Approximate Option Pricing

As increasingly large volumes of sophisticated options are traded in world financial markets, determining a "fair" price for these options has become an important and difficult computational problem. Many valuation codes use the binomial pricing model, in which the stock price is driven by a random walk. In this model, the value of an n-period option on a stock is the expected time-discounted value of the future cash flow on an n-period stock price path. Path-dependent options are particularly difficult to value since the future cash flow depends on the entire stock price path rather than on just the final stock price. Currently such options are approximately priced by Monte Carlo methods with error bounds that hold only with high probability and which are reduced by increasing the number of simulation runs. In this paper we show that pricing an arbitrary path-dependent option is #-P hard. We show that certain types of path-dependent options can be valued exactly in polynomial time. Asian options are path-dependent options that are particularly hard to price, and for these we design deterministic polynomial-time approximate algorithms. We show that the value of a perpetual American put option (which can be computed in constant time) is in many cases a good approximation to the value of an otherwise identical n-period American put option. In contrast to Monte Carlo methods, our algorithms have guaranteed error bounds that are polynomially small (and in some cases exponentially small) in the maturity n. For the error analysis we derive large-deviation results for random walks that may be of independent interest.

Authors

• Prasad Chalasani
• Somesh Jha
• Isaac Saias

Keywords

• Pricing
• Security
• Investments
• Laboratories
• Cost accounting
• Polynomials
• Algorithm design and analysis
• Error analysis
• Contracts
• Stochastic processes
• Option Pricing
• Monte Carlo Simulation
• Important Problem
• Random Walk
• Estimation Algorithm
• Binomial Model
• Stock Price
• Valuable Option
• Pricing Model
• Error Bounds
• Future Cash Flows
• Estimation Error
• Random Variables
• Markov Chain
• Dynamic Programming
• Financial Problems
• Head And Tail
• Stopping Rule
• Partial Sums
• Payoff Function
• Risk-free Interest Rate
• Almost Surely
• Stochastic Differential Equations
• Pricing Problem
• Pricing Theory
• Asset Pricing
• Number Of Time Periods