Bewley's Lattice model uses a binomial tree to represent the possible future price movements of the underlying asset. The model starts by defining the current price of the asset and the possible up and down price movements. The up and down price movements are represented by the following equations:
P(u) = S * u
P(d) = S * d
Where:
P(u) is the price of the asset at the next time step in the up movement
P(d) is the price of the asset at the next time step in the down movement
S is the current price of the asset
u is the up movement factor
d is the down movement factor
The up and down movement factors are determined by the expected return of the underlying asset and the risk-free rate of return. They are calculated as:
u = e^(rdt)
d = e^(-rdt)
Where:
e is the base of the natural logarithm
r is the risk-free rate of return
dt is the time step
Once the up and down price movements are determined, the model can be used to calculate the price of the option at each node of the lattice. The option price at each node is calculated using the following equation:
V(S, t) = e^(-rdt) * (pV(S*u, t+1) + (1-p)V(Sd, t+1))
Where:
V(S, t) is the option price at the current node
p is the probability of the up movement
V(Su, t+1) is the option price at the next node in the up movement
V(Sd, t+1) is the option price at the next node in the down movement
The model works backwards from the final node of the lattice, where the option's payoff is known, to the current price of the asset. It takes into account the probability of each price movement, the expected return of the underlying asset, and the risk-free rate of return.
In the case of American options, the model introduces the concept of early exercise by comparing the option price at each node with the intrinsic value of the option. The intrinsic value is the difference between the option strike price and the underlying asset's price. The option holder will exercise the option if the intrinsic value is greater than the option price.
It is a powerful tool to price derivatives, particularly in complex situations like American options, and it is widely used in the financial industry to analyze the pricing of derivatives.
0 Comments