140 F. D’Ippoliti et al.
Ta b le 2 .Comparison among prices of European options with spot priceS( 0 )= 4116 .40, time
to maturity 1 year, riskless rater= 3 .78% and dividend yieldd= 3 .37% on November 23,
2006
Moneyness % Strike mkt vol Closed ESA ESA MC
K σMKT formula( 3 ) (rejection) (B&K) price
90. 0 3704. 76 0. 1780 530. 43 530. 65 531. 18 529. 42
95. 0 3910. 58 0. 1660 383. 14 383. 65 383. 31 382. 38
97. 5 4013. 49 0. 1600 316. 19 316. 61 316. 76 315. 50
100. 0 4116. 40 0. 1550 255. 95 255. 98 256. 22 255. 29
102. 5 4219. 31 0. 1500 201. 77 201. 45 201. 42 201. 07
105. 0 4322. 22 0. 1450 154. 22 154. 62 153. 92 153. 43
110. 0 4528. 04 0. 1380 83. 71 83. 45 83. 32 82. 60
115. 0 4733. 86 0. 1320 40. 13 40. 03 38. 84 38. 59Valuation of barrier options and Greeks
We provide some numerical results on the valuation of discrete “up-and-out” barrier
call options whose prices are given by (12), withM =2 monitoring times, and
two different barriersH={ 5000 ; 5500 }.Tohaveasortofbenchmark,weusethe
same data as the European case. Barrier option prices are computed simulating 1200
volatility paths and 40 price paths conditional oneach variance path and jumps.
By comparing the prices of European and barrier options (H=5000) with the
same moneyness (see Tables 2 and 3), the relative change in price ranges from 24%
(moneyness 90%) to 70% (moneyness 115%). The higher the moneyness, the less
likely the option will expire with a positive payoff, either because theunderlying hits
the barrier before the maturity, knocking-down the option, or because the option is,
at expiration, out-of-the-money. This feature is also present when the barrier level
changes, as in Table 3.
Ta b le 3 .Prices of barrier options with two different barriers, spot priceS( 0 )= 4116 .40, time
to maturity 1 year, riskless rater= 3 .78% and dividend yieldd= 3 .37% on November 23,
2006
Moneyness % StrikeK mkt volσMKT H= 5000 H= 5500
90. 0 3704. 76 0. 1780 415. 0204 519. 1443
95. 0 3910. 58 0. 1660 298. 2722 374. 2468
97. 5 4013. 49 0. 1600 239. 5962 308. 4141
100. 0 4116. 40 0. 1550 189. 5186 249. 8215
102. 5 4219. 31 0. 1500 142. 6677 196. 8798
105. 0 4322. 22 0. 1450 103. 9032 149. 7763
110. 0 4528. 04 0. 1380 45. 5808 79. 8648
115. 0 4733. 86 0. 1320 12. 1330 35. 8308