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This study proposes a lattice option pricing model for
multiple underlying assets under the complete market assumption.
In an economy with n-risky assets and a riskless bond, the
number of states can be only (N + 1) to form a dynamically
complete market. Since there are always fewer equations,
N means, N variances plus (N(N - l))/2 correlations than
the number of unknowns, N(N + 1), the nodes cannot be identified
uniquely. This study introduces the concept of rotation
in higher dimensional space to solve the mismatch problem.
Cox-Rubinstein-Ross's (CRR) binomial model, (Cox et al.,1979)
has induced many researchers to generalize the CRR model
to approximate a multi-asset lognormal process.For instance, Evnine (1983) has developed a multiple binomial
model, which approximates the increments of two lognormal
processes by three sequential moves. Boyle (1988) has proposed
a five-jump model to approximate a joint bivariate lognormal
process and showed its use by valuing American options that
depend on prices of two state variables. Unfortunately,
extending this procedure to three or more state variables
is cumbersome because sets of parameters yielding non-negative
probabilities for the jumps have to be first obtained, and
the difficulty of selecting suitable parameters results
in implementation problem.
To overcome these problems, Boyle et al. (1989) have considered
an alternative approximating procedure. Specifically, for
the two state variable problem they have used a four-jump
multiperiod lattice to approximate the logarithmic return
process.Intuitively, one would think that if one log normal process
can be approximated by one binomial tree, then two lognormal
processes can also be approximated by two binomial processes.
This would lead to a multinomial process with four uncertain
states. However, with four unknown states and only two stocks
and one bond available for trading, markets cannot be completed
by dynamic trading, and options cannot be replicated dynamically.Those models mentioned above are not developed under the
complete market assumption. Therefore those lattice models
that price American style options with multiple risky assets
use various optimization procedures to solve the mismatch
problem between the number of states and the number of assets.
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