Exact and approximated option pricing 1396 Numerical results
In this section, we apply our model to the DJ Euro Stoxx 50 market (data provided by
Banca IMI, Milan), using the set of parameters reported in Table 1. These parameters
have been chosen in order to test the efficiency of our algorithm and obtain a good
approximation of market volatilities. Model calibration is beyond the scope of this
paper and is left for further research.
Ta b le 1 .Values of parameters of the models (1) and (2)θ∗ k∗ σS σv ξλjS δS jv
0. 175 0. 25 0. 08 0. 2 − 0. 40. 05 0. 025 0. 02 0. 03The Dow Jones Euro Stoxx 50 (DJ50) ‘blue-chip’ index covers the fifty EuroZone
largest sector leaders whose stocks belong to the Dow Jones Euro Stoxx Index. DJ50’s
option market is very liquid and ranges widely in both maturities (from one month to
ten years) and strike prices (moneyness from 90% up to 115%). It is worth noting that
indexes carry dividends paid by companies so that a dividend yielddhas to be properly
considered by subtracting it from the drift term in the dynamics ofS. Volatilities in
Table 2 (column 2) represent the term
√
σS^2 +ξ^2 v( 0 )(the instantaneous variance of
spot return att=0, and not simplyσSas in the B&S model), wherev( 0 )is the initial
value of the stochastic volatility dynamics. It follows that we can obtainv( 0 )from
v( 0 )=
(
σMKT^2 −σS^2)
/ξ^2 ,whereσMKTis the market volatility.Exact versus approximated pricing
We present some numerical comparisons of theESAdescribed in Section 4 and other
simulation methods. For this purpose, we use European call options on November 23,
2006; relevant data are shown in Table 2. We compare prices derived with different
methods: the closed formula (3) (column 4), theESAmodified with the rejection
sampling (column 5), theESAproposed in [5, 6] (column 6), and a Monte Carlo
estimator (see (5) in [6]) (column 7). For theESAs, we simulate 100,000 variance
paths and 1000 price paths conditional oneach variance path and jumps. Prices in
column 5 are very similar to those obtained with the closed formula (3) and improve
the approximation obtained using the originalESA. Our results also confirm thatESA
is more efficient than a standard Monte Carlo approach, as stated in [5, 6]. The time
needed to obtain each price with theESAin column 5 is about 545 seconds with a
FORTRAN code running on an AMD Athlon MP 2800+, 2.25 GHz processor. This
computational time is shorter than that reported in [5, 6] for a comparable number of
simulations.
This is an encouraging result for pricing options that do not have closed-form
formulæ such as barrier options and Greeks.