This chapter explains the recursive computation algorithm of SCOs and illustrates the computing results of 3-fold SCOs.
§6.1 Computation Algorithm
The computation differences between European options and compound options (2-fold or more) lie in the EAP and the dimension of normal integrals. By definition, the EAP is the asset price which makes the (compound) option price equivalent to a specific strike price. Similar to the concept of implied volatility, the EAP can be regarded as the "implied asset price", solving by the known (compound) option price (given as the strike price) and other conventional option parameters except the asset price itself. Thus there is no EAP concern in the 1-fold option computation and it is calculated only for the 2 or more fold compound options. It seems that i-1 EAPs ( ) are calculated during the i-fold SCO price computation. However, more EAPs are calculated because they are solving by the bisection method in this study and the higher-fold EAPs are obtained based on the lower-fold EAPs. Many EAPs are figured just for another and are not used straight for the SCOs price calculation. Hence the nested algorithms, using the lower-fold SCO pricing formula for EAPs while seeking for the higher-fold one, are time-consuming.
i g S#g,i,∀1≤ <
The other computing dissimilarity between European and compound options is the normal integrals. The highest dimension of normal integrals of the SCO equals its fold number. Precise computation of the multivariate normal integration needs more work than that of a univariate case. Besides, the precise approximation of multivariate normal integrals with arbitrary dimension and integration range is neither easy nor convenient, although the univariate, bivariate and trivariate cases are disclosed explicitly (Denz, 2004). Lin (2004) compares 3 computing methods for the multivariate normal integral, including the improved Gauss quadrature method, Monte Carlo method and Lattice method, to evaluating the 4-fold SCCs.The Monte Carlo integration is applied here for normal integral computation in case the higher fold SCOs are adopted. Casimon et al. (2004) even use the SCCs up to 6 fold!
The recursive computing algorithm of the SCO price, calculating from the first fold to the last fold, include 5 looped steps and are exhibited in Figure 6.1. The computing algorithm do not encompass any estimation or calibration of parameters, which should be ready when the algorithm begin. In the flow chart, the rhombuses represent decision symbols where a decision must be made, while the rectangles symbolize the actions. The details of the chart are explained as follows.
1. EAP Existence
6. Go to Next Fold
4. Current Fold SCO Price Calculation (Monte Carlo Integration in this study)
3. The EAP Calculation (Bisection Method in this study) YES
YES
5. Last Fold SCO
End NO
NO YES
2. EAP Availability
NO SCOs Price Calculation Start
(From the First Fold)
Figure 6.1 The Nested Computing Algorithm of the SCOs
Step 1: Check the EAP existence of the current fold. If EAP exists, go to Step 2, otherwise terminate the algorithm. The EAP may not exist because of the non-negative range limitation of the decreasing SCO price. There is no need to calculate EAP for the 1-fold option because it is for compound options only.
Step 2: Check the EAP availability. If the desired EAP is available, skip to Step 4, otherwise go to Step 3. The EAP calculation is time-consuming, thus it can be used
repeatedly to save time if the same one was solved before.
Step 3: The EAP Calculation. Since the EAP is like "implied asset price", it is solved according to Theorem 3.2 (d) or Theorem 3.3 (f) to by the bisection method in this study. Within this step, it is necessary to calculate the lower-fold SCO prices, which is the main target of the computation algorithm. Hence it causes the processes to be nested and sophisticated.
Step 4: The Current Fold SCO Price Calculation. The SCO price is computed according to Equation (3.2.1) or (3.3.1) if all the EAPs are available. The cumulative probabilities of multivariate normal density are acquired by Monte Carlo integration in this study.
Step 5: Check whether the current fold is last fold. If yes, the last SCO price is the final result, otherwise go to Step 6.
Step 6: Go to the next fold. If the current fold is not the last one, go to the next fold and results so far are bases to calculate the next fold SCO price. Compared with the current fold case, the dimension and fold number are increased by one to enter the next loop.
In the SCOs evaluation algorithm, there are one recursive loop and three decision nodes. The recursive loop occurs in the EAP calculation (Step 3), while the decision nodes take place in determining whether the current fold is last (Step 5), EAP existence (Step 1) and availability (Step 2), respectively. The recursive loop involved in the bisection method together with decision nodes makes the computation sophisticated.
The numerical methods mentioned above, such as the bisection method for EAPs in Step 3 and Monte Carlo integration for multivariate normal integrals in Step 4, can be substituted by other suitable methods. The conventional options just need the "Step 4" to calculate the option price straightforwardly. By contrast, the looped and nested computation algorithm of SCO prices, involving some numerical techniques, are more complicated.
§6.2 Three-Fold SCOs Illustration
This subsection illustrates the 8 cases of 3-fold sequential compound options, including the call on call on call (CCC), call on call on put (CCP), call on put on call (CPC), call on put on put (CPP), put on call on call (PCC), put on call on put (PCP), put on put on call (PPC), put on put on put (PPP). The parameters of these SCOs are identical for comparison in the numerical examples. The time to maturity of 3 folds are all equal to one, and the strikes K1, K2, K3 are 10, 100, 500 respectively. Assume the volatility and dividend rate keeps constant in these three folds. Figure 6.2 exhibits the SCOs price along the volatility and asset price.
0 Price Surface of the 3-fold SCO: call on call on call
Stock Price
0 Stock Price
Option Price
riceOption P
(a) Call on call on call (b) Call on call on put Stock Price
Option Price
0 Stock Price
Option Price
(c) Call on put on call (d) Call on put on put Stock Price
Option Price
(h) Put on put on put Stock Price
Option Price
(e) Put on call on call Stock Price
Option Price
0 Stock Price
Option Price
(f) Put on call on put
Figure 6.2: The Price Surface of 3-fold SCOs
The price surface of PCC is presented in Table 6.1 and Figure 6.2 (e). The follows explain the PCC to understand the feature of SCOs. The max price of PCC is about 9.51 because the last fold put option strikes with 10. The PCC price drops as the stock price hikes under the same volatility due to the put feature of the underlying asset. Although with different underlying assets, the PPP also has a similar phenomenon due to the same reason. This reason also supports the fact that the PCC price descends with the volatility (sigma) increasing under the same stock price.
Theoretically, the SCOs are monotone with respect to the asset prices (Thomassen and Van Wouwe, 2002; Lee et al., 2007). However, the integrals evaluated by Monte Carlo simulation result in subtle non-monotonicity.
Table 6.1: Prices of the 3-fold SCO (Put on Call on Put) Volatility of Asset Price
SCOs
Price 0.05 0.10 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
1 9.51 9.51 9.51 9.51 9.51 9.51 9.51 9.51 9.51 9.51 9.51 9.51 9.51 9.51 9.51 9.51 9.51 9.51 9.51 9.51 26 9.51 9.51 9.51 9.51 9.51 9.51 9.51 9.51 9.51 9.51 9.51 9.50 9.49 9.46 9.41 9.34 9.24 9.13 9.00 8.86 51 9.51 9.51 9.51 9.51 9.51 9.51 9.51 9.51 9.51 9.49 9.45 9.36 9.23 9.05 8.85 8.62 8.38 8.12 7.88 7.65 76 9.51 9.51 9.51 9.51 9.51 9.51 9.51 9.50 9.46 9.35 9.18 8.93 8.62 8.29 7.98 7.65 7.33 7.02 6.76 6.51 101 9.51 9.51 9.51 9.51 9.51 9.51 9.49 9.43 9.27 9.01 8.66 8.23 7.79 7.37 7.02 6.65 6.32 6.02 5.78 5.55 126 9.51 9.51 9.51 9.51 9.51 9.50 9.42 9.23 8.90 8.44 7.94 7.38 6.87 6.43 6.09 5.74 5.43 5.16 4.95 4.75 151 9.51 9.51 9.51 9.51 9.51 9.44 9.23 8.85 8.32 7.71 7.10 6.48 5.97 5.55 5.25 4.94 4.67 4.43 4.26 4.10 176 9.51 9.51 9.51 9.51 9.48 9.30 8.85 8.28 7.59 6.88 6.24 5.62 5.14 4.76 4.51 4.25 4.02 3.83 3.69 3.56 201 9.51 9.51 9.51 9.50 9.38 9.00 8.28 7.56 6.78 6.03 5.41 4.82 4.40 4.08 3.88 3.66 3.48 3.32 3.21 3.11 226 9.51 9.51 9.51 9.46 9.17 8.52 7.55 6.74 5.94 5.21 4.65 4.11 3.76 3.49 3.34 3.17 3.02 2.89 2.81 2.74 251 9.51 9.51 9.51 9.35 8.78 7.86 6.70 5.89 5.13 4.45 3.96 3.48 3.20 2.99 2.88 2.75 2.63 2.53 2.47 2.42 276 9.51 9.51 9.48 9.09 8.19 7.05 5.81 5.06 4.38 3.78 3.36 2.95 2.73 2.56 2.49 2.39 2.30 2.22 2.19 2.15 301 9.51 9.51 9.39 8.61 7.40 6.16 4.93 4.29 3.70 3.18 2.84 2.49 2.33 2.20 2.16 2.08 2.02 1.96 1.94 1.92 326 9.51 9.50 9.14 7.88 6.48 5.25 4.10 3.59 3.10 2.67 2.40 2.11 1.99 1.90 1.88 1.83 1.78 1.74 1.73 1.72 351 9.51 9.46 8.64 6.93 5.49 4.37 3.37 2.97 2.59 2.24 2.03 1.78 1.70 1.64 1.64 1.60 1.57 1.55 1.55 1.54 376 9.51 9.28 7.82 5.82 4.52 3.57 2.72 2.44 2.15 1.87 1.71 1.51 1.46 1.42 1.44 1.41 1.39 1.38 1.39 1.39 401 9.51 8.74 6.68 4.69 3.62 2.87 2.18 2.00 1.78 1.56 1.44 1.28 1.26 1.23 1.26 1.25 1.24 1.23 1.25 1.26 426 9.49 7.61 5.36 3.62 2.82 2.27 1.73 1.62 1.47 1.30 1.22 1.09 1.08 1.07 1.11 1.11 1.11 1.11 1.13 1.15 451 9.20 5.91 4.03 2.68 2.15 1.77 1.36 1.32 1.21 1.08 1.03 0.92 0.93 0.93 0.98 0.98 0.99 1.00 1.03 1.05 476 7.67 3.99 2.83 1.92 1.61 1.37 1.07 1.06 1.00 0.90 0.87 0.79 0.81 0.82 0.86 0.88 0.89 0.90 0.93 0.96 501 4.37 2.32 1.87 1.33 1.18 1.05 0.83 0.86 0.82 0.75 0.74 0.67 0.70 0.71 0.77 0.78 0.80 0.82 0.85 0.87 526 1.41 1.16 1.17 0.89 0.86 0.79 0.65 0.69 0.68 0.63 0.63 0.57 0.61 0.63 0.68 0.70 0.72 0.74 0.78 0.80 551 0.24 0.50 0.69 0.59 0.61 0.60 0.50 0.55 0.56 0.53 0.53 0.49 0.53 0.55 0.61 0.63 0.65 0.67 0.71 0.74 576 0.02 0.19 0.39 0.38 0.43 0.45 0.39 0.44 0.46 0.44 0.45 0.42 0.46 0.49 0.54 0.57 0.59 0.62 0.65 0.68 601 0.00 0.07 0.21 0.24 0.30 0.33 0.30 0.36 0.38 0.37 0.39 0.36 0.40 0.43 0.48 0.51 0.54 0.56 0.60 0.63 626 0.00 0.02 0.11 0.15 0.21 0.25 0.23 0.29 0.31 0.31 0.33 0.31 0.35 0.38 0.43 0.46 0.49 0.51 0.55 0.58 651 0.00 0.01 0.05 0.09 0.14 0.18 0.18 0.23 0.26 0.26 0.28 0.27 0.31 0.34 0.39 0.42 0.45 0.47 0.51 0.54 676 0.00 0.00 0.03 0.05 0.10 0.14 0.14 0.19 0.21 0.22 0.24 0.23 0.27 0.30 0.35 0.38 0.41 0.43 0.47 0.50 701 0.00 0.00 0.01 0.03 0.07 0.10 0.10 0.15 0.18 0.18 0.21 0.20 0.24 0.27 0.32 0.35 0.37 0.40 0.44 0.47 726 0.00 0.00 0.01 0.02 0.05 0.07 0.08 0.12 0.15 0.16 0.18 0.17 0.21 0.24 0.29 0.31 0.34 0.37 0.40 0.43 751 0.00 0.00 0.00 0.01 0.03 0.05 0.06 0.10 0.12 0.13 0.15 0.15 0.19 0.22 0.26 0.29 0.31 0.34 0.37 0.40 776 0.00 0.00 0.00 0.01 0.02 0.04 0.05 0.08 0.10 0.11 0.13 0.13 0.17 0.19 0.23 0.26 0.29 0.31 0.35 0.38 801 0.00 0.00 0.00 0.00 0.01 0.03 0.04 0.06 0.08 0.09 0.11 0.11 0.15 0.17 0.21 0.24 0.27 0.29 0.32 0.35 826 0.00 0.00 0.00 0.00 0.01 0.02 0.03 0.05 0.07 0.08 0.10 0.10 0.13 0.16 0.19 0.22 0.25 0.27 0.30 0.33 851 0.00 0.00 0.00 0.00 0.01 0.02 0.02 0.04 0.06 0.07 0.09 0.09 0.12 0.14 0.18 0.20 0.23 0.25 0.28 0.31 876 0.00 0.00 0.00 0.00 0.00 0.01 0.02 0.03 0.05 0.06 0.07 0.08 0.11 0.13 0.16 0.19 0.21 0.23 0.26 0.29 901 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.03 0.04 0.05 0.07 0.07 0.09 0.11 0.15 0.17 0.19 0.22 0.25 0.27 926 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.02 0.03 0.04 0.06 0.06 0.08 0.10 0.13 0.16 0.18 0.20 0.23 0.26 951 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.02 0.03 0.04 0.05 0.05 0.08 0.09 0.12 0.14 0.17 0.19 0.22 0.24
Asset Price
976 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.02 0.03 0.04 0.05 0.07 0.09 0.11 0.13 0.15 0.18 0.20 0.23