Academic journal article International Journal of Business

An Exponential Extrapolation Approach for the Valuation and Hedging of American Options

Academic journal article International Journal of Business

An Exponential Extrapolation Approach for the Valuation and Hedging of American Options

Article excerpt

ABSTRACT

In this paper, we propose an extrapolation approach for the valuation and hedging of American options. The approach developed in this paper is very simple and computationally efficient. In particular, it can be used to value the long-term American options and American exotic options, such as lookback options and Asian options, to a high degree of accuracy.

JEL classification: G13

Keywords: Exponential extrapolation approaches; Long-term American options; American exotic options

I. INTRODUCTION

In an important contribution, Geske and Johnson [11] showed that it was possible to value an American option by using a series of compound options. They demonstrated that the estimated values using the compound option approach with Richardson extrapolation were very close to the values obtained using other numerical techniques. The Geske and Johnson [11] approach is of interest since it allows evaluation of options for which the possible exercise dates are relatively large or the values of the options are affected by more than one variable.

There have been numerous extensions of the Geske and Johnson [11] approach. These extensions can be divided into three broad categories. The first extension is the one first proposed by Kim [17] and extended by Jacka [16], Car, Jarrow and Myneni [7], Kim and Yu [18] and Huang, Yu, and Subrahmanyam [14]. In a sense, this is the continuous-time extension of the Geske and Johnson [11] approach. The second extension is the one proposed by Breen [3]. His approach is a hybrid of the Cox, Ross, and Rubinstein [9] binomial method (henceforth referred to as CRR) and the Geske and Johnson [11] approach.

The third extension is the one proposed by Bunch and Johnson [5] and Ho, Stapleton and Subrahmanyam [13] (henceforth referred to as HSS). Bunch and Johnson [5] modified the Geske and Johnson [11] approach to approximate the American option values, using only the value of a European option ([P.sub.1]) permitting exercise at maturity date, T, and the value of a twice exerciseable option ([P.sub.2]) permitting exercise at time T/2 and T. However, they used the maximum value of [P.sub.1] and [P.sub.2] rather than the simple [P.sub.1] and [P.sub.2] of Geske and Johnson [11]. Bunch and Johnson [5] showed that their approach worked well for the non-deep in-the-money options. Moreover, they showed that their approach was much more computationally efficient than the Geske and Johnson [11] method. Following similar lines, HSS [13] derived an exponential formula that was based on an observed relationship between the American option value and the number of exercise points allowed up to the maturity date. Although their method made significant improvements on the valuation of futures options and currency options with low carrying cost, it incurred large pricing biases for deep in-the-money American put options with short-term maturities and for long-term American options. (2)

In this paper, we propose an exponential extrapolation approach that not only is very simple and computationally efficient but also significantly reduces the large pricing biases incurred by the Bunch and Johnson [5] approach and the HSS [13] method for the deep in-the-money American put options with short-term maturities and for long-term American options. (3) Additionally, we show that our exponential extrapolation approach can compute the values of American exotic options to a high degree of accuracy.

The rest of this paper is organized as follows. In Section 2, we review the Bunch and Johnson [5] approach and the HSS [13] method. In Section 3, we propose an exponential extrapolation approach for the valuation and hedging of American options. In Section 4, we compare the American put and call values estimated by our exponential extrapolation approach with those estimated by the Bunch and Johnson [5] approach and the HSS [13] method. We also demonstrate that our exponential extrapolation approach can compute the values of American exotic options and the option hedging parameters accurately. …

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