VRM 14. Binomial Trees

Introduction

No-arbitrage argument –
Risk-neutral valuation –
Consider a three-month call option on the stock with a strike price of USD 52. If the stock price turns out to be USD 60, the payoff for the option will be USD 8. If it turns out to be USD 40, the payoff will be zero.

Consider a portfolio consisting of a long position in ∆ shares of the stock and a short position in one call option ( ∆ is the Greek capital letter for delta).

Generalization

Suppose that the price of a non-dividend paying stock is currently S, and that during a time T it will either move up to $S_u$ (providing a return of u – 1) or down to $S_d$ (providing a return of d – 1). Consider a derivative that provides a payoff of $f_u$ if the stock price increases, and a payoff of $f_d$ if the stock price decreases.

Now a portfolio is formed consisting of:
- A short position in one unit of the derivative; and
- A position of ∆ in the stock

where:

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

The value of the portfolio at time T is

$S_u \Delta - f_u$ if the stock price increases, and

$S_d \Delta - f_d$ if the stock price decreases.

Put-call parity states that when a European put option and a European call option on a non-dividend paying stock have the same strike price and time to maturity, then

Call Price + PV of Strike Price = Put Price + Stock Price

Multi-Step Trees

Of course, it is quite unrealistic to model stock price changes using a tree with a single step. To create a more realistic valuation model, the life of an option is divided into many steps.

Let’s define the length of a tree step as ∆t. Hence

$f=e^{-r\Delta t} \left[pf_u+(1-p) f_d\right]$

where:

$p=\frac{(e^{r\Delta t}-d)}{(u-d)}$

The parameters u and d should be chosen to reflect the volatility of the stock price. If annual volatility is σ then appropriate values for the parameters are

$u=e^{\sigma\sqrt{\Delta t}}$

$d=e^{-\sigma\sqrt{\Delta t}}$

where ∆t is measured in years.

A Two-Step Tree For European Call Option

American Options

European options can only be exercised at maturity. The calculations change while dealing with American options (i.e., options that can be exercised at any time).

At each node, two calculations need to be carried out to determine:

How much the option is worth if it is exercised at the node, and
How much the option is worth if it is not exercised.

The value at the node is the greater of these two.

Since ______________________, the option should _______________

at node C, and thus the value at this node is _______________.

Since ______________________, the option should _______________

at node B, and thus the value at this node is _______________.

Multi-Step Trees

Delta

Delta was calculated as

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

Other Assets – Options On Dividend Paying Stocks

A dividend yield of q implies that the formulas that have been presented so far must be adjusted slightly. The total return in a risk-neutral world is r. Dividends provide a return of q. The expected growth of the stock price must therefore be r – q. In the case where dividends are paid at rate q,

$pS_u+(1-p)S_d=Se^{(r-q)\Delta t}$

so that:

$p=\frac{e^{(r-q)\Delta t}-d}{u-d}$

Everything else about the tree, including the calculation of u and d and the roll back procedure, is the same as before.

Other Assets – Call On Stock Index

For an option on a stock index, it is usually assumed the index provides a dividend yield

EXAMPLE –

Consider a stock index standing at 2,500. Suppose the dividend yield on the index is 2% while the risk-free rate is 3%. Suppose further that the volatility of the index is 15% per annum. If a three step-tree is used to value a European call option with a strike price of 2,500 and a time to maturity of six months, then $\Delta t=\frac{0.5}{3}=0.1667$

In this case:

$u=e^{0.15\sqrt{0.1667}}=1.0632$

$d=e^{-0.15\sqrt{0.1667}}$

$p=\frac{e^{(0.03-0.02)\times0.1667}-0.9406}{1.0632-0.9406}$

and the value of the option given by the three-step tree is 119.579.

This figure shows a three-step tree for valuing a European call option on an index with a strike price of 2,500. The upper number at each node is the stock price, and the lower number is the option price.

Other Assets – Options On Currency

A currency can be considered as an asset providing a yield at the foreign risk-free rate. Therefore, the analysis that has been presented for a stock paying a continuous dividend yield applies, with q equal to the foreign risk-free rate (r_x). This means that:

$p=\frac{e^{(r-r_x)\Delta t}-d}{u-d}$

The rest of the analysis is as explained earlier in this chapter for non-dividend paying stocks.

Other Assets – Call On Currency

EXAMPLE

As an example, consider a four-step tree for a one-year American option to buy a foreign currency for 0.8000 when the current exchange rate is 0.7800. The volatility of the exchange rate is 12%, while the domestic and foreign risk-free rates are 2% and 6% (respectively). In this case:

$u=e^{0.12\sqrt{0.25}}=1.0618$

$d=e^{-0.12\sqrt{0.25}}=0.9418$

$p=\frac{e^{(0.02-0.06)\times0.25}-0.9418}{1.0618-0.9418}=0.4021$

The tree is shown in this figure. (The option is exercised early at nodes A and B. The value of the option to buy one unit of the foreign currency is 0.0188.)

Other Assets – Options On Futures

Because it costs nothing to enter into a futures contract, the return on a futures contract in a risk-neutral world must be zero. This means that a futures contract can be treated like a stock, paying a continuous dividend yield equal to r. This is because when q = r, the expected growth rate of the stock is zero.

Hence

$p=\frac{1-d}{u-d}$

The rest of the analysis is as explained earlier in this chapter for non-dividend paying stocks.

Other Assets – Put On Futures

EXAMPLE

Consider a three-step tree to value an American nine-month put option on a futures contract when the current futures price is 38, and the strike price is 40. assume the volatility to be 20%, and the risk-free rate to be 4%. In this case:

$u = e^{0.2 \times \sqrt{0.25}} = 1.1052$

$d = e^{-0.2 \times \sqrt{0.25}} = 0.9048$

$p = \frac{1 - 0.9048}{1.1052 - 0.9048} = 0.4750$

The value of the option is 3.828.

This figure shows a three-step tree for valuing an American put option on a futures contract with a strike price of 40. The upper number at each node is the futures price, and the lower number is the option price.

Frequently Asked Questions

What is the binomial tree model ?

The binomial tree model is a method for pricing options by modeling possible price changes of the underlying asset over time.

How does the binomial tree model differ for European and American options ?

For American options, the binomial tree model checks at each node whether early exercise is more beneficial.

What is put-call parity ?

Put-call parity is a relationship that connects the prices of European call and put options with the same strike price and maturity.

How is volatility incorporated in the binomial tree model ?

Volatility is used to determine the up and down factors, which represent possible price changes in each step of the tree.

What is delta in options trading ?

Delta measures the sensitivity of an option's price to changes in the price of the underlying asset.

How are options on dividend-paying stocks modeled in the binomial tree ?

For dividend-paying stocks, the model adjusts the expected stock price growth by subtracting the dividend yield.

What is risk-neutral valuation ?

Risk-neutral valuation is a technique that values options by assuming all investors are indifferent to risk.

How are options on futures priced using the binomial tree model ?

Options on futures are priced by treating the futures contract as a stock with a continuous dividend yield equal to the risk-free rate.

How does the binomial tree model handle currency options ?

Currency options are treated similarly to dividend-paying stocks, with the foreign risk-free rate acting as the dividend yield.

What is the significance of a multi-step tree in option pricing ?

A multi-step tree provides a more accurate representation of price movements and option valuation over time.

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Introduction

No-arbitrage argument –
Risk-neutral valuation –
Consider a three-month call option on the stock with a strike price of USD 52. If the stock price turns out to be USD 60, the payoff for the option will be USD 8. If it turns out to be USD 40, the payoff will be zero.

Consider a portfolio consisting of a long position in ∆ shares of the stock and a short position in one call option ( ∆ is the Greek capital letter for delta).

Generalization

Suppose that the price of a non-dividend paying stock is currently S, and that during a time T it will either move up to $S_u$ (providing a return of u – 1) or down to $S_d$ (providing a return of d – 1). Consider a derivative that provides a payoff of $f_u$ if the stock price increases, and a payoff of $f_d$ if the stock price decreases.

Now a portfolio is formed consisting of:
- A short position in one unit of the derivative; and
- A position of ∆ in the stock

where:

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

The value of the portfolio at time T is

$S_u \Delta - f_u$ if the stock price increases, and

$S_d \Delta - f_d$ if the stock price decreases.

Put-call parity states that when a European put option and a European call option on a non-dividend paying stock have the same strike price and time to maturity, then

Call Price + PV of Strike Price = Put Price + Stock Price

Multi-Step Trees

Of course, it is quite unrealistic to model stock price changes using a tree with a single step. To create a more realistic valuation model, the life of an option is divided into many steps.

Let’s define the length of a tree step as ∆t. Hence

$f=e^{-r\Delta t} \left[pf_u+(1-p) f_d\right]$

where:

$p=\frac{(e^{r\Delta t}-d)}{(u-d)}$

The parameters u and d should be chosen to reflect the volatility of the stock price. If annual volatility is σ then appropriate values for the parameters are

$u=e^{\sigma\sqrt{\Delta t}}$

$d=e^{-\sigma\sqrt{\Delta t}}$

where ∆t is measured in years.

A Two-Step Tree For European Call Option

American Options

European options can only be exercised at maturity. The calculations change while dealing with American options (i.e., options that can be exercised at any time).

At each node, two calculations need to be carried out to determine:

How much the option is worth if it is exercised at the node, and
How much the option is worth if it is not exercised.

The value at the node is the greater of these two.

Since ______________________, the option should _______________

at node C, and thus the value at this node is _______________.

Since ______________________, the option should _______________

at node B, and thus the value at this node is _______________.

Multi-Step Trees

Delta

Delta was calculated as

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

Other Assets – Options On Dividend Paying Stocks

A dividend yield of q implies that the formulas that have been presented so far must be adjusted slightly. The total return in a risk-neutral world is r. Dividends provide a return of q. The expected growth of the stock price must therefore be r – q. In the case where dividends are paid at rate q,

$pS_u+(1-p)S_d=Se^{(r-q)\Delta t}$

so that:

$p=\frac{e^{(r-q)\Delta t}-d}{u-d}$

Everything else about the tree, including the calculation of u and d and the roll back procedure, is the same as before.

Other Assets – Call On Stock Index

For an option on a stock index, it is usually assumed the index provides a dividend yield

EXAMPLE –

In this case:

$u=e^{0.15\sqrt{0.1667}}=1.0632$

$d=e^{-0.15\sqrt{0.1667}}$

$p=\frac{e^{(0.03-0.02)\times0.1667}-0.9406}{1.0632-0.9406}$

and the value of the option given by the three-step tree is 119.579.

This figure shows a three-step tree for valuing a European call option on an index with a strike price of 2,500. The upper number at each node is the stock price, and the lower number is the option price.

Other Assets – Options On Currency

A currency can be considered as an asset providing a yield at the foreign risk-free rate. Therefore, the analysis that has been presented for a stock paying a continuous dividend yield applies, with q equal to the foreign risk-free rate (r_x). This means that:

$p=\frac{e^{(r-r_x)\Delta t}-d}{u-d}$

The rest of the analysis is as explained earlier in this chapter for non-dividend paying stocks.

Other Assets – Call On Currency

EXAMPLE

$u=e^{0.12\sqrt{0.25}}=1.0618$

$d=e^{-0.12\sqrt{0.25}}=0.9418$

$p=\frac{e^{(0.02-0.06)\times0.25}-0.9418}{1.0618-0.9418}=0.4021$

The tree is shown in this figure. (The option is exercised early at nodes A and B. The value of the option to buy one unit of the foreign currency is 0.0188.)

Other Assets – Options On Futures

Because it costs nothing to enter into a futures contract, the return on a futures contract in a risk-neutral world must be zero. This means that a futures contract can be treated like a stock, paying a continuous dividend yield equal to r. This is because when q = r, the expected growth rate of the stock is zero.

Hence

$p=\frac{1-d}{u-d}$

The rest of the analysis is as explained earlier in this chapter for non-dividend paying stocks.

Other Assets – Put On Futures

EXAMPLE

$u = e^{0.2 \times \sqrt{0.25}} = 1.1052$

$d = e^{-0.2 \times \sqrt{0.25}} = 0.9048$

$p = \frac{1 - 0.9048}{1.1052 - 0.9048} = 0.4750$

The value of the option is 3.828.

This figure shows a three-step tree for valuing an American put option on a futures contract with a strike price of 40. The upper number at each node is the futures price, and the lower number is the option price.

Frequently Asked Questions

What is the binomial tree model ?

The binomial tree model is a method for pricing options by modeling possible price changes of the underlying asset over time.

How does the binomial tree model differ for European and American options ?

For American options, the binomial tree model checks at each node whether early exercise is more beneficial.

What is put-call parity ?

Put-call parity is a relationship that connects the prices of European call and put options with the same strike price and maturity.

How is volatility incorporated in the binomial tree model ?

Volatility is used to determine the up and down factors, which represent possible price changes in each step of the tree.

What is delta in options trading ?

Delta measures the sensitivity of an option's price to changes in the price of the underlying asset.

How are options on dividend-paying stocks modeled in the binomial tree ?

For dividend-paying stocks, the model adjusts the expected stock price growth by subtracting the dividend yield.

What is risk-neutral valuation ?

Risk-neutral valuation is a technique that values options by assuming all investors are indifferent to risk.

How are options on futures priced using the binomial tree model ?

Options on futures are priced by treating the futures contract as a stock with a continuous dividend yield equal to the risk-free rate.

How does the binomial tree model handle currency options ?

Currency options are treated similarly to dividend-paying stocks, with the foreign risk-free rate acting as the dividend yield.

What is the significance of a multi-step tree in option pricing ?

A multi-step tree provides a more accurate representation of price movements and option valuation over time.

Introduction

Generalization

Multi-Step Trees

A Two-Step Tree For European Call Option

American Options

Multi-Step Trees

Delta

Other Assets – Options On Dividend Paying Stocks

Other Assets – Call On Stock Index

Other Assets – Options On Currency

Other Assets – Call On Currency

Other Assets – Options On Futures

Other Assets – Put On Futures

Frequently Asked Questions

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Introduction

Generalization

Multi-Step Trees

A Two-Step Tree For European Call Option

American Options

Multi-Step Trees

Delta

Other Assets – Options On Dividend Paying Stocks

Other Assets – Call On Stock Index

Other Assets – Options On Currency

Other Assets – Call On Currency

Other Assets – Options On Futures

Other Assets – Put On Futures

Frequently Asked Questions

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