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 .