For European-style Asian basket options, we apply our approach to options with different basket compositions, and then compare the option prices obtained by our tree approach with those obtained by Monte Carlo simulation. However, for American-style Asian basket options, there are few methods except least-squares Monte Carlo for pricing the American-style Asian basket options. Therefore, there are no publicly available benchmarks.
Consider the following baskets for the Asian basket call and put options:
Basket 1: F0
>
50;50@
; V>
0.3; 0.2@
; U1,2 0.6; a>
0.3; 0.7@
; X 50.Basket 2: F0
>
100;120@
; V>
0.2; 0.3@
; U1,2 0.9; a>
1;1@
; X 20.Basket 3: F0
>
150;100@
; V>
0.3; 0.2@
; U1,2 0.7; a>
1;1@
; X .50Basket 4: F0
>
95;90;105@
; V>
0.2; 0.3; 0.25@
; U1,2 U1,3 0.9 ; U2,3 0.8 ;>
1; 0.8; 0.5@
a ; X .30
Basket 5: F0
>
100;90;95@
; V>
0.25; 0.3; 0.2@
; U1,2 U1,3 0.9 ; U2,3 0.8 ;>
0.6; 0.8; 1@
a ; X 40.
We assume the risk-free interest rate is 5% per annum and the time to expiry ( T ) is 1 year. Furthermore, the number of time steps for the tree is 100 and the bucket size k is 300 . For Monte Carlo simulation, the number of simulation runs is 100000 .
Table 1: Parameters of GLN obtained by moment matching.
Table 1 shows the parameters of GLN obtained by moment matching. W may be positive or negative, P* must be zero and V* may be higher or lower than the volatilities of the assets in the basket.
The European-style Asian basket call and put option prices obtained by our tree approach are given in column 3 and column 5 in Table 2. Column 4 and column 6 in Table 2 are the corresponding option prices obtained by Monte Carlo simulation; and the standard errors of the option prices are given in parenthesis after the option prices by Monte Carlo simulation.
Basket Approx.
distribution
W P* V*
Basket 1 Shifted 1.0668 0 0.2115
Basket 2 Shifted 35.7071 0 0.3602
Basket 3 Neg. shifted 59.0260 0 0.3141
Basket 4 Neg. shifted 31.9634 0 0.3149
Basket 5 Shifted 30.1925 0 0.3133
Table 2: European-style Asian basket option prices
Table 2 shows that the prices obtained by our tree approach are within the 95%
confidence intervals of those obtained by Monte Carlo simulation for both positive and negative skewness, with both positive and negative weights. Our approach is attractive for two reasons. First, the time complexity of Hull-White methodology is
( 2)
O kn . Therefore, when n is small, the CPU time of our approach is less than Monte Carlo simulation with 100000 runs. Second, Monte Carlo simulation can only generate the paths of the baskets with a positive definite correlation matrix. In contrast, our approach can price not only options with a positive definite correlation matrix for the assets but also options with a non-positive definite correlation matrix, which cannot be processed by Monte Carlo simulation without further adjustments.
For American-style Asian basket options, Monte Carlo simulation is not Basket Approx.
distribution
European-style call option European-style put option Our
Basket 1 Shifted 2.2666 2.2532
(0.0119)
2.2666 2.2581 (0.0101)
Basket 2 Shifted 4.3862 4.3509
(0.0247)
4.3862 4.3470 (0.0184) Basket 3 Neg. shifted 7.4898 7.4286
(0.0321)
7.4898 7.4369 (0.0416) Basket 4 Neg. shifted 4.4959 4.4615
(0.0187)
4.0203 3.9989 (0.0229)
Basket 5 Shifted 3.4354 3.4105
(0.0223)
6.2891 6.2490 (0.0229)
appropriate because it cannot handle early exercise. Therefore, we compare our approach with the least-squares Monte Carlo (LSMC) method (Longstaff and Schwartz, 2001). There are two methods for LSMC. First, we approximate the basket by GLN, then use a single asset to represent the whole basket. After that, we use LSMC on the single asset to obtain the option prices (we call it LSMC-1). Or, we can run LSMC directly on the basket paths generated by the correlated geometric Brownian motion to obtain the option prices (given in LSMC-2). LSMC-2 should serve as the benchmark theoretically.
Table 3: American-style Asian basket option prices
Table 3 shows the prices of American-style Asian basket options with the same baskets as before. It implies GLN approximation is suitable to handle the early
Basket Approx . distribu
tion
American-style call option American-style put option Our
exercise feature because the option prices obtained by LSMC-1 are close to those obtained by LSMC-2. We find that the American-style option prices obtained by LSMC-1 and LSMC-2 in Table 3 are also close to the European-style ones by Monte Carlo simulation in Table 2 mainly because less than 3% paths in LSMC-2 and no path in LSMC-1 are early exercised. We observe that the option prices obtained by our tree approach are too high compared to both LSMC-1 and LSMC-2. In fact, nearly 5% of the states are early exercised in the tree, and they are mainly located in the middle of the tree. But, we recall, no path is early exercised in LSMC-1 and less than 3% paths are early exercised in LSMC-2. Therefore, we conjecture that it is the Hull-White methodology of our tree approach that results in overpricing, not the approximation by GLN.
We use the Asian basket call option for Basket 2 to analyze the convergence of the tree with the Hull-White methodology. When we choose a larger bucket size k per node to obtain more precise option prices, the option prices would converge to the prices obtained by Monte Carlo simulation for European-style Asian basket options (see Fig.4). However, when we do the same comparison for American-style Asian basket options with LSMC-2, the option prices obtained by our tree approach would overprice no matter how large bucket size we choose (see Fig. 5).
Fig. 4: Convergence of European-style Asian Basket Option Price with k .
Fig. 5: Convergence of American-style Asian Basket Option Price with k .