Efficient Path-Dependent Valuation Using Lattices: Fixed and Floating Strike Asian Options

A lattice-based method is advanced for evaluating functionals of sequences of path-wise values of a lattice's state variable. For the Asian call valuations in this paper, the lattices discretely replicate the stochastic future states of conventionally prescribed, lognormally distributed, equity values. The Asian call valuations have the same level of precision as do valuations arising from numerical solutions based on the derivatives' governing partial differential equations or from high-confidence Monte Carlo, but are accomplished without the significant computer time and sophisticated software which attend those calculations. The method is termed SCEV induction, for "State Conditional Expected Value." By rolling forward through the lattice, expected values of prescribed functionals of the path-wise levels attained by the state variable are defined for all paths to every state individually. For Asian options, the method establishes the first few moments of the arithmetic average of a stock price, both conditionally for each expiry state, and unconditionally as well. These moments are used to define a proxy for the unspecifiable conditional distributions of the average, and applying the payoff rule numerically to the proxy ultimately provides the valuation. The results are compared with published values for options with continuous averaging over a range of strike, volatility, and riskless rate. To affect convergence of value from discrete-step lattices to the limiting case, an extrapolation method provides rapid convergence to the results in the continuous dynamic. Since state-conditional valuations are an intermediate step, then appropriate expiry state-dependent modification of the payoff rules provides floating strike Asian call valuation in the same framework, and the same precision, as for the fixed strike valuations. Application of SCEV induction to path-dependent cash flows of fixed income securities is discussed, particularly with regard to the valuation issues entailed in mortgage backed securities.