INTRODUCTION
A binomial tree model is extremely important, popular and useful technique
for pricing an option (Cox et al., 1979; Rendleman
and Bartter, 1979). In the binomial model, the market is composed of a nonrisky
asset B (bond), corresponding to the investment into a savings account in a
bank and of a risky asset S (stock), corresponding for example, to a quoted
stock in the exchange. For the sake of simplicity, we suppose that the time
intervals have the same length Δt = t_{n}t_{n1}, i.e.,
NxΔt = T and the interest rate is constant over the period (0, T), that
is r_{n} = r/N for every interval Δ = t_{n}t_{n1}.
Then the dynamics of the bond is given by:
So that:
It is evident that B_{n} converges to B_{o}×e^{r}.
The details of compound interest are available in study of Hull
(2011) and Pascucci (2011).
For the risky asset, we assume that the dynamics is stochastic: In particular, we assume that when passing from time t_{n1} to time t_{n} the stock can only increase or decrease its value with constant increase and decrease rates. Let u indicates the increase rate of the stock over the period [t_{n1}, t_{n}] and d indicates the decrease rate, where 0<d<1<u.
We point out that we have:
Hence, a “trajectory” of the stock is vector such as (for example, in the case N = 6):
It is wellknown (for example (3, 4) that the binomial tree model is arbitrage –free and complete if and only if condition:
The binomial tree model is a natural bridge, overture to continuous models
for which it is possible to derive the BlackScholes option pricing formula
(Black and Scholes, 1973; Georgiadis,
2011; Merton, 1973).
There are two types of options: calls and puts. A call option gives the holder
the right to buy the underlying asset for a certain price by a certain date.
A put option gives the holder the right to sell the underlying asset by a certain
date for a certain price. There are four possible positions in option markets:
A long position in a call, a short position in a call, a long position in a
put and a short position in a put. Taking a short position in an option is known
as writing it. Options are currently traded on stocks, stock indices, foreign
currencies, futures contracts and other assets. Options can be either American
or European, a distinction that has nothing to do with geographical location.
American options can be exercised at any time up to the expiration date, whereas
European options can be exercised only on the expiration date itself. European
options are generally easier to analyze.
Random binomial tree model
Random walks in a random environment: Random walks in a random environment
on the integers Z was introduced and studied by Solomon
(1975). Let (α_{n}) be a sequence of independent, identically
distributed random variables with 0≤α_{n}≤1 for all n. The
random walk in a random environment on the integers Z is the sequence {X_{n}}
where X_{0 }= 0 and inductively X_{n+1 }= X_{n}+1 (X_{n}1),
with probability αX_{n}, (1αX_{n}). Solomon have
proved that randomizing the environment in some sense slows down the random
walk. Later, Menshikov and Peteritis (2001) studied random
walks in a random environment on a regular, rooted, colored tree and the asymptotic
behavior of the walks was classified for ergodicity or transience in terms of
the geometric properties of the matrix describing the random environment.
Random binomial tree model: As mentioned above for binomial tree model during each time step the value of stock either moves up with a certain probability by u times or moves down by d times with a certain probability. Let us call the pair (u, d) environment of binomial tree model. Note that in this model pair (u, d) is the same for any moment of time.
Now we define random environment and random binomial branch model as follows. Let {U_{n}} and {D_{n}} be the sequences of independent, identically distributed random variables with U_{n}>1 and 0<D_{n}<U_{n} for all n. This pair (U_{n}, D_{n}) is called a random environment.
The random binomial tree model in a random environment (U_{n}, D_{n}) is defined as model such that during nth moment of time the value of stock either moves up with a certain probability by U_{n} times or moves down by D_{n} times with a certain probability for some realization of these random variables.
Simplest random binomial tree model: Let {U_{n}} and {D_{n}} be the sequences of independent, identically distributed random variables such that the random variable U_{n }takes only two values u_{1} and u_{2}, respectively the random variable D_{n} takes two values d_{1} and d_{2}. Thus the pair of random variables (U_{n}, D_{n}) describes two possible environments (u_{1}, d_{1} ) and (u_{2}, d_{2}).
To avoid complicated mathematical formulations let introduce and explore simplest example of random environment in binomial tree model using tossing a coin.

Fig. 1: 
Stock shares of Rolls Royce for period from 6th April till
28th June 2011 
Let us consider two environments (u_{1}, d_{1}) and (u_{2}, d_{2}). We will toss repeatedly a coin and if the result is “Head” we will apply first environment, in opposite case the second. That is we will choose environment randomly and we will call such model simplest random binomial tree model.
To justify necessity consideration of such models consider Fig. 1 which represents the movements of stock shares.
By examining the graph on the Fig. 1, one can see that there are two environments instead of one, namely (u_{1}, d_{1}) = (1.04; 0.98) and (u_{2}, d_{2}) = (1.02; 0.96). For this case we have two environments and they chosen randomly:
H, T, H, T, H, H, H, T, H, T, T, T, H, H, T, T, T….
One can see that this model more exactly reflects the trajectory of stock.
Theorem 1: The simplest random binomial tree model with two possible environments (u_{1}, d_{1}) and (u_{2}, d_{2}) is arbitragefree and complete if and only if the following conditions holds:
where, r is the interest rate.
Proof: The proof immediately follows from similar proposition for binomial
tree models with single environment. For example proposition available in study
of Hull (2011) and Pascucci (2011).
Options on simplest random binomial tree models: Below we will consider simplest random binomial tree model where the randomness of environment is defined by tossing a coin (non necessarily fair) N times. Let Ω_{N} be a sample space with outcome ω = (ω_{1}, ω_{2},…, ω_{N}), where ω_{i}∈{H,T}, i = 1, 2, …, N, observing “Heads” or “Tails”. Note that below T is a period of time and T is Tail. Let Pr {H} = α and Pr {T} = 1α, where 0≤α≤ 1. For any outcome ω∈Ω_{N} we compute price of an option, i.e., construct a random variable and option price for random binomial tree model we define as expectation of corresponding random variable. In this study we consider European call option for simplest random binomial tree model. Other options are considered similarly.
Recall that the objective of the analysis is to calculate the option price at the initial node of the tree.
Options on singleperiod binomial tree: Firstly, we consider European
call option for simplest random singleperiod binomial tree model. In this case
N = 1, i.e., Δt = T and Ω_{1} = {H, T} and random environments
(u_{1}, d_{1}) and (u_{2}, d_{2}) correspond
to outcomes “Head” and “Tail’ respectively. Then we can
compute option prices f (H) and f (T) assuming that condition (4) holds using
well known formula (Cox et al., 1979; Pascucci,
2011):
Where:
Then option price f for simplest random singleperiod binomial tree model we define as expectation of random variable f: Ω_{1}→R, i.e.,
Example 1: Let us consider simplest random binomial tree model with two environment u_{1}= 1.1, d_{1} =0.9 and u_{2} = 1.4, d_{2} = 0.7. If a stock price is currently $20 and the riskfree interest rate is 12% per annum with continuous compounding, find the value of 3months of European call option with a strike price of $21 for simplest random singleperiod binomial tree model.
We consider the probability of turning out of the first environment as α = 0.9 and of the second 1α = 0.1. Such as the second environment has high u and low d, this kind of situation indicates to a crisis or too good economical state which happens seldom. So the price of an option calculated by using two environments, considering casual and extraordinary situations of market, is a more realistic one.
Using formulas from Eq. 57 one can compute
that:
f (H) = 0.633; f (T) = 3.206
and
f = α f (H) +(1α) f (T) = 0.890
Options on twoperiod binomial tree: In this case N = 2, i.e., 2xΔt = T and Ω_{2} = {(H, H); (H, T), (T, H); (T, T)} and random environments (u_{1}, d_{1}) and (u_{2}, d_{2}) correspond to outcomes “Head” and “Tail’ respectively.
For outcomes (H, H) and (T, T) the value if corresponding options one can compute directly using well known formula for the case of two time steps of usual binomial tree model:
where, f_{uu} the value of the stock after two up movements, f_{ud
}after one up and one down and f_{dd} after two down movements.
For outcome (H, T) repeated application of equation (5) gives:
Substituting from Eq. 9 and 10 into 5,
we get:
f = α^{2}f (H, H)+2α(1α) f (H, T)+(1α)^{2}f
(T, T)
and it is evident that f (H, T) = f (T, H).
Then the option price f for simplest random twostep binomial tree model is defined as expectation of random variable f: Ω_{2}→R, i.e.,
f = α^{2}f (H, H)+2α(1α) f (H, T)+(1+α)^{2}
f (T, T)
Example 2: Now let us look at the example 1 in a two steps model. Using formulas (8) and (11) one can compute that f (H, H) = 1.282; f (H, T) = f (T, H) = 3.490; f (T, T) = 3.819 and the value of our option will be f = α^{2 }f (H, H) + 2α(1α) f (H, T)+(1α)^{2} f (T, T) = 1.705.
Options on nperiod binomial tree: Suppose that a tree with Ntime steps
is used to value a European call option with strike price K and life T. Each
step is of length T/N, i.e., NxΔt = T. For usual binomial tree model, if
there have been j upward movements and Nj downward movements on the tree, the
final stock price is S_{0 }u^{j }d^{Nj}, where u is
the proportional up movement, d is the proportional down movement and S_{0}
is the initial stock price. Then option price f is computed as follows:
Now consider simplest random binomial tree model where the randomness of environment
is defined by tossing a coin (non necessarily fair) N times. Let Ω_{N}
be a sample space with outcome ω = (ω_{1}, ω_{2},…,
ω_{N}), where ω_{i}∈{H,T}, i = 1, 2,…, N,
observing “Heads” or “Tails”. Let Pr {H} = α and Pr{T}=
1α, where 0≤α≤ 1. For any outcome ω∈Ω_{N}
let M (ω) be the number of tails in tossing of coin N times. If M (ω)
= 0 or M (ω) = N, then one can compute option price for European call using
previous formula. Now we show how to compute option price for 0<M(ω)<N.
Let M (ω) = 1. Then using trivial identity (a+b)^{N} = (a+b)^{N1}
(a+b), we can find option price f_{1} (ω) as following:
Similarly for M (ω) = m, where 1≤m≤N1, we can find option price f_{m}(ω) as following:
where, p_{1} and p_{2} one can compute using formula (6) for corresponding environments (u_{1}, d_{1}) and (u_{2}, d_{2}), respectively.
Let A_{m}={ω∈Ω_{N}; M (ω) = m}. It is well known that:
Now option price f for simplest random Nperiod binomial tree model we define as expectation of random variable f: Ω_{N}→R, i.e.,:
where, m = 0, 1,…, N.
CONCLUSIONS
One way of deriving the famous BlackScholesMerton result for valuing a European
option on a nondividendpaying stock is by allowing the number of time steps
in a binomial tree to approach infinity. This study has introduced random binomial
tree model and derived formula for computing European call options for such
models. Similarly one can define and study other options. To derive formula
for the valuation of a European call option for random binomial tree models
similar to the BlackScholesMerton formula for usual binomial tree models,
we have to investigate:
by allowing the number of time steps to approach infinity. Since in the case of random binomial tree models we have “doubled” randomness. Also it is an interesting problem to derive the BlackScholesMerton differential equation with random coefficients. All these problems will be subjected in coming papers.
ACKNOWLEDGMENTS
This work was supported by the IIUM Grant EDW B 124030881. The authors also acknowledges useful remarks from referee.