Chapter 4. Research Design and Analytical Framework
4.4. Monte Carlo Simulation
4.4.2. Simulating the SDEs De…ning Value
The SDEs describing the evolution of downstream value are based on the ob- servations discussed in Section 4.3.2. Maximum Likelihood Estimation is used to calibrate the SDEs and the value of the underlying contingent claims are estimated using risk-neutral valuation principles. An intensity-based approach is used to account for the e¤ect of the stochastic dynamics of exchange line activation. The price of market risk and the price of the risk of default are used as handles that de…ne the martingale equivalents of the process that generates value. To illustrate, taking the example of the analogue platform, the evolution of downstream value of an activated line is de…ned as follows4
dSa(t) = as( as(t) Sa(t) )dt+ asdW(t)as (4.2)
where Sa(t) represents the log of the average downstream value of an ac-
tivated exchange line. The postscript/subscript a distinguishes the analogue platform from the ADSL platform and the postscript/subscript sdistinguishes
the parameters ofSa(t) from those of other processes in the analogue platform.
Now here as(t) is the level around which the process ‡uctuates, as the speed
of reversion to the mean, as the volatility of the noise term and W(t)as is a
Wiener process. The seasonal variation exhibited by Sa(t) is described as an
ordinary trigonometric function, as follows
as(t) = s+ s(t) + ssin(!st+ s) (4.3)
where s, s, s, !s and s are constants. Here s(t) captures the drift of
Sa(t) through time. Now s, s, !s and s capture the other usual parameters
of a trigonometric function. The process describing the evolution of Sa(t)
under risk-neutral expectations is given by
dSa(t) = as as(t) Sa(t)
1 as as
dt+ asdfW(t) (4.4)
where 1 is the market price of risk anddWf(t)is aQ Weiner process. The
expectation with respect to Eqn. 4.4 is
EQ[Sa(t) jzs] = (Sa(s) as(s))e as (t s)+ as(t) (4.5) where as(t) = s+ s(t) + ssin(!st+ s) 1 as as Or as(t) = as(t) 1 as as (4.6) The variance of Sa(t) is V ar[Sa(t) jzs] = 2 as 2 as(1 e 2 as(t s)) (4.7)
Now the risk-neutral dynamics of Sa(t) can be simulated based on knowl-
edge of Eqn. 4.5 and Eqn. 4.7 where the noise associated with the process, in the interval [s; t], is represented as follows
es(t)jzs = as
s
(1 e 2 as(t s))
2 as "(t) (4.8)
where "(t) is a random variable described byN(0;1): Similarly the process
F(Sa(t);t; t) can be simulated based on knowledge of its mean and variance.
Turning to model calibration, to illustrate, the parameters of Sa(t) are
calibrated using a two-step procedure.5 The parameters of the trigonometric
function s, s, s, !s and s are …rst determined through least squares esti-
mation using Matlab. These parameters are estimated such that the sum of squares i.e.
n
X
t=1
k(Sa(t) as(t))2 k (4.9)
is minimized.6 Letting a(t) = Sa(t) as(t), we observe data a = f a(t0); a(t1)::::::: a(tn)g drawn from a population where a(ti) are indepen-
dent and identically distributed. Next the parameters that de…ne mean re- version i.e. a and a are estimated using Maximum Likelihood Estimation
(MLE). We use MLE to solve for a =f a ; a g such that the likelihood of
observing the sample data is maximized. This is achieved by maximizing the following likelihood function. The log-likeliwood function, the development of which is explained more fully in Chapter 5, is as follows
5This has been done to avoid over-…tting given the relatively low number observations,
thereby giving emphasis to the overall trend. The one-step alternative approach is subse- quently discussed in Chapter 5.
L( a(t0); a(ti)::::::: a(tn); a ; a ; a ) = a 2 log 2 a 2 a 1 2 Pn i=0log(1 e 2 a (t s) a 2 a Pn i=0 ( a(ti) a ( a(ti 1) a )e (ti ti 1))2 (1 e 2 a (ti ti 1)) (4.10)
With respect to the analogue platform, based on evidence from the period September 1999 to June 2007, we describe the evolution of exchange line acti- vation as an Ito process, as follows
d a(t)
a(t)
= a dt+ a dW(t)a (4.11)
where a is the drift of the process, a its volatility and dW(t)a , is a
Wiener process. The expectation with respect to Eqn. 4.11 above is as follows
E[ a(t)]jzs] = a(s) exp ( a
1 2
2
a )(t s) (4.12)
and the variance of the process is
V ar[ a(t))jzs] = 2a (t s) (4.13)
Now Eqn. 4.12 and Eqn. 4.13 form the basis of simulating a(t):From Eqn. 4.13 the noise term of the process, in the interval [s; t], is
e (t)jzs= a p
(t s)"(t) (4.14)
where "is a random variable from a standard normal distribution. Turning to the calibration of a(t) , we observe data a =f a(t0) ; a(t1) ::::: :: a(tn) g
drawn from a population where a(ti) are independent and identically distrib-
uted. We use the Maximum Likelihood Estimation to solve for a =f a ; a g such that the likelihood of observing the sample data is maximized. The rele- vant log-likeliwood function, the development of which is explained more fully in Chapter 5, is as follows
logp( a(t0) ; a(t1):::::::::: a(tn); ap; 2a ) = 1 2 Pn n=1 2 4log[2 2a (ti ti 1)] + a(ti) a(ti 1) ( a 2 a 2 ) (ti ti 1) !2 2 a (ti ti 1) 3 5 + logp( a(t0); a ; 2a ) Pn i=0log a(ti) (4.15)
Under the assumption that the risk associated withpa(t)is incorporated in
Sa(t), knowing that a rational access seeker will align entry and exit such that
for any time interval F(Sa(t);t; t)> Ka, the value of a contingent claim on the
Sa(t) process incorporating the stochastic intensity of line activation, is
C(s; F(s; t);Ka; t) =
EQ[(S
a(t);t; t) Ka;0]+ E[ a(t)] e r(t s)
(4.16) where E[ a(t)]is the intensity of exchange line inactivation under objective measures. With respect to the ADSL platform, the SDEs describing the evo- lution of downstream value are based on observations from January 2000 to December 2008. Here too Maximum Likelihood Estimation is used to calibrate the SDEs, and the value of the underlying contingent claims are estimated us- ing risk-neutral valuation principles. And an intensity-based approach is used to value contingent claims on the process generating value. The price of market risk and the price of the risk of default are used as the handles that de…ne the martingale equivalents of the process that generates value.