Cox-Ross-Rubinstein model

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One-period model with parameters d = 0.5, u = 2 and the probabilities p and 1-p

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

The binomial model is a method for determining fair option prices . The duplication principle is applied, which in its simplest form evaluates the price of the option when the share price rises and the price of the option when the share price falls.

The call value is independent of the probability of the price increase or decrease as well as independent of the risk attitude of the 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 .

Example of determining an option price

To evaluate an option, the repayments in the subsequent period are first considered. If a purchase option is purchased (= so-called long call), the option is exercised when the price rises; then the buyer receives a repayment (if a cash settlement was agreed) or he receives the share at the subscription price and can sell it at the higher price. If, on the other hand, the share price has fallen (below the subscription price), the buyer lets the option lapse; he then does not receive any return flow.

Numerical example: A share costs today . In one period she can ${\displaystyle S_{0}=10}$

• either the value 11 (option value is then 1)
• or assume the value 9 (option value is then zero).

A portfolio (Δ shares long, 1 call short) is created. The amount Δ shares for which the portfolio assumes the same value in both options is risk-free - regardless of their probability of occurrence. 1 Call short here means that a call option is sold (the writer position of a call is taken).

${\displaystyle 11\Delta -1=9\Delta \quad \Leftrightarrow \quad 2\Delta =1\quad \Leftrightarrow \quad \Delta =1/2\quad \Rightarrow }$ In both situations the portfolio value at time T is 4.5.

The present value of the portfolio in (assuming a risk-free interest rate of 3% and a period of one year) is: ${\displaystyle t=0}$

${\displaystyle 4{,}5e^{-0{,}03}=4{,}367}$

Determination of the option price today: ${\displaystyle f}$

${\displaystyle 10\cdot {\frac {1}{2}}-f=5-f{\;{\overset {!}{=}}\;}4{,}367}$

${\displaystyle \Rightarrow {\text{Optionspreis }}f=0{,}633.}$

Option delta

The delta factor is important when evaluating and hedging. It is the sensitivity of the option price to a change in the share price by one unit.

Call Delta = ${\displaystyle \Delta _{C}={\frac {C_{u}-C_{d}}{S_{u}-S_{d}}}}$

Put delta = ${\displaystyle \Delta _{P}={\frac {P_{u}-P_{d}}{S_{u}-S_{d}}}}$

Change in the option price by changing the underlying share price.

The delta factor of a call option is positive, the delta factor of a put option is negative. In the case of two-level binomial trees, the delta is specified for the two time steps, with the upward and downward movement being taken into account in the second time step.

Duplication

A call option (purchase option on a share) can be duplicated using a portfolio of shares and a loan (fixed-income securities). From the arbitrage-free condition it follows that the value of this portfolio corresponds to the current option value. The option is duplicated as a partially credit-financed share purchase.

${\displaystyle C_{0}=S_{0}\Delta _{C}-B_{0}\ }$

${\displaystyle B_{0}=(S^{u}\Delta -C^{u})(1+r)^{-T}\ }$

${\displaystyle P_{0}=S_{0}\Delta _{P}-B_{0}\ }$

${\displaystyle B_{0}=(S^{u}\Delta +P^{u})(1+r)^{-T}\ }$

1. ${\displaystyle S^{u}\cdot x-(1+r)^{T}\cdot y=C^{u}}$
2. ${\displaystyle S^{d}\cdot x-(1+r)^{T}\cdot y=C^{d}}$

where x is the number of long stocks per call (corresponds to the delta ) and y is the amount of credit per call.

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

After some transformations, we get the value of a current European call, which is the discounted expected value with regard to the pseudo-probabilities. The risk-free interest rate and volatility are used.

The result is independent of the probability of the price drop or increase. The risk attitude of market participants is also irrelevant.

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

Hedging principle

The idea of ​​the hedging principle is to build a risk-free position from stocks and a short call or long put. It follows from no-arbitrage that the return on this portfolio must match the risk-free interest rate.

When Hedgingprinzip results in the as , with a risk-free portfolio of Delta shares long and a call is made short. The current value of this portfolio is the product of the delta and the current share price minus the call price. If you discount this amount, you get the future risk-free value. ${\displaystyle \Delta _{C}}$ ${\displaystyle {\tfrac {C^{u}-C^{d}}{S^{u}-S^{d}}}}$

${\displaystyle \Delta _{P}=\Delta _{C}-1\ }$

Risk-neutral probabilities

The third method used in the binomial model are the risk-neutral probabilities (equivalent martingale measures ). The assessment is made as if the market participants were risk-neutral. The current share price is understood as the discounted expected value of future share prices.

${\displaystyle S_{0}={\frac {pS^{u}+(1-p)S^{d}}{(1+r)^{T}}}\ }$

• ${\displaystyle S^{u}=uS_{0}\ }$
• ${\displaystyle S^{d}=dS_{0}\ }$
• ${\displaystyle p={\frac {(1+r)^{T}-d}{u-d}}\ }$
• ${\displaystyle (1-p)={\frac {u-(1+r)^{T}}{u-d}}\ }$

This is carried over to the call and put value:

${\displaystyle C_{0}={\frac {pC^{u}+(1-p)C^{d}}{(1+r)^{T}}}\ }$

${\displaystyle P_{0}={\frac {pP^{u}+(1-p)P^{d}}{(1+r)^{T}}}\ }$

Multi-level binomial model for European options

Multi-period model with a recombining tree and a constant bank account process

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

A multi-level binomial model takes into account the fact that stock prices can change more than once. A trading interval (day, hour, etc.) is given with delta t. A distinction is made between European and American options. Stock prices can change more than once. To do this, we divide the time into several trading intervals . The multi-level binomial model is the discretization of the Black-Scholes model . It is one of the most widely used models in financial mathematics today .

In the multi-level binomial model, a distinction is made between recombining and non-recombining trees . Non-recombining trees are required with path-dependent options.

In order to achieve the duplication characteristic, the reallocation must take place within the framework of a self-financed strategy.

example

Binary option, with payouts in of 1 in the up state and 0 in the down state. ${\displaystyle t=1}$

Several methods can be used here: duplication or hedging.

Choice of using risk-neutral probabilities that can be further used.

${\displaystyle (54q_{0}+49(1-q_{0}))/(1+r)=50}$ (corresponds to the share price)

Each instrument can be assessed using the risk-neutral probability.

${\displaystyle q_{0}={\frac {50(1+r)-49}{54-49}}=0{,}7}$

Calculation of and ${\displaystyle q_{1}u}$ ${\displaystyle q_{1}d}$

${\displaystyle q_{1},u={\frac {54(1+r)-52}{57-52}}=0{,}94}$

${\displaystyle q_{1},d={\frac {49(1+r)-48}{52-48}}=0{,}8625}$

Decimal places 2: for percentages 4.

${\displaystyle BC_{0}=(1q_{0})/1+r=0{,}67}$Weighting of the payoffs with probabilities (can also be interpreted as state price )

Exercise characteristics

Principle of the dynamic reallocation strategy

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

Discounting

Cash flows subject to risk must be discounted using the risk-adjusted interest rate (e.g. using the CAPM interest rate). However, the risk characteristic of an option depends on the level of the share price and the remaining term. The risk-adjusted interest rate is ; the exact function is unknown. ${\displaystyle f(S_{t},T-t)}$

From the completeness of the markets it follows that one can locally generate a risk-free portfolio of long stocks and short calls in each node over time. The present value results from the risk-free interest rate, which is the appropriate interest rate here.

Exercising Options

In the case of American options, the value depends on the time of exercise and the then given share price.

If dividends are also paid, the question of exercising before or after the dividend date arises. The prerequisite is that the share price exceeds the base price shortly before the dividend date. The exercise value is the share price before dividend payment minus the base value, which is identical to the share price after dividend payment plus the dividend minus the base price.

If the call is not exercised, the value of the American call corresponds to that of the European call. The reason is that the call after dividend payment is only exercised at the end (Why? The lower limit for the European call value (after dividend payment) is known). It is the ex-dividend price less the base price discounted over the remaining term.

If one compares the first possibility with the calculated lower bound of the second possibility ...

No jump in option values

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

The dividend discount is no surprise and is therefore already included in the call prices prior to the payout date.

This can be shown by contradiction proof:

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

By entering into a short position of the European call before the distribution and closing out the position after the distribution, a profit greater than zero can be realized. It consists of the call before the payout minus the call after the payout, which, according to the assumption, must be greater than zero. Thus, no dividend effect can be observed with the call . In the case of the share, however, there is a dividend discount.

Exercising American puts

The prerequisite for this is the validity of the Black-Scholes model. The anticipation that it will be possible in the future.

Dividends

With discrete dividends paid proportionally to the price, the tree remains recombining. Although this does not model the normal case, the binomial tree can still be numerically controlled.

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

Smooth pasting condition

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

See also

• Korn-Kreer-Lenssen model

literature

• Stefan Reitz: Mathematics of the modern financial world. Derivatives, portfolio models and rating procedures. Vieweg + Teubner Verlag, Wiesbaden 2011, ISBN 978-3-8348-0943-8 , chapter 3.
• Steven E. Shreve: Stochastic Calculus for Finance I. The Binomial Asset Pricing Model. Springer, New York 2005, ISBN 0-387-24968-0 .

Individual evidence

1. ^ John C. Cox, Stephen Ross, Mark Rubinstein: Option Pricing: A Simplified Approach. In: Journal of Financial Economics. No. 7, 1979, pp. 229-263.