Abstract:
Interest Rate Derivatives (IRDs) are generally jump-diffusion processes which
are usually modelled with L´evy processes. Brownian motion has been used ex tensively for modelling IRDs, however, this does not capture the jumps inherent
in the IRDs. To hedge risks in a L´evy market, it is important to consider the
presence of jumps. This work was therefore designed to model IRDs driven by
some subordinated L´evy processes that consider jumps.
The classical Vasicek short rate model drt = a(b − rt)dt + σdWt (where rt
,
a, b, σ and Wt denote interest rate, speed of mean-reversion, long-term mean
rate, volatility of the short rate model and Brownian motion, respectively) was
extended to a model driven by subordinated L´evy processes using Itˆo formula
for semimartingales. Using the extended Vasicek model, expressions for the
price of IRDs: zero-coupon bond, with Variance Gamma (VG) and Normal In verse Gaussian (NIG) as the underlying sources of uncertainties, were derived.
Expressions for the greeks were derived by means of Skorohod integral, Ornstein Uhlenbeck operator and the Malliavin calculus. Consequently, the greeks ob tained were used to determine the sensitivities of the parameters of the model.
Monthly dataset of the Nigerian Interbank Offer Rate from 2007 to 2017 was
obtained from the Central Bank of Nigeria website and used to validate the
model.
The greek expressions that measure the price sensitivities to interest rate, namely,
the delta 4V G associated with the VG process and the delta 4NIG associated
with the NIG process were obtained as
4V G = e
−r0T
−TE[Φ(P)]+E
Φ(P)
σ
2
a
(e
−aT −e
−at)K
−2
