890     62.237   60.349  
248.92117   44.328   40.818   38.921
230.00000 30.378   26.175   20.951  
212.51708   16.200   11.238   2.517
196.36309     6.010   1.275  
181.43700       0.646   0.000
167.64549         0.000  
154.90230           0.000
Time 0.00 0.10 0.20 0.30 0.40 0.50
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Table 7.2: Convergence of the price implied by the binomial model to the price implied by the Black-Scholes formula
Time steps 5 10 20 50 100 150 Black-Scholes
Option value 30.378 30.817 30.724 30.751 30.769 30.740 30.741


The numerical procedure to price an option described above does not change if the underlying pays a continuous dividend at rate $ d.$ It is sufficient to set $ b=r-d$ instead of $ b=r$ for the cost of carry. Dividends paid at discrete points of time, however, require substantial modifications in the recursive option price computation, which we going to discuss in the following section.

Example 8.2  
We consider a call on US-Dollar with a time to maturity of $ 4$ months, i.e.  $ \tau = 1/3$ years, a current exchange rate of $ S=1.50$ EUR/USD and an exercise price $ K=1.50$ EUR/USD. The continuous dividend yield, which corresponds to the US interest rate, is assumed to be $ 1\%$, and the domestic interest rate is $ 9\%$. It follows that the cost of carry being the difference between the domestic and the foreign interest rate is equal to $ b = r - d = 8\%.$ Table 7.3 gives as in the previous example the option prices implied by the binomial model.


Table 7.3: Evolution of option prices (with continuous dividends)
Current EUR/ USD-price $ S_t$ 1.50
Exercise price $ K$ 1.50
Time to maturity $ \tau$ 0.33
Volatility $ \sigma$ 0.20
Risk free interest rate $ r$ 0.09
Continuous dividend $ d$ 0.01
Time steps 6
Option type European call
Price Option prices
1.99034 0.490
1.89869 0.405
1.81127 0.324 0.311
1.72786 0.247 0.234
1.64830 0.180 0.161 0.148
1.57240 0.127 0.105 0.079
1.50000 0.087 0.067 0.042 0.000
1.43093 0.041 0.022 0.000
1.36504 0.012 0.000 0.000
1.30219 0.000 0.000
1.24223 0.000 0.000
1.18503 0.000
1.13046 0.000
Time 0.00 0.06 0.11 0.17 0.22 0.28 0.33
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