Efficient Pricing of Asian and Asian Barrier Options

Path-dependent derivatives have payoffs that depend strongly on the price history of the underlying asset. In pricing such derivatives, the historical information needs to be encoded as part of the state. Asian options are strongly path-dependent derivatives. Although efficient numerical methods and approximate closed-form formulas are available, most lack convergence guarantees.

Asian options can be priced on the lattice. Let the time to maturity be partitioned into $n$ periods. The best exact convergent lattice algorithm runs in 2^{O(sqrt{n},)} time. (An exact algorithm is one that does not employ approximations beyond the discretization of the continuous-time model.) All efficient lattice algorithms that are not exact keep only a polynomial number of states and use interpolation to compensate for the less than full representation of the states. Furthermore, we have extended our results in pricing discretely monitored Asian options. This is particularly important because majority of Asian options traded are discretely monitored. This work presents the first $O(n^2)$-time convergent lattice algorithm for pricing European-style Asian options; it is the most efficient lattice algorithm with convergence guarantees. The algorithm relies on the Lagrange multipliers to choose optimally the number of states for each node of the lattice. The algorithm is also memory efficient.

Another important type of path-dependent derivative is the barrier options. The price history decides whether a specified event has been triggered or not, and it influences the final payoff. We have combined the barrier feature into Asian options, and provided an extension of the $O(n^2)$-time convergent lattice algorithm to price this difficult option.

Extensive numerical experiments and comparison with existing PDE, analytical, and lattice methods confirm the performance claims and the competitiveness of our algorithm. Theses results place the problem of European-style Asian option and Asian barrier option pricing in the same complexity class as that of the vanilla option on the lattice.