Binomial Models. Dr. San-Lin Chung Department of Finance National Taiwan University. In this lecture, I will cover the following topics: 1. Brief Review of Binomial Model 2. Extensions of the binomial models in the literature 3. Fast and accurate binomial option models
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Dr. San-Lin Chung
Department of Finance
National Taiwan University
1. Brief Review of Binomial Model
2. Extensions of the binomial models in the literature
3. Fast and accurate binomial option models
4. Binomial models for pricing exotic options
5. Binomial models for other distributions or processes
The main idea of binomial option pricing theory is pricing by arbitrage. If one can formulate a portfolio to replicate the payoff of an option, then the option price should equal to the price of the replicating portfolio if the market has no arbitrage opportunity.Binomial model is a complete market model, i.e. options can be replicated using stock and risk-free bond (two states next period, two assets).On the other hand, trinomial model is not a complete market model.
S0 uD – ƒu
S0dD – ƒd
ƒ = [ p ƒu + (1 – p )ƒd ]e–rT
(1– p )
When we are valuing an option in terms of the underlying stock the expected return on the stock is irrelevant
1 – p
erdt= pu + (1– p )d
s2dt = pu2 + (1– p )d 2 – [pu + (1– p )d ]2
When dt is small, a solution to the equations is
S0= 50; X = 50; r =10%; s = 40%;
T = 5 months = 0.4167;
dt = 1 month = 0.0833
The parameters imply
u = 1.1224; d = 0.8909;
a = 1.0084; p = 0.5076
There have been many extensions of the CRR model. The extensions can be classified into five directions.
The first direction consists in modifying the lattice to improve the accuracy and computational efficiency.
Broadie and Detemple (1996)
Figlewski and Gao (1999)
Heston and Zhou (2000)
The second branch of the binomial OPM literature has incorporated multiple random assets.
Boyle, Evnine, and Gibbs (1989)
Madan, Milne, and Shefrin (1989)
Ho, Stapleton, and Subrahmanyam (1995)
Chen, Chung, and Yang (2002)
The third direction of extensions consists in showing the convergence property of the binomial OPM.
Cox, Ross, and Rubinstein (1979)
Amin and Khanna (1994)
Nelson and Ramaswamy (1990)
The fourth direction of the literature generalizes the
binomial model to price options under stochastic volatility and/or stochastic interest rates.
Stochastic interest rate: Black, Derman, and Toy (1990), Nelson and Ramaswamy (1990), Hull and White (1994), and others.
Stochastic volatility: Amin (1991) and Ho, Stapleton, and Subrahmanyam (1995) Ritchken and Trevor (1999)
The fifth extension of the CRR model focus on adjusting the standard multiplicative-binomial model to price exotic options, especially path-dependent options.
Asian options: Hull and White (1993) and Dai and Lyuu (2002).
Barrier options: Boyle and Lau(1994), Ritchken (1995), Boyle and Tian (1999), and others.
Instead of setting u = 1/d we can set each of the 2 probabilities to 0.5 and
The Binomial Black & Scholes (BBS) method is
proposed by Broadie and Detemple (1996). The
BBS method is identical to the CRR method, except
that at the time step just before option maturity the
Black and Scholes formula replaces at all the nodes.
The second method was put forward by Tian(1999) termed
“flexible binomial model”. To construct the so-called
flexible binomial model, the following specification is
where λ is an arbitrary constant, called the “tilt
parameter”. It is an extra degree of freedom over the
standard binomial model. In order to have “nonnegative
probability”, the tilt parameter must satisfy the inequality (8)
after jumps, u and d, are redefined.
Heston and Zhou (2000) show that the accuracy or rate of
convergence of binomial method depend, crucially on the
smoothness of the payoff function. They have given an
approach that is to smooth the payoff function. Intuitively,
if the payoff function at singular points can be smoothing,
the binomial recursion might be more accurate. Hence they
let G(x) be the smoothed one;
where g(x) is the actual payoff function.
WAND (2002) showed that the binomial option pricing errors are related to the node positioning and they defined a ratio for node positioning.
The relationship between the errors and node positioning.
Theorem 1.In the GCRR model, the three parameters are as
where is a stretch parameter which determines the shape of
the binomial tree. Moreover, when , i.e., the number of time
steps n grows to infinity, the GCRR binomial prices will converge
to the Black-Scholes formulae for European options.
Various Types of GCRR models:
1. Path dependent options using trees
2. Options where there are two stochastic variables (exchange option, maximum option, etc.)
S = 50, s = 40%, r = 10%, dt = 1 month & the life of the option is 3 months
(This example is presented to illustrate the methodology. A more efficient ways of handling American lookbacks is in Section 20.6.)
14.64Example: An American Lookback Put Option (Figure 20.2, page 463)
S0 = 50, s = 40%, r = 10%, dt = 1 month,
This approach works for lookback options because
S = 50.00
S = 45.72
ZPart of Tree to Calculate Value of an Option on the Arithmetic Average
S=50, X=50, s=40%, r=10%, T=1yr, dt=0.05yr. We are at time 4dt
Consider Node X when the average of 5 observations is 51.44
Node Y: If this is reached, the average becomes 51.98. The option price is interpolated as 8.247
Node Z: If this is reached, the average becomes 50.49. The option price is interpolated as 4.182
Node X: value is
(0.5056×8.247 + 0.4944×4.182)e–0.1×0.05 = 6.206
Barrier assumed by tree
Barrier assumed by tree
Cheuk and Vorst (1996)
Time varying barrier
In all cases a trinomial tree is preferable to a binomial tree
Consider a two-asset case:
Under the first approach: Transform variables so that they are not correlated & build the tree in the transformed variables
We define two new uncorrelated variables:
These variables follow the processes:
where and are uncorrelated Wiener processes.
At each node of the tree, and can be calculated from and using the inverse relationships
Take the correlation into account by adjusting the position of the nodes:
Take the correlation into account by adjusting the probabilities
Chen, R. R., S. L. Chung, and T. T. Yang, 2002, Option Pricing in a Multi-Asset, Complete-Market Economy, Journal of Financial and Quantitative Analysis, Vol. 37, No. 4, 649-666.
With two uncorrelated Brownian motions with equal variances, the three points, A, B, and C, are best to be “equally” apart from each other. This can be achieved most easily by choosing 3 points, located 120 degrees from each other, on the circumference of a circle, as shown in Exhibit 2.
To incorporate the correlation between the two Brownian motions, we then rotate the axes, as shown in Exhibit 3.
The rotation of the axes is defined as follows:
where is the rotation angle of the x-axis counterclockwise and y-axis clockwise. After rotation, we have:
Finally, for any given time, t, the next period stock prices are:
Ref : Amin (1995, pp.39-40) has a very nice discussion on this issue.
Amin, 1995, Option Pricing Trees, Journal of Derivatives,
Assume that the underlying asset price follows
dS = rSdt + (t)Sdz
then the annual variance of the asset price over the period
[0, T] is
Let N be the number of time steps desired, then .
The time step for each period is denoted as h(t), h(2t), …, h(nt). Amin let
In this case, the tree is recombining because
where u(t), u(2t), …, u(nt) are size of up movement at
can move into three possible states, up, down, or middle, in
the next period. If S denotes the asset price at time t, then at
time t + dt, the prices will be Su, Sd, or Sm. The parameters
are defined as follows
where 1, the dispersion parameter, is chosen freely as
long as the resulting probabilities are positive. Let i
In this case the tree is recombining and the probability of
each branch is of course time varying.
must carefully choose dt and . Roughly speaking, dt must
be small enough such that
For , as discussed in Boyle (1988), its values must be larger
than 1. Denote the maximum and minimum of the
instantaneous volatility for the period from time 0 to T as
max and min. Then
larger than 1 automatically.
Boyle, P. (1988), A Lattice Framework for Option Pricing with Two State Variables, Journal of Financial and Quantitative Analysis, 23, 1-12.
1. Hilliard, J. E., and A. Schwartz, Pricing Options on Traded Assets under Stochastic Interest Rates and Volatility: A Binomial Approach, Journal of Financial Engineering, 6, 281-305.
2. Hilliard, J. E., A. L. Schwartz, and A. L. Tucker, 1996, Bivariate Binomial Options Pricing with Generalized Interest Rate Processes, Journal of Financial Research, 14, 585-602.
3. Nelson, D. B., and K. Ramaswamy, 1990, Simple Binomial Processes as Diffusion Approximations in Financial Models, Review of Financial Studies, 3, 393-430.
4. Ritchken, P., and R. Trevor, 1999, Pricing Option under Generalized GARCH and Stochastic Volatility Processes, Journal of Finance, 54, 377-402.
5. Hillard, J. E., and A. Schwartz, 2005, “Pricing European and American Derivatives under a Jump-Diffusion Process: A Bivariate Tree Approach,” Journal of Financial and Quantitative Analysis, 40, 671-691.
6. Camara, A., and S. L. Chung, 2006, Option Pricing for the Transformed-Binomial Class, Journal of Futures Markets, Vol. 26, No. 8, 759-788.
Nelson and Ramaswamy (1990) proposed a general tree method to approximate diffusion processes.
Generally a binomial or trinomial tree is not recombined because the volatility is not a constant. Nelson and Ramaswamy (1990) suggested a transformation of the variable such that the transformed variable has a constant volatility.
For example, under the CEV model:
For example, under the CIR model:
The asset price and return volatility are assumed to follow:
dS = msdt + f(S)h(V)dZs
dV = mvdt + bVdZv (1)
Under Q measure ms＝S(r - d).
First of all, make the following transformation to obtain a unit variance variable Y:
Ritchken, P., and R. Trevor, 1999, Pricing Option under Generalized GARCH and Stochastic Volatility Processes, Journal of Finance, 54, 377-402.
The main idea is to keep the spanning of the tree flexible, i.e. the size of up or down movements can be adjusted to match the conditional variance.
AntÓnio Câmara1 and San-Lin Chung2
This paper generalizes the seminal Cox-Ross-Rubinstein (1979) binomial option pricing model (OPM) to all members of the class of transformed-binomial pricing processes. Our investigation addresses issues related with asset pricing modeling, hedging strategies, and option pricing. We derive explicit formulae for (1) replicating or hedging portfolios; (2) risk-neutral transformed-binomial probabilities; (3) limiting transformed-normal distributions; and (4) the value of contingent claims. We also study the properties of the transformed-binomial class of asset pricing rocesses. We illustrate the results of the paper with several examples.
multiplicative-binomial option pricing model: Cox, Ross, and Rubinstein (1979), Rendlemen and Bartter (1979), and Sharpe (1978)
pricing by arbitrage: According to this rule, when there are no arbitrage opportunities, if a portfolio of stocks and bonds replicates the payoffs of an option then the option must have the same current price as its replicating portfolio.
Third, this paper provides a class of distributions that may explain observed option prices (or implied volatilities).
This M-binomial model assumes that u = 2 and d = 0.5.
For example, if r = 1.25 and = 10 then this SL-binomial model assumes that u = 2.1429 and d = 0.3571.
In this SU-binomial model, it is assumed that u = 1.4107 and d = 0.7106.
For example, if = 300 then this SB-binomial model
assumes that u = 2.5 and d = 0.4545.
Following Johnson (1949), the transformation for the SL-binomial is defined as the following in this article:
The transformation for the SU-binomial model is defined as:
The third example considered in this paper is the SB-
binomial model, corresponding to the SB-normal model of Johnson (1949). The transformation for the SB-binomial model is as follows:
This figure shows the convergence pattern resulting from option price calculations with the SL-binomial model. We use the following selection of parameters: S = 100, K = 100, r = 0.1, = 20, t = 1.0, = 0.25.
This figure shows the convergence pattern resulting from option price calculations with the SU-binomial model. We use the following selection of parameters: S = 100, K = 100, r = 0.1, t = 1.0, = 0.25.
option price calculations with the SB-binomial model. We
use the following selection of parameters: S = 100, K =100, r = 0.1, = 300, t = 1.0, = 0.25.