The pu and d calculated from Equation 4 may then be used in a similar fashion to those discussed in the Jaarrow Model tutorial. This is a modification of the original Jarrow-Rudd model that incorporates a risk-neutral probablity rather than jrarow equal probability. This is shown in Figure 3 of the Binomal Model tutorial. Since there are three unknowns in the binomial model pu and d a third equation is required to calculate unique values for them.

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Option Pricing - Alternative Binomial Models This tutorial discusses several different versions of the binomial model as it may be used for option pricing. A discussion of the mathematical fundamentals behind the binomial model can be found in the Binomal Model tutorial.

As introduced in that tutorial there are primarily three parameters -- p, u and d -- that need to be calculated to use the binomial model. The Binomal Model tutorial discusses the way that p, u and d are chosen in the formulation originally proposed by Cox, Ross, and Rubinstein. In the tutorials presented here several alternative methods for choosing p, u and d are presented. The methods discussed here are those proposed by, Jarrow-Rudd : This is commonly called the equal-probability model.

Tian : This is commonly called the moment matching model. Jarrow-Rudd Risk Neutral : This is a modification of the original Jarrow-Rudd model that incorporates a risk-neutral probablity rather than an equal probability. Cox-Ross-Rubinstein With Drift : This is a modification of the original Cox-Ross-Runinstein model that incorporates a drift term that effects the symmetry of the resultant price lattice. Leisen-Reimer : This uses a completely different approach to all the other methods, relying on approximating the normal distrbution used in the Black-Scholes model.

Jarrow-Rudd For reasons that will become self-evident, the binomial model proposed by Jarrow and Rudd is often refered to as the equal-probability model. In the Binomal Model tutorial two equations are given that ensure that over a small period of time the expected mean and variance of the binomial model will match those expected in a risk neutral world.

Since there are three unknowns in the binomial model p, u and d a third equation is required to calculate unique values for them. The third equation proposed by Jarrow and Rudd is Equation 1: Third Equation for the Jarrow-Rudd Binomial Model and hence there is an equal probability of the asset price rising or falling.

This leads to the equations, Equation 2: Parameters for the Jarrow-Rudd Binomial Model The p, u and d calculated from Equation 2 may then be used in a similar fashion to those discussed in the Binomal Model tutorial to generate a price tree and use it for pricing options.

Note that a consequence of Equation 1 is that the Jarrow-Rudd model is no longer risk neutral. The alternative Jarrow-Rudd Risk Neutral model, discussed shortly, addresses this drawback. Tian In the Binomal Model tutorial two equations are given that ensure that over a small period of time the expected mean and variance of the binomial model will match those expected in a risk neutral world.

Note that the mean an variance are called the first and second moments of a distribution. The model proposed by Tian exactly matches the first three moments of the binomial model to the first three moments of a lognormal distribution. Hence the three equations used by Tian are Equation 3: Three Equation for the Tian Binomial Model This leads to the parameters, Equation 4: Parameters for the Tian Binomial Model The p, u and d calculated from Equation 4 may then be used in a similar fashion to those discussed in the Binomal Model tutorial.

This gives the following parameters, Equation 5: Parameters for the Jarrow-Rudd Risk Neutral Binomial Model The p, u and d calculated from Equation 4 may then be used in a similar fashion to those discussed in the Binomal Model tutorial.

Cox-Ross-Rubinstein With Drift The derivation of the original binomial model equations as discussed in the Binomal Model tutorial holds even when an arbitrary drift is applied to the u and d terms. This is shown in Figure 3 of the Binomal Model tutorial. However, with the original model, if the option is a long way out of the money then only a few of the resulting lattice points may have a non-zero payoff associated with them at expiry. The drift term can be used to drift or skew the lattice upwards or downwards to obtain a lattice where more of the nodes at expiry are in the money.

A drawback of that particular drift is that the underlying price tree is a function of the strike and hence must be recalculated for options with different strikes, even if all other factors remain constant.

Leisen-Reimer Leisen and Reimer developed a model with the purpose of improving the rate of converegence of their binomial tree. However the convergence is not smooth. There are several ways this can be calculated. One suggested by Leisen and Reimer is to use, Equation 8: Cummulative Distribution Approximation for the Leisen-Reimer Binomial Model where n is the number of time points in the model including times 0 and T which must be odd, and d1 and d2 are their usual definitions from the Black-Scholes formulation.

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## Jarrow-Rudd Model Formulas

The code may be used to price vanilla European or American, Put or Call, options. Given appropriate input parameters a full lattice of prices for the underlying asset is calculated, and backwards induction is used to calculate an option price at each node in the lattice. Creating a full lattice is wasteful of memory and computation time when only the option price is required. However the code could easily be modified to show how the price evolves over time in which case the full lattices would be required. Note that the primary purpose of the code is to show how to implement the Jarrow-Rudd Risk Neutral binomial model. The code contains no error checking and is not optimized for speed or memory use. As such it is not suitable for inclusion into a larger application without modifications.

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## The Jarrow-Rudd model

All three models supported by the calculator — this one, Cox-Ross-Rubinstein and Leisen-Reimer — follow the same logic for constructing binomial trees that part is explained in underlying price tree and option price tree. The models only differ in sizes and probabilities of underlying price up and down moves in the underlying price tree. Jarrow-Rudd calculations of these are explained below. You can find these calculations in the Model Calculations section at the bottom of the Main sheet, in cells BC Up and Down Move Sizes With the probabilities fixed, all the inputs that normally affect price moves — volatility, interest rate and yield — must be reflected in the move sizes.