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gram which uses the EM algorithm to estimate the ¬ve 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 µ denote standard
normal random variables, independently generated at every occurrence:
p
(a) Generate X0 from an arbitrary distribution. Put Y0 = ρX0 + 1 ’ ρ2 µ
(b) Repeat for n = 1, 2, ....10, 000
p
i. De¬ne Xn = ρYn’1 + 1 ’ ρ2 µ
p
ii. De¬ne Yn = ρXn + 1 ’ ρ2 µ
(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 = ±f (t)dt + σf (t)dWt
dg = ±g (t)dt + σg (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

σg (t)
πt =
σg (t) ’ σf (t)

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

dVt = Vt {πt ±f (t) + (1 ’ πt )±g (t)}dt.

Therefore
πt ±f (t) + (1 ’ πt )±g (t) = r(t)
and this implies a relationship between the drift and di¬usion terms:

±g (t) ’ r(t) ±g (t) ’ r(t)
= »(t), say,
=
σg (t) σg (t)

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

df = [r(t) + σf (t)»(t)]dt + σf (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 di¬usion coe¬cients σf (t), σg (t) do not depend on
time and »(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 σf < σg and a constant value for »(t) and compare your
estimates with the true values.

5. Assume a di¬usion model for interest rates with time-varying coe¬cient;

drt = a(rt , t)dt + σ(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{’ 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 ’ t) = ’
T ’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 a¬ne or
a linear function of the interest rate, i.e. f (r, t) = exp{c(T ’t)+d(T ’t)r}
for some functions c(.), d(.). Generally this linearity occurs provided that
both the drift term and the square of the di¬usion coe¬cient σ 2 (x, t)are
linear in r. Suggest a graphicsl method for calibrating the parameters
c(T ’ t), d(T ’ 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 .

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