Pricing Asian Options with the Binomial Model

1. Da-Yoon Chung Daniel Lu Pricing Asian Options with the Binomial Model

2. Options • An option is a contract that can be bought or sold, its value is a function of the value of the underlying stock • An Asian option is an option whose terminal value is based on the average prices of stock at certain points in time.

3. Pricing Options • RecombinantBinomial Tree • Problem with Asian Options: • Non-recombinant (2^N) paths we have to consider

5. (Serial) Algorithm Step 1: generate the average price tree • for each (i,j), store 2*N out of iCj possible values of the running average up to that point • Requires 2*N random paths of length O(N) consisting of up or down for each (i,j) (O(N^3) storage overhead) • Step 2: generate the option prices tree • Each level of the tree depends only on the next • use backwards inductive approach starting at the leaves of the tree • use linear interpolation to find an estimate of the option price at each node

6. CUDA Algorithm (global memory) Step 1: generate the average price tree • Compute all random paths required using Thrust • For each (i,j), the 2*N average values are computed in parallel (one thread per node) • Write all updates immediately to the global tree • Step 2: generate the option price tree • Compute each level of the tree in parallel • Write all updates immediately to the global tree

7. CUDA Algorithm (shared memory) Step 1: generate the average price tree • Again, for each (i,j), the 2*N average values are computed in parallel (one thread per node) • Store all intermediate values in shared memory to minimize global memory accesses • Use a hash function to generate the random path within the kernel (reduce memory overhead) Step 2: generate the option price tree • Divide the tree into subtreesat the same depth in the original tree which can be computed independently • Compute one level of subtrees per kernel call • Store all computations for subtrees in shared memory

9. Limitations • Size of shared memory • N = 64, Tree occupies N * N * (2 * N) * sizeof(float) = 2^6 * 2^6 * (2 * 2^6) * 4 = 2^21 = 2m (48k shared memory) • Increasingly sequential as N increases • Nature of the algorithm • Step 2 of the algorithm is inherently sequential

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12. Results