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
and hence there is an equal probability of the asset price rising or falling.
This leads to the equations,
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
This leads to the parameters,
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.
Jarrow-Rudd Risk Neutral
The Jarrow-Rudd Risk Neutral model is a minor modification to the standard Jarrow-Rudd model. The same u and d are used, however instead of p=1/2 the standard risk neutral value for p is chosen.
This gives the following parameters,
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 model is a special case of the Cox-Ross-Rubinstein With Drift model.
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. Hence, for an arbirary η, the following parameters give a valid binomial model
When the drift term η is set to the zero this model collapses to the original Cox-Ross-Rubinstein binomial model which generates a lattice of prices that is centred around the current asset price S_{0}. 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.
The Jarrow-Rudd Risk Neutral model is a specific case of the Cox-Ross-Rubinstein With Drift model. A common drift is η = (ln(X)-ln(S_{0}))/T, (where ln(·) is the natural logarithm and T is the time to expiry in years) which makes the resulting lattice symmetric about the strike at expiry. 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. (All of the models discussed above converge to the Black-Scholes solution in the limit as the size of the time step Δt is reduced to zero. However the convergence is not smooth.)
The Leisen-Reimer tree is generated using the parameters,
where h^{-1}(·) is a discrete approximation to the cummulative distribution function for a normal distribution. There are several ways this can be calculated. One suggested by Leisen and Reimer is to use,
where n is the number of time points in the model (including times 0 and T) which must be odd, and d_{1} and d_{2} are their usual definitions from the Black-Scholes formulation.