11.3 The Binomial Pricing Model

The binomial pricing model arises from discrete random walk models of the underlying asset. This method is only a reasonable approximation of the evolution of the stock prices when the number of trading intervals is large and the time between trades is small (Jarrow and Turnbull; 1996, pp. 213). It is particularly useful for pricing American options numerically, since it can deal with the possibility of early option exercise. An exact analytical solution with the Black-Scholes model for the American options is not possible, because of the complexity of the boundary conditions (see subsection 11.2.4).

The binomial model breaks down the time to expiration of an option into potentially very large number of time intervals, or steps. A tree of stock prices is initially produced, moving forward from the present to expiration. At each interval, the asset price $S$ can branch upwards to the value $Su$ (Figure 11.3) or downwards to the value $Sd$, by an amount calculated using the volatility and time to expiration. A binomial distribution of prices, for the underlying asset is thus produced. The tree represents all the possible paths that the stock price can take during the life of the option.

$\parbox{5.95cm}{ \ {\includegraphics[wi... ...ree at time $t_0$\ with respective probabilities $p$\ and $q$, where $p+q=1$.}}$ $\parbox{5.95cm}{ {\includegraphics[wid... ...6\defpicwidth]{BT.ps}} \caption{\small CCR binomial tree. }\cite{de:ka:ch:96}}$

At the end of the tree, i.e. at expiration of the option, the option values for each possible stock price are known, as they are equal to their intrinsic values. Assuming that the payoff function of the option is determined only by the value of the underlying asset at expiration, the model then works backwards through each time interval, calculating the option value at each step. The final step is at current time and stock price, where the theoretical fair value of the option is calculated. This recursive pricing procedure is based on the assumption of risk neutrality. In a risk neutral world all individuals require no compensation for risk, so that the option can be priced as though the underlying asset's expected return is risk-free.

The most popular binomial tree is that from Cox, Ross and Rubinstein (1979), also known as the Cox-Ross-Rubinstein (CRR) binomial tree. In this approach, the underlying asset evolves along a risk-neutral binomial tree with constant logarithmic price spacing, corresponding to constant volatility, as illustrated in Figure (11.4).

The CRR binomial tree is a discrete version of the Black-Scholes constant volatility process. Any higher multinomial tree, for example a trinomial tree, can used as a discrete development of the geometric Brownian motion. However, all of them converge, as the time interval tends to zero, to the same continuous constant volatility process. The CRR tree is discussed in subsection (11.3.2) and illustrated in the following computational examples in XploRe .

11.3.1 Software Application

XploRe offers the folllowing quantlets to calculate European and American option prices with the Cox-Ross-Rubinstein binomial tree:

asset (vers)

uses the quantlet bitree to calculate the option price and the price process of the underlying asset, specifying interactively the input parameters.

{s,ow,op} = bitree (vers,task)

{s,ow,op} = bitree (S,K,r,sigma,tau,n,vers,opt,typeofdiv,div)

calculates the option price and the price process of the underlying asset using the CRR binomial tree. The input parameters are specified either interactively or directly.

IBTcrr (S,K,r,sigma,level,delta,task)

calculates the European option price on non-dividend paying underlying asset, specifying directly the input parameters.

optstart ()

calculates the prices of either European or American options, or their implied volatilities, specifying interactively the input parameters. For American options the McMillan formula or binomial trees can be used.

asset opens different interactive menus for input parameters. It uses the quantlet bitree to price European and American options.

The input parameter $\tt {vers}$ in the quantlets asset and bitree specifies the type of option. It has the value 1 for a call and 0 for a put. $\tt {task}$ is a scalar that specifies the type of dividend payment(s): for $\tt {task}=1$ no dividend, for $\tt {task}=2$ a continuously paid dividend, for $\tt {task}=3$ a dividend as a percentage of the value of the underlying asset and for $\tt {task}=4$ a fixed dividend at the end of T is assumed. If $\tt {task}=5$, then an exchange rate is assumed as underlying. In this case, $\tt {S}$ is replaced by the exchange rate, i.e. the domestic currency price of a unit foreign currency.

When the quantlet bitree is used interactively, the first five input parameters follow the usual notation: $\tt {S}$ for the price of the underlying asset, $\tt {K}$ for the strike price, $\tt {r}$ for the annualized risk-free interest rate in %, $\tt {sigma}$ for the annualized volatility in % and $\tt {tau}$ for time to expiration. The other parameters are: $\tt {n}$ for the number of intervals in the tree, $\tt {opt}$ for the type of option, which has the value 1 for a call (default) or 0 for a put. The input parameter $\tt {typeofdiv}$ specifies the type of dividend payments. It has the values 0 to 4 for the same cases as in $\tt {task}$, with $\tt {type of dividend=0}$ as default. If $\tt {type of dividend\neq0}$, then the value(s) of dividend(s) must be specified in $\tt {div}$. For more than one dividend payment, $\tt {div}$ is a (m x 2) dimensional matrix, where the first column contains the time points when dividends should be paid and the second, the corresponding dividend values.

Both quantlets, asset and bitree output the tree of possible prices of the underlying asset, which is contained in a ${(n+1)\times(n+1)}$ dimensional matrix $\tt s$, the tree of option prices, which is contained in a $(n+1)\times(n+1)$ dimensional matrix $\tt {ow}$ and the price of the option $\tt {op}$.

In the following example, the European put price on a dividend paying underlying asset $S$ is computed through quantlet betree :

library ("finance")
 S=100               ; spot price
 K=100               ; strike price
 r=10.517            ; annualized risk-free interest rate in %
 sigma=30            ; annualized volatility in %
 tau=1               ; time to expiration in years
 n=5                 ; number of intervals
 vers=0              ; European option
 opt=0               ; put
 typeofdiv=3         ; fixed dividend
 t=0.25|0.5|0.75|1   ; dividends paid quarterly
 f=10|10|10|15       ; dividend amounts
 div=t~f
 bitree(S,K,r,sigma,tau,n,vers,opt,typeofdiv,div)

XLGfindex8.xpl

 
The output contains the tree of possible stock prices, the tree of option prices and the computed option price:

Contents of _tmp
 [1,] " tree of possible stock prices "

Contents of s
 [1,]      100   93.583   77.945   62.968   48.548   29.596
 [2,]      100   109.16   91.568    74.88   58.965   38.705
 [3,]      100   100.85   109.38   90.459   72.588   50.617
 [4,]      100   100.85   91.568   110.83   90.403   66.196
 [5,]      100   100.85   91.568   82.148    113.7   86.569
 [6,]      100   100.85   91.568   82.148   72.588   113.21

Contents of _tmp
 [1,] " tree of option prices "

Contents of ow
 [1,]   33.197   41.725   49.935   57.386    64.18   70.404
 [2,]        0   27.201   36.318   45.477   53.765   61.295
 [3,]        0        0   20.481   29.903   40.144   49.383
 [4,]        0        0        0   13.252   22.331   33.804
 [5,]        0        0        0        0   6.0428   13.431
 [6,]        0        0        0        0        0        0

Contents of _tmp
 [1,] " the option price "

Contents of op
 [1,]   33.197

When binomial trees are used in practice, the life of the option is typically divided into 30 or more time steps, of length $\Delta t$. This computation can be easily carried out with XploRe . With 30 time steps, 31 possible stock prices and $2^{30}$, or about one billion, possible stock prices are considered. The asset returns in one step of the tree, $u$ and $d$, are chosen to match the stock price volatility. A popular way of doing this is by setting

$$u=exp(\sigma \sqrt{ \Delta t}) \hspace{0.5cm} \textrm{and} \hspace{0.5cm} d=\frac{1}{u},$$ (11.39)

as explained in subsection (11.3.2).

IBTcrr calculates the price of a European option on a non-dividend paying underlying asset. $\tt {level}$ specifies the number of intervals in the tree and $\tt {deltat}$ is the length of the discrete time interval. $\tt {task}$ is a scalar, which has the value $\tt {1}$ for call and $\tt {0}$ for put. The other parameters follow the usual notation. The output window shows the calculated European option price. The same price results when the quantlet betree is used. The last quantlet is recommended for computation, since it yields not only the option price as in IBTcrr , but also the whole tree.

A second example illustrates how to price a European call with IBTcrr :

library ("finance")
 S=100         ; spot price
 K=100         ; strike price
 r=0.1057      ; annualized risk-free interest rate
 sigma=0.3     ; annualized volatility
 level=5       ; number of intervals
 deltat=0.20   ; length of time interval
 task=1        ; European call
 C=IBTcrr(S, K, r, sigma, level, deltat,task)
 C             ; call price

XLGfindex9.xpl

 
Then the output window shows the European call price:

Contents of C
 [1,]   17.583

optstart asks the user to specify the model, which will compute the option price. It offers the Black-Scholes and the MacMillan formulae as an analytical approach and the binomial tree model as a numerical method. In the latter, the quantlet bitree is used for building the tree and pricing the option.

11.3.2 CRR Binomial Tree

The Cox, Ross and Rubinstein (CRR) binomial tree can be interpreted as a numerical procedure to solve the Black-Scholes equation. There are two main ideas underlying the tree. First, a continuous random walk (11.12) may be modelled by a discrete random walk with the following properties:

• The price of the underlying asset $S$ changes only at discrete times $t_0=0$, $t_1=\Delta t$, $t_2=2\Delta t$, ..., $t_n=n\Delta t$, ..., $t_N=N\Delta t=T$, where $T$ is the expiration date of the option and $\Delta t=\frac{T}{N}$ denotes the one time step.
• If the price of the underlying asset is $S_{n,i}$ at state $i$ and time $t_n$, then at $t_{n+1}$ it may take only one of two possible values, $S_{n+1,i+1}=uS_{n,i}>S_{n,i}$ or $S_{n+1,i}=dS_{n,i}<S_{n,i}$ (Figure 11.5). This is equivalent to assuming that there are only two returns possible at each time step, $u-1>0$ and $d-1<0$. These two returns are the same for all time steps.
• The probability, $p$ of $S_{n,i}$ moving up to $S_{n+1,i+1}=uS_{n,i}$ is known. The same results for the probability $q$ of $S_{n,i}$ moving down to $dS_{n+1,i}$, since $p+q=1$.

The second assumption underlying a binomial tree is that of a risk-neutral world, i.e. the investor risk preferences are irrelevant to option valuation. This has two implications. First, the expected return from all traded securities is the risk-free interest rate. This means that the drift term $\mu$ in the stochastic differential equation for the asset return (11.4) is replaced by the risk-free interest rate $r$ whenever it appears

$$\frac{dS_t}{S_t} = r\, dt + {\sigma}\, d W_t.$$ (11.40)

Second, the option value $V_n$, at $t_n=n\Delta t$, is its expected value at $t_{n+1}=(n+1)\Delta t$, discounted by the risk-free interest rate $r$

$$V_n = E[exp(-r\Delta t)V_{n+1}].$$ (11.41)

Within this framework, the probabilities $p$, $q$ and the returns $u$, $d$ should reflect the important statistical properties of the continuous random walk (11.40), which means that they have to insure that for $\Delta t \rightarrow 0$ the underlying asset $S$ follows the Brownian motion. In other words, the parameters $p$, $q$, $u$, $d$ should give the correct values for the mean and the variance of the underlying asset, i.e. $lnS_{n+1}\sim N\left(lnS_n+(b-\frac{\sigma^2}{2}\Delta t),\sigma^2\Delta t\right)$, during a time interval $\Delta t$. Consequently, these parameters must solve the following equations:

$$p+q = 1$$ (11.42)

$$E=p\,ln(u\;S_n)+q\,ln(d\;S_n) = ln(S_n)+\left(b-\frac{\sigma^2}{2}\right)\Delta t$$ (11.43)

$$p\left(ln(u\;S_n)-E\right)^2+q\left(ln(d\;S_n)-E\right)^2 = \sigma^2\Delta t$$ (11.44)

Substituting $q=p-1$ in (11.43) and (11.44), there are three unknown parameters and two non-linear equations to solve. To obtain a unique solution, a supplementary restriction for the parameters is needed. Cox, Ross and Rubinstein (1979) chose the restriction $u\;d=1$, since it drastically simplifies the tree. At time point $t_n$ there are only $i=1,..., n+1$ possible nodes and

$$S_{n,i}=u^nd^{n-i}S_0,$$ (11.45)

where $S_0$ is the asset price in $t_0$ (Figure 11.5).

$\parbox{12cm}{\caption{\small CRR binomial tree for $S$, with the restriction $ud=1$}}$

Solving the equations (11.42), (11.43) and (11.44) for $p$, $u$, and $d$ and neglecting the terms smaller then $\Delta t$ results in:

$$\ p=\frac{1}{2}+\frac{1}{2} \left(b-\frac{1}{2}\sigma^2\right)\fr... ...ma},\hspace{0.5cm} u=exp(\sigma \sqrt{ \Delta t}), \hspace{0.5cm} d=\frac{1}{u}$$ (11.46)

The time steps are of equal length, so that the risk-neutral probability $p$ as calculated by 11.46 is the same at each node. The option price $V_{n,i}=V(S_{n,i},t_n)$, at node $i$ and time $t_n$, is the expected payoff at $t_{n+1}$ discounted at the risk-free interest rate:

$$V_{n,i}=exp(-r\Delta t)\left[pV_{n+1,i+1}+(1-p)V_{n,i+1}\right]$$ (11.47)

At the end of the tree the option price is known. It equals the option value at expiration. For a call option, it is:

$$V_{N,i}=max\left\{0,S_{N,i}-K\right\}, \hspace{0.5cm} k=0,...,n$$ (11.48)

The option values at each node, $V_{n,i}$, $i=0,...,n$ and $n=N-1,...,0$ will be then determined recursively by working backwards through the tree.

When the underlying asset is a stock, which pays dividend(s), then the reduction of the stock prices by the dividend(s) amount must be considered. Details on the use of binomial trees for fixed or percentage dividend(s) are given in Franke et al. (2001, pp. 87).

In the case of an American put, or a European call on dividend paying underlying asset, the option price will be checked at each node to decide whether or not the early exercise would be optimal. If the option is held until expiration, its value at the final node is the same as for the European option. This is the case for an American call, since there is always the chance that until expiration the underlying price increases. Hence, the price of an American call equals the price of its European counterpart.