134 F. D’Ippoliti et al.
jumps that contains two diffusion terms: the first has constant volatility, as in the Black
and Scholes (B&S) model [4], while the latter is of the Heston type. The dynamics
of the volatility follow a square-root process with jumps. We suppose that the arrival
times of both jumps are concurrent, hence we will refer to our model as a stochastic
volatility with contemporaneous jumps (SVCJ) model. We claim that two diffusion
terms in the dynamics of spot returns make our model more flexible than the Heston
one.
Valuation of non-European options usually requires numerical techniques; in most
cases some kind of discretisation is necessary so that a pricing bias is present. To avoid
this flaw, we opt for the “exact simulation” approach developed by Broadie and Kaya
(B&K) [5, 6] for stochastic volatility and other affine jump-diffusion models. This
method is based on both a Fourier inversion technique and some conditioning argu-
ments so to simulate the evolution of the value and the variance of an underlying
asset. Unlike B&K’s algorithm, to determine the integral of the variance, we replace
the inverse transform method with a rejection sampling technique. We then compare
the results of the closed-form expressionfor European-style option prices with their
approximated counterparts using data from the DJ Euro Stoxx 50 derivative market.
Having found that the modified algorithm returns reliable values, we determine prices
and Greeks for barrier options for which no explicit formula exists.
2 Stochastic volatility jump-diffusion model
Let( ,F,Q)be a complete probability space whereQis a risk-neutral probability
measure and considert∈[0,T]. We suppose that a bidimensional standard Wiener
processW=(W 1 ,W 2 )and two compound Poisson processesZSandZvare defined.
We assume thatW 1 ,W 2 ,ZSandZvare mutually independent. We suppose that
dS(t)=S(t−)[
(r−λjS)dt+σSdW 1 (t)+ξ√
v(t−)dW 2 (t)+dZS(t)]
, (1)
dv(t)=k∗(θ∗−v(t−))dt+σv√
v(t−)dW 2 (t)+dZv(t), (2)whereS(t)is the underlying asset,
√
v(t)is the volatility process, and parameters
r,σS,ξ,k∗,θ∗andσvare real constants (ris the riskless rate). The processesZS(t)
andZv(t)have the same constant intensityλ>0 (annual frequency of jumps). The
processZS(t)has log-normal distribution of jump sizes; ifJSis the relative jump size,
then log( 1 +JS)is distributed according to theN
(
log( 1 +jS)−^12 δ^2 S,δ^2 S)
law, where
jSis the unconditional mean ofJS. The processZv(t)has an exponential distribution
of jump sizesJv>0 with meanjv. Note thatJS∈(− 1 ,+∞)implies that the stock
price remains positive for allt∈[0,T]. The variancev(t)is a mean reverting process
with jumps wherek∗,θ∗andσvare, respectively, the speed of adjustment, the long-
run mean and the variation coefficient. Ifk∗,θ∗,σv>0, 2k∗θ∗≥σv^2 ,v( 0 )≥0and
Jv>0, then the processv(t)is positive for allt∈[0,T] with probability 1 (see [12]
in the no-jump case) and captures the large positive outliers in volatility documented
in [3]. Jumps in both asset price and variance occur concurrently according to the