4.1 Binomial Trees
31
Note that the absolute value of the share price S drops out of the equation and we
can solve for the probability p
p =
e
ρT
− e
−σ
√
T
e σ
√
T − e −σ
√
T
,
(4.3)
which then determines the probabilities for up ( p) and for downward (1 − p) moves
as a consequence of the share volatility σ and the average expected growth rate ρ.
Since the call option participates in the same market, it is on average subjected to
the same probabilities for up and down movement and we can estimate the expectation
value ¯
c T of the option at the later time T to be
¯
c T = pc 1 + (1 − p)c 2 ,
(4.4)
but we wanted to calculate the value of the option at the initial time and therefore
need to back-propagate the value using the discount factor e
−ρT and determine the
value of the option c at the initial time
c = e
−ρT [ pc 1 + (1 − p)c 2 ] .
(4.5)
Keep in mind that the values for c 1 and c 2 depend on the strike price of the option
K . For example, the value of the call option at node i is c i and is given by
c i = max(S
i
T − K , 0) ,
(4.6)
where S
i
T is the value of the underlying share S at node i at time T.
Obviously the development of the price of the option can be analyzed further by
subdividing any interval of time in many, say n, sub-sections. This results in a tree
where each section has 2
n nodes. Progressing from one section to the next one branch
describes the probability for the share value to increase and the other to decrease. In
the first pass through the tree, we fill in the expected values of the share price S at all
later times by multiplying the previous share value with the appropriate of move-up
factors e
σ
√
T and move-down factors e
−σ
√
T
. In this way we fill the entire tree with
share value prices, including the values at the final nodes. These values we compare
with the strike price K using (4.6) and calculate the value of the option on each of the
final nodes. In a second pass through the tree, now backwards in time, we calculate
the value of the option on each of the preceeding nodes using the single-branch (4.5).
In this way we can trickle down backwards through the tree to arrive at the option
price c at the initial time—now.
Up to this point the reasoning was based on discretized time steps, but in the limit
of reducing the step length to zero, we approach a time-continuous model, which
can be shown [1] to be equivalent to the model we discuss in subsequent sections.
31
Note that the absolute value of the share price S drops out of the equation and we
can solve for the probability p
p =
e
ρT
− e
−σ
√
T
e σ
√
T − e −σ
√
T
,
(4.3)
which then determines the probabilities for up ( p) and for downward (1 − p) moves
as a consequence of the share volatility σ and the average expected growth rate ρ.
Since the call option participates in the same market, it is on average subjected to
the same probabilities for up and down movement and we can estimate the expectation
value ¯
c T of the option at the later time T to be
¯
c T = pc 1 + (1 − p)c 2 ,
(4.4)
but we wanted to calculate the value of the option at the initial time and therefore
need to back-propagate the value using the discount factor e
−ρT and determine the
value of the option c at the initial time
c = e
−ρT [ pc 1 + (1 − p)c 2 ] .
(4.5)
Keep in mind that the values for c 1 and c 2 depend on the strike price of the option
K . For example, the value of the call option at node i is c i and is given by
c i = max(S
i
T − K , 0) ,
(4.6)
where S
i
T is the value of the underlying share S at node i at time T.
Obviously the development of the price of the option can be analyzed further by
subdividing any interval of time in many, say n, sub-sections. This results in a tree
where each section has 2
n nodes. Progressing from one section to the next one branch
describes the probability for the share value to increase and the other to decrease. In
the first pass through the tree, we fill in the expected values of the share price S at all
later times by multiplying the previous share value with the appropriate of move-up
factors e
σ
√
T and move-down factors e
−σ
√
T
. In this way we fill the entire tree with
share value prices, including the values at the final nodes. These values we compare
with the strike price K using (4.6) and calculate the value of the option on each of the
final nodes. In a second pass through the tree, now backwards in time, we calculate
the value of the option on each of the preceeding nodes using the single-branch (4.5).
In this way we can trickle down backwards through the tree to arrive at the option
price c at the initial time—now.
Up to this point the reasoning was based on discretized time steps, but in the limit
of reducing the step length to zero, we approach a time-continuous model, which
can be shown [1] to be equivalent to the model we discuss in subsequent sections.