Binomial Trees

After completing this reading you should be able to:

• Calculate the value of an American and a European call or put option using a one-step and two-step binomial model.
• Describe how volatility is captured in the binomial model.
• Describe how the value calculated using a binomial model converges as time periods are added.
• Define and calculate the delta of a stock option.
• Explain how the binomial model can be altered to price options on stocks with dividends, stock indices, currencies, and futures.

Pricing Options Using the Binomial Model

The binomial option pricing model is a simple approximation of returns which, upon refining, converges to the analytic pricing formula for vanilla options. The model is also useful for valuing American options that can be exercised before expiry.

The model can be represented as:

$$
\begin{array}
\hline
{} & {\small P } & { S }_{ 0 }u \\
{ S }_{ 0 } & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\
{} & {\small 1-P} & { S }_{ 0 }d \\
\end{array} $$

The notation used is as follows:

\({ S }_{ 0 }\)=stock price today

\(P\)=probability of a price rise

\(u\)=The factor by which the price rises

\(d\)=The factor by which the price falls

Over a small time interval \(\Delta t\), the price today rises or falls to one of only two potential future values: \({ S }_{ 0 }u\), and \({ S }_{ 0 }d\).

The underlying price is assumed to follow a random walk.

Risk-neutral Valuation

The following formula are used to price options in the binomial model:

\(u\)=size of the up move factor=\({ e }^{ \sigma \sqrt { t } }\), and

\(d\)=size of the down move factor=\({ e }^{ -\sigma \sqrt { t } }=\frac { 1 }{ { e }^{ \sigma \sqrt { t } } } =\frac { 1 }{ u } \)

\(\sigma\) is the annual volatility of the underlying asset’s returns and \(t\) is the length of the step in the binomial model.

\({ \pi }_{ u }=\)probability of an up move=\(\frac { { e }^{ rt }-d }{ u-d } \)

\({ \pi }_{ d }\)=probability of a down move=\(1-{ \pi }_{ u }\)

Let \(f_u\) be the value of an option when the price goes up and \(f_d\) the value when the price goes down. 

The value,\(f\) of the option, for one step-binomial is given by:

$$f = e^{-rt}\left(\pi f_u + (1-\pi)f_d\right)$$

Where,

$${ \pi }=\frac { { e }^{ rt }-d }{ u-d }$$

Example: Risk-neutral Valuation

The price of an exchange-quoted zero-dividend share is $30. Over the past year, the stock has exhibited a standard deviation of 17%. The continuously compounded risk-free rate is 5% per annum. Compute the value of a 1-year European call option with a strike price of $30 using a one-period binomial model:

The up- and down-move factors are:

$$\begin{align*} u&={ e }^{ 0.17\times \sqrt { 1 } }=1.1853\\  d&=\frac { 1 }{ 1.1853 } =0.8437 \end{align*}$$

The risk-neutral probabilities of an up- and down-move are:

$$ \begin{align*}{ \pi }_{ u }&=\frac { \left( { e }^{ 0.05\times 1 } \right) -D }{ U-D } =\frac { 1.0513-0.8437 }{ 1.1853-0.8437 } =0.61\\  { \pi }_{ d }&=1-0.61=0.39 \end{align*}$$

Exhibit 1: Binomial Tree – Stock

$$
\begin{array}
\hline
{} & {} & 1.1853\times $30=$35.60 \\
$30 & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\
{} & {} & 0.8437\times $30=$25.30 \\
\end{array} $$

Exhibit 2: Binomial Tree – Option

$$
\begin{array}
\hline
{} & {} & \max\left( 0,$35.6-$30 \right) =$5.6 \\
{ c}_{ 0 } & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\
{} & {} & \max\left( 0,$25.3-$30 \right) =$0 \\
\end{array} $$

The expected value of the option in one year is given by:

$$ { c }_{ u}\times { \pi }_{ u }+{ c }_{ d }\times { \pi }_{ d }=$5.6\times 0.61+$0\times 0.39=$3.42 $$

The expected value of the option at present is given by:

$$ { c }_{ 0 }=$3.42{ e }^{ \left( -0.05\times 1 \right) }=$3.25 $$

Two-step Binomial model

In the two-period model, the tree is expanded to create room for a greater number of potential outcomes. Exhibit 3 below presents the two-period stock price tree:

$$
\begin{array}
\hline
{} & {} & {} & {} & { S }_{ uu } \\
{} & {} & { S }_{ u } & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\
{ S }_{ 0 } & {\begin{matrix} \\ \begin{matrix} \begin{matrix} \quad \quad \quad \Huge \diagup \\ \end{matrix} \\ \quad \quad \quad \Huge \diagdown \end{matrix} \\ \end{matrix} } & {} & {} & { S }_{ ud} \quad or \quad { S }_{ du } \\
{} & {} & { S }_{ d} & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\
{} & {} & {} & {} & { S }_{ dd} \\
\end{array} $$

The two-step model uses the same formulae used in the one-step version to calculate the value of an option.However, here, we replace \(t\) with \(\Delta t\), which is the length of one-step. If we have say, an option that matures in one year period, then for a two-step binomial model, \(\Delta t=1/2=0.5\)

Thus, the value of an option is given by:

$$f = e^{-r\Delta t}\left(\pi f_u + (1-\pi)f_d\right)$$

and 

$${ \pi }=\frac { { e }^{ r\Delta t }-d }{ u-d }$$

Example: Two-Step Binomial

The price of an exchange-quoted zero-dividend share is $30. Over the past year, the stock has exhibited a standard deviation of 17%. The continuously compounded risk-free rate is 5% per annum. Compute the value of a 1-year European call option with a strike price of $30 using a two-step binomial model

The up- and down-move factors are:

$$\begin{align*} u&={ e }^{ 0.17\times \sqrt { 0.5 } }=1.1277\\  d&=\frac { 1 }{ 1.1277 } =0.8867 \end{align*}$$

So that,

$$
\begin{array}\hline
{} & {} & {} & {} & { S }_{ uu }=38.15 \\
{} & {} & { S }_{ u }=33.83 & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\{ S }_{ 0 }=30 & {\begin{matrix} \\ \begin{matrix} \begin{matrix} \quad \quad \quad \Huge \diagup \\ \end{matrix} \\ \quad \quad \quad \Huge \diagdown \end{matrix} \\ \end{matrix} } & {} & {} & { S }_{ ud}= 30\quad or \quad { S }_{ du }=30 \\
{} & {} & { S }_{ d}=26.60 & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\{} & {} & {} & {} & { S }_{ dd}= 23.59\\
\end{array} $$

The option values are

$$\begin{array}{l|l|l}
{ S }_{ uu }=$38.15 & { f }_{ uu }=\max\left( ⁡$38.15-$30,0 \right) & { f }_{ uu }=$8.15 \\ \hline
{ S }_{ ud }=$30& { f }_{ ud }=\max\left( ⁡$30-$30,0 \right) & { f }_{ ud }=$0 \\ \hline
{ S }_{ du }=$30 & { f }_{ du }=\max\left( ⁡$30-$30,0 \right) & { f }_{ du }=$0 \\ \hline
{ S }_{ dd }=$23.59 & { f }_{ dd }=\max\left( ⁡$23.59-$30,0 \right) & { f }_{ dd }=$0 \\
\end{array}$$

The risk-neutral probability is:

$$\pi =\frac { \left( { e }^{ 0.05\times 0.5 } \right) -d }{ u-d } =\frac { 1.0253-0.8867 }{ 1.1277-0.8867 } =0.58$$

Thus,

$$f_u = e^{-r \Delta t} \left(\pi f_{uu} + (1-\pi)f_{ud}\right)=e^{-0.05\times 0.5}\left(0.58\times 8.15 + 0.42\times 0\right)=4.6103$$

and

$$f_d = e^{-r\Delta t}\left(\pi f_{ud} + (1-\pi)f_{dd}\right)=e^{-0.05\times 0.5}\left(0.58\times 0 + 0.42\times 0\right)=0$$

Thus, the value of the option is 

$$f = e^{-r\Delta t}\left(\pi f_u + (1-\pi)f_d\right)=e^{-0.05\times 0.5}\left(0.58\times 4.6102 + 0.42\times 0\right)=2.6079$$

Note: The value of a put can be calculated once the value of the call has been determined, using the put-call parity relationship.

$$ \text{Call Price} + \text{PV of Strike Price} = \text{Put Price} + \text{Stock Price}$$

Increasing the Number of Steps in the Binomial Model

Binomial models with one or two steps are unrealistically simple. Assuming only one or two steps would yield a very rough approximation of the option price. In practice, the life of an option is divided into 30 or more time steps. In each step, there is a binomial stock price movement.

As the number of time steps is increased, the binomial tree model makes the same assumptions about stock price behavior as the Black– Scholes–Merton model. When the binomial tree is used to price a European option, the price converges to the Black–Scholes–Merton price as the number of time steps is increased.

Delta

The delta, Δ, of a stock option, is the ratio of the change in the price of the stock option to the change in the price of the underlying stock. It is the number of units of the stock an investor/trader should hold for each option shorted in order to create a riskless portfolio. This process is called delta-hedging.

The delta of a call option is always between 0 and 1 because as the underlying asset increases in price, call options increase in price.  The delta of a put option, on the other hand, is always between -1 and 0 because as the underlying security increases, the value of put options decrease.

For instance, suppose that when the price of a stock change from $20 to $22, the call option price changes from $1 to $2. We can calculate the value of delta of the call as:

$$ \frac { 2-1 }{ 22-20 } =0.5 $$

This means that if the underlying stock increases in price by $1 per share, the option on it will rise by $0.5 per share, all else being equal.

Suppose that an investor is long one call option on the stock above (with a delta of 0.5, or 50 since options have a multiplier of 100). The investor could delta hedge the call option by shorting 50 shares of the underlying stock. Conversely, if the investor is long one put on the stock (with a delta of -0.5, or -50), they would maintain a delta neutral position by purchasing 50 shares of the underlying stock.

Generally,

$$\Delta= \frac { f_u-f_d }{ S_u-S_d } $$

How Volatility is Captured in the Binomial Model

As the standard deviation increases, so does the divide (dispersion) between stock prices in up and down states (\({ S }_{ u }\) and \({ S }_{ d }\), respectively). Suppose there was no deviation at all. Would we have a binomial tree in the first place? The answer is no.

With zero standard deviation, (\({ S }_{ u }\) would be equal to \({ S }_{ d }\), and instead of a tree, we would have a straight line. But provided there’s some deviation, the gap between stock prices in the upstate and stock prices in the downstate increasingly widens as the deviation increases.

To capture volatility, therefore, it would be paramount to evaluate stock prices at each time period present in the tree.

How the Modified Binomial Model can be Altered to Price Options on Non-zero Dividend Stocks, Stock Indices, Currencies, and Futures

Given a stock that pays a continuous dividend yield \(q\), the following formula can be used to price the resulting option:

$$\begin{align*}\text{ Probability of an up move}&={ \pi }_{ u }=\frac { { e }^{ \left( r-q \right) t }-d }{ u-d }\\ \text{Probability of a down move}=1-{ \pi }_{ u }\end{align*} $$

\(u\)=size of the up move factor=\({ e }^{ \sigma \sqrt { t } }\), and

\(d\)=size of the down move factor=\({ e }^{ -\sigma \sqrt { t } }=\frac { 1 }{ { e }^{ \sigma \sqrt { t } } } =\frac { 1 }{ u } \)

Note: The sizes of the up move factor and down move factor are the same as in the zero-dividend model.

Sometimes it may also be necessary to price options constructed with a stock index as the underlying, for instance, an option on the S&P 500 index. Such an option would be valued in a manner similar to that of the dividend-paying stock. It’s assumed that the stocks forming part of the index pay a dividend yield equal to \(q\).

The binomial model can also be modified to incorporate the unique characteristics of options on futures. Of note is the fact that futures contracts are largely considered cost-free to initiate, and therefore in a risk-neutral environment, they are zero-growth instruments. The only formula that changes is that of the probability of an up move, where:

$$ { \pi }_{ u }=\frac { 1-d }{ u-d } $$

When dealing with options on currencies, a plausible assumption is that the return earned on a foreign currency asset is equal to the foreign risk-free rate of interest. As such, the probability of an up move is given by:

$$ { \pi }_{ u }=\frac { { e }^{ \left( { r }_{ DC }-{ r }_{ FC } \right) t }-d }{ u-d } $$

American Options

To value an American option, we check for early exercise at each node. If the value of the option is greater when exercised, we assign that value to the node. If that’s not the case, we assign the value of the option unexercised. We then work backward through the tree as usual.

The Binomial Model When Time is Continuous

The binomial model is essentially a discrete-time model where we evaluate option values at discrete times, say, intervals of one year, intervals of six months, intervals of three months, etc.

However, if we were to shrink the length of time intervals to arbitrarily small values, we’d end up with a continuous-time model where the price can move at non-discrete times. The binomial model converges to the continuous-time model when time periods are made arbitrarily small.

Questions

Question 1

Suppose we have a 6-month European call option with \(K = $23\). Suppose the stock price is currently $22 and in two-time steps of three months, the stock can go up or down by 10%. The up move factor is 1.1 while the down move factor is 0.9. The risk-free rate of interest is 12%.

Compute the value of the call today.

1. $2
2. $1.54
3. $1.45
4. $0

The correct answer is C.

$$
\begin{array}
{} & {} & {} & { S }_{ uu }=$26.62 \\
{} & { S }_{ u }=$24.2 & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\
{ S }_{ 0 }=$22 \begin{matrix} & {} & \\ &\Huge \diagup & \\ &\Huge \diagdown & \\ & { } & \end{matrix} & {} & \begin{matrix} {} \\ \\ { } \end{matrix} & \begin{matrix} { S }_{ ud }=$21.78 \\ \\ { S }_{ du }=$21.78 \end{matrix} \\
{} & { S }_{ d }=$19.8 & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\
{} & {} & {} & { S }_{ dd }=$17.82 \\
\end{array} $$

\( { S }_{ u }=22\times 1.1=24.2,\)

\({ S }_{ uu }=22\times 1.1\times 1.1 \)

Other values at other nodes are calculated using the relevant up/down factors.

$$\begin{align*} { \pi }_{ u }&=\frac { { e }^{ rt }-d }{ u-d } =\frac { { e }^{ 0.12\times 0.25 }-0.9 }{ 1.1-0.9 } =0.6523, \\ { \pi }_{ d }&=1-0.6523=0.3477 \end{align*}$$

Let \(f\) represent the value of the call:

$$
\begin{array}{l|l|l}
{ S }_{ uu }=26.62 & { f }_{ uu }=\max\left( ⁡$26.62-$23,0 \right) & { f }_{ uu }=$3.62 \\ \hline
{ S }_{ ud }=$21.78 & { f }_{ ud }=\max\left( ⁡$21.78-$23,0 \right) & { f }_{ ud }=$0 \\ \hline
{ S }_{ du }=$21.78 & { f }_{ du }=\max\left( ⁡$21.78-$23,0 \right) & { f }_{ du }=$0 \\ \hline
{ S }_{ dd }=$17.82 & { f }_{ dd }=\max\left( ⁡$17.82-$23,0 \right) & { f }_{ dd }=$0 \\
\end{array}
$$

The expected value of the call six months from now is given by:

$$ \begin{align*} &0.6523\times 0.6523\times $3.62+0.6523\times 0.3477\times $0\\&+0.3477\times 0.6523\times $0+0.3477\times 0.3477\times $0\\ &=$1.54 \end{align*} $$

$$\text{Value of the call today} =\(\frac { $1.54 }{ { e }^{ 0.12\times 0.5 } } =$1.45$$

Question 2

A 1-year $50 strike European call option exists on \(ABC\) stock currently trading at $49. \(ABC\) pays a continuous dividend of 3% and the current continuously compounded risk-free rate is 4%. Assuming an annual standard deviation of 3%, compute the value of the call today.

1. $0.31
2. $0.30
3. $0.47
4. $0

The correct answer is B.

$$\begin{align*} u &={ e }^{ \sigma \sqrt { t } }={ e }^{ 0.03\times 1 }=1.03\\ d &=\frac { 1 }{ 1.03 } =0.97 \end{align*} $$

Note that the stock is dividend-paying, and therefore the formula for the probability of an up move is given by:

Probability of an up move=\({ \pi }_{ u }=\frac { { e }^{ \left( r-q \right) t }-d }{ u-d } =\frac { { e }^{ \left( 0.04-0.03 \right) 1 }-0.97 }{ 1.03-0.97 } =0.67 \)

Probability of a down move =\(1-0.67=0.33\)

Let \(S\) represent the price of the stock and \(f\) represent the value of the call. This is a one-step binomial process.

$$
\begin{array}{l|l}
{ S }_{ u }=$49\times 1.03=$50.47 & { f }_{ u }=\max\left( $50.47-$50,0 \right) =$0.47 \\ \hline
{ S }_{ d }=$49\times 0.97=$47.53 & { f }_{ d }=\max\left( $47.53-$50,0 \right) =$0 \\
\end{array}
$$

$$ \begin{align*}\text{Value of the call option one year from today}&=\left($0.47\times 0.67+$0\times 0.33\right)=$0.31\\ \text{Value of the call today}&=\frac { $0.31 }{ { e }^{ 0.04 } } =$0.30 \end{align*}$$