Da-Yoon Chung
Daniel Lu
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.

Pricing Options

Recombinant
Binomial Tree

Problem with Asian Options:
 Non-recombinant (2^N) paths we have to
consider
(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
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
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 subtrees at 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
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
Results
performance of Step 1
20
18
16
14
Speedup
12
10
8
6
Speedup (Shared to serial)
4
Speedup(Global to Serial)
2
0
0
50
100
150
Depth of Tree (N)
200
250
Results
performance of Step 2
0.35
0.3
Speedup
0.25
0.2
0.15
0.1
Speedup (Global to Serial)
0.05
Speedup (Shared to Serial)
0
0
20
40
60
Depth of Tree (N)
80
100
120
Results
performance of hybrid (CUDA shared step 1 + serial step 2)
4.5
4
3.5
Speedup
3
2.5
2
1.5
1
0.5
0
0
50
100
150
Depth of Tree (N)
200
250