Chapter 5: Models and Equations 281
where
k=E[J−1], λ′=λ(1+k), σn^2 =σ^2 +
nσ′^2
T−t
and
rn=r−λk+
nln(1+k)
T−t
,
andVBSis the Black–Scholes formula for the option
value in the absence of jumps.
Fixed Income
In the following we use the continuously compounded
interest convention. So that one dollar put in the bank
at a constant rate of interestrwould grow exponen-
tially,ert. This is the convention used outside the fixed-
income world. In the fixed-income world where interest
is paid discretely, the convention is that money grows
according to
(
1 +r′τ
)n
,
wherenis the number of interest payments,τis the
time interval between payments (here assumed con-
stant) andr′is the annualized interest rate.
To convert from discrete to continuous use
r=
1
τ
ln(1+r′τ).
The yield to maturity (YTM) or internal rate of return (IRR)
Suppose that we have a zero-coupon bond maturing
at timeTwhen it pays one dollar. At timetit has a
valueZ(t;T). Applying a constant rate of return ofy
betweentandT, then one dollar received at timeThas
apresentvalueofZ(t;T)attimet,where
Z(t;T)=e−y(T−t).
