Accurate & Efficient Pricing of Arithmetic Average Asian Options within the Hull-White Method

Abstract

Simulation or other approximate methods have traditionally been required to price most exotic options as there have existed no closed-form expressions for their prices. By contrast, simulation of vanilla options without dividends is not necessary, as Black-Scholes expected prices are represented by closed-form expressions. Relying on analytical closed-form results developed by Market Memory Trading, L.L.C. (MMT), described in recent publication http://ssrn.com/abstract=2546430, errors of the widely used Hull-White Method (HWM), based on binomial trees, for valuing arithmetic average Asian options are studied here as functions of the number of time steps n, grid-spacing h, moneyness and other option parameters. It is demonstrated that: As expected, the HWM approximation error with respect to exact reference prices provided by MMT is a decreasing function of the number of time steps n for a reasonably small grid spacing h, e.g. for h = 0.01 and n = 200 the error is within 0.1% for In-The-Money (ITM) options, around 5% for At-The-Money (ATM) options and a few tens of percent for deeply Out-of-The-Money (OTM) options for a wide range of options parameters. The remarkable accuracy of the HWM approximation for ITM options is due to the fact that approximation errors are associated with estimates of the options’ time value. Time value of deeply ITM options is relatively small, making relative value of errors in its estimates even smaller. For ATM and, especially, for OTM options time value is the only value and errors are substantial. The availability of exact prices provided by MMT has allowed discovery of the important and surprising fact that for any number of time steps n, even as small as 20, there exists an optimal grid spacing h* which reproduces exact prices within just 0.5% error margin for the entire spectrum of moneyness and for any choices of expiry T, volatility σ and interest rate r. For the most liquid ATM and close to ATM options, an empirical expression for h* was found in a remarkably simple form.The opportunity provided by this formula for practitioners is that as few as n = 20 time steps can be used to price an ATM Asian option with error less than 0.5% within milliseconds for any choice of option parameters, rather than running extremely long simulations (say, 500 or more time steps) with sub-optimal grid-spacing. In this sense, knowledge of h* is practically equivalent to having the analytical closed-form pricing of Asian options. All computations were performed using the multi-threaded C code (8 threads) on a Windows 8.1, 2.4GHz (8 virtual cores) Intel i7 machine. With suboptimal grid-spacing h=0.01 evaluation of each price for n=200 time steps has taken on average 450 seconds. On a similar platform MMT produces exact prices in a millisecond per price for any set of option parameters.