Binomial Option Valuation Model

Binomial Option Valuation Model

One-Period Binomial Option Valuation Model

In the one-period binomial model, we start today (at time t=0) when the stock price is \(S_{0}\). Then, the stock price can either jump upwards or downwards over the one-period time interval to t=1. This is illustrated below:

$$S_1=\{\begin{align*}S_{0}u& , \text{if the stock price jumps up} \\S_{0}d& , \text{if the stock price jumps down}\end{align*}$$

This can be shown in the following binomial tree:

One-Period Binomial TreeWhere:

$$u=\frac{S_{0}u}{S_{0}}$$

$$d=\frac{S_{0}d}{S_{0}}$$

One period Binomial option payoffs

Consider a call option that pays \(c_{u}\) if the price of the underlying asset jumps up and  \(c_{d}\) if the price of the underlying asset jumps down.

The value of the call option at expiry is expressed as:

\(c_{u}=Max(0,S_{0}u-K)\), if the price of the underlying jumps up

and 

\(c_{d}=Max(0,S_{0}d-K), \)if the stock price jumps down

Where K is the strike price.

This is shown in the following binomial tree:

One period Binomial Call Option PayoffsSimilarly, the value of a put option at expiration is given by:

\(p_{u}=Max(0,K-S_{0}u)\), if the price of the underlying asset jumps up

and

\(p_{d}=Max(0, K- S_{0}d)\), if the price of the underlying asset jumps down.

One period Binomial Put Option PayoffsOne period Binomial option values

The initial values of call and put options with a one period to expiry are determined using the following formulas:

$$C_{0}=\frac{qc_{u}+(1-q)c_{d}}{1+r}$$

And

$$p_{0}=\frac{qp_{u}+(1-q)p_{d}}{1+r}$$

Where: 

$$q=\frac{(1+r)-d}{u-d}$$

Where: 

\(r\) is the risk-free rate for a single period.

\(q\) gives the risk-neutral probability of an upward move in price

\(1-q\) gives the risk-neutral probability of a downward move

Example: Calculating the price of an option using the one-period binomial option valuation model

Consider a European put option with a strike price of $50 on a stock whose initial price is $50. The risk-free rate of interest is 4%, the up-move factor u = 1.20, and the down move factor d =0.83. The price of the put option can be determined using the one-period binomial model as follows:

$$S_{0}u=50\times1.20=$60$$

$$S_{0}d=50\times0.83=$41.50$$

Recall that put payoff is given by:

\(p_{u}=\text{Max}(0, K-S_{0}u)\), if the price of the underlying asset jumps up

and 

\(p_{d}=\text{Max}(0, K-S_{0}d)\), if the price of the underlying asset jumps down.

$$p_{u}=Max(0,50-60)=$0$$

$$p_{d}=\text{Max}(0,60-41.50)=$18.50$$

The value of the put is then calculated using the formula: 

$$p_{0}=\frac{qp_{u}+(1-q)p_{d}}{1+r}$$

Where:

$$q=\frac{(1.04)-0.83}{1.20-0.83}=0.5676$$

$$p_{0}=\frac{0.5676\times$0+0.4324\times$18.50}{1.04}=$7.69$$

Two-Period Binomial Option Valuation Model

The one-period binomial model can be extended into a multi-period context. The two-period binomial lattice can be seen as three-one period binomial lattices as shown below:

Two-Period Binomial Option Valuation ModelThe underlying asset can result in only three possible values:

\(S_{0}uu=\) When price moves up twice

\(S_{0}ud=\) When price either moves up then down or down then up

\(S_{0}dd=\) When price moves down twice

Call Payoffs

A call option under the two-period binomial option model will have three possible payoffs at expiry as follows:

$$C_{uu}=\text{max}(0, S_{0}u^{2}-K)$$

$$C_{ud}=Max(0,S_{0}ud-K)$$

$$C_{dd}=Max(0,S_{0}d^{2}-K)$$

Put Payoffs

Similar to a call option, a put option will have three possible payoffs:

$$p_{uu}=Max(o,K-S_{0}u^{2})$$

$$p_{ud}=Max(0,K-S_{0}ud)$$

$$p_{dd}=Max(0,K-S_{0}d^{2})$$

Option Values

A European call option’s value can be determined using the two-step binomial valuation model using the following formula.

$$c_{0}=\frac{q^{2}c_{uu}+2q(1-q)c_{ud}+(1-q)^{2}c_{dd}}{(1+r)^{2}}$$

The two-period European put value is given as:

$$p_{o}=\frac{q^{2}p_{uu}+2q(1-q)p_{ud}+(1-q)^{2}p_{dd}}{(1+r)^{2}}$$

These concepts will be explained more with examples in the sections that follow.

Question

Suppose that a one-year European call option has a strike price of £60. The underlying non-dividend-paying stock is currently trading at £60. Over one year, the stock price can either jump up to £90 or jump down to £50. The annual risk-free interest rate is 4%. Using a one-period binomial option valuation model, the price of the call option is closest to:

  1. £2.44
  2. £9.04
  3. £15.64

Solution

The correct answer is B:

The payoff of a European call option at expiration is given by:

\(c_{u}=Max(0,S_{0}u-K)\), if the price of the underlying jumps up

and

\(c_{d}=Max(0,S_{0}d-K)\), if the stock price jumps down

$$u=\frac{90}{60}=1.5$$

$$d=\frac{50}{60}=0.83$$

$$c_{u}=Max(0,90-60)=£30$$

$$c_{d}=Max(0,50-60)=£0$$

The value of a call option is then calculated using the formula:

$$c_{0}=\frac{qc_{u}+(1-q)c_{d}}{1+r}$$

Where:

$$q=\frac{1.04-0.83}{1.5-0.83}=0.3134$$

$$c_{0}=\frac{0.3134\times£30+0.6866\times£0}{1.04}=£9.04$$

Reading 38: Valuation of Contingent Claims

LOS 38 (a): Describe and interpret the binomial option valuation model and its component terms

Shop CFA® Exam Prep

Offered by AnalystPrep

Featured Shop FRM® Exam Prep Learn with Us

    Subscribe to our newsletter and keep up with the latest and greatest tips for success
    Shop Actuarial Exams Prep Shop Graduate Admission Exam Prep


    Daniel Glyn
    Daniel Glyn
    2021-03-24
    I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way!
    michael walshe
    michael walshe
    2021-03-18
    Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.
    Nyka Smith
    Nyka Smith
    2021-02-18
    Every concept is very well explained by Nilay Arun. kudos to you man!
    Badr Moubile
    Badr Moubile
    2021-02-13
    Very helpfull!
    Agustin Olcese
    Agustin Olcese
    2021-01-27
    Excellent explantions, very clear!
    Jaak Jay
    Jaak Jay
    2021-01-14
    Awesome content, kudos to Prof.James Frojan
    sindhushree reddy
    sindhushree reddy
    2021-01-07
    Crisp and short ppt of Frm chapters and great explanation with examples.