<< стр. 3(всего 3)СОДЕРЖАНИЕ
gram which uses the EM algorithm to estimate the п¬Ѓve parameters in this
model and test it on simulated data for sample sizes n = 100, 1000, 2000.
Comment on bias and variance of the estimators.

3. Consider the following algorithm: the random variables Оµ denote standard
normal random variables, independently generated at every occurrence:
p
(a) Generate X0 from an arbitrary distribution. Put Y0 = ПЃX0 + 1 в€’ ПЃ2 Оµ
(b) Repeat for n = 1, 2, ....10, 000
p
i. Deп¬Ѓne Xn = ПЃYnв€’1 + 1 в€’ ПЃ2 Оµ
p
ii. Deп¬Ѓne Yn = ПЃXn + 1 в€’ ПЃ2 Оµ
(c) Output (Xj , Yj ), j = 5000, ...10000
Plot the points (Xj , Yj ), j = 5000, ...10000 and explain what this
algorithm is designed to provide.

4. Suppose two interest rate derivatives (F and G) have price processes de-
pending on the spot interest rate r(t)

exp{f (t, r(t))) and exp{g(t, r(t)} respectively,
392 ESTIMATION AND CALIBRATION

where, under the risk-neutral distribution,

df = О±f (t)dt + Пѓf (t)dWt
dg = О±g (t)dt + Пѓg (t)dWt .

In this case both derivatives are driven by the same Brownian motion
process Wt which drives the interest rates r(t). Show that if we maintain
a proportion of our investment

Пѓg (t)
ПЂt =
Пѓg (t) в€’ Пѓf (t)

(a negative value corresponds to short selling) in derivative F and the
remainder 1 в€’ ПЂt in derivative G, then our investment is risk-free and has
value Vt satisfying

dVt = Vt {ПЂt О±f (t) + (1 в€’ ПЂt )О±g (t)}dt.

Therefore
ПЂt О±f (t) + (1 в€’ ПЂt )О±g (t) = r(t)
and this implies a relationship between the drift and diп¬Ђusion terms:

О±g (t) в€’ r(t) О±g (t) в€’ r(t)
= О»(t), say,
=
Пѓg (t) Пѓg (t)

with О»(t) independent of the particular derivative. In other words all
interest rate derivatives can be expressed in the form

df = [r(t) + Пѓf (t)О»(t)]dt + Пѓf (t)dWt .

Assume you have available observations on the price of two interest rate
derivatives taken daily over a period of 100 days as well as current interest
rates r(t). Assume the diп¬Ђusion coeп¬ѓcients Пѓf (t), Пѓg (t) do not depend on
time and О»(t) is a linear function of t. Explain how you would calibrate
the parameters in this model to market data. Simulate data using con-
stant values for Пѓf < Пѓg and a constant value for О»(t) and compare your
estimates with the true values.

5. Assume a diп¬Ђusion model for interest rates with time-varying coeп¬ѓcient;

drt = a(rt , t)dt + Пѓ(rt , t)dWt .

Consider a 0-coupon bond which, if invested today at time t returns 1
dollar at time T . If the current short rate is rt , the value of this bond can
be written as a function
ZT
f (rt , t) = EQ [exp{в€’ rs ds}]
t
7.8. PROBLEMS 393

where EQ denotes expectation under the risk-neutral measure. The yield
curve describes the current expectations for average interest rates;

log(f (rt , t))
Yield(T в€’ t) = в€’
T в€’t
The more common models such as the Vasicek, the CIR and the Merton
models for interest rate structure are such that the yield curve is aп¬ѓne or
a linear function of the interest rate, i.e. f (r, t) = exp{c(T в€’t)+d(T в€’t)r}
for some functions c(.), d(.). Generally this linearity occurs provided that
both the drift term and the square of the diп¬Ђusion coeп¬ѓcient Пѓ 2 (x, t)are
linear in r. Suggest a graphicsl method for calibrating the parameters
c(T в€’ t), d(T в€’ t) to market data if we are provided with the prices of zero
coupon bonds with a variety of maturities T at a number of time points
t1 < t2 < ...tn .

 << стр. 3(всего 3)СОДЕРЖАНИЕ