Binomial options pricing model

The Cox-Ross- Rubinstein model (short CRR model, often also: binomial model ) is a discrete model for the modeling of investment and equity price developments. Here, several development possibilities are postulated and each assigned with a positive probability for each time step. The restriction to only two possibilities for development is also called the binomial model.

The binomial model is a method for the determination of fair option prices. Here, the duplication principle is applied, which evaluates the price of the option at climbing the share price and the price of the option when the stock price fall in its simplest form.

The call value is independent of the probability of price increase or decrease, and regardless of the risk attitude of market participants.

The binomial model is easier to use than the Black- Scholes model. It was developed in 1979 by John C. Cox, Stephen Ross and Mark Rubinstein.

• 4.1 Example

• 5.1 Principle of dynamic redeployment strategy
• 5.2 discounting
• 5.3 Exercise of Options 5.3.1 No jump in the option values

Example for determining an option's price

In order to evaluate a first option the repayments in the following period are considered. The purchase of a call option ( = so-called long call) the option is exercised due to a rise rate; then the buyer will receive a refund (if a cash settlement was agreed) or he receives the shares at the subscription price and can sell at the higher rate. However, if the share price fell ( below the reference price), can the buyer the option to expire; he then receives no reflux.

Numerical example: A stock today costs. In a period they can

• Either the value 11 (option value is then 1)
• Or the value 9 (option value is then zero) accept.

It is a portfolio ( Δ shares long, 1 short call ) is formed. The quantity Δ shares in the portfolio takes the same value in both ways is - regardless of their probability - risk free. Call 1 short here means that a call option is sold ( it is the short position of a call -related).

The present value of the portfolio (assuming a risk- free interest rate of 3% and a period length of one year) is:

Determination of the option price today:

Option Delta

The delta factor is important in the evaluation and hedging. It is the sensitivity of the option price to changes in the share price by one unit.

Change in the option price due to change in the underlying share price.

The delta of a call option is positive, the delta of a put option is negative. In two-stage Binomialbäumen the delta is given for the two time steps, wherein the second time step which is taken into account the up and down movement.

Duplication

A call option ( call option on a share) can be duplicated by a portfolio of shares and a loan ( fixed income securities ). From No- arbitrage implies that the value of this portfolio corresponds to the current option value. The Option will be duplicated as partially debt -financed purchase of shares.

Where x is the number of shares per call is long ( corresponding to the delta) and y is the volume of credit per call.

These are two equations with two unknowns. Equation 1 minus equation 2 gives the x, what is the difference between call up and call down, divided by the value of the stock up and down the value of the stock.

After some transformations we obtain the value of a current European call, which is calculated as the discounted expectation with respect to the pseudo- probabilities. Here, the risk-free rate and the volatility will be used.

The result is independent of the likelihood of Kursab or gain. The risk attitude of market participants does not matter.

An intuitive explanation for this could be that if S ^ u occurs with high probability, which would be higher share price in t = 0 and the call value.

Hedgingprinzip

The idea of ​​Hedgingprinzips is to build a risk-free position from shares and a short call or long put. From No- arbitrage implies that the return of this portfolio must match the risk-free rate.

When Hedgingprinzip results in the as, with a riskless portfolio of delta shares long and short a call is made. The current value of this portfolio is the product of delta and current stock price minus the call price. Zinst If this amount is from, so you get the future riskless.

Risk-neutral probabilities

The third method, which finds application in the binomial model, the risk-neutral probabilities ( equivalent martingale ). The evaluation is carried out so as if market participants are risk- neutral. The current share price is understood as the discounted expected value of future stock prices.

This is transferred to the call and the Putwert:

Multilevel binomial model for European options

This model can of course be refined by shortening the time intervals and considered several time points. This is a multi-period model. Moreover, a number of possible states can be considered.

In a multi-step binomial model is included that stock prices more than once may change. A trading interval ( day, hour, etc.) is given by delta t. There is a difference between European and American options. Stock prices may change more than once. We divide the time into several trading intervals (trading interval). The multi-stage binomial model is the discretization of the Black- Scholes model. It is now in financial mathematics is one of the most widely used models available.

In the multi-stage binomial model, a distinction is not recombining recombining of trees. Non- recombining trees are required for path-dependent options.

To achieve the Duplikationseigenschaft the redeployment must take place within the framework of a self-financing strategy.

Example

Binary option, with disbursements of 1 in the up - state and 0 in the down state.

Here, several methods can be used: duplication or hedging.

Optionally using risk-neutral probability, which can be further used.

Using the risk-neutral probability can be evaluated each instrument.

Calculation of and

2 decimal places: in percentages 4

Weighting of payoffs with probabilities ( interpretable as state price)

Exercise properties

Principle of dynamic redeployment strategy

With a dynamic reallocation strategy with only two instruments each payment profile at the time of settlement can be generated. About dynamic trading strategies, a complete market is created.

Discounting

Risky cash flows must be discounted at the risk-adjusted interest rate (eg with the CAPM rate). However, the risk characteristic of an option depends on the amount of the share price and the time to maturity. The risk-adjusted interest rate is; the exact functional form is unknown.

From the completeness of markets follows that one can locally produce a riskless portfolio of equity long and short call over time in each node. The present value is obtained here thus the risk-free interest rate, which is the appropriate rate of interest here.

Exercise of options

For American options, the value is dependent on the date of exercise and then from the given height of the stock price.

If additional dividends paid so the question arises of exercise before or after the record date. The prerequisite is that the stock price exceeds the strike price just before the dividend date. The exercise value is the share price prior to the dividend payment minus the base value, which is identical to the stock price after the dividend payment plus the dividend less the strike price is.

If not exercised, the value of an American call of the European call. The reason is that the call is exercised after the dividend payment only at the end (Why?. , The lower bound on the European call value ( after dividend payments ) is known). It is the ex-dividend price minus the over the remaining term of the discounted strike price.

Comparing the first option with the calculated lower bound of the second possibility ...

No dip in the option values

We want to show that on the record date is no jump in the option values ​​present: The call value of the share before dividends is equal to the call value after the dividend payment.

The dividend discount is no surprise and is therefore included in the call price before the distribution date already.

This is proved using proof by contradiction:

The call price is greater than the call before dividend price after distribution. Then an arbitrage strategy:

Thus, a gain can be achieved by taking a short position of the European calls before the dividend and liquidating the position after the distribution is greater than zero realize. It consists of the call before dividend payments minus the calls after distribution, which by assumption must be greater than zero, yes. Thus, no dividend effect can be observed with the call. In terms of stock, however, there is a dividend discount.

Exercise of American puts

Prerequisite is the validity of the Black- Scholes model. The anticipation that it is possible in the future.

Dividends

For discrete dividends which are paid in proportion to the course, the tree remains repackaging. While this does not model the normal case, leaves the binomial tree but more dominate numerically.

One result is that the option value depends on the exercise strategy.

Smooth Pasting Condition

The value of a European put is always smaller than that of the corresponding American put. The value of an American put must also be above its intrinsic value. The smooth pasting condition is a condition that guarantees that the first derivatives of the two functions equated at the optimum time of exercise have the same slope.