Chapter 9. Project: The Binomial Lattice Model
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338
CHAPTER 9
. Project: The binomial lattice model
SMM
S
Snm
Snm––11
S22
S11
Snm– 1
S12
S00
S01
SjM
S02
S0M
t0
t1
t2
tm – 1
tM
tm
t
FIGURE 9.1 A binomial lattice originating at the current time t = t0 with stock level S00 to final
m−1
m−1
gives rise to two points, Snm = uSn−1
and
time tM = T . At every time slice tm−1 a grid point Sn−1
m
m−1
Sn−1 = dSn−1 , at a later time tm = tm−1 + t.
on the stock must equal the return on the prevailing risk-free rate r. Assuming r constant, we
find that the condition
puS + 1 − p dS = er t S
(9.2)
must be satisfied at all nodes S = Snm . Hence,
pu + 1 − p d = er
Let us introduce a lattice volatility parameter
t
(9.3)
by means of the following equation:
pu2 + 1 − p d2 = e 2r+
2
t
(9.4)
Proposition 9.1. In the limit as t → 0, the lattice volatility converges to the continuous-time
lognormal volatility in the Black–Scholes model.
Proof. For a lognormal distribution, we have
S i+1 = S i exp r − 21
t+
2
√
x∼N 0 1
tx
(9.5)
where S i denotes a stock price at time ti and t = ti+1 − ti . Conditional on the stock price
being S i at time ti , the following expected values at a later time ti+1 = ti + t obtain using
equation (9.5):
E S i+1 = S i er
E S i+1
2
t
= S i 2 e 2r+
(9.6)
2
t
(9.7)
Within the binomial lattice we have instead:
Eb S i+1 = pu + 1 − p d S i
Eb S
i+1 2
= pu + 1 − p d
Equating the variances E S i+1 2 − E S i+1
tion (9.3), gives equation (9.4).
2
2
2
and Eb S i+1
(9.8)
S
i 2
2
− Eb S i+1
(9.9)
2
, and using equa-
9.2 Lattice Calibration and Pricing
339
Equations (9.3) and (9.4) allow one to parameterize the three unknowns d u p in the
binomial lattice by means of the risk-free rate r, the lattice volatility , and a third degree
of freedom. To resolve the indeterminacy we are at liberty to choose another constraining
equation. Two choices are popular:
p=
1
2
(9.10)
u=
1
d
(9.11)
and
For the case p = 21 , the lattice parameters can be expressed as follows in terms of a lattice
volatility and drift r:
d = er
t
1− e
2
t
−1
(9.12)
u = er
t
1+ e
2
t
−1
(9.13)
p=
1
2
(9.14)
This is a recombining binomial tree that drifts upward in the stock price direction.
If u = d1 , the tree is symmetric about the line S = S0 with zero drift and the lattice
parameters are given as follows:
√
d = a − a2 − 1
(9.15)
u = 1/d
(9.16)
p = er t − d / u − d
(9.17)
where
a = e−r t + e r+
2
t
/2
(9.18)
9.2 Lattice Calibration and Pricing
Prices of European-style options are computed iteratively, starting from the maturity date T ,
where the payoff function S is ascribed to the terminal nodes SnM , n = 0 1
M. Let
fnm = V Snm tm be the option price at the node Snm . For a call option with strike K, the final
time condition is
fnM =
SnM = max SnM − K 0
(9.19)
For the put struck at the same level, the condition is instead
fnM =
SnM = max K − SnM 0
(9.20)
The risk-neutral condition on the option price process applied to each node yields the following
recurrence relation (i.e., valuation formula):
fnm = e−r
t
m+1
pfn+1
+ 1 − p fnm+1
(9.21)
The price of the option at current time t0 and spot S0 is given by the last iterate, V S0 t0 = f00 .
340
CHAPTER 9
. Project: The binomial lattice model
option price
f00 (σ2)
f00 (σ) = Cref
f00 (σ1)
t0
T
t
FIGURE 9.2 A schematic of upper and lower bands of option prices (i.e., the outer node values
M) for two different lattice geometries corresponding to lattice volatilities 1
f0m fmm m = 0
(dashed lines) and 2 (solid lines). The lower lattice volatility value 1 gives a lower estimate of the
reference market option price, while the higher value 2 gives an upper estimate of the market option
price. The lattice volatility (for given time step t and interest rate r) that prices the market option
value exactly lies in the interval 1 < < 2 .
To price American options, the method is similar, except an adjustment is made to account
for the possibility of early exercise. Namely, the risk-neutral valuation formula is now (see
Section 1.14.1 on dynamic programming):
fnm = max
Snm e−r
m+1
pfn+1
+ 1 − p fnm+1
t
(9.22)
In the lattice calibration step, one has to adjust the lattice volaility to match the price
of the single at-the-money option used as calibration target. Figure 9.2 shows a schematic
representation of the lattice calibration procedure. Notice that the optimal value for the lattice
volatility
does not necessarily coincide with the Black–Scholes implied volatility I of
the option, but it converges to this value in the limit of time steps of vanishing length. The
calibration procedure requires the use of a root-finding algorithm. The existence of a root is
guaranteed with both choices p = 21 and u = d1 , for in both cases the resulting families of
binomial models allow for arbitrarily large or small values of the volatility. The worksheet
bin contains an at-the-money European call as the calibration target or reference call option
contract. The option is quoted in terms of a Black–Scholes implied volatility I . The market
price results from the Black–Scholes formula
Cref = C S0 Kref r
I
Tref − t0
(9.23)
The current time is denoted by t0 and the spot is S0 . To determine the lattice volatility , one
has to find a root of the equation
f00 = f00
r
t = Cref
(9.24)
for a given choice of r and lattice geometry. Here we have explicitly written the dependence
of the binomial approximation to the price, i.e., f00 , in terms of the lattice parameters. The
value of f00 is found iteratively using equation (9.21) or equation (9.22), depending on
whether the option is European or American, respectively. The final time condition is given
by equation (9.19) for calls and equation (9.20) for puts. The value for the strike is set
as K = Kref . Having finally obtained a value for , the model can be used to price other
American or European options.
CHAPTER
. 10
Project: The Trinomial Lattice Model
The main task in this project is to build a trinomial lattice model to price European and
American claims within an explicit finite-difference scheme. Both drifted and nondrifted
types of lattice geometries are considered. For the drifted lattice model, the drift is adjusted
to account for the prevailing interest rate so as to maintain risk neutrality. As with binomial
models, the model is parameterized by means of a suitably defined lattice volatility, which
is then calibrated to match the price of a given at-the-money European option. Option
prices are obtained for all single-barrier and plain-vanilla European as well as Americanstyle claims. Extensions to Derman–Kani (i.e., local volatility) trinomial trees are left to the
interested reader.
Worksheets: pded1, pded2
Required Libraries: MFioxl, MFBlas, MFFuncs, MFZero, MFStat
10.1 Building the Lattice
Trinomial lattices are normally based on lattices of fixed geometry and parameterized by
the nodal transition probabilities. Consider a recombining two-dimensional tree with a total
number of time steps M ≥ 1. The nodes of the tree are placed along the time lines tm ,
m=0 1
M, where the initial (e.g., present) calendar time is denoted by t0 . We will
denote the time to expiry by T, which defines a time step of size t = T − t0 /M (i.e.,
tM = T ) for the lattice. At the mth time step, there are 2m + 1 nodes in a standard trinomial
lattice.
The nodes are chosen on a log-rectangular grid and can be generally expressed as follows:
Snm = S00 em
t+n x
(10.1)
for n = −m −m + 1
0
m − 1 m. The spot (i.e., initial stock price) is S0 = S00 . The
choice of the parameters and x is discussed shortly. Note that by taking logarithms of
equation 10.1, x gives a measure for the change in log S within a given time slice. Namely,
m
m
m
n log Sn ≡ log Sn+1 − log Sn = x gives the node spacing for a fixed value of time. Changes
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342
C H A P T E R 10
. Project: The trinomial lattice model
due to a possible drift can arise from the difference m log Snm ≡ log Snm+1 − log Snm =
t.
In the stochastic process underlying the trinomial tree model, stock prices can jump from a
node Snm to the nodes Snm+1 , with n = n n ± 1. There are three transition probabilities, p+ ,
p0 , and p− , that correspond to an upward move, middle move (i.e., no move for zero drift),
and downward move, respectively, for any trinomial tree. These risk-neutral probabilities are
subject to two constraints; the first is that of probability conservation,
p+ + p 0 + p − = 1
(10.2)
A trinomial tree is recombining, and the nodes span a cone within a rectangular grid
arrangement in log-stock and time space (see Figure 10.1). Notice, though, that, as we discuss
later, to price several options of different strikes at once, it is useful to extend the trinomial
lattice to cover the complete rectangular grid of 2M + 1 M + 1 points, so at every time
step m we have 2M + 1 points Snm , where n = −M
0
M.
In what follows we present three different geometric constructions of trinomial lattices.
The first two, Cases 1 and 2, assume = 0, while the third asks for an additional constraint on
the probability amplitudes and adjusts the drift in such a way as to achieve risk neutrality.
10.1.1 Case 1 ( = 0)
= 0, the risk-neutrality constraint E Stm+1 Stm = S = er t S gives:
Since
p+ e x + p 0 + p − e −
x
= er
t
(10.3)
Probability conservation (10.2) allows one to eliminate the variable p0 = 1 − p+ + p− ,
and gives
p+ e x − 1 + p− e− x − 1 = er t − 1
m+1
S22
2
1
S1
S1
S01
S0
(10.4)
Sn + 1
m
Sn
m+1
Sn
S
0
S0
m+1
2
Sn – 1
2
1
S–1
S–1
2
S–2
t0
t1
t2
tm
tm + 1
t
FIGURE 10.1 A schematic of the nondrifted ( = 0) trinomial lattice originating at current time t = t0
with stock level S00 . At every time slice tm , a stock at level Snm can change to Snm+1 , with n = n n ± 1.
The drifted lattice has a similar geometry, except all nodes are shifted by an amount exp
t after
every time step.