Exact and approximated option pricing 141Finally, Table 4 reports delta and gamma for European and barrier options for
different strikes. The overall time required to obtain each barrier option price (Table 3)
along with its Greeks (Table 4) is about 1600 seconds.
Ta b le 4 .Simulation estimates of Greeks for European and barrier options with the following
option parameters: barrierH=5000, spot priceS( 0 )= 4116 .40, time to maturityT−t= 1
(year), riskless rater= 3 .78% and dividend yieldd= 3 .37% on November 23, 2006
Moneyness Strike Delta Delta Gamma Gamma
% K (European) (barrier) (European) (barrier)
97. 50 4013. 49 0. 61919 0. 263209 0. 00054727 0. 0006277
100. 00 4116. 40 0. 56006 0. 241053 0. 00060312 0. 0005138
102. 50 4219. 31 0. 49546 0. 210052 0. 00065031 0. 0003766
105. 00 4322. 22 0. 42850 0. 182986 0. 00068271 0. 00021497 Conclusions
An alternative stochastic volatility jump-diffusion model for option pricing is pro-
posed. To capture all empirical features of spot returns and volatility, we introduce a
jump component in both dynamics and we suppose that jumps occur concurrently. This
pricing model admits, in the spirit of Heston, a closed-form solution for European-
style options. To evaluate path-dependent options, we propose a modified version of
the numerical algorithm developed in [5, 6] whose major advantage is the lack of
discretisation bias. In particular, we replace the inversion technique proposed by the
authors with a rejection sampling procedure to improve the algorithm efficiency. We
firstly apply our methodology to price options written on the DJ Euro Stoxx 50 index,
and then we compare these prices with values obtained applying the closed-form ex-
pression, the Broadie and Kaya algorithm and a standard Monte Carlo simulation (see
Table 2). The numerical experiments confirm that prices derived with theESAmodi-
fied by the rejection sampling provide the most accurate approximation with respect
to the closed formula values. On the basis of this result, we perform the valuation of
barrier options and Greeks whose values cannot be expressed by explicit expressions.
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