r−q)2= 0.9768d1 = e 0pm = 1 − 4   = 0.150422σ̂1 + (r − q)p0,d = 1 − p0,u − pm = 0.4219p1,d = 1 − p1,u = 0.503248During the period of analysishttp://www.londonstockexchange.com.
the
dividend
yield
was
nearly
constant.
For
details,
see
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Given u1 , u0 , d0 , d1 , the possible one-step-ahead values for the FTSE100 index can becalculated as follows:→ u1 St = 6741.01→ u0 St = 6662.45St = 6584.82St+1 →→ d0 St = 6508.11→ d1 St = 6432.25and the tree is recursively generated.The payoffs at maturity are calculated with the usual formula:CT = max(ST − K, 0)and the values of the option in the previous periods are obtained operating backwards.In particular:• At time T − 1, the conditional values of the option at the ith node, CTi −1 (sT −1 = 0),CTi −1 (sT −1 = 1), are given by iCT −1 (sT −1 = 0) = [(1 − p0,u − pm )CTi−1 + pm CTi + p0,u CTi+1 ]e−rCTi −1 (sT −1 = 1) = [(1 − p1,u )CTi−2 + p1,u CTi+2 ]e−rNote that the nodes ±1 consider the option values obtained with u0 and d0 , while thenodes ±2 consider u1 and d1 . The calculation is repeated for all the nodes at time T − 1and we obtain two sets of conditional option values CTi −1 (sT −1 = 0), CTi −1 (sT −1 = 1).• At time T − 2– for each node i, the values of the option are obtained conditional on the regime inT − 1: i
i+1iCT −2/T −1 (sT −1 = 0) = [(1 − p0,u − pm )CTi−1−1 (sT −1 = 0) + pm CT −1 (sT −1 = 0) + p0,u CT −1 (sT −1 = 0)]i−1iiCT −2/T −1 (sT −1 = 1) = [(1 − p0,u − pm )CT −1 (sT −1 = 1) + pm CT −1 (sT −1 = 1) + p0,u CTi+1−1 (sT −1 = 1)]
– using the estimated transition probabilities (p̂00 = 0.992889, p̂11 = 0.963762), theyare then discounted considering the possibility that a switch occurs between timeT − 1 and T − 2: iCT −2 (sT −2 = 0) = [p̂00 CTi −2/T −1 (sT −1 = 0) + (1 − p̂00 )CTi −2/T −1 (sT −1 = 1)]e−rCTi −2 (sT −2 = 1) = [(1 − p̂11 )CTi −2/T −1 (sT −1 = 0) + p̂11 CTi −2/T −1 (sT −1 = 1)]e−rAgain we obtain two sets of conditional option values CTi −2 (sT −2 = 0), CTi −2 (sT −2 = 1).• This computation is iterated for T − 3, T − 4, . . . , t.At the evaluation time t, we obtain two conditional values Ct (st = 0), Ct (st = 1), whichare respectively 1047.61 and 1093.54. Finally, the value of the option is calculated as a
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weighted average of these two values, where the weights depend on the knowledge ofthe current regime. If the regime is unknown, the estimated unconditional probabilitiesp̂0 =
1 − p̂112 − p̂00 − p̂11
p̂1 =
1 − p̂002 − p̂00 − p̂11
can be used.The pentanomial option value is therefore 1055.14 while the market value was 1047.5.Another possibility, probably better from a methodological point of view as it uses allthe available information, is to consider the ﬁltered probabilities obtained as output of theestimation step, i.e. P (st = 0|Y t ) = 0.98636043 and P (st = 1|Y t ) = 0.01363957. In thatcase, the pentanomial option value is 1048.24.As an exercise, readers may wish to replicate the following examples:1. European put option quoted on 23/10/2000, strike price: 7000, FTSE100: 6315.9, Libor:6.1475%, days to maturity: 39, option price: 700.Results: Conditional option values: 756.15 and 679.89. Option value consideringunconditional probabilities: 692.4.2. European put option quoted on 03/11/2000, strike price: 5500, FTSE100: 6385.44,Libor: 6.12625%, days to maturity: 160, option price: 117.Results: Conditional option values: 143.29 and 101.08. Option value consideringunconditional probabilities: 108.
APPENDIX B: APPLICATION TO VALUE AT RISKConsider a portfolio which perfectly replicates the composition of the FTSE100 index.Given the estimated volatility of the stochastic volatility models, the VaR of this portfoliocan be obtained following the procedure proposed in Barone-Adesi et al. (1998).The historical portfolio returns are rescaled by the estimated volatility series to obtainthe standardised residuals ut = yt /σt , t = 1, . . . , T (in our case T = 898, see footnote 33).The historical simulation can be performed by bootstrapping the standardised returns toobtain the desired number of residuals u∗j , j = 1, . . . , M, where M can be arbitrarily large.To calculate the next period returns, it is sufﬁcient to multiply the simulated residuals bythe forecasted volatility σ̂T +1/T :yj∗ = u∗j σ̂T +1/TThe VaR for the next day, at the desired level of conﬁdence h, is then calculated as theMhth element of these returns sorted in ascending order.
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