Thomas S. Y. Ho
Brownlee O. Currey Visiting
Professor
Owen School
Vanderbilt University
and
President
Thomas Ho Company
New York
212-571-0121
tom.ho@thomasho.com
September 2000
This paper proposes a closed form binomial interest
rate model. The model extends the
Ho-Lee model to incorporate the term structure of volatility. The model
exhibits a mean reversion process. It is a market model that takes the Black
volatility as given. The paper also derives the drift of the forward rate
process, as proposed by Heath, Jarrow and Morton. Given the model computational efficiency, the model can be used
for Asian options with optimal early exercise option valuation.
Ho-Lee (1986) proposes an arbitrage-free binomial
interest rate model. Extending
Cox-Ross-Rubinstein arbitrage-free binomial lattice model for stock, Ho-Lee
provides a closed form solution to the binomial movement of the yield curve
(the discount function). This model has
the following properties: (1) at each node, that denoted by the state (i) and
future time (n), the model specifies the yield curve as a function of the
initial yield curve and the initial interest rate volatility ( a closed form
solution), (2) there is no arbitrage possibility by holding any portfolio of
bonds along this yield curve over a one step binomial movement at any node point
(binomial model.) These two properties
ensure that the valuation of interest rate contingent claims is arbitrage-free
and is efficient in computation.
A limitation of Ho-Lee is its assumption of constant
interest rate volatility. The model assumes that the one-period interest rate
volatility is constant over time and, as a result, the model cannot accept a
term structure of volatilities. The
generalization of the model was suggested but not reported in Ho and Lee. Ho
(1999) describes a model that takes the term structure of interest rates and
volatilities as given and specifies the arbitrage-free yield curve movements in
a closed form solution. The purpose of
this paper is to describe the behavior of this model.
Specifically, this paper uses the model Ho (1999) to
derive the “money market account” model for deriving the pricing kernal
(Harrison and Kreps), the forward drift of Heath-Jarrow-Morton approach (HJM)
(1990). In so doing, this paper shows
that a closed form binomial model can capture the term structure of
volatilities as in Black-Derman-Toy. But unlike Black Derman and Toy model,
this model has a closed form solution of the yield curve at each node point and
hence a closed form solution for a broad range of interest rate contingent
claims at any future time (n) and state (i). This model is similar to Hull –
White, being a normal model and has mean reversion of interest rates. But this
model is a market model in the sense of Brace, Gatarek and Musiela ( 1997)/
Jamshidian (1997), where we take the market observed Black volatilities directly as inputs to the model.
HJM provides a general approach to interest rate
modeling. In contrast, this paper
provides a specific, but restrictive, model.
In so doing, the value of the model is based on its practical
applications. The model can efficiently price an early exercise option on a
long dated fixed rate instrument over a short expiration period. For example, consider an American swaption. The holder of the option can terminate the
swap any time up till the expiration date (say one year), where one party pays
a fixed rate with long dated tenor (say 10 year) and receives a daily floating
rate. To value such a security, a daily step-size binomial model and the
recursive valuation procedure over option period are used. At each node point,
a closed form solution of the long dated fixed rate bond is calculated. Other binomial interest rate models (
example Black Derman Toy) requires calibrating first the daily step size
binomial model and then recursively calculates a ten year bond value at each
node point over the expiration period would required extensive computations.
The ability to calculate any long dated interest
rate contingent claims at any node point and can apply the recursive
methodology to value an American option is important to an interest rate
model. This property is important to
determine the cost of convexity of a constant maturity swap, the prepayment
options of mortgagors, lapsing behavior of insurance policyholders, withdrawal
behavior of the customers of the savings and deposits accounts. More generally,
portfolio strategies in asset liability management require optimal short-term
multi-period strategies on long-term assets and liabilities. The proposed model can provide a practical solution.
The paper proceeds as follows. Section A describes the model. Section B describes the yield curve
movements of the model. We show that the yield curve movements are related directly
to the shape of the term structure of volatilities, including reversion for a
downward sloping volatility curve. Section C analyzes the behavior of the term
structure of volatilities. The spot volatilities is related to the forward
volatilities. A direct relationship ( and not a calibration) of Black
volatilities is given to the input volatilities of the model. Section D derives
the money market rate model and the forward drifts along the forward curve.
This establishes the relationship of the simulated interest rates to the term
structure of interest rates and volatilities.
Section E contains the conclusions, which summarizes the attributes of
the model.
.
The model is a binomial lattice model with the basic
assumptions similar to Ho-Lee. We
assume a multi-period discrete time model where the market is perfect and
complete. At each future time, n, and
state i, we define a discount bond with
maturity T to be the bond that pays $1 at the end of the Tth period counting
from time n, with no other payments to the bondholder. The price of this discount bond is denoted
by Pni (T).
The function Pni (.) where the time to
maturity T is the input parameter is called the discount function. P00(T) is the initial
discount function, which is observed and is given to the model.
We note that Pni (0)
must equal to 1 since it is the value of $1 at time n and state i. Pni (1) is
the discount factor for one period at time n and state i. At each node of the
binomial model, we assume that the discount function can shift up or down. We
assume that the risk neutral probability for the upstate and for the downstate
is the same, 0.5. At the initial node
of time n and state i, the binomial outcome for the upstate is denoted by the
time (n+1) and the state (i + 1).
Similarly, for the down state, the node is denoted by time (n+1) and
state (i).
The term structure movement is arbitrage-free if
there is no arbitrage opportunity by holding a particular portfolio of the
discount bonds at each node point of the binomial lattice. Such is the case, if the expected return of
holding a discount bond of any maturity T over any one period is the one period
return.
Specifically, according to Harrison and Kreps, we
require that:
Pni
(T) = 0.5 Pni
(1) ( Pn+1i+1 (T - 1) + Pn+1i
(T –1 ) ) (1)
Equation (1) should be satisfied for any n = 0,1,….
and i = 0, 1, 2,…, n.
Pni (1) is the
discount factor at the node time n and state i that determines the present
value of the expected value of the discount bond with maturity (T – 1) at time
(n+1). Pni (T) is the term structure at the node of time n and state i. -ln Pni (1)
is the “money market rate.” And therefore it ensures that the term structure
follows a martingale process, as expressed in equation (1).
The model is discrete time and recombining, in the
framework of Cox, Ross Rubinstein.
Indeed, CRR derives the stock option model assuming the stock price to
follow a binomial movement. This model
assumes that the yield curve follows a binomial movement, with the constraint
that the movement is arbitrage free.
From equation (1), we can then specify the model of the term structure
at each node point given the initial term structure of interest rates and
inputs related to the term structure of volatilities (which will be discussed
later.) This is similar to CRR solution
for stock lattice where the stock movements are specified by the stock
volatilities and risk free rate. The difference here is that we are not
specifying one price (the stock) but a function of prices (the term structure
of interest rates) at each node point.
Proposition 1. The Closed Form Solution
Let 1 > di > 0 be some real numbers for i = 1, 2, 3, … and let P(T) be the
initial discount function. Define d n,
m = dn dn-1 dn-2 … dm+1 dm . ( Section C will show that d n, m are related to the volatilities.) For the
purpose of this proposition, di are input data to the model.
The arbitrage-free movement of the discount function
is given by:
Pni (T) = P(T+n)( 1+dn-1 … d1) (1 + dn-1 … d2) … (1+ dn-1)_____ 2__(dT+n-1 … dn)i (2)
P ( n) (1 +dT+n-1 … d1)(1+dT+n-1 …d2) …(1+dT+n-1…dn-1)(1+dT+n-1…dn)
Alternatively, it can be expressed as
Pni (T) =
[ P(T+n)/P(n)] ( P G-1Tnj ) (dT+n-1 … dn)i – 0.5n
(2a)
where G Tnj = (d T+n-1, n –0.5+ d T+n-1,
n 0.5 * d n-1, j) /( 1 + d n-1,
j )
The result shows that that the T- period discount
bond price n-period hence is the forward bond price (determined by the initial
discount function), adjusted by a factor G n-times the forward price.
The bond price at the ith state is then a multiple (higher and lower) of
this base value.
Proof:
The proof is straightforward by substituting
Equation (2) to Equation (1) to verify the arbitrage free conditions holds.
QED.
We will show later that d n, m is related to the term structure of
volatilities, which are taken as given in this model. Equation (2) provides that the complete description of the
movements of the discount function. It
shows that any future discount function has three factors. The first factor is the forward price
implied from the initial discount function.
Then there is an adjustment term, which depends on the volatilities of
the yield curve movements. In the
limiting case when there is no volatility, (we will show when di equals 1 for all i) the adjustment factor is one. Both of these two terms are independent of
the state i. Therefore the uncertainty
of the interest rate movement is specified only by the last factor.
The importance of Equation (2) is that the model is
closed form. The complete information of the discount function at each node
point is given by the input data of the term structure of interest rates and
volatilities. Consider for example
Black-Derman- Toy, to derive the value of a long bond at a node, we must first
calibrate an extended lattice of one period rate that fits the term structure
of interest rates and volatilities. Then a backward substitution model is used
to price the zero-coupon bond at a particular node. This process can be inefficient and may have significant
numerical estimation errors.
Proposition 2. Extension to the Ho – Lee Model
When dn is a constant d for all n, then the model
is the same as Ho – Lee.
Proof:
Note that when dn are constant, then
d n, m = d n-m+1
(3)
If we substitute Equation (3) to Equation (2), we
have
Pni (T) =[ P( T+
n)/P(n)] [2( 1 + dn-1) ( 1 + dn-2) ( 1 + dn-3)…( 1 + d)/{ (1+ d T +n-1) ... (1+dT)} ]d Ti
(4)
Equation (4) is Ho-Lee model.
QED.
The main difference of the model proposed is the
introduction of the time dependent d of the Ho- Lee model such
that the interest rate movement can be modeled in a binomial lattice. In the continuous time model, Ho-Lee model
can be expressed as
dr (t) = q(t) dt
+ s dz
where r is the instantaneous interest rate, q(t) a function of time,
initial discount function, dz is the wiener process, and s the
constant volatility. The model proposed
in this paper takes the form in continuous time as follows:
dr (t) = q(t) dt
+ s(t) dz (4)
The main difference is that the volatility is time dependent but not state dependent. An important aspect of the model described by Equation (4) is that the model is completely specified by the term structured of interest rates and volatilities. The model has no parameters to be estimated or calibrated from observed prices. We will show the Black caplet prices can be used as direct input to the model.
Section B describes the term structure of interest
rate movements entirely by the discount function. This section will relate the results to the yield curve movements
and term structure of volatilities of the bond yields, which are market
conventions. At the initial date, we have the discount function P(T). We define
the yield of a discount bond with maturity T initially to be:
r (T) = - ln
P(T)/T (5)
More generally, we can define the yield curve at
each node point to be
rni (T) = -ln Pni (T)/T
(6)
Equations (5) and (6) assume continuously
compounding rate of returns for the bonds.
At the end of the first period, the yield curve is either r10 (T) or r11(T).
We now define spot volatility of term T. That is:
s (T) =
0.5 ( r11 (T)
- r10
(T) )
(7)
Since this is a binomial model, the spread between
the two binomial outcomes is twice the standard deviation. Therefore Equation
(7) is the standard deviation of the shifts of the yields for a binomial model.
We also define the forward volatility at time n and
state i for the T period bond yield. It
is defined as:
sn i f
(T) =
0.5 ( rni+1 (T)
- rni (T)) (8)
Therefore rni =
-ln Pni (1) can be interpreted as the
one period rate, analogous to the instantaneous rate in a continuous time
interest rate model. Now substitute Equation (1) into Equation (6) and (8), we
show that the forward volatility
sn f (T) = 0.5 ln
d T+n-1, n / T
(5)
We note that the volatility is independent of the
state i, since the RHS of equation (5) does not contain any index i. At any time n, the yield of a discount bond
has the same volatility at any state i. For this reason the model is normal. The yields have a binomial
distribution ( and becomes a normal distribution in the limit of increasing
smaller step size of the lattice.) Note
that ln d T+n-1, n has T terms. Therefore Equation (5) states that
the volatilities is related to 0.5 of the log of the geometric means of the di
This normal model belongs to the class of models
that was first proposed in Vasicek (1977).
The model assumes that the interest rate rises and falls over the next
instant (or next period ) is independent of the interest rate level. But the standard deviation depends on the
time n. The next proposition relates
the model to input data d to
the spot volatilities.
The spot volatility of term T is given by
s (T) = - 0.5 (ln dT dT-1 dT-2 … d1 )/T.
(9)
Proof:
According to Equation (7), we can compute the value
of the discount function for the upstate and the downstate after one period.
P11 (T) = 2 [P(T+1)/P(1)][1/(
1 + dT,1)] d T,1
P10(T) = 2 [P(T+1)/P(1)][1/( 1
+ dT,1)]
Now
s (T) = - 0.5 (r10
(T) - r11(T))
= -
0.5 ln d T,1 / T
QED
The importance of Proposition 3 is that the observed
spot volatilities provide the complete information to determine all the input
data to the interest rate model.
However, the observed volatilities are the forward volatilities implied
by the Black model from the observed prices of the caplets and floorlets. We therefore first determine the
relationship of the forward volatilities and the spot volatilities.
Proposition 4. Forward and Spot
Volatilities Relationship
The relationship between the term structure of
forward volatilities and spot volatilities
( T + n) s (T + n) =
n s (n ) + Tsfn (T)
(10)
Proof:
From Equation (5), we have
T.sn f (T) = 0.5 ln
d T+n-1, n
From Equation (9), we have
n s (n) = 0.5 ln d n,1
QED
Caplets are call options on a future rate. Consider
a caplet that resets at n, on a rate of term T. Then the payoff is the maximum
of the realized rate of term T net the strike rate and zero. Specifically, we
have max ( rni
(T) – r, 0 ) as the payoff of a caplet.
But Equation (6) gives
rni (T) = -ln Pni (T)/T,
and therefore, we have the boundary conditions to price the caplet. A backward recursive procedure would enable
us the price the caplet. A series of
caplet prices can provide us a corresponding series of forward
volatilities sn f (T) = 0.5 ln
d T+n-1, n / T .
These forward volatilities are then used to
determine the spot volatilities according to equation (10). A smooth function
is then used to fit this series of spot volatilities. From this smooth spot
volatilities, we can now determine all the d by equation (9). Then Equation (1) shows that the entire term
structure movement can be specified.
In essence, Black volatilities of caplets can be
expressed as the forward volatilities based on the model. We can use
proposition 4 to measure the volatilities in terms of the spot volatilities,
which is not dependent on the reset rate.
Then the term structure Pni
(T) can be determined at each node point in closed form. Since the spot
volatilities are not calibrated as in Black Derman and Toy or estimated from
any economic parameters as in Hull and White, this model is similar to the
market model of Brace, Gatarek, and Musiela.
This section analyzes the rates movements of the
model. To isolate the movement from the shape of the initial yield curve, we
consider the rate distribution around the forward rates. Then we provide the model for the money
market rates and the model for the one period forward rates, both for used in
literature in deriving interest rate models.
Proposition 5. Forward Rate and Expected Interest Rate
The expected yield of a T period discount bond over
an n period horizon is the forward rate with a drift. The drift D is given by:
D( n, i, T) = (1/T) ln ( Gn1T Gn2T
… Gn n-1T)
where GniT = ( a -0.5 + a 0.5
d n-1,i)/( 1 + d n-1,i), for a = d T+n-1, n
Proof:
From Equation (2), we have:
Pni (T) = [
P(T+n)/P(n)] ( P G-1Tnj ) (dT+n-1 … dn)i – 0.5n
where G Tnj = (d T+n-1, n –0.5+ d T+n-1,
n 0.5 * d n-1, j) /( 1 + d n-1,
j )
From Equation (6), we have:
rni (T) = -ln Pni (T)/T
= -(1/T) ln [P(T+n)/P(n)] +
(1/T)ln ( P GTnj
) – (i- 0.5n)ln (dT+n-1 … dn) (11)
where G Tnj = (d T+n-1, n –0.5+ d T+n-1,
n 0.5 * d n-1, j) /( 1 + d n-1,
j )
But the expectation of (1/T) i ln d T+n-1, n
over the states i is (0.5 n/T) ln d T+n-1, n, since
the random variable i has a binomial distribution. The first term of the RHS is the yield of the forward discount
bond. Therefore the drift term is given by:
D = (1/T) ln ( Gn1T Gn2T … Gn
n-1T)
(12)
Where GniT = ( a -0.5 + a 0.5
d n-1,i)/( 1 + d n-1,i), for a = d T+n-1, n (13)
QED
Proposition 5 provides a clear description of the
yield curve movements. For each term (T) on the yield curve, we can construct
the forward yield curve for that term.
The yield curve is
-(1/T) ln [P(T+n)/P(n)] for each n = 1, 2, …. The forward rate at on each date is a
binomial distribution. The spread
between two nodes is constant (1/T)ln d
T+n-1, n. That is, the
spread is twice the forward volatility for reset term T and option expiration
date n, according to Equation (5).
Now, if we have a downward sloping term structure of
volatilities, then di is increasing with i.
According to the last term of equation (7), the spread of between two nodes of
the binomial distribution tightens, as n increases. In another words, the yield curve movements follow a mean
reverting process. Further, for longer
the term T, lower is the volatility ( consider a = d T+n-1, n )
, which is consistent with market observations.
The mean of this binomial distribution is not on the
forward curve however. The mean is
drifted above the forward curve, which is to be proved in the next proposition.
Proposition 6 Convexity Drift
The drift is always positive and increases with
volatilities. That is, the expected forward rate of any time period of delivery
T and future date n is higher than the forward rate implied by the initial
yield curve. And this spread increases with volatilities.
Proof:
To show D > 0, according to equation (12), it
suffices to show that GniT is greater than 1 for all i.
Note that 1 >
a 0.5 , d n-1,i > 0
Therefore,
it follows that
( 1 - d n-1,i a 0.5) ( 1
- a 0.5 ) > 0 (14)
In expanding Equation (14) and simplify, we get
1 + d n-1,i
a > a 0.5 ( 1 + d n-1,i)
Then it follows that GniT > 1
Note that with higher volatilities, both
d n-1,i , a become
smaller and Gi becomes larger.
QED.
The positive drift can be explained as the convexity
effect. Since the stochastic future
rates are distributed as a binomial, the price behavior of a zero coupon bond
has a positive convexity. As a result,
the expected return of holding a bond over a period of time would have a higher
return than the risk free rate over that same period, if there were no drift in
the expected forward rate. Therefore,
to assure the arbitrage-free condition to hold in Equation (1), we must have a
positive drift.
Proposition 6 provides a description of the forward
curve movements. To specify an interest
rate movement in a one-factor model, in fact it is necessary and sufficient
only to specify the one period rate, which is the shortest rate on the forward
curve.
Vasicek, Cox-Ingersoll- Ross and others have
specified interest rate models using the short rate movement. Black Derman and Toy, Hull and White are
examples of using the short rate only to specify the arbitrage-free movements
of the yield curve. In principle, once
we have a lattice of one period rate, we can always calculate any bond value at
any node point. In practice, this can be computational intensive.
The one period rate is also important for
determining the arbitrage-free condition in Equation (1). This rate, often
called the “money market rate” is used as a numeraire to specify the movements
of other securities in the market. The
following proposition analyzes this money market rate of the model.
The money market rate is the one period rate of the
binomial model. The process is given by
rni = - ln [P(n +1)/P(n)] +
ln [Gn1 Gn2 … Gn n-1 ] +
(n/2 - i ) ln d n (15)
where Gni is defined as
Gni
= (d n -0.5 + d n 0.5 d
n-1,i )/ ( 1 + d n-1,i )
(16)
Proof:
The proof is quite straightforward. We let T =1, as we are only concerned with
the one period rate, and substitute it in Equation (14). Equation (16) is the same as Equation (13),
except that we suppress the subscript T, since it is always one.
QED
Equation (5) provides a clear description of the
model. Both the first term and the
third term are quite simple to explain.
The first term is the forward one period rate into the future. The third term is the binomial distribution
of the interest rate with mean zero, with the volatility (which is the standard
deviation of a binomial distribution) of n0.5 .sn f (1), where sn f (1) is the one period forward volatility at time n
(see equation (5).) Equation (15)
shows that the factors Gni are
important to the specification of the model.
The following proposition provides insight into these factors.
Consider a node of time n and state i. The one
period forward rate as seen at time n for a contract maturing at time T (as
measured from time n) is denoted as f ( n, i, T).
Proposition 8 The Forward Rates Model
The movement of one period forward rate of the
forward date is given by:
f1i
(T) = - ln [P(T+1)/P(T)] + ln GT - (i – 0.5) ln d T for i = 0 or 1
where
GT
= (d T -0.5 + d T 0.5 d
T-1,1)/ ( 1 + d T-1,1 )
And
d T-1,1 =
d T-1 …d 1
Proof:
The one period forward rate with forward delivery
date T is given by:
f (n, i, T ) =
- ln Pni (T+1)/ Pni
(T).
For simplicity, we consider the case when we are at
the initial position, n = i =0. The
general case can be similarly calculated and the analogous result can be
derived.
At the following one period, at state 1, the bond P(T+1)
P11 (T) = 2 [P( T+1)/P(1)] (d T-1 …d 1)/ ( 1 + d T-1 …d 1).
Then the forward price at state 1 is given by
P11 (T)/ P11
(T-1 ) = [P(T+1)/P(T)][ ( 1 + d T-1 …d 1)/( 1 + d T …d 1 )] – ln d T
Similarly, at state 0, we have
P10 (T)/ P10
(T-1 ) = [P(T+1)/P(T)][ (1 + d T-1 …d 1)/( 1 + d T …d 1 )]
Then, forward rates are given by – ln Pni. The drift is given by the forward rate
net of the expected forward rates. That is:
Drift = d
= - ln[P(T+1)/P(T)] - 0.5 (- ln [P11 (T)/
P11 (T-1 )] – ln[ P10 (T)/ P10
(T-1 )])
The result follows by a direct computation.
QED
Heath-Jarrow-Morton approach is to specify the
movement of the interest rate by the movement of the short term forward
rate. Proposition 8 shows that the
drift of the one period forward rate is simply ln Gni.
It is also interesting to note that previous
research has shown that the continuous time version of Ho- Lee model as an
instantaneous drift of sT.
Such is the case when we consider the limiting case with the step size
approach zero. However, the value of
the Ho –Lee model is not based on the continuous time formulation, but on its
binomial lattice formulation for practical implementation reasons. For this reason, Proposition 8 provides a
useful specification of the drift.
This paper analyzes a closed form binomial interest
rate model. We can compare the model
with other interest rate models in terms of their attributes.
1.
Spot
yield curve.
Similar to other
arbitrage-free model, the model takes the initial yield curve is given. Unlike Vasicek, Cox-Ingersoll-Ross, and some
extent Hull White, where the short term interest rates are postulated to follow
a certain process, and the model is calibrated in fit the observed data.
Equation (2) shows that the pricing model and the rates movements take the
observed spot curve as given. For this
reason, this model can accept any yield curve shape and for this reason, the
model can be used to calculate key rate durations or other yield curve
analytics without any economic justifications of the yield curve movements.
By taking the initial yield
curve as given without other constraints have other implications. In capital markets, liquidity risk and sovereign
risk (particularly for developing economies) are also important consideration
for the determination of the term structure of interest rates. Since the
strengths of the arbitrage-free interest rate model is to enable us to relative
value other securities to the term structure of interest rates, we need to take
these risks into account, beyond the fundamental time value of money
concept. However, there is no
constraint ensuring the forward rates to be positive, given other market
technicalities. Indeed, there were
times that the forward rates in the US Treasury market were negative. Or in
recent times, the floors of Yen rates with zero strike rate has positive value,
since major institutions still have to deposit their significant holdings with
banks or capital markets, even the if market rates were zero. In these
instances, lognormal model like Black Derman and Toy would not accept the spot
curve as given.
2.
Term
Structure of Volatilities
Similar to Black Derman and
Toy, this model takes the term structure of volatilities as given. The model accepts any shape of the term
structure of volatilities, specified by the one period forward volatilities,
Equation (5). There are constraints to
the shape of the spot volatilities only because the spot volatilities may infer
extremely high or other unacceptable values of the forward volatilities ,
violating the constraints 1 > di > 0.
Further, the model forward
volatility can be specified by the Black volatility. Black volatilities assume that the forward rate follows a
lognormal diffusion process, and the caplet price is assumed to be priced by a
stock option model. These assumptions
can be eliminated by re-specifying the Black volatilities in terms of this
arbitrage-free interest rate model. In so doing, we can have a more reasonable
term structure of volatilities to analyze the market and at the same time a
relatively transparent model to translate the volatility value to the caplet
price.
3.
Binomial,Discrete
Time, and Continuous Time Models
Recently, Grant and Vora
(1999) and Das and Sundaram (1999) have pointed out that for practical
implementation, discrete time models (as opposed to continuous time) can
provide computational efficiency. While
they derive their models in discrete time, they do not require the paths to
recombine. This model requires the
paths to recombine. Binomial model can
be efficient for most American option
valuation.
Heath Jarrow Morton
continuous time approach offers flexibility and generality in specifying the
interest rate model, particularly in simulating the forward movements of
interest rates. But the approach often
lacks the computational efficiency and as the result, while the approach may
offer a more realistic model in capturing the market behavior, the valuation
may have larger model errors.
The paper provides a model that can be extended to
value Asian options and multi-factor models.
Given its closed form solution, the model can be used for small step
sizes and allow more computing time to accept other factors ( like time
dependency in Asian options or other risk factors.) These issues will be left for future research.
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